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SOUTHERN  BRANCH, 

OF  C7VLIF0RNIA, 


HISTORY  OF  THE  THEORY  OF  NUMBERS 


VOLUME  I 


DIVISIBILITY  AND  PRIMALITY 


By  Leonard  Eugene  Dickson 

Professor  of  Mathematics  in  the  University  of  Chicago 


Published  by  the  Carnegie  Institution  of  Washington 
Washington,  1919 

/  1  i  5  'J 


CARNEGIE  INSTITUTION  OF  WASHINGTON 
Publication  No.  256,  Vol.  I 


PRESS  OF  GIBSON  BROTHERS 
WASHINGTON.  D.  C. 


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,  _       &!£::  steering* 

'^vi-^  r7'       Mathenrstical 
2^H-j  Sciences 


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PREFACE. 


libraiy 


The  efforts  of  Cantor  and  his  collaborators  show  that  a  chronological 
history  of  mathematics  down  to  the  nineteenth  century  can  be  written  in 
four  large  volumes.  To  cover  the  last  century  with  the  same  elaborateness, 
it  has  been  estimated  that  about  fifteen  volumes  would  be  required,  so 
extensive  is  the  mathematical  literature  of  that  period.  But  to  retain  the 
chronological  order  and  hence  devote  a  large  volume  to  a  period  of  at  most 
seven  years  would  defeat  some  of  the  chief  purposes  of  a  history,  besides 
making  it  very  inconvenient  to  find  all  of  the  material  on  a  particular  topic. 
In  any  event  there  is  certainly  need  of  histories  which  treat  of  particular 
branches  of  mathematics  up  to  the  present  time. 

The  theory  of  numbers  is  especially  entitled  to  a  separate  history  on 

account  of  the  great  interest  which  has  been  taken  in  it  continuously  through 

the  centuries  from  the  time  of  Pythagoras,  an  interest  shared  on  the  one 

extreme  by  nearly  every  noted  mathematician  and  on  the  other  extreme 

^    by  numerous  amateurs  attracted  by  no  other  part  of  mathematics.     This 

v   history  aims  to  give  an  adequate  account  of  the  entire  literature  of  the 

\     theory  of  numbers.     The  first  volume  presents  in  twenty  chapters  the 

material  relating  to  divisibility  and  primality.     The  concepts,  results,  and 

Jl    authors  cited  are  so  numerous  that  it  seems  appropriate  to  present  here  an 

introduction  which  gives  for  certain  chapters  an  account  in  untechnical 

language  of  the  main  results  in  their  historical  setting,  and  for  the  remaining 

•   chapters  the  few  remarks  sufficient  to  clearly  characterize  the  nature  of  their 

v^,    contents. 

J'    '  '   Perfect  numbers  have  engaged  the  attention  of  arithmeticians  of  every 
*»>•    century  of  the  Christian  era.     It  was  while  investigating  them  that  Fermat 
discovered  the  theorem  which  bears  his  name  and  which  forms  the  basis 
of  a  large  part  of  the  theory  of  numbers.     A_perfect  number  is  one,  like 
6  =  1+2+3,  which  equals  the  sum  of  its  divisors  other  than  itself .     Euclid 
,.   proved  that  2^~'^{2^  —  \)  is  a  perfect  numbeflf  2^  —  1  is  a  prime.     For  p  =  2, 
3,  5,  7,  the  values  3,  7,  31,  127  of  2''-l  are  primes,  so  that  6,  28,  496,  8128 
are  perfect  numbers,  as  noted  by  Nicomachus  (about  A.  D.  100).     A  manu- 
script dated  1456  correctly  gave  33550336  as  the  fifth  perfect  number;  it  cor- 
* !  responds  to  the  value  13  of  p.    Very  many  early  writers  believed  that  2^  —  1 
I  is  a  prime  for  every  odd  value  of  p.     But  in  1536  Regius  noted  that 

2^-1  =  511  =  7-73,  211-1=2047  =  23-89 

are  not  primes  and  gave  the  above  fifth  perfect  number.     Cataldi,  who 
founded  at  Bologna  the  most  ancient  known  academy  of  mathematics, 


IV  PREFACE. 

noted  in  1603  that  2''  —  1  is  composite  if  p  is  composite  and  verified  that  it  is 
a  prime  for  p  =  13,  17,  and  19;  but  he  erred  in  stating  that  it  is  also  a  prime 
for  p  =  23,  29,  and  37.  In  fact,  Fermat  noted  in  1640  that  2'-'-l  has  the 
factor  47,  and  2^'-l  the  factor  223,  while  Euler  observed  in  1732  that 
2''  — 1  has  the  factor  1103.  Of  historical  importance  is  the  statement  made 
by  Mersenne  in  1644  that  the  first  eleven  perfect  numbers  are  given  by 
2P-i(2P_i)  for  p  =  2,  3,  5,  7,  13,  17,  19,  31,  67,  127,  257;  but  he  erred  at 
least  in  including  67  and  excluding  61,  89,  and  107.  That  2"  — 1  is  com- 
posite was  proved  by  Lucas  in  1876,  while  its  actual  factors  were  found  by 
Cole  in  1903.  The  primality  of  2^^  — 1,  a  number  of  19  digits,  was  estab- 
lished by  Pervusin  in  1883,  Seelhoff  in  1886,  and  Hudelot  m  1887.  Both 
Powers  and  Fauquembergue  proved  in  1911-14  that  2^^  — 1  and  2^°^  — 1  are 
primes.  The  primality  of  2'^  —  1  and  2™  —  1  had  been  estabhshed  by  Euler 
and  Lucas  respectively.  Thus  2^—  1  is  known  to  be  a  prime,  and  hence  lead 
to  a  perfect  number,  for  the  twelve  values  2,  3,  5,  7,  13,  17,  19,  31,  61,  89, 
107  and  127  of  p.  Since  2^'  — 1  is  known  (pp.  15-31)  to  be  composite  for  32 
primes  p  ^257,  only  the  eleven  values  p  =  137,  139,  149,  157,  167,  193,  199, 
227,  229,  241,  257  now  remain  in  doubt. 

Descartes  stated  in  1638  that  he  could  prove  that  every  even  perfect 
number  is  of  Euclid's  type  and  that  every  odd  perfect  number  must  be  of  the 
form  ps^,  where  p  is  a  prime.  Euler's  proofs  (p.  19)  were  published  after  his 
death.  Xd.  immediate  proof  of  the  former  fact  was  given  by  Dickson  (p.  30). 
According  to  Sylvester  (pp.  26-27),  there  exists  no  odd  perfect  number  with 
fewer  than  six  distinct  prime  factors,  and  none  with  fewer  than  eight  if  not 
divisible  by  3.  But  the  question  of  the  existence  of  odd  perfect  numbers 
remains  unanswered. 

A  multiply  perfect  number,  like  120  and  672,  is  one  the  sum  of  whose 
divisors  equals  a  multiple  of  the  number.  They  were  actively  investigated 
during  the  years  1631-1647  by  IMersenne,  Fermat,  St.  Croix,  Frenicle,  and 
Descartes.  Many  new  examples  hav^e  been  found  recently  by  American 
writers. 

Two  numbers  are  called  amicable  if  each  equals  the  sum  of  the  aliquot 
divisors  of  the  other,  where  an  aliquot  divisor  of  a  number  means  a  divisor 
other  than  the  number  itself.  The  pair  220  and  284  was  known  to  the 
Pythagoreans.  In  the  ninth  century,  the  Arab  Thabit  ben  Korrah  noted 
that  2"/!«  and  2"s  are  amicable  numbers  if  /j=3-2''-l,  t  =  2>'2'^^-l  and  s  = 
9.22"-!  _i  are  all  primes,  and  n>  1.  This  result  leads  to  amicable  numbers 
for  n  =  2  (giving  the  above  pair),  n  =  4  and  n  =  7,  but  for  no  further  value 
^  200  of  n.  The  chief  investigation  of  amicable  numbers  is  that  by  Euler 
who  listed  (pp.  45,  46)  62  pairs.  At  the  age  of  16,  Paganini  announced  in 
1866  the  remarkable  new  pair  1184  and  1210.  A  few  new  pairs  of  very 
large  numbers  have  been  found  by  Legendre,  Seelhoff,  and  Dickson. 


PREFACE.  V 

Interesting  amicable  triples  and  amicable  numbers  of  higher  order  have 
been  recently  found  by  Dickson  and  Poulet  (p.  50). 

Although  it  had  been  employed  in  the  study  of  perfect  and  amicable 
numbers,  the  explicit  expression  for  the  sum  a{n)  of  all  the  divisors  of  n  is 
reserved  for  Chapter  II,  in  which  is  presented  the  history  of  Fermat's  two 
problems  to  solve  (T(x^)=y^  and  <t{x^)  =y^  and  John  Wallis's  problem  to  find 
solutions  other  than  a;  =  4  and  y  =  5  oi  (T{x^)=o-{y^). 

Fermat  stated  in  1640  that  he  had  a  proof  of  the  fact,  now  known  as 
Fermat's  theorem,  that,  if  p  is  any  prime  and  x  is  any  integer  not  divisible 
by  p,  then  x^~^  —  l  is  divisible  by  p.  This  is  one  of  the  fundamental  theo- 
rems of  the  theory  of  numbers.  The  case  x  =  2  was  known  to  the  Chinese  as 
early  as  500  B.  C.  The  first  published  proof  was  given  by  Euler  in  1736. 
Of  first  importance  is  the  generalization  from  the  case  of  a  prime  p  to  any 
integer  n,  published  by  Euler  in  1760:  if  (/)(n)  denotes  the  munber  of  positive 
integers  not  exceeding  n  and  relatively  prime  to  n,  then  x*^"^  —  1  is  divisible 
.  by  n  for  every  integer  x  relatively  prime  to  n.  Another  elegant  theorem 
states  that,  if  p  is  a  prime,  l+jl-2-3. . .  .{p  —  l)\  is  divisible  by  p;  it  was 
first  pubUshed  by  Waring  in  1770,  who  ascribed  it  to  Sir  John  Wilson.  This 
theorem  was  stated  at  an  earlier  date  in  a  manuscript  by  Leibniz,  who  with 
Newton  discovered  the  calculus.  But  Lagrange  was  the  first  one  to  publish 
(in  1773)  a  proof  of  Wilson's  theorem  and  to  observe  that  its  converse  is 
true.  In  1801  Gauss  stated  and  suggested  methods  to  prove  the  generali- 
zation of  Wilson's  theorem:  if  P  denotes  the  product  of  the  positive  integers 
less  than  A  and  prime  to  A,  then  P+1  is  divisible  by  A  if  A  =4,  p""  or  2p"*, 
where  p  is  an  odd  prime,  while  P  —  1  is  divisible  by  A  if  A  is  not  of  one  of 
these  three  forms.  A  very  large  number  of  proofs  of  the  preceding  theorems 
are  given  in  the  first  part  of  Chapter  III.  Various  generalizations  are  then 
presented  (pp.  84-91).  For  instance,  if  iV  =  p/' . . .  p/*,  where  Pi, ...,  p« 
are  distinct  primes, 

a^-(a^/P'+  .  .  .  +a^/PO  +  (a^/P'P'+  ...)-•••  .  +(-l)''a^/^---P'' 

is  divisible  by  N,  a  fact  due  to  Gauss  for  the  case  in  which  a  is  a  prime. 

Many  cases  have  been  found  in  which  o"~^  — 1  is  divisible  by  n  for  a 
composite  number  n.  But  Lucas  proved  the  following  converse  of  Fermat's 
theorem :  if  a^  —  1  is  divisible  by  n  when  x  =  n  —  l,  but  not  when  x  is  a  divisor 
|<n  —  1  of  71  —  1,  then  w  is  a  prime. 

Any  integral  symmetric  function  of  degree  d  of  1,  2, . . .,  p  — 1  with 
I  integral  coefficients  is  divisible  by  the  prime  p  if  c^  is  not  a  multiple  of  p  —  1. 
A  generalization  to  the  case  of  a  divisor  p"  is  due  to  Meyer  (p.  101) .  Nielsen 
proved  in  1893  that,  if  p  is  an  odd  prime  and  if  k  is  odd  and  l<fc<p  —  1,  the 
sum  of  the  products  of  1,  2, . . .,  p  — 1  taken  A;  at  a  time  is  divisible  by  p^. 
Taking  fc  =  p  —  2,  we  see  that  if  p  is  a  prime  >  3  the  numerator  of  the  fraction 


A 


VI  PREFACE. 

equal  to  1  +  1/2+1/3+  . . .  +l/(p  —  1)  is  divisible  by  p'^,  a  result  first  proved 
by  Wolstenholme  in  1862.  Sylvester  stated  in  1866  that  the  sum  of  all 
products  of  n  distinct  numbers  chosen  from  1,  2, . .  . ,  w  is  divisible  by  each 
prime  >  n + 1  which  is  contained  in  any  term  of  the  set  m — n + 1 , .  . . ,  w,  m + 1 . 
There  are  various  theorems  analogous  to  these. 

In  Chapter  IV  are  given  properties  of  the  quotient  {uP~^  —  l)/p,  which 
plays  an  important  role  in  recent  investigations  on  Fermat's  last  theorem 
(the  impossibiUty  of  x'^-\-y^  =  z^  if  p>2),  the  history  of  which  will  be  treated 
in  the  final  chapter  of  Volume  II.  Some  of  the  present  papers  relate  to 
(w*^"^  — 1)/«,  where  n  is  not  necessarily  a  prime. 

TMiile  Euler's  ^-function  was  defined  above  in  order  to  state  his  general- 
ization of  Fermat's  theorem,  its  numerous  properties  and  generalizations 
are  reserved  for  the  long  Chapter  V.  In  1801  Gauss  gave  the  result  that 
4>{d^  +  .  . .  -\-<i>{dk)  =  n,  if  di, .  .  . ,  d^  are  the  divisors  of  n;  this  was  generalized 
by  Laguerre  in  1872,  H.  G.  Cantor  in  1880,  Busche  in  1888,  Zsigmondy  in 
1893,  Vahlen  in  1895,  Elliott  in  1901,  and  Hammond  in  1916.  In  1808 
Legendre  proved  a  simple  formula  for  the  number  of  integers  ^  n  which  are 
divisible  by  no  one  of  any  given  set  of  primes.  The  asymptotic  value  of 
(f>{l)-\- .  .  .  +0(G)  for  G  large  was  discussed  by  Dirichlet  in  1849,  Mertens  in 
1874,  Perott  in  1881,  Sylvester  in  1883  and  1897,  Cesaro  in  1883  and  1886-8, 
Berger  in  1891,  and  Kronecker  in  1901.  The  solution  of  4>{x)=g  was  treated 
by  Cayley  in  1857,  Mmin  in  1897,  Pichler  in  1900,  Carmichael  in  1907-9, 
Ranum  in  1908,  and  Cunningham  in  1915.  H.  J.  S.  Smith  proved  in  1875 
that  the  m-rowed  determinant,  ha\ing  as  the  element  in  the  ith  row  and 
ji\i  column  any  function  fib)  of  the  greatest  common  divisor  5  of  i  and  j, 
equals  the  product  of  F{\),  F{2),. . .,  F(m),  where 

F(m)=/(m)-2/g)+2/(^J-....,     m  =  py 

In  particular,  F{m)=<t>{m)  if  f{8)=8.  In  several  papers  (pp.  128-130) 
Cesaro  considered  analogous  determinants.  The  fact  that  30  is  the  largest 
number  such  that  all  smaller  numbers  relatively  prime  to  it  are  primes  was 
first  proved  by  Schatunowsky  in  1893. 

A.  Thacker  in  1850  evaluated  the  sum  4>k{n)  of  the  kth.  powers  of  the 
integers  ^n  which  are  prime  to  n.  His  formula  has  been  expressed  m^ 
various  symbolic  forms  by  Ces^o  and  generalized  by  Glaisher  and  Nielsen./ 
Crelle  had  noted  in  1845  that  <piin)  =  |n0(  n).  In  1869  Schemmel  considered 
the  number  of  sets  of  n  consecutive  integers  each  <  m  and  prime  to  m.  In 
connection  with  linear  congruence  groups,  Jordan  evaluated  the  number  of 
different  sets  of  k  positive  integers  ^?i  whose  greatest  common  divisor  is 
prime  to  n.  This  generalization  of  Euler's  (^-function  has  properties  as 
simple  as  the  latter  function  and  occurs  in  many  papers  under  a  variety  of 
notations.     It  in  turn  has  been  generalized  (pp.  151-4). 


PREFACE.  VII 

The  properties  of  the  set  of  all  irreducible  fractions,  arranged  in  order  of 
magnitude,  whose  numerators  are  ^  m  and  denominators  are  ^  n  (called  a 
Farey  series  if  m  =  n),  have  been  discussed  by  many  writers  and  applied  to 
the  approximation  of  numbers,  to  binary  quadratic  forms,  to  the  composi- 
tion of  linear  fractional  substitutions,  and  to  geometry  (pp.  155-8). 

Some  of  the  properties  of  periodic  decimal  fractions  are  already  familiar 
tq  the  reader  in  view  of  his  study  of  arithmetic  and  the  chapter  of  alge- 
bra dealing  with  the  sum  to  infinity  of  a  geometric  progression.  For  the 
generalization  to  periodic  fractions  to  any  base  h,  not  necessarily  10,  the 
length  of  the  period  of  the  periodic  fraction  for  1/d,  where  d  is  prime  to  h, 
is  the  least  positive  exponent  e  such  that  h^  —  \  is  divisible  by  d.  Hence  this 
Chapter  VI,  which  reports  upon  more  than  160  papers,  is  closely  related  to 
the  following  chapter  and  furnishes  a  concrete  introduction  to  it. 

The  subject  of  exponents  and  primitive  roots  is  one  of  the  important 
topics  of  the  theory  of  numbers.  To  present  the  definitions  in  the  customary, 
compact  language,  we  shall  need  the  notion  of  congruence.  If  the  differ- 
ence of  two  integers  a  and  6  is  divisible  by  m,  they  are  called  congruent 
modulo  m  and  we  write  a=&  (mod  m).  For  example,  8=2  (mod  6).  If 
n'=  1  (mod  m),  but  n*^  1  (mod  m)  for  0<s<e,  we  say  that  n  belongs  to  the 
exponent  e  modulo  m.  For  example,  2  and  3  belong  to  the  exponent  4 
modulo  5,  while  4  belongs  to  the  exponent  2.  In  view  of  Euler's  generaliza- 
tion of  Fermat's  theorem,  stated  above,  e  never  exceeds  0(m).  If  n  belongs 
to  this  maximum  exponent  ^(n)  modulo  m,  n  is  called  a  primitive  root  of  m. 
For  example,  2  and  3  are  primitive  roots  of  5,  while  1  and  4  are  not.  Lam- 
bert stated  in  1769  that  there  exists  a  primitive  root  of  any  prime  p,  and 
Euler  gave  a  defective  proof  in  1773.  In  1785  Legendre  proved  that  there 
are  exactly  4>{e)  numbers  belonging  modulo  p  to  any  exponent  e  which 
divides  p  — 1.  In  1801  Gauss  proved  that  there  exist  primitive  roots  of  m 
if  and  only  if  m  =  2,  4,  p*  or  2p*,  where  p  is  an  odd  prime.  In  particular,  for 
a  primitive  root  a  of  a  prime  modulus  p  and  any  integer  N  not  divisible 
by  p,  there  is  an  exponent  ind  N,  called  the  index  of  N  by  Gauss,  such  that 
N=a''"^^  (mod  p).  Indices  play  a  role  similar  to  logarithms,  but  we  re- 
quire two  companion  tables  for  each  modulus  p.  The  extension  to  a  power 
of  prime  modulus  is  immediate.  For  a  general  modulus,  systems  of  indices 
were  employed  by  Dirichlet  in  1837  and  1863  and  by  Kronecker  in  1870. 
Jacobi's  Canon  Arithmeticus  of  1839  gives  companion  tables  of  indices  for 
each  prime  and  power  of  a  prime  <  1000.  Cunningham's  Binary  Canon  of 
1900  gives  the  residues  of  the  successive  powers  of  2  when  divided  by  each 
prime  or  power  of  a  prime  <  1000  and  companion  tables  showing  the  powers 
of  2  whose  residues  are  1,  2,  3, .  .  ..  In  1846  Arndt  proved  that,  if  ^  is  a 
primitive  root  of  the  odd  prime  p,  g  belongs  to  the  exponent  p"~"^(p  — 1) 
modulo  p'*  if  and  only  ii  G  =  g^~^  —  1  is  divisible  by  p^,  but  not  by  p^'^\  where 


VIII  PREFACE. 

X<n;  taking  X=  1,  we  see  that,  if  G  is  not  divisible  by  p^,  g'  is  a  primitive 
root  of  p^  and  of  all  higher  powers  of  p.  This  Chapter  VII  presents  many 
more  theorems  on  exponents,  primitive  roots,  and  binomial  congruences,  and 
cites  various  lists  of  primitive  roots  of  primes  <  10000. 

Lagrange  proved  easily  that  a  congruence  of  degree  n  has  at  most  n  roots 
if  the  modulus  is  a  prime.  Lebesgue  found  the  number  of  sets  of  solutions  of 
01^1"*+  •  •  •  -\-akXk"=a  (mod  p),  when  p  is  a  prime  such  that  p  —  1  is  divisible 
by  m.  Konig  (p.  226)  employed  a  cyclic  determinant  and  its  minors  to  find 
the  exact  number  of  real  roots  of  any  congruence  in  one  unknown;  Gegen- 
bauer  (p.  228)  and  Rados  (p.  233)  gave  generalizations  to  congruences  in 
several  unknowns. 

Galois's  introduction  of  imaginary  roots  of  congruences  has  not  only 
led  to  an  important  extension  of  the  theory  of  numbers,  but  has  given  rise 
to  wide  generalizations  of  theorems  which  had  been  obtained  in  subjects 
like  linear  congruence  groups  by  applying  the  ordinary  theory  of  numbers. 
Instead  of  the  residues  of  integers  modulo  p,  let  us  consider  the  residues  of 
polynomials  in  a  variable  x  with  integral  coefficients  with  respect  to  two 
moduH,  one  being  a  prime  p  and  the  other  a  polynomial  f{x)  of  degree  n 
which  is  irreducible  modulo  p.  The  residues  are  the  p"  polynomials  in  x  of 
degree  n  —  1  whose  coefficients  are  chosen  from  the  set  0, 1, . .  . ,  p  —  1 .  These 
residues  form  a  Galois  field  within  which  can  be  performed  addition,  sub- 
traction, multiplication,  and  division  (except  by  zero) .  As  a  generahzation 
of  Fermat's  theorem,  Galois  proved  that  the  power  p"  — 1  of  any  residue 
except  zero  is  congruent  to  unity  with  respect  to  our  pair  of  moduli  p  and 
f{x).  He  avoided  our  second  modulus  f{x)  by  introducing  an  undefined 
imaginary  root  i  of  f{x)  =  0  (mod  p)  and  considering  the  residues  modulo  p 
of  polynomials  in  i;  but  the  above  use  of  the  two  moduH  affords  the  only 
logical  basis  of  the  theory.  In  view  of  the  fullness  of  the  reports  in  the  text 
(pp.  233-252)  of  the  papers  on  this  subject,  further  comments  here  are 
unnecessary.  The  final  topics  of  this  long  Chapter  VIII  are  cubic  congru- 
ences and  miscellaneous  results  on  congruences  and  possess  little  general 
interest. 

In  Chapter  IX  are  given  Legendre's  expression  for  the  exponent  of  the 
highest  power  of  a  prime  p  which  divides  the  factorial  1-2. .  .m,  and  the 
generalization  to  the  product  of  any  integers  in  arithmetical  progression; 
many  theorems  on  the  divisibility  of  one  product  of  factorials  by  another 
product  and  on  the  residues  of  multinomial  coefficients ;  various  determina- 
tions of  the  sign  in  1-2...  (p  — l)/2==tl  (mod  p);  and  miscellaneous 
congruences  involving  factorials. 

In  the  extensive  Chapter  X  are  given  many  theorems  and  formulas 
concerning  the  sum  of  the  kth.  powers  of  all  the  divisors  of  n,  or  of  its  even  or 
odd  divisors,  or  of  its  divisors  which  are  exact  sth  powers,  or  of  those  divisors 


PREFACE.  IX 

whose  complementary  divisors  are  even  or  odd  or  are  exact  sth  powers,  and 
the  excess  of  the  sum  of  the  A;th  powers  of  the  divisors  of  the  form  4m +1  of 
a  number  over  the  sum  of  the  A;th  powers  of  the  divisors  of  the  form  4m -f  3, 
as  well  as  more  technical  sums  of  divisors  defined  on  pages  297,  301-2,  305, 
307-8,  314-5  and  318.  For  the  important  case  k  =  0,  such  a  sum  becomes 
the  number  of  the  divisors  in  question.  There  are  theorems  on  the  number 
of  sets  of  positive  integral  solutions  of  UiU^. .  .Uk  =  n  or  of  x''y^  =  n.  Also 
Glaisher's  cancellation  theorems  on  the  actual  divisors  of  numbers  (pp. 
^'  310-11,  320-21).  Scattered  through  the  chapter  are  approximation  and 
asymptotic  formulas  involving  some  of  the  above  functions. 

In  Chapter  XI  occur  Dirichlet's  theorem  on  the  number  of  cases  in  the 

division  of  n  by  1,  2, . . . ,  p  in  turn  in  which  the  ratio  of  the  remainder  to  the 

divisor  is  less  than  a  given  proper  fraction,  and  the  generalizations  on  pp. 

330-1;  theorems  on  the  number  of  integers  ^n  which  are  divisible  by  no 

,(  exact  sth  power  >  1 ;  theorems  on  the  greatest  divisor  which  is  odd  or  has 

/  specified  properties;  many  theorems  on  greatest  coromon  divisor  and  least 

I  common  multiple ;  and  various  theorems  on  mean  values  and  probability. 

'        The  casting  out  of  nines  or  of  multiples  of  11  or  7  to  check  arithmetical 

computations  is  of  early  origin.     This  topic  and  the  related  one  of  testing 

the  divisibility  of  one  number  by  another  have  given  rise  to  the  numerous 

elementary  papers  cited  in  Chapter  XII. 

The  frequent  need  of  the  factors  of  numbers  and  the  excessive  labor 
required  for  their  direct  determination  have  combined  to  inspire  the 
construction  of  factor  tables  of  continually  increasing  limit.  The  usual 
method  is  essentially  that  given  by  Eratosthenes  in  the  third  century  B.  C. 
A  special  method  is  used  by  Lebon  (pp.  355-6).  Attention  is  called  to 
Lehmer's  Factor  Table  for  the  First  Ten  Millions  and  his  List  of  Prime 
Numbers  from  1  to  10,006,721,  published  in  1909  and  1914  by  the  Carnegie 
Institution  of  Washington.  Since  these  tables  were  constructed  anew  with 
the  greatest  care  and  all  variations  from  the  chief  former  tables  were  taken 
account  of,  they  are  certainly  the  most  accurate  tables  extant.  Absolute 
accuracy  is  here  more  essential  than  in  ordinary  tables  of  continuous  func- 
tions. Besides  giving  the  history  of  factor  tables  and  lists  of  primes,  this 
Chapter  XIII  cites  papers  which  enumerate  the  primes  in  various  intervals, 
prime  pairs  (as  11,  13),  primes  of  the  form  4n+l,  and  papers  listing  primes 
written  to  be  base  2  or  large  primes. 

Chapter  XIV  cites  the  papers  on  factoring  a  number  by  expressing  it  as 
a  difference  of  two  squares,  or  as  a  sum  of  two  squares  in  two  ways,  or  by  use 
of  binary  quadratic  forms,  the  final  digits,  continued  fractions.  Pell  equa- 
tions, various  small  moduli,  or  miscellaneous  methods. 

Fermat  expressed  his  belief  that  Fn  =  2^"+l  is  a  prime  for  every  value  of  n. 
While  this  is  true  if  n  =  1,  2,  3,  4,  it  fails  forn  =  5  as  noted  by  Euler.     Later, 


X  PREFACE. 

Gauss  proved  that  a  regular  polygon  of  m  sides  can  be  constructed  by  ruler 
and  compasses  if  m  is  a  product  of  a  power  of  2  and  distinct  odd  primes  each 
of  the  form  Fn,  and  stated  correctly  that  the  construction  is  impossible  if  m 
is  not  such  a  product.  In  view  of  the  papers  cited  in  Chapter  XV,  F„  is 
composite  if  n  =  5,  6,  7,  8,  9,  11,  12,  18,  23,  36,  38  and  73,  while  nothing  is 
known  for  other  values  >4  of  n.  No  conoment  will  be  made  on  the  next 
chapter  which  treats  of  the  factors  of  numbers  of  the  form  o"±6"  and  of 
certain  trinomials. 

In  Chapter  XVII  are  treated  questions  on  the  divisors  of  terms  of  a 
recurring  series  and  in  particular  of  Lucas'  functions 

a  —  o 

where  a  and  h  are  roots  oi  x'^  —  Px-\-Q  =  Q,  P  and  Q  being  relatively  prime 
integers.  By  use  of  these  functions,  Lucas  obtained  an  extension  of  Euler's 
generaUzation  of  Fermat's  theorem,  which  requires  the  correction  noted  by 
Carmichael  (p.  406),  as  well  as  various  tests  for  primality,  some  of  which 
have  been  emploj^ed  in  investigations  on  perfect  numbers.  Many  papers  on 
the  algebraic  theory  of  recurring  series  are  cited  at  the  end  of  the  chapter. 

Euchd  gave  a  simple  and  elegant  proof  that  the  number  of  primes  is  infi- 
nite. For  the  generalization  that  every  arithmetical  progression  n,  n+m, 
n-\-2m,.  .  .,  in  which  n  and  m  are  relatively  prime,  contains  an  infinitude 
of  primes,  Legendre  offered  an  insufficient  proof,  while  Dirichlet  gave  his 
classic  proof  by  means  of  infinite  series  and  the  classes  of  binary  quadratic 
forms,  and  extended  the  theorem  to  complex  integers.  Mertens  and  others 
obtained  simpler  proofs.  For  various  special  arithmetical  progressions,  the 
theorem  has  been  proved  in  elementary  ways  by  many  writers.  Dirichlet 
also  obtained  the  theorems  that,  if  a,  26,  and  c  have  no  common  factor, 
ax'^+2hxy-\-cy^  represents  an  infinitude  of  primes,  while  an  infinitude  of  these 
primes  are  representable  by  any  given  linear  form  Mx+N  with  M  and  N 
relatively  prime,  pro\^ded  a,  h,  c,  M,  N  are  such  that  the  quadratic  and  linear 
forms  can  represent  the  same  number. 

No  complete  proof  has  been  found  for  Goldbach's  conjecture  in  1742  that 
every  even  integer  is  a  sum  of  two  primes.  One  of  various  analogous  un- 
proved conjectures  is  that  every  even  integer  is  the  difference  of  two  consec- 
utive primes  in  an  infinitude  of  ways  (in  particular,  there  exists  an  infinitude 
of  pairs  of  primes  differing  by  2).  No  comment  will  be  made  on  the  further 
topics  of  this  Chapter  XVIII:  polynomials  representing  numerous  primes, 
primes  in  arithmetical  progression,  tests  for  primality,  number  of  primes 
between  assigned  limits,  Bertrand's  postulate  of  the  existence  of  at  least  one 
prime  between  x  and  2x  — 2  for  x>3,  miscellaneous  results  on  primes, 
diatomic  series,  and  asymptotic  distribution  of  primes. 


PREFACE.  XI 

If  F(m)=2/(d),  summed  for  all  the  divisors  d  of  m,  we  can  express 
/(m)  in  terms  of  F  by  an  inversion  formula  given  in  Chapter  XIX  along  with 
generalizations  and  related  formulas.  Bougaief  called  F{m)  the  numerical 
integral  of /(m). 

The  final  Chapter  XX  gives  many  elementary  results  involving  the  digits 
of  numbers  mainly  when  written  to  the  base  10. 

Since  the  history  of  each  main  topic  is  given  separately,  it  has  been 
possible  without  causing  confusion  to  include  reports  on  minor  papers  and 
isolated  problems  for  the  sake  of  completeness.  In  the  cases  of  books  and 
journals  not  usually  accessible,  the  reports  are  quite  full  with  some  indication 
of  the  proofs.  In  other  cases,  proofs  are  given  only  when  necessary  to 
differentiate  the  paper  from  others  deriving  the  same  result. 

The  references  were  selected  mainly  from  the  Subject  Index  of  the  Royal 
Society  of  London  Catalogue  of  Scientific  Papers,  volume  1, 1908  (with  which 
also  the  proof-sheets  were  checked),  and  the  supplementary  annual  volumes 
forming  the  International  Catalogue  of  Scientific  Literature,  Jahrbuch 
iiber  die  Fortschritte  der  Mathematik,  Revue  semestrielle  des  publications 
math^matiques,  Poggendorff's  Handworterbuch,  Kliigel's  Mathematische 
Worterbuch,  Wolffing's  Mathematischer  Biicherschatz  (a  list  of  mathemat- 
ical books  and  pamphlets  of  the  nineteenth  century),  historical  journals,  such 
as  Bulletino  di  bibliografia  e  di  storia  delle  scienze  matematiche  e  fisiche, 
Bolletino . .  . . ,  BibUotheca  Mathematica,  Abhandlungen  zur  Geschichte 
der  mathematischen  Wissenschaften,  various  histories  and  encyclopedias, 
including  the  Enclyclop^die  des  sciences  mathematiques.  Further,  the 
author  made  a  direct  examination  at  the  stacks  of  books  and  old  journals 
in  the  libraries  of  Chicago,  California,  and  Cambridge  Universities,  and 
Trinity  College,  Cambridge,  and  the  excellent  John  Crerar  Library  at  Chi- 
cago. He  made  use  of  G.  A.  Plimpton's  remarkable  collection,  in  New 
York,  of  rare  books  and  manuscripts.  In  1912  the  author  made  an 
extended  investigation  in  the  libraries  of  the  British  Museum,  Kensington 
Museum,  Royal  Society,  Cambridge  Philosophical  Society,  Bibliotheque 
Nationale,  Universite  de  Paris,  St.  Genevieve,  I'lnstitut  de  France,  Uni- 
versity of  Gottingen,  and  the  Konigliche  Bibhothek  of  Berlin  (where  there 
is  a  separate  index  of  the  material  on  the  theory  of  numbers).  Many 
books  have  since  been  borrowed  from  various  libraries;  the  Ladies'  and 
other  Diaries  were  loaned  by  R.  C.  Archibald. 

At  the  end  of  the  volume  is  a  separate  index  of  authors  for  each  of  the 
twenty  chapters,  which  will  facilitate  the  tracing  of  the  relation  of  a  paper 
to  kindred  papers  and  hence  will  be  of  special  service  in  the  case  of  papers 
inaccessible  to  the  reader.  The  concluding  volume  will  have  a  combined 
index  of  authors  from  which  will  be  omitted  minor  citations  found  in  the 
chapter  indices. 


XII  PREFACE. 

The  subject  index  contains  a  list  of  symbols;  while  [x]  usually  denotes 
the  greatest  integer  ^x,  occasionally  such  square  brackets  are  used  to 
inclose  an  addition  to  a  quotation.  The  symbol  *  before  an  author's 
name  signifies  that  his  paper  was  not  available  for  report.  The  symbol  f 
before  a  date  signifies  date  of  death.  Initials  are  given  only  in  the  first  of 
several  immediately  successive  citations  of  an  author. 

Although  those  volumes  of  Euler's  Opera  Omnia  which  contain  his  Com- 
mentationes  Arithmeticae  CoUectse  have  been  printed,  they  are  not  yet 
available;  a  table  showing  the  pages  of  the  Opera  and  the  corresponding 
pages  in  the  present  volume  of  this  history  will  be  given  in  the  concluding 
volume. 

The  author  is  under  great  obligations  to  the  following  experts  in  the 
theory  of  numbers  for  numerous  improvements  resulting  from  their  reading 
the  initial  page  proofs  of  this  volume:  R.  D.  Carmichael,  L.  Chanzy,  A. 
Cunningham,  E.  B.  Escott,  A.  Gerardin,  A.  J.  Kempner,  D.  N.  Lehmer,  E. 
Maillet,  L.  S.  Shively,  and  H.  J.  Woodall;  also  the  benefit  of  D.  E.  Smith's 
accurate  and  extensive  acquaintance  with  early  books  and  writers  was  for- 
tunately secured ;  and  the  author's  special  thanl<:s  are  due  to  Carmichael  and 
Kempner,  who  read  the  final  page  proofs  with  the  same  critical  attention 
as  the  initial  page  proofs  and  pointed  out  various  errors  and  obscurities. 
To  these  eleven  men  who  gave  so  generously  of  their  time  to  perfect  this 
volume,  and  especially  to  the  last  two,  is  due  the  gratitude  of  every  devotee 
of  number  theory  who  may  derive  benefit  or  pleasure  from  this  history.  In 
return,  such  readers  are  requested  to  further  increase  the  usefulness  of  this 
work  by  sending  corrections,  notices  of  omissions,  and  abstracts  of  papers 
marked  not  available  for  report,  for  insertion  in  the  concluding  volume. 

Finally,  this  laborious  project  would  doubtless  have  been  abandoned  soon 
after  its  inception  seven  years  ago  had  not  President  Woodward  approved 
it  so  spontaneously,  urged  its  completion  with  the  greatest  thoroughness, 
and  given  continued  encouragement. 

L.  E.  Dickson. 
November,  1918. 


^  TABLE  OF  CONTENTS. 

Chapter.  page. 

I.  Perfect,  multiply  perfect,  and  amicable  numbers 3 

\  11.  Formulas  for  the  number  and  simi  of  divisors,  problems  of  Fermat 

and  Wallis 51 

III.  Fermat's  and  Wilson's  theorems,  generalizations  and  converses; 

symmetric  functions  of  1,  2, . . . ,  p—  1,  modulo  p 59 

IV.  Residue  of  (wp~^  —  l)/p  modulo  p 105 

V.  Euler's  (^function,  generalizations;  Farey  series 113 

VI.  Periodic  decimal  fractions;  periodic  fractions;  factors  of  10"  =•=!...  .  159 

VII.  Primitive  roots,  exponents,  indices,  binomial  congruences 181 

VIII.  Higher  congruences 223 

IX.  Divisibility  of  factorials  and  multinomial  coefficients 263 

X.  Sum  and  number  of  divisors 279 

XI.  Miscellaneous  theorems  on  divisibility,  greatest  common  divisor, 

least  common  multiple 327 

XII.  Criteria  for  divisibility  by  a  given  number 337 

XIII.  Factor  tables,  lists  of  primes 347 

v^IV.  Methods  of  factoring 357  • 

XV.  Fermat  numbers  F„  =  22"+l 375 

XVI.  Factors  of  a"±6« 381 

XVII.  Recurring  series;  Lucas'  Un,  Vn 393 

^VIII.  Theory  of  prime  numbers 413 

XIX.  Inversion  of  functions;  Mobius'  function  ix{n);  numerical  integrals 

and  derivatives 441 

XX.  Properties  of  the  digits  of  numbers 453 

Author  index 467 

Subject  index 484 

1 


(. 


CHAPTER  I. 

PERFECT.  MULTIPLY  PERFECT.  AND  AMICABLE  NUMBERS. 
Perfect,  Abundant,  and  Deficient  Numbers. 

By  the  aliquot  parts  or  divisors  of  a  number  are  meant  the  divisors, 
including  unity,  which  are  less  than  the  number.  A  number,  like  6  =  1  -h 
2+3,  which  equals  the  sum  of  its  aliquot  divisors  is  called  perfect  (voll- 
kommen,  vollstandig) .  If  the  sum  of  the  aliquot  divisors  is  less  than  the 
number,  as  is  the  case  with  8,  the  number  is  called  deficient  (diminute, 
defective,  unvollkommen,  unvollstandig,  mangelhaft).  If  the  sum  of  the 
aliquot  divisors  exceeds  the  number,  as  is  the  case  with  12,  the  number  is 
called  abundant  (superfluos,  plus  quam-perfectus,  redundantem,  exc^dant, 
iibervollstandig,  iiberflussig,  iiberschiessende) . 

Euclid^  proved  that,  if  p  =  1+2+2^+  •  •  •  +2"  is  a  prime,  2"p  is  a  perfect 
number.  He  showed  that  2"p  is  divisible  by  1,  2, .  . . ,  2",  p,  2p, . . . ,  2'*~^p, 
but  by  no  further  number  less  than  itself.  By  the  usual  theorem  on 
geometrical  progressions,  he  showed  that  the  sum  of  these  divisors  is  2"^. 

The  early  Hebrews^"  considered  6  to  be  a  perfect  number. 

Philo  Judeus^''  (first  century  A.  D.)  regarded  6  as  the  most  productive 
of  all  numbers,  being  the  first  perfect  number. 

Nicomachus^  (about  A.  D.  100)  separated  the  even  numbers  (book  I, 
chaps.  14,  15)  into  abundant  (citing  12,  24),  deficient  (citing  8,  14),  and 
perfect,  and  dwelled  on  the  ethical  import  of  the  three  types.  The  perfect 
(I,  16)  are  between  excess  and  deficiency,  as  consonant  sound  between 
acuter  and  graver  sounds.  Perfect  numbers  will  be  found  few  and  arranged 
with  fitting  order;  6,  28,  496,  8128  are  the  only  perfect  numbers  in  the 
respective  intervals  between  1,  10,  100,  1000,  10000,  and  they  have  the 
property  of  ending  alternately  in  6  and  8.  He  stated  that  Euclid's  rule 
gives  all  the  perfect  numbers  without  exception. 

Theon  of  Smyrna^  (about  A.  D.  130)  distinguished  between  perfect 
(citing  6,  28),  abundant  (citing  12)  and  deficient  (citing  8)  numbers. 

^Elementa,  liber  IX,  prop.  36.    Opera,  2,  Leipzig,  1884,  408. 
^"S.  Rubin,  "Sod  Hasfiroth"  (secrets  of  numbers),  Wien,  1873,  59;    citation  of  the  Bible, 

Kings,  II,  13,  19. 
**Treatise  on  the  account  of  the  creation  of  the  world  as  given  by  Moses,  C.  D.  Young's 

transl.  of  Philo's  works,  London,  1854,  vol.  1,  p.  3. 
'Nicomachi  Gerasini  arithmeticse  Ubri  duo.     Nunc  primdm  typis  excusi,  in  lucem  eduntur. 

Parisiis,  1538.     In  officina  Christian!  WecheU.     (Greek.) 
Theologumena  arithmeticae.     Accedit  Nicomachi  Gerasini  institutio  arithmetica  ad  fidem 

codicum  Monacensium  emendata.     Ed.,  Fridericus  Astius.     Lipsiae,  1817.     (Greek.) 
Nicomachi  Geraseni  Pythagorei  introductionis  arithmeticae  libri  ii.     Recensvit  Ricardus 

Hoche.     Lipsiae,  1866.     (Greek.) 
'Theonis   Smymaei   philosophi   Platonici   expositio   rerum   mathematicarum   ad  legendum 

Platonem  utiHum.     Ed.,  Ed.  Hiller,  Leipzig,  1878,  p.  45. 
Theonis  Smymaei  Platonici,  Latin  by  Ismaele  BuUialdo.     Paris,  1644,  chap.  32,  pp.  70-72. 

3 


4  History  of  the  Theory  of  Numbers.  [Chap.  I 

lamblichus*  (about  283-330)  repeated  in  effect  the  remarks  by  Nico- 
machus  on  perfect,  abundant,  and  deficient  numbers,  but  made  erroneous 
additions.  He  stated  that  there  is  one  and  but  one  perfect  number  in  the 
successive  intervals  between  1,  10,  100,...,  100000,  etc.,  to  infinity. 
"Examples  of  a  perfect  number  are  6,  and  28,  and  496,  and  8128,  and  the  like 
numbers,  alternately  ending  in  6  and  8."  He  remarked  that  the  Pythag- 
oreans called  the  perfect  number  6  marriage,  and  also  health  and  beauty 
(on  account  of  the  integrity  of  its  parts  and  the  agreement  existing  in  it). 

Aurelius  Augustinus^  (354-430)  remarked  that,  6  being  the  first  perfect 
number,  God  effected  the  creation  in  6  daj's  rather  than  at  once,  since  the 
perfection  of  the  work  is  signified  by  the  number  6.  The  sum  of  the  aUquot 
parts  of  9  falls  short  of  it;  likewise  for  10.  But  the  sum  of  the  aliquot 
parts  of  12  exceeds  it. 

Anicius  Manhus  Severinus  Boethius^  (about  481-524),  in  a  Latin  exposi- 
tion of  the  arithmetic  of  Nicomachus,  stated  that  perfect  numbers  are  rare, 
easily  counted,  and  generated  in  a  very  regular  order,  while  abundant 
(superfluos)  and  deficient  (diminutos)  numbers  are  found  to  an  unlimited 
extent  and  not  in  regular  order.  The  perfect  numbers  below  10000  are 
6,  28,  496,  8128.     And  these  numbers  alwaj^s  end  alternately  in  6  and  8. 

Munyos^  stated  that  Boethius  added  to  EucUd's  idea  of  perfect  number 
that  of  deficient  (diminute)  and  abundant  (redundantem)  numbers. 

Isidorus  of  Seville^  (570-636)  distinguished  even  and  odd  numbers, 
perfect  and  abundant  numbers,  linear,  flachen  and  Korper  Zahlen  (primes, 
products  of  two,  products  of  three  factors). 

Alcuin^  (735-804) ,  of  York  and  Tours,  explained  the  occurrence  of  the 
number  6  in  the  creation  of  the  universe  on  the  ground  that  6  is  a  perfect 
number.  The  second  origin  of  the  human  race  arose  from  the  deficient 
number  8;  indeed,  in  Noah's  ark  there  were  8  souls  from  which  sprung  the 
entire  human  race,  showing  that  the  second  origin  was  more  imperfect  than 
the  first,  which  was  made  according  to  the  number  6. 

^lamblichus  Chalcidensis  ex  Coele-Syria  in  Nicomachi    Geraseni    arithmeticam    introduc- 

tionem,  et  de  Fato.     Accedit  Joachimi  Camerarii  explicatio  in  duos  libros  Nicomachi. 

Ed.,  Samuel  Tennulius.     Amhemiae,  1668,  pp.  43-47.    (Greek  text  and  Latin  translation 

in  parallel  columns.) 
lamblichi  in  Nicomachi  arithmeticam  introductionem  Uber  ad  fidem  codicis  Florentini. 

Ed.,  H.  Pistelli.     Lipsiae,  1894.     (Greek.) 
*De  Civitate  Dei,  hber  XI,  cap.  XXX,  ed.,  B.  Dombart,  Lipsiae,  1877, 1,  p.  504.    The  reference 

by  Frizzo"'  i'  to  lib.  II,  cap.  39. 
"Arithmetica  boetij,  Augsburg,  1488;  Cologne,  1489;  Leipzig,  1490;  Venice,  1491-2,  1499; 

Paris,  [1496,  1501],  1503,  etc.;  lib.  1,  cap.  20.  "De  generatione  numeri  perfecti." 
Opera  Boetii,  Venice,  1491-2,  etc.;  ed.,  Friedlein,  Leipzig,  1867. 
"Institvtiones  arithmeticae  ad  percipiendam  astrologiam  et  mathematicas  facultates  neces- 

sariae.     Auctore  Hieronymo  Mimj'os,  Valentiae,  1566,  f.  5,  verso. 
*Incipit  epistola  Isidori  iunioris  hispalensis  .  .  .  Finit  Uber  etymologiarum  .  .  .  [Augsburg, 

1472];  Venice,  1483,  etc.     In  this  book  of  etymologies,  arithmetic  is  treated  very  briefly 

in  Book  3,  beginning  f .  15. 
•Bibhotheca  Rerum  Germanicarum,  tomus  sextus:  Monumenta  Alcuiniana,  Berlin,   1873, 

epistolae  259,  pp.  818-821.     Cf.  Migne,  Patrologiae,  vol.  100,  1851,  p.  665;  Hankel, 

Geschichte  Math.,  p.  311. 


Chap.  I]     PERFECT,  MULTIPLY  PERFECT,  AND   AmICABLE   NUMBERS.  5 

Thabit  ben  Korrah,^°  in  a  manuscript  composed  the  last  half  of  the 
ninth  century,  attributed  to  Pythagoras  and  his  school  the  employment  of 
perfect  and  amicable  numbers  in  illustration  of  their  philosophy.  Let 
s  =  1+2+  ...  +2".  Then  (prop.  5),  2'*s  is  a  perfect  number  if  s  is  a  prime; 
2"p  is  abundant  if  p  is  a  prime  <s,  deficient  if  p  is  a  prime  >s,  and  the 
excess  or  deficiency  of  the  sum  of  all  the  divisors  over  the  number  equals 
the  difference  of  s  and  p.  Let  (prop.  6)  p'  and  p"  be  distinct  primes  >2; 
the  sum  of  the  divisors  <N  oi  N  =  p'p"2"  is 

a  =  (2"+i-l)(l+p'+p")  +  (2"-l)py. 

Hence  N  is  abundant  or  deficient  according  as 

a-iV=(2"+^-l)(l+p'+p")-py>0or  <0. 

Hrotsvitha,^^  a  nun  in  Saxony,  in  the  second  half  of  the  tenth  century, 
mentioned  the  perfect  numbers  6,  28,  496,  8128. 

Abraham  Ibn  Ezra^^"  (tll67),  in  his  commentary  to  the  Pentateuch, 
Ex.  3,  15,  stated  that  there  is  only  one  perfect  number  between  any  two 
successive  powers  of  10. 

Rabbi  Josef  b.  Jehuda  Ankin^^'',  at  the  end  of  the  twelfth  century,  recom- 
mended the  study  of  perfect  numbers  in  the  program  of  education  laid  out 
in  his  book  "Healing  of  Souls." 

Jordanus  Nemorarius^^  (tl236)  stated  (in  Book  VII,  props.  55,  56)  that 
every  multiple  of  a  perfect  or  abundant  number  is  abundant,  and  every 
divisor  of  a  perfect  number  is  deficient.  He  attempted  to  prove  (VII,  57) 
the  erroneous  statement  that  all  abundant  numbers  are  even. 

Leonardo  Pisano,  or  Fibonacci,  cited  in  his  Liber  Abbaci^^  of  1202, 
revised  about  1228,  the  perfect  numbers 

1  2^(2^-1)  =6,  i  2^(2^-1)  =28,  |  2^(2^-1)  =496, 

excluding  the  exponent  4  since  2^  —  1  is  not  prime.     He  stated  that  by  pro- 
ceeding so,  you  can  find  an  infinitude  of  perfect  numbers. 

i^Manuscript  952,  2,  Suppl.  Arabe,  Bibliotheque  imperiale,  Paris.  Textual  transl.,  except 
of  the  proofs  which  are  given  in  modem  algebraic  notation  as  foot-notes  [as  numbers 
were  represented  by  line,  in  the  manuscript],  by  Franz  Woepcke,  Journal  Asiatique, 
(4),  20,  1852,  420-9. 

"See  Ch.  Magnin,  Theatre  de  Hrotsvitha,  Paris,  1845. 

""Mikrooth  Gedoloth,  Warsaw,  1874  ("Large  Bible"  in  Hebrew).  Samuel  Ben  Sdadias  Ibn 
Motot;  a  Spaniard,  wrote  in  1370  a  commentary  on  Ibn  Ezra's  commentary,  Perush  ai 
Perush  Ibn  Ezra,  Venice,  1554,  p.  19,  noting  the  perfect  numbers  6,  28,  496,  8128,  and 
citing  EucUd's  rule.  Steinschneider,  in  his  book  on  Ibn  Ezra,  Abh.  Geschichte  Math. 
Wiss.,  1880,  p.  92,  stated  that  Ibn  Ezra  gave  a  rule  for  finding  all  perfect  numbers. 
As  this  rule  is  not  given  in  the  Mikrooth  Gedoloth  of  1874,  Mr.  Ginsburg  of  Columbia 
University  infers  the  existence  of  a  fuller  version  of  Ibn  Ezra's  commentary. 
"^Quoted  by  Giideman,  Das  Jiidische  Unterrichtswesen  wahrend  der  Spanish  Arabischen 
Periode,  Wien,  1873. 

*^In  hoc  opere  contenta.  Arithmetica  decern  libris  demonstrata  ....  Epitome  i  libros 
arithmeticos  diui  Seuerini  Boetij  .  .  .  ,  Paris,  1496,  1503,  etc.  It  contains  Jordanus' 
"Elementa  arithmetica  decern  libris,  demonstrationibus  Jacobi  Fabri  Stapulensis,"  and 
"Jacobi  Fabri  Stapulenais  epitome  in  duos  Hbros  arithmeticos  diui  Seuerini  Boetij." 

i^Il  Liber  Abbaci  di  Leonardo  Pisano.  Roma,  1857,  p.  283  (Scritti,  vol.  1). 


6  History  of  the  Theory  of  Numbers.  [Chap.  I 

In  the  manuscripts^  Codex  lat.  Monac.  14908,  a  part  dated  1456  and  a 
part  1461,  the  first  four  perfect  numbers  are  given  (J.  33')  as  usual  and  the 
fifth  perfect  number  is  stated  correctly  to  be  33550336. 

Nicolas  Chuquet^^  defined  perfect,  deficient,  and  abundant  numbers, 
indicated  a  proof  of  EucHd's  rule  and  stated  incorrectly  that  perfect  num- 
bers end  alternately  in  6  and  8. 

Luca  Paciuolo,  de  Borgo  San  Sepolcro,^^  gave  (f.  6)  Euclid's  rule,  saying 
one  must  find  by  experiment  whether  or  not  the  factor  1+2+4+.  .  .  is 
prime,  stated  (f.  7)  that  the  perfect  numbers  end  alternately  in  6  and  8,  as 
6,  28,  496,  etc.,  to  mfinity.  In  the  fifth  article  (ff.  7,  8),  he  illustrated  the 
finding  of  the  aliquot  divisors  of  a  perfect  number  by  taking  the  case  of  the 
fourteenth  perfect  number  9007199187632128.  He  gave  its  half,  then  the 
half  of  the  quotient,  etc.,  until  after  26  divisions  by  2,  the  odd  number 
134217727,  marked  "  Indi^dsibilis "  [prime].  Dividing  the  initial  number 
by  these  quotients,  he  obtained  further  factors  [1,2,...,  2'^,  but  written  at 
length].  The  proposed  number  is  said  to  be  evidently  perfect,  since  it  is  the 
sum  of  these  factors  [but  he  has  not  employed  all  the  factors,  since  the  above 
odd  number  equals  2'-'^  —  1  and  has  the  factor  2^  —  1  =  7] .  Although  Paciuolo 
did  not  list  the  perfect  numbers  between  8128  and  90 .  .  .8,  the  fact  that  he 
called  the  latter  the  fourteenth  perfect  number  imphes  the  error  expressly 
committed  bj^  Bo^illus.^" 

Thomas  Bradwardin^"  (1290-1349)  stated  that  there  is  only  one  perfect 
number  (6)  between  1  and  10,  one  (28)  up  to  100,  496  up  to  1000,  8128  up 
to  10000,  from  which  these  numbers,  taken  in  order,  end  alternately  in  6 
and  8.     He  then  gave  EucUd's  rule. 

Faber  Stapulensis^^  or  Jacques  Lefevre  (born  at  Etaples  1455,  tl537) 
stated  that  all  perfect  numbers  end  alternately  in  6  and  8,  and  that  Euclid's 
rule  gives  all  perfect  numbers. 

Georgius  Valla^^  gave  the  first  four  perfect  numbers  and  observed  that 

"The  manuscript  is  briefly  described  by  Gerhardt,  Monatsber.     Berlin  Ak.,   1870,   141-2. 

See  Catalogus  codicum  latinorum  bibliothecae  regiae  Monacensis,  Tomi  II,  pars  II, 

codices  nuna.  11001-15028  complectus,  Munich,  1876,  p.  250.     An  extract  of  ff.  32-34 

on  perfect  numbers  was  published  by  MaximiUan  Curtze,  BibUotheca  Mathematica, 

(2),  9,  1895,  39-42. 
"Triparty  en  la  science  des  nombres,  manuscript  No.  1436,  Fonds  Fran^ais,  BibliothSque 

Nationale  de  Paris,  written  at  Lyons.  1484.     Published  by  Aristide  Marre,  Bull.  Bibl. 

Storia  Sc.  Mat.  et  Fis..  13  (1880),  593-659,  693-814;  14  (1881),  417-460.     See  Part  1, 

Ch.  Ill,  3,  619-621,  manuscript,  ff.  20-21. 
"Summa  de  Arithmetica  geometria  proportioni  et  proportionalita.     [Suma     .  .  ,  Venice,  1494.] 

Toscolano,  1523  (two  editions  substantially  the  same). 
"Arithmetica  thome  brauardini.     Tractatus  perutilis.     In  arithmetica  speculativa  a  magistro 

thoma  Brauardini  ex  libris  eucUdis  boecij  &  ahorum  qua  optimne  excerptus.     Parisiis, 

1495,  7th  unnumbered  page. 
Arithmetica  Speculativa  Thome  Brauardini  nuper  mendis  Plusculis  tersa  et  diligenter  Impressa, 

Parisiis  [1502],  6th  and  7th  unnumbered  pages.    Also  undated  edition  [1510],  3d  page. 
"Epitome  (iii)  of  the  arithmetic  of  Boethius   in   Faber's   edition  of  Jordanus,"  1496,  etc. 

Also  in  Introductio  Jacobi  fabri  Stapulesis  in  Arithmecam  diui  Seuerini  Boetij  pariter 

Jordani,   Paris,   1503,   1507.     Also  in  Stapulensis,  Jacobi  Fabri,  Arithmetica  Boethi 

epitome,  Basileae,  1553,  40. 
"De  expetendis  et  fvgiendis  rebvs  opvs,  Aldus,  1501.     Liber  I  (  =  Arithmeticae  I),  Cap.  12. 


Chap.  I]      PERFECT,  MULTIPLY  PERFECT,  AND   AmICABLE   NuMBERS.  7 

"these  happen  to  end  in  6  or  8. .  .and  these  terminal  numbers  will  always 
be  found  alternately." 

Carolus  Bovillus^"  or  Charles  de  Bouvelles  (1470-1553)  stated  that  every 
perfect  number  is  even,  but  his  proof  applies  only  to  those  of  Euclid's  type. 
He  corrected  the  statement  of  Jordanus^^  that  every  abundant  number  is 
even,  by  citing  45045  [  =  5-9-7-ll-13]  and  its  multiples.  He  stated  that 
2"  — 1  is  a  prime  if  n  is  odd,  expUcitly  citing  511  [  =  7-73]  as  a  prime.  He 
listed  as  perfect  numbers  2"~^(2'*  — 1),  n  ranging  over  all  the  odd  numbers 
^  39  [Cataldi^  later  indicated  that  8  of  these  are  not  perfect].  He  repeated 
the  error  that  all  perfect  numbers  end  alternately  in  6  and  8.  He  stated 
(f.  175,  No.  25)  that  if  the  sum  of  the  digits  of  a  perfect  number  >6  be 
divided  by  9,  the  remainder  is  unity  [proved  for  perfect  numbers  of  Euclid's 
type  by  Cataldi,^^  p.  43].  He  noted  (f.  178)  that  any  divisor  of  a  perfect 
number  is  deficient,  any  multiple  abundant.  He  stated  (No.  29)  that  one  or 
both  of  6n=i=l  are  primes  and  (No.  30)  conversely  any  prime  is  of  the  form 
6n=t  1  [Cataldi,^  p.  45,  corrects  the  first  statement  and  proves  the  second]. 
He  stated  (f.  174)  that  every  perfect  number  is  triangular,  being  2"  (2''  —  l)/2. 

Martinus^^  gave  the  first  four  perfect  numbers  and  remarked  that  they 
end  alternately  in  6  and  8. 

Gasper  Lax'^^  stated  that  the  perfect  numbers  end  alternately  in  6  and  8. 

V.  Rodulphus  Spoletanus^^  was  cited  by  Cataldi,'*^  with  the  implication 
of  errors  on  perfect  numbers.     [Copy  not  seen.] 

Girardus  Ruffus^^  stated  that  every  perfect  number  is  even,  that  most 
odd  numbers  are  deficient,  that,  contrary  to  Jordanus,^^  the  odd  number 
45045  is  abundant,  and  that  for  n  odd  2^*  — 1  always  leads  to  a  perfect  num- 
ber, citing  7,  31,  127,  511,  2047,  8191  as  primes  [the  fourth  and  fifth  are 
composite]. 

Feliciano^^  stated  that  all  perfect  numbers  end  alternately  in  6  and  8. 

Regius^^  defined  a  perfect  number  to  be  an  even  number  equal  to  the 
sum  of  its  aliquot  divisors,  indicated  that  511  and  2047  are  composite,  gave 
correctly  33550336  as  the  fifth  perfect  number,  but  said  the  perfect  numbers 

^''Caroli  Bouilli  Samarobrini  Liber  De  Perfectis  Numeris  (dated  1509  at  end),  one  (ff.  172-180) 
of  13  tracts  in  his  work,  Que  hoc  volumine  continetur:  Liber  de  intellectu,  .  .  .  De 
Numeris  Perfectis,  .  .  .  ,  dated  on  last  page,  1510,  Paris,  ex  ofEcina  Henrici  Stephani. 
Biography  in  G.  Maupin,  Opinions  et  Curiosit^s  touchant  la  Math.,  Paris,  1,  1901, 186-94. 

"Ars  Arithmetica  loannis  Martini,  Silicei:  in  theoricen  &  praxim.  1513,  1514.  Arithmetica 
loannis  Martini,  Scilicei,  Paris,  1519. 

"Arithmetica  speculatiua  magistri  Gasparis  Lax.     Paris,  1515,  Liber  VII,  No.  87  (end). 

*3De  proportione  proportionvm  dispvtatio,  Rome,  1515. 

"Divi  Severini  Boetii  Arithmetica,  dvobvs  discreta  hbris,  Paris,  1521;  ff.  40-44  of  the  commen- 
tary by  G.  Ruffus. 

"Libro  di  Arithmetica  &  Geometria  speculatiua  &  praticale:  Composto  per  maestro  Fran- 
cesco FeUciano  da  Lazisio  Veronese  Intitulato  Scala  Grimaldelli:  Nouamente  stampato. 
Venice,  1526  (p.  3),  1527,  1536  (p.  4),  1545, 1550,  1560,  1570,  1669,  Padoua,  1629,  Verona, 
1563,  1602. 

*Vtrivsqve  Arithmetices,  epitome  ex  uariis  authoribus  concinnata  per  Hvdalrichum  Regium. 
Strasburg,  1536.  Lib.  I,  Cap.  VI:  De  Perfecto.  Hvdalrichvs  Regius,  Vtrivsque.  .  . 
ex  variis  .  .  .  ,  Friburgi,  1550  [and  1543],  Cap.  VI,  fol.  17-18. 


8  History  of  the  Theory  of  Numbers.  [Chap.  I 

end  alternately  in  6  and  8.  A  multiple  of  an  abundant  or  perfect  number 
is  abundant,  a  divisor  of  a  perfect  number  is  deficient. 

Cardan^^  (1501-1576)  stated  that  perfect  numbers  were  to  be  formed 
by  Euclid's  rule  and  always  end  with  6  or  8;  and  that  there  is  one  between 
any  two  successive  powers  of  ten. 

De  la  Roche-^  stated  in  effect  that  2""^ (2"  —  1)  is  perfect  for  every  odd  n, 
citing  in  particular  130816  and  2096128,  given  by  n  =  9,  n  =  ll.  This 
erroneous  law  led  him  to  believe  that  the  successive  perfect  numbers  end 
alternately  in  6  and  8. 

Noviomagus-^  or  Neomagus  or  Jan  Bronckhorst  (1494-1570)  gave 
Euclid's  rule  correctly  and  stated  that  among  the  first  10  numbers,  6  alone  is 
perfect, .  .  . ,  among  the  first  10000  numbers,  6,  28,496,  8128  alone  are  perfect, 
etc.,  etc.  [implying  falsely  that  there  is  one  and  but  one  perfect  number 
with  any  prescribed  number  of  digits].  In  Lib.  II,  Cap.  IV,  is  given  the 
sieve  (or  crib)  of  Eratosthenes,  with  a  separate  column  for  the  multiples 
of  3,  a  separate  one  for  the  multiples  of  5,  etc. 

WilUchius^'^  (tl552)  listed  the  first  four  perfect  numbers  and  stated  that 
to  these  are  to  be  added  a  very  few  others,  whose  nature  is  that  they  end 
either  in  6  or  8. 

Michael  StifeP^  (1487-1567)  stated  that  all  perfect  numbers  except  6 
are  multiples  of  4,  while  4(8-1),  16(32-1),  64(128-1),  256(512-1),  etc., 
to  infinity,  are  perfect  [error,  Kraft^°].  He  later^-  repeated  the  latter  error, 
listing  as  perfect 

2X3,  4X7,  16X31,  64X127,  256X511,  1024X2047, 

"&  so  fort  an  ohn  end."     Every  perfect  munber  is  triangular. 

Peletier^^  (1517-1582)  stated  (1549,  V  left;  1554,  p.  20)  that  the  perfect 
numbers  end  in  6  or  8,  that  there  is  a  single  perfect  number  between  any 
two  successive  powers  of  10,  and  (1549,  C  III  left;  1554,  pp.  270-1)  that 
4(8-1),  16(32-1),  64(128-1),  256(511),.  .  .are  perfect.  The  first  two 
statements  were  also  given  later  by  Peletier.^ 

^'Hieronimi  C.  Cardani  Medici   Mediolanensis,  Practica  Arithmetice,  &   Mensurandi  singu- 

laris.     Milan,  1537,  1539;  Xiirnberg,  1541,  1542,  Cap.  42,  de  proprietatibus  numerorum 

mirificis.     Opera  IV,  Lyon,  1663. 
-*Larismetique  &  Geometrie  de  maistre  Estienne  de  la  Roche  diet  Ville  Franche,  Nouuelle- 

ment  Imprimee  &  des  fautea  corrigee,  Lyon,  1538,  fol.  2,  verso.     Ed.  1,  1520. 
'"De  Nvmeris  libri  dvo   ....  authore  loanne  Nouiomago,  Paris,  1539,  Lib.  II,  Cap.  III. 

Reprinted,  Cologne,  1544;  Deventer,  1551.     Edition  by  G.  Frizzo,  Verona,  1901,  p.  132. 
'°Iodoci  Vvillichii  Reselliani,  Arithmeticae  libri  tres,  Argentorati,  1540,  p.  37. 
'^Arithmetica  Integra,  Norimbergae,  1544,  ff.  10,  11. 
"Die  Cosa  Cbristoffs  Rudolffs  Die  schonen  Exempeln  der  Coss  Durch  Michael  Stifel  Gebessert 

vnd  sehr  gemehrt,  Konigsperg  in  Preussen,  1553,  Anbang  Cap.  I,  f.  10  verso,  f.  11  (f. 

27  v.),  and  1571. 
"L'Arithmetiqve  de  lacqves  Peletier  dv  Mans,  departie  en  quatre  Liures,  Poitiers,  1549, 

1550,  1553.  .  .  .  ,  ff.  77  v,  78  r.  Reviie  e  augmentee  par  1'  Auteur,  Lion,  1554 

Troisieme  edition,  reucue  et  augmentee,  par  lean  de  Tovmes,  1607. 
"Arithmeticae  Practicae  methodvs  facilis,  per  Gemmam  Frisivm,  Medicvm,  ac  Mathematicum 

conscripta  ....   In  eandem  loannis  Steinii  &  lacobi  Peletarii  Annotationes.    Antver- 

piae,  1581,  p.  10. 


Chap.  I]     PERFECT,  MULTIPLY   PERFECT,  AND   AmICABLE   NUMBERS.  9 

Postello^^  stated  erroneously  that  130816  [  =  256-511]  is  perfect. 

Lodoico  Baeza^^  stated  that  Euchd's  rule  gives  all  perfect  numbers. 

Pierre  Forcadel"  (tl574)  gave  130816  as  the  fifth  perfect  number, 
implying  incorrectly  that  511  is  a  prime. 

Tartaglia^^  (1506-1559)  gave  an  erroneous  [Kraft ^^J  list  of  the  first 
twenty  perfect  numbers,  viz.,  the  expanded  forms  of  2"~^(2'*  — 1),  for  n  =  2 
and  the  successive  odd  numbers  as  far  as  n  =  39.  He  stated  that  the  sums 
1+2+4,  1+2+4+8, ..  .are  alternately  prime  and  composite;  and  that 
the  perfect  numbers  end  alternately  in  6  and  8.  The  third  ''notable  prop- 
erty" mentioned  is  that  any  perfect  number  except  6  yields  the  remainder  1 
when  divided  by  9. 

Robert  Recorde^^  (about  1510-1558)  stated  that  all  the  perfect  numbers 
under  6-10^  are  6,  28,  496,  8128,  130816,  2096128,  33550336,  536854528  [the 
fifth,  sixth,  eighth  of  these  are  not  perfect]. 

Petrus  Ramus^°  (1515-1572)  stated  that  in  no  interval  between  succes- 
sive powers  of  10  can  you  find  more  than  one  perfect  number,  while  in  many 
intervals  you  will  find  none.  At  the  end  of  Book  I  (p.  29)  of  his  Arith- 
meticae  libri  tres,  Paris,  1555,  Ramus  had  stated  that  6,  28,  496,  8128  are 
the  only  perfect  numbers  less  than  lOOpOO. 

Franciscus  Maurolycus*^  (1494-1575)  gave  an  argument  to  show  that 
every  perfect  number  is  hexagonal  and  hence  triangular. 

Peter  Bungus^^  (fieoi)  gave  (1584,  pars  altera,  p.  68)  a  table  of  20 
numbers  stated  erroneously  to  be  the  perfect  numbers  with  24  or  fewer 
digits  [the  same  numbers  had  been  given  by  Tartaglia^^].  In  the  editions 
of  1591,  etc.,  p.  468,  the  table  is  extended  to  include  a  perfect  number  of 
25  digits,  one  of  26,  one  of  27,  and  one  of  28.  He  stated  (1584,  pp.  70-71 ; 
1591,  pp.  471-2)  that  all  perfect  numbers  end  alternately  in  6  and  28; 
employing  Euclid's  formula,  he  observed  that  the  product  of  a  power  of  2 
ending  in  4  by  a  number  ending  in  7  itself  ends  in  28,  while  the  product  of 
one  ending  in  6  by  one  ending  in  1  ends  in  6.     He  verified  (1585,  pars 

^"Theoricae  Arithmetices  Compendium  h  Guilielmo  Postello,  Lutetiae,  1552,  a  syllabus  on  one 

large  sheet  of  arithmetic  definitions. 
"Nvmerandi  Doctrina,  Lvtetiae,  1555,  fol.  27-28. 

''L'Arithmeticqve  de  P.  Forcadel  de  Beziers,  Paris,  1556-7.     Livre  I  (1556),  fol.  12  verso. 
3*La  seconda  Parte  del  General  Trattato  di  Nvmeri,  et  Misvre  di  Nicolo  Tartagha,  Vinegia, 

1556,  f.  146  verso. 
L'  Arithmetiqve  de  Nicolas  Tartagha  Brescian   ....    Recueillie,  &  traduite  d'ltalien  en 

FranQois,  par  Gvillavme  Gosselin  de  Caen,  ....  Paris,  1578,  f.  98  verso,  f.  99. 
'®The  Whetstone  of  witte,  whiche  is  the  seconde  parte  of  Arithmetike,  London,  1557,  eighth 

unnumbered  page. 
^''Petri  Rami  Scholarum  Mathematicarum,  Libri  unus  et  triginta,  k  Lazaro  Schonero  recog- 

niti  &  emendati,  Francofvrti,  1599,  Libr.  IV  (Arith.),  p.  127,  and  Basel,  1578. 
"Arithmeticorvm  hbri  dvo,  Venetiis,  1575,  p.  10;  1580.     Published  with  separate  paging,  at 

end  of  Opuscula  mathematica. 
*^Mysticae  nvmerorvm  significationis  liber  in  dvas  divisvs  partes,  R.  D.  Petro  Bongo  Canonico 

Bergomate  avctore.     Bergomi.     Pars  prior,  1583,  1585.     Pars  altera,  1584. 
Petri  Bungi  Bergomatis  Numerorum  mysteria,  Bergomi,  1591,  1599,  1614,  Lutetiae  Parisio- 

rum,  1618,  all  four  with  the  same  text  and  paging.     Classical  and  biblical  citations  on 

numbers  (400  pages  on  1,  2,  .  .  ,  12).     On  the  1618  edition,  see  Font^s,  M6m.  Acad.  So. 

Toulouse,  (9),  5,  1893,  371-380. 


10  History  of  the  Theory  of  Numbers.  [Chap,  i 

prior,  p.  238;  1591,  p.  343)  for  the  first  seven  numbers  of  his  table  [two 
being  imperfect,  however]  that  the  sum  of  the  digits  of  a  perfect  number 
exceeds  by  unity  a  multiple  of  9.  Every  perfect  number  is  triangular 
(1591,  p.  270).  Every  multiple  of  a  perfect  number  is  abundant,  every 
divisor  deficient  (1591,  p.  464). 

Unicornus^^  (1523-1610)  cited  Bungus  and  repeated  his  error  that 
2"- 1  (2^  —  1)  is  always  perfect  for  n  odd  and  that  all  perfect  numbers  end 
alternately  in  6  and  8. 

Cataldi"^  (1548-1626)  noted  in  his  Preface  that  Paciuolo's^^  fourteenth  per- 
fect number  90.  .  .8  is  in  fact  abundant  since  it  arose  from  1+2+4+  •  ■  • 
+2^^  =  134217727, which  is  divisible  by  7,whereas  Paciuolo  said  it  was  prime. 
Citing  the  error  of  the  latter,  Bovillus,^°  and  others,  that  all  perfect  num- 
bers end  alternately  in  6  and  8,  Cataldi  observed  (p.  42)  that  the  fifth  per- 
fect number  is  33550336  and  the  sixth  is  8589869056,  from  8191  =2'^- 1  and 
131071=2^^  —  1,  respectively,  proved  to  be  primes  (pp.  12-17)  by  actually 
trying  as  possible  divisor  every  prime  less  than  their  respective  square  roots. 
He  gave  (pp.  17-22)  the  corresponding  work  showing  2^^  —  1  to  be  prime.  He 
stated  (p.  11)  that  2'*-!  is  a  prime  forn  =  2,  3,  5,  7,  13, 17, 19,  23,  29,  31,  37, 
remarking  that  the  prime  n  =  ll  does  not  yield  a  perfect  number  since 
(p.  5)  2^^  —  1=2047  =  23*89,  while  it  is  composite  if  n  is  composite.  He 
proved  (p.  8)  that  the  perfect  numbers  given  by  Euclid's  rule  end  in  6  or  8. 
He  gave  (pp.  28-40,  48)  a  table  of  all  divisors  of  all  even  and  odd  numbers 
^  800,  and  a  table  of  primes  <  750. 

Georgius  Henischiib^^  (1549-1618)  stated  that  the  perfect  numbers  end 
alternately  in  6  and  8,  and  that  one  occurs  between  any  two  successive 
powers  of  10.  He  applied  Euclid's  formula  without  restricting  the  factor 
2"—!  to  primes. 

Johan  Rudolff  von  Graff enried"*^  stated  that  all  perfect  numbers  are 
given  by  Euclid's  rule,  which  he  applied  without  restricting  2"  —  1  to  primes, 
expressly  citing  256X511  as  the  fifth  perfect  number.  Every  perfect 
number  is  triangular. 

Bachet  de  Mezirac''^  (1581-1638)  gave  (f.  102)  a  lengthy  proof  of 
Euclid's  theorem  that  2'*p  is  perfect  if  p  =  l+2+ . .  .  +2^*  is  a  prime,  but 

"De  I'arithmetica  vniversale  del  Sig.  loseppo  Vnicorno,  Venetia,  1598,  f.  57. 

"Trattato  de  nvmeri  perfetti  di  Pierto  Antonio  Cataldo,  Bologna,  16C3.  According  to  the 
Preface,  this  work  was  composed  in  1588.  Cataldi  founded  at  Bologna  the  Academia 
Erigende,  the  most  ancient  known  academy  of  mathematics;  his  interest  in  perfect 
numbers  from  early  youth  is  shown  by  the  end  of  the  first  of  his  "due  lettioni  fatte  nell' 
Academia  di  Perugia"  (G.  Libri,  Hist.  Sc.  Math,  en  ItaUe,  2d  ed.,  vol.  4,  Halle,  1865,  p. 
91).    G.  Wertheim,  BibHotheca  Math.,  (3),  3, 1902, 76-83,  gave  a  summary  of  the  Trattato. 

"Arithmetica  Perfecta  et  Demonstrata,  Georgii  Henischiib,  Augsburg  [1605],  1609,  pp.  63-64. 

"Arithmeticae  Logistica  Popularis  Librii  IIII.  Jn  welchen  der  Algorithmus  in  gantzen  Zahlen 
u.  Fracturen  .  .  .  .  ,  Bern,  1618,  1619,  pp.  236-7. 

*^Elementorum  arithmeticorum  libri  XIII  auctori  D  .  .  .  ,  a  Latin  manuscript  in  the  Biblio- 
thfeque  de  I'lnstitut  de  France.  On  the  inside  of  the  front  cover  is  a  comment  on  the 
sale  of  the  manuscript  by  the  son  of  Bachet  to  DaUbert,  treasurer  of  France.  A  general 
account  of  the  contents  of  the  manuscript  was  given  by  Henry,  Bull.  Bibl.  Storia  Sc.  Mat. 
e  Fis.,  12, 1879,  pp.  619-641.  The  present  detailed  account  of  Book  4,  on  perfect  numbers, 
was  taken  from  the  manuscript. 


Chap.  I]      PERFECT,  MULTIPLY   PERFECT,  AND   AMICABLE   NUMBERS.  11 

(f.  103,  verso)  is  abundant  if  p  is  composite.  Every  multiple  of  a  perfect 
or  abundant  number  is  abundant,  every  divisor  of  a  perfect  number  is 
deficient  (ff.  104  verso,  105).  The  product  of  two  primes,  other  than  2X3, 
is  deficient  (f.  105  verso).  The  odd  number  945  is  abundant,  the  sum  of  its 
ahquot  divisors  being  975  (f.  107).  Commenting  (f.  Ill  verso,  f.  112)  on 
the  statement  of  Boethius^  and  Cardan^^  that  the  perfect  numbers  end 
alternately  in  6  and  8,  he  stated  that  the  fourth  is  8128  and  the  fifth  is 
2096128  [an  error],  the  fifth  not  being  130816  =  256X511,  since  511  =7X73. 

Jean  Leurechon^^  (about  1591-1670)  stated  that  there  are  only  ten 
perfect  numbers  between  1  and  10^^,  listed  them  (noting  the  admirable 
property  that  they  end  alternately  in  6  and  8)  and  gave  the  twentieth  per- 
fect number.     [They  are  the  same  as  in  Tartaglia's^^  list.] 

Lantz^^  stated  that  the  perfect  numbers  are  2(4-1),  4(8-1),  16(32-1), 
64(128-1),  256(512-1),  1024(2048-1),  etc. 

Hugo  Sempilius^°  or  Semple  (Scotland,  1594-Madrid,  1654)  stated  that 
there  are  only  seven  perfect  numbers  up  to  40,000,000;  they  end  alternately 
in  6  and  8. 

Casper  Ens^^  stated  that  there  are  only  seven  perfect  numbers  <4-10'', 
viz.,  6,  28, 496, 8128, 130816,  1996128  [for  2096128],  33550336,  and  that  they 
end  alternately  in  6  and  8. 

Daniel  Schwenter^^  (1585-1636)  made  the  same  error  as  Casper  Ens.^^ 

Erycius  Puteanus^^  quoted  from  Martiano  Capella,  lib.  VII,  De  Nuptiis 
Philologiae,  to  the  effect  that  the  perfect  number  6  is  attributed  to  Venus; 
for  it  is  made  by  the  union  of  the  two  sexes,  that  is,  from  triad,  which  is 
male  since  it  is  odd,  and  from  diad,  which  is  feminine  since  it  is  even. 
Puteanus  said  that  the  perfect  numbers  in  order  are  6,  28,  496,  8128, 
130816,  2096128,  33550336,  and  gave  all  their  divisors  [implying  that  511, 
2047,  8191  are  primes],  and  stated  that  these  seven  and  all  the  remaining 
end  alternately  in  6  and  8.  Between  any  two  successive  powers  of  10  is  one 
perfect  number.     That  they  are  all  triangular  adds  perfection  to  the  perfect. 

Joannes  Broscius^"^  or  Brocki  remarked  that  there  is  no  perfect  number 
between  10000  and  10000000,  contrary  to  Stifel,^^  Bungus,^^  Sempilius.^" 
Puteanus,^^  and  the  author  of  Selectarum  Propositionum  Mathematicarum, 
quas  propugnavit,  Mussiponti,  Anno  1622,  Maximilianus  Willibaldus,  Baro 

^^Recreations  math^matiques,  Pont-^-Mousson,  1624;  London,  1633,  1653,  1674  (these  three 
EngUsh  editions  by  Wm.  Oughtred),  p.  92.  The  authorship  is  often  attributed  to 
Leurechon's  pupil  Henry  Van  Etten,  whose  name  is  signed  to  the  dedicatory  epistle. 
Cf.  Poggendorff,  Handworterbuch,  1863,  2,  p.  250  (under  C.  Mydorge);  Bibliotheque 
des  6crivains  de  la  compagnie  de  Jesus,  par  A.  de  Backer,  2,  1872,  731;  Biograpliie 
Generale,  31,  1872,  10. 

"Institutionum  Arithmeticarum  hbri  quatuor  h  loanne  Lantz,  Coloniae  Agrippinae,  1630,  p.  54. 

"De  Mathematicis  Disciphnis  hbri  Duodecim,  Antverpiae,  1635,  Lib.  2,  Cap.  3,  N.  10,  p.  46. 
There  is  (pp.  263-5)  an  index  of  writers  on  geometry  and  one  for  arithmetic. 

"Thaumaturgus  Math.,  Munich,  1636,  p.  101;  Coloniae.  1636,  1651;  Venice,  1706. 

"Dehciae  Physico-Mathematicae  oder  Mathemat:  vnd  Philosophische  Erquickstunden, 
part  I  (574  pp.),  Numberg,  1636,  p.  108. 

"De  Bissexto  Liber:  nova  temporis  facula  qua  intercalandi  arcana  ....  Lovanii,  1637; 
1640,  pp.  103-7.  Reproduced  by  J.  G.  Graevius,  Thesaurus  Antiquitatum  Romanarum 
(12  vols.,  1694-9),  Lugduni  Batavorum,  vol.  8. 


12  History  of  the  Theory  of  Numbers.  [Chap,  i 

in  Waldpurg.  WTiile  they  considered  511X256  and  2047X1024  as  perfect, 
511  has  the  factor  7,  and  (as  pointed  out  to  him  by  Stanislaus  Pudlowski) 
2047  has  the  factor  23.     Broscius  stated  that 

2^-1  has  the  factor  3   5  7   11    13    17   19   23   29   31 
if  n  is  a  multiple  of  2   4   3    10   12     8    18   11   28     5. 

The  contents  of  the  second  dissertation  are  given  below  under  the  date  1652. 

Ren^  Descartes,^^  in  a  letter  to  Mersenne,  November  15,  1638,  thought 
he  could  prove  that  every  even  perfect  number  is  of  Euclid's  type,  and  that 
every  odd  perfect  number  must  have  the  form  ps^,  where  p  is  a  prime.  He 
saw  no  reason  why  an  odd  perfect  number  may  not  exist.  For  p  =  22021, 
s  =  3'7-ll-13,  ps^  would  be  perfect  if  p  were  prime  [but  p  =  61-19^].  In  a 
letter  to  Frenicle,  January  9, 1639,  Oeuvres,  2,  p.  476,  he  expressed  his  belief 
that  an  odd  perfect  number  could  be  found  by  replacing  7,  11,  13  in  s  by 
other  values. 

Fermat^^  stated  that  he  possessed  a  method  of  solving  all  questions 
relating  to  aliquot  parts.  Citing  this  remark,  Frenicle^'  challenged  Fermat 
to  find  a  perfect  number  of  20  or  21  digits.  Fermat^^  replied  that  there  is 
none  with  20  or  21  digits,  contrary  to  the  opinion  of  those  who  believe 
that  there  is  a  perfect  number  between  any  two  consecutive  powers  of  10. 

Fermat,^^  in  a  letter  to  Mersenne,  June  (?),  1640,  stated  three  proposi- 
tions which  he  had  proved  not  without  considerable  trouble  and  which  he 
called  the  basis  of  the  discovery  of  perfect  numbers:  if  n  is  composite,  2"  —  1 
is  composite;  if  n  is  a  prime,  2"  — 2  is  divisible  by  2n,  and  2"  — 1  is  divisible 
by  no  prime  other  than  those  of  the  form  2kn-\-l  [cf .  Euler^'].  For  example, 
2"-l  =  23-89,  2^^-l  has  the  factor  223.  Also  2"^^-!  has  the  factor  47, 
Oeuvres,  2,  p.  210,  lett-er  to  Frenicle,  October  18,  1640. 

Mersenne^°  (1588-1648)  stated  that,  of  the  28  numbers*  exhibited  by 

"De  numeris  perfectis  disceptatio  qua  ostenditur  a  decern  millibus  ad  centies  centena  millia, 
nullum  esse  perfectum  numenim  atque  ideo  ab  unitate  usque  ad  centies  centena  millia 
quatuor  tantum  perfectos  numerari,  Amsterdam,  1638.  Reproduced  as  the  first  (pp. 
115-120)  of  two  dissertations  on  perfect  numbers,  they  forming  pp.  111-174  of  Apologia 
pro  Aristotele  &  Evchde,  contra  Petrvm  Ramvm,  &  aUos.  Addititiae  sunt  Dvae  Discep- 
tationes  de  Nvmeris  Perfectis.  Authore  loanne  Broscio,  Dantiaci,  1652  (with  a  some- 
what different  title,  Amsterdam,  1699). 

"Oeuvres  de  Descartes,  II,  Paris,  1898,  p.  429. 

s«Oeuvres  de  Fermat,  2,  Paris,  1894,  p.  176;  letter  to  Mersenne,  Dec.  26,  1638. 

*'Oeuvres  de  Fermat,  2,  p.  185;  letter  to  Mersenne,  March,  1640. 

osQeuvres,  2,  p.  194;  letter  to  Mersenne,  May  (?),  1640. 

"Oeuvres  de  Fermat,  2,  pp.  198-9;  Varia  Opera  Math.  d.  Petri  de  Fermat,  Tolosae,  1679,  p. 
177;  Precis  des  Oeuvres  math,  de  P.  Fermat  et  de  1'  Arithmdtique  de  Diophante,  par  E. 
Brassinne,  M6m.  Ac.  Imp.  Sc.  Toulouse,  (4),  3,  1853,  149-150. 

•°F.  Marini  Mersenni  minimi  Cogitata  Physico  Mathematica,  Parisiis,  1644.  Praefatio 
Generahs,  No.  19.  C.  Henry  (Bull.  Bibl.  Storia  Sc.  Mat.  e  Fis.,  12,  1879,  524-6)  beheved 
that  these  remarks  were  taken  from  letters  from  Fermat  and  Frenicle,  and  that  Mersenne 
had  no  proof.  A  similar  opinion  was  expressed  by  W.  W.  Rouse  Ball,  Messenger 
Math.,  21,  1892,  39  (121).  On  documents  relating  to  Mersenne  see  Tinterm^diaire  des 
math.,  2,  1895,  6;  8,  1901,  105;  9,  1902,  101,  297;  10,  1903,  184.     Cf.  Lucas."* 

*Only  24  were  given  by  Bungus.  While  his  table  has  28  lines,  one  for  each  number  of  digits, 
there  are  no  entry  of  numbers  of  5,  11,  17,  23  digits. 


Chap.  I]     PERFECT,  MULTIPLY  PERFECT,  AND   AMICABLE   NUMBERS.  13 

Bungus,^^  chap.  28,  as  perfect  numbers,  20  are  imperfect  and  only  8  are 
perfect : 

6,     28,    496,     8128,     23550336  [for  33. .  .],     8589869056, 
137,438691328,     2305843008139952128, 

which  occur  at  the  lines  marked  1,  2,  3,  4,  8,  10,  12  and  29  [for  19]  of 
Bungus'  table  [indicating  the  number  of  digits].  Perfect  numbers  are  so 
rare  that  only  eleven  are  known,  that  is,  three  different  from  those  of 
Bungus;  norf  is  there  any  perfect  number  other  than  those  eight,  unless 
you  should  surpass  the  exponent  62  in  1+2+2^+ ..  .  The  ninth  perfect 
number  is  the  power  with  the  exponent  68  less  1;  the  tenth,  the  power  128 
less  1 ;  the  eleventh,  the  power  258  less  l,i.e.,  the  power  257,  decreased  by 
unity,  multiplied  by  the  power  256.  [The  first  11  perfect  numbers  are 
thus  said  to  be  2"-'(2"-l)  for  n  =  2,  3,  5,  7,  13,  17,  19,  31,  67,  127,  257,  in 
error  as  to  n  =  61,  67,  89,  107  at  least.]  He  who  would  find  11  others  will 
know  that  all  analysis  up  to  the  present  will  have  been  exceeded,  and  will 
remember  in  the  meantime  that  there  is  no  perfect  number  from  the  power 
17000  to  32000,  and  no  interval  of  powers  can  be  assigned  so  great  but 
that  it  can  be  given  without  perfect  numbers.  For  example,  if  the  exponent 
be  1050000,  there  is  no  larger  exponent  n  up  to  2090000  for  which  2"  — 1 
is  a  prime.  One  of  the  greatest  difficulties  in  mathematics  is  to  exhibit  a 
prescribed  number  of  perfect  numbers;  and  to  tell  if  a  given  number  of 
15  or  20  digits  is  prime  or  not,  all  time  would  not  suffice  for  the  test,  what- 
ever use  is  made  of  what  is  already  known. 

Mersenne"  stated  that  2^  —  1  is  a  prime  if  p  is  a  prime  which  exceeds 
by  3,  or  by  a  smaller  number,  a  power  of  2  with  an  even  exponent.  Thus 
2^-1  is  a  prime  since  7  =  2^3;  again,  since  67  =  3+2*^,  2^^  +  1  =  1... 7 
[for  2®^  — 1]  is  a  prime  and  leads  to  a  perfect  number  [error  corrected  by 
Cole^^^].  Understand  this  only  of  primes  2^  —  1.  Wherefore  this  property 
does  not  belong  to  the  prime  5,  but  to  3,  7,  31,  127,  8191,  131071,  524287, 
2147483647,  and  all  such.  Numbers  expressible  as  the  sum  or  difference 
of  two  squares  in  several  ways  are  composite,  as  65  =  1+64  =  16+49.  As 
he  speaks  of  Frenicle's  knowledge  of  numbers,  at  least  part  of  his  results 
are  doubtless  due  to  the  latter. 

In  1652,  J.  Broscius  (Apologia,^^  p.  121)  observed  that  while  perfect 
numbers  were  deduced  by  Euclid  from  geometrical  progressions,  they  may 
be  derived  from  arithmetical  progressions: 

6  =  1+2+3,   28  =  1+2+3+4+5+6+7,   496  =  1+2+3+ ... +31. 

fNeque  enim  vllus  est  alius  perfectus  ab  illis  octo,  nisi  superes  exponentem  numerum  62, 
progressionis  duplae  ab  1  incipientis.  Nonus  enim  perfectus  est  potestas  exponentis  68, 
minus  1.  Decimus,  potestas  exponentis  128,  minus  1.  Vndecimus  denique,  potestas 
258,  minus  1,  hoc  est  potestas  257,  unitate  decurtata,  multiplicata  per  potestatem  256. 

*T.  Marini  Mersenni  Novarvm  Observationvm  Physico-Mathematicarum^  Tomvs  III,  Parisiis, 
1647,  Cap.  21,  p.  182.  The  Reflectiones  Physico-Math.  begin  with  p.  63;  Cap.  21  is 
quoted  in  Oeuvres  de  Fennat,  4,  1912,  pp.  67-8. 


14  History  of  the  Theory  of  Numbers.  [Chap,  i 

He  stated  that  while  perfect  numbers  end  with  6  or  28,  the  proof  by  Bungus*' 
does  not  show  that  they  end  alternately  with  6  and  28,  since  Bungus 
included  imperfect  as  well  as  perfect  numbers.  The  numbers  130816  and 
2096128,  cited  as  perfect  by  Puteanus,^^  are  abundant.  After  giving  a 
table  of  the  expanded  form  of  2"  forn  =  0,  1, .  .  . ,  100,  Broscius  (p.  130,  seq.) 
gave  a  table  of  the  prime  divisors  of  2"  —  1  (n  =  1, .  . . ,  100),  but  showing  no 
prime  factor  when  n  is  any  one  of  the  primes,  other  than  11  and  23,  less 
than  100.  For  n  =  ll,  the  factors  are  23,  89;  for  n  =  23,  the  factor  47  is 
given.  Thus  omitting  unity,  there  remain  only  23  numbers  out  of  the  first 
hundred  which  can  possibly  generate  perfect  numbers.  Contrary  to  Car- 
dan,^^  but  in  accord  with  Bungus,^^  there  is  (p.  135)  no  perfect  number 
between  10*  and  10\  Of  Bungus'  24  numbers,  only  10  are  perfect  (pp. 
135-140):  those  with  1,  2,  3,  4,  8,  10,  12,  18,  19,  22  digits,  and  given  by 
2'-i(2'*-l)  for  n  =  2,  3,  5,  7,  13,  17,  19,  29,  31,  37,  respectively.  The  pri- 
maUty  of  the  last  three  was  taken  on  the  authority  of  unnamed  predecessors. 

There  are  only  21  abundant  numbers  between  10  and  100,  and  all  of 
them  are  even;  the  only  odd  abundant  number  <1000  is  945,  the  sum  of 
whose  aliquot  di\isors  is  975  (p.  146).  The  statement  by  Lucas,  Th^orie 
des  nombres,  1,  Paris,  1891,  p.  380,  Ex.  5,  that  3^-5-79  [deficient]  is  the 
smallest  abundant  number  is  probably  a  misprint  for  945  =  3^-5-7.  This 
error  is  repeated  in  Encyclopedic  Sc.  Math.,  I,  3,  Fas.  1,  p.  56. 

Johann  Jacob  Heinlin^-  (1588-1660)  stated  that  the  only  perfect  num- 
bers <4-10'  are  6,  28,  496,  8128,  130816,  2096128,  33550336,  and  that  all 
perfect  numbers  end  alternately  in  6  and  8. 

Andrea  Tacquet^^  (Antwerp,  1612-1660)  stated  (p.  86)  that  Euclid's 
rule  gives  all  perfect  numbers.  Referring  to  the  11  numbers  given  as 
perfect  by  Mersenne,^^  Tacquet  said  that  the  reason  why  not  more  have 
been  found  so  far  is  the  greatness  of  the  numbers  2^  —  1  and  the  vast  labor 
of  testing  their  primaUty. 

Frenicle^  stated  in  1657  that  EucUd's  formula  gives  all  the  even  perfect 
numbers,  and  that  the  odd  perfect  numbers,  if  such  exist,  are  of  the  form 
p/c^,  where  p  is  a  prime  of  the  form  4n+l  [cf.  Euler^^]. 

Frans  van  Schooten^^  (the  younger,  1615-1660)  proposed  to  Fermat 
that  he  prove  or  disprove  the  existence  of  perfect  numbers  not  of  Euclid's 
type. 

Joh.  A.  Leuneschlos^^  remarked  that  the  infinite  multitude  of  numbers 
contains  only  ten  perfect  numbers;  he  who  will  find  ten  others  will  know 

'*Joh.  Jacobi  Heinlini,  Synopsis  Math,  praecipuas  totius  math ....  Tubmgae,  1653.     Synopsis 

Math.  Universalis,  ed.  Ill,  Tubingae,  1679,  p.  6.     English  translation  of  last  by  Venterus 

Mandey,  London,  1709,  p.  5. 
"Arithmeticae  Theoria  et  Praxis,  Lovanii,  1656  and  1682  (same  paging),  [1664,  1704].     Hia 

opera  math.,  Antwerpiae,  1669,  does  not  contain  the  Arithmetic. 
"Correspondence  of  Chr.  Huygens,  No.  389;  Oeuvres  de  Fermat,  3,  Paris,  1896,  p.  567. 
"Oeuvres  de  Huygens,  II,  Correspondence,  No.  378,  letter  from  Schooten  to  J.  Wallis,  Mar.  18, 

1658.     Oeuvres  de  Fermat,  3,  Paris,  1896,  p.  558. 
"Mille  de  Quantitate  Paradoxa  Sive  Admiranda,  Heildelbergae,  1658,  p.  11,  XLVI,  XLVII. 


Chap.  I]     PERFECT,  MULTIPLY  PERFECT,  AND   AmICABLE   NUMBERS.  15 

that  he  has  surpassed  all  analysis  up  to  the  present.  Goldbach"  called 
Euler's  attention  to  these  remarks  and  stated  that  they  were  probably 
taken  from  Mersenne,  the  true  sense  not  being  followed. 

Wm.  Leybourn^^  hsted  as  the  first  ten  perfect  numbers  and  the  twentieth 
those  which  occur  in  the  table  of  Bungus.^^  "The  number  6  hath  an  emi- 
nent Property,  for  his  parts  are  equal  to  himself." 

Samuel  Tennulius,  in  his  notes  (pp.  130-1)  on  lamblicus,^  1668,  stated 
that  the  perfect  numbers  end  alternately  in  6  and  8,  and  included  130816  = 
256-511  and  2096128  =  1024-2047  among  the  perfect  numbers. 

Tassius®^  stated  that  all  perfect  numbers  end  in  6  or  8.  Any  multiple 
of  a  perfect  or  abundant  number  is  abundant,  any  divisor  of  a  perfect 
number  is  deficient.  He  gave  as  the  first  eight  known  perfect  numbers  the 
first  eight  listed  by  Mersenne.^" 

Joh.  Wilh.  Pauli^°  (Philiatrus)  noted  that  if  2"  — 1  is  a  prime,  n  is,  but 
not  conversely.  For  n  =  2,  3,  5,  7,  13,  17,  19,  2"-l  is  a  prime;  but  2^^-l 
is  divisible  by  23,  2^^  — 1  by  47,  and  2*^  —  1  by  83,  the  three  divisors  being 
2n+l. 

G.  W.  Leibniz'^^  quoted  in  1679  the  facts  stated  by  Pauli  and  set  himself 
the  problem  to  find  the  basis  of  these  facts.  Returning  about  five  years 
later  to  the  subject  of  perfect  numbers,  Leibniz  implied  incorrectly  that 
2^^  —  1  is  a  prime  if  and  only  if  p  is. 

Jean  Prestet^^  (tl690)  stated  that  the  fifth, .  .  . ,  ninth  perfect  numbers  are 

23550336  [for  33. . .],  8589869056,  137438691328,  238584300813952128  [for 

2305. .  .39952128],     2''^-2^'\ 

[Hence  2'*-^(2'*-l)  for  7i  =  13,  17,  19,  31,  257.  The  numerical  errors  were 
noted  by  E.  Lucas,i24  p  7g4  j 

Jacques  Ozanam^^  (1640-1717)  stated  that  there  is  an  infinitude  of  perfect 
numbers  and  that  all  are  given  by  Euclid's  rule,  which  is  to  be  applied  only 
when  the  odd  factor  is  a  prime. 

Charles  de  Neuveglise^^  proved  that  the  products  3-4, . . .,  8-9  of  two 
consecutive  numbers  are  abundant.  All  multiples  of  6  or  an  abundant 
number  are  abundant. 

"Correspondence  Math.  Phy8.,ed.,Fus8, 1,  1843;  letters  to  Euler,  Oct.  7,  1752  (p.  584),  Nov.  18 

(p.  593). 
'^Arithmetical  Recreations;  or  Enchriridion  of  Arithmetical  Questions  both  Delightful  and 

Profitable,  London,  1667,  p.  143. 
"Arithmeticae  Empiricae  Compendium,  Johannis  Adolfi  Tassii.     Ex  recensione  Henrici  Siveri, 

Hamburgi,  1673,  pp.  13,  14. 
^"De  nvmiero  perfecto,  Leipzig,  1678,  Magister-disputation. 
"Manuscript  in  the  Hannover  Library.     Cf.  D.  Mahnke,  Bibhotheca  Math.,  (3),  13,  1912-3, 

53-4,  260. 
"Nouveaux  elemens  des  Mathematiques,  ou  Principes  generaux  de  toutes  les  sciences,  Paris, 

1689,  I,  154-5. 
"Recreations  mathematiques  et  physiques,  Paris  and  Amsterdam,  2  vols.,  1696,  I,  14,  15. 
"Traits  methodique  et  abreg6  de  toutes  les  mathematiques,  Trevoux,  1700,  tome  2  (L'arith- 

m^tique  ou  Science  des  nombres),  241-8. 


16  History  of  the  Theory  of  Numbers.  [Chap,  i 

John  Harris,"^  D.  D.,  F.  R.  S.,  stated  that  there  are  but  ten  perfect 
numbers  between  unity  and  one  million  of  millions. 

John  Hiir^  stated  that  there  are  only  nine  perfect  numbers  up  to  a 
hundred  thousand  million.  He  gave  (pp.  147-9)  a  table  of  values  of  2" 
forn  =  l,. . .,  144. 

Christian  Wolf"  (1679-1754)  discussed  perfect  numbers  of  the  form 
y"x  [where  x,  y  are  primes].     The  sum  of  its  aliquot  parts  is 

l+y+  .  .  .  +i/"+a:+?/x+  .  .  .  +i/""'x, 
which  must  equal  y'^x.     Thus 

x  =  {l-\-y+  . .  .  +2/")M  d  =  y^-l-y-  .  .  .  -7/""^ 

He  stated*  that  x  is  an  integer  only  when  d  =  l,  and  that  this  requires 
y  =  2,  x  =  l  +2+  .  .  .  +2".  Then  if  this  x  is  a  prime,  2"x  is  a  perfect  number. 
This  is  said  to  be  the  case  forn  =  8  and  n  =  10,  since  2^  —  1  =  51 1  and  2^'  —  1  = 
2047  are  primes,  errors  pointed  out  bj^  Euler.^^  A.  G.  Kastner^^  was  not 
satisfied  with  the  argument  leading  to  the  conclusion  y  =  2. 
Jacques  Ozanam^^  listed  as  perfect  numbers 

2(4-1),  4(8-1),  16(32-1),  64(128-1),  256(512-1),  1024(2048-1),. . . 

without  expUcit  mention  of  the  condition  that  the  final  factor  shall  be  prime, 
and  stated  that  perfect  numbers  are  rare,  only  ten  being  known,  and  all 
end  in  6  and  8  alternately.     [Criticisms  by  Montucla,®^  Gruson.-^°°] 

Johann  Georg  Liebnecht^°  said  there  were  scarcely  5  or  6  perfect  num- 
bers up  to  4.10";  they  always  end  alternately  in  6  and  8. 

Alexander  ]Malcolm^^  observed  that  it  is  not  yet  proved  that  there  is  no 
perfect  number  not  in  Euclid's  set.  He  stated  that,  if  pA  is  a  perfect 
number,  where  p  is  a  prime,  and  if  M<p  and  M  is  not  a  factor  of  A,  then 
MM  is  an  abundant  number  [probably  a  misprint  for  MA,  as  the  condi- 
tions are  satisfied  when  p  =  7,  .4=4,  M  =  5,  and  MA  =20  is  abundant,  while 
Af^  =  25  is  deficient]. 

Christian  Wolf^-  made  the  same  error  as  Casper  Ens.^^ 

^'Lexicon  Technicum,  or  an  Universal  English  Dictionarj'  of  Arts  and  Sciences,  vol.  I,  London, 
1704;  ed.  5,  vol.  2,  London,  1736. 

"Arithmetik,  London,  ed.  2,  1716,  p.  3. 

^'Elementa  Matheseos  Universae,  Halae  Magdeburgicae,  vol.  I,  1730  and  1742,  pp.  383-^,  of 
the  five  volume  editions  [first  printed  1713-41];  vol.  I,  1717,  315-6,  of  the  two  volume 
edition.  Quoted,  with  other  errors,  Ladies'  Diary,  1733,  Q.  166;  Leybourn's  ed.,  1, 
1817,  218;  Button's  ed.,  2,  1775,  10;  Diarian  Repository,  by  Soc.  Math.,  1774,  289. 

*"Jam  ut  X  sit  numerus  integer,  nee  in  casu  speciali,  si  y  per  numerum  explicetur,  numerus 
partium  aliquotarum  diversus  sit  a  numero  earundem  in  formula  general!;  necesse  est  ut 
d  =  l." 

"Math.  Anfangsgrlinde,  I,  2  (Fortsetzung  der  Rechnenkunst,  ed.  2,  1801,  546-8). 

"Recreations  math.,  new  ed.  of  4  vols.,  1723,  1724,  1735,  etc.,  I,  29-30. 

'"Grund-Satze  der  gesammten  Math.  Wiss.  u.  Lehren,  Giessen  u.  Franckfurt,  1724,  p.  21. 

*iA  new  system  of  arithmetik,  theoretical  &  practical,  London,  1730,  p.  394. 

"Mathematisches  Lexicon,  I,  1734  (under  Vollkommen  Zahl). 


Chap.  Il      PERFECT,  MULTIPLY  PERFECT,  AND   AmICABLE   NUMBERS.  17 

Leonard  Euler^^  (1707-1783)  noted  that  2''  — 1  may  be  composite  for  n 
a  prime;  for  instance,  2^'  — 1  =  23-89,  contrary  to  Wolf.'^^  If  n  =  4m  — 1 
and  8m  — 1  are  primes,  2"  — 1  has  the  factor  8m  — 1,  so  that  2^-1  is  com- 
posite for  n  =  ll,  23,  83,  131,  179,  191,  239,  etc.  [Proof  by  Lucas.^^sj 
Furthermore,  2^^-l  has  the  factor  223,  2^^-l  the  factor  431,  22^-1  the 
factor  1103,  2"^  — 1  the  factor  439,  etc.  ''However,  I  venture  to  assert 
that  aside  from  the  cases  noted,  every  prime  less  than  50,  and  indeed  than 
100,  makes  2"~^(2"  — 1)  a  perfect  number,  whence  the  eleven  values  1,  2,  3, 
5,  7,  13,  17,  19,  31,  41,  47  of  n  yield  perfect  numbers.  I  derived  these 
results  from  the  elegant  theorem,  of  whose  truth  I  am  certain,  although  I 
have  no  proof:  aJ^  —  V  is  divisible  by  the  prime  n+1,  if  neither  a  nor  h  is." 
[For  later  proofs  by  Euler,  see  Chapter  III  on  Fermat's  theorem.]  Euler's 
errors  as  to  n  =  41  and  47  were  corrected  by  Winsheim,®^  Euler^^  himself, 
and  Plana.i^o 

Michael  Gottlieb  Hansch^  stated  that  2^*— 1  is  a  prime  if  n  is  any  of 
the  twenty- two  primes  ^79  [error,  Winsheim,^^  Kraft^^]. 

George  Wolfgang  Kraft^^  corrected  Stifel's^^  error  that  511-256  is  per- 
fect and  the  error  of  Ozanam  (Elementis  algebrae,  p.  290)  that  the  sum  of 
all  the  divisors  of  2*"  is  a  prime,  by  noting  that  the  sum  forn  =  2  is  511  =  7-73 ; 
and  n6ted  that  false  perfect  numbers  were  listed  by  Ozanam.'^^  Kraft 
presented  (pp.  9-11)  an  incomplete  proof,  communicated  to  him  by  Tobias 
Maier  [cf.  Fontana^^^],  that  every  perfect  number  is  of  Euclid's  type. 
Let  1,  m,  n, . .  .,p,  A,. .  .be  the  aliquot  parts  of  any  perfect  number  pA, 
where  p  and  A  are  the  middle  factors  [as  4  and  7  jn  28].     Then 

q        r        n       m 

Solving  for  A,  he  stated  that  the  denominator  must  be  unity,  whence 
'p  =  2q/D,  D  =  q  —  l—q/r  —  q/n  —  q/m.  Again,  D  =  l,  whence  g  =  2r/D', 
D'  =  r  —  l—r/n—r/m.  From  I>'  =  1,  r  =  2n/I)",  D"  =  n  —  l—n/m.  From 
D"  =  l,  n  =  2m/(m  — 1),  m  — 1  =  1,  m  =  2,  n  =  4,  r  =  8,  etc.  Thus  the  aliquot 
parts  up  to  the  middle  must  be  the  successive  powers  of  2,  and  A  must  be 
a  prime,  since  otherwise  there  would  be  new  divisors.  For  p  =  2"~\  we 
get  A  =2"  — 1.  Kraft  observed  that  if  we  drop  from  Tartaglia's^^  list  of  20 
numbers  those  shown  to  be  imperfect  by  Euler's^^  results,  we  have  left  only 
eight  perfect  numbers  2"-^(2"-l)  for  n^39,  viz.,  those  for  n  =  2, 3,  5,  7,  13, 
17,  19,  31.  For  these,  other  than  the  first,  as  well  as  for  the  false  ones  of 
Tartaglia,  if  we  add  the  digits,  then  add  the  digits  of  that  sum,  etc.,  we 
finally  get  unity  (p.  14)  [proof  by  WantzeP^^].  All  perfect  numbers  end 
in  6  or  28. 

*3Comm.  Acad.  Petropol.,  6,  1738,  ad  annos  1732-3,  p.  103.     Commentationes  Arithmeticae 

Collectae,  I,  Petropoli,  1849,  p.  2. 
^Epistola  ad  mathematicos  de  theoria  arithmetices  nouis  a  se  inuentis  aucta,  Vindobonae 

[Vienna],  1739. 
"De  numeris  perfectis,  Comm.  Acad.  Petrop.,  7,  1740,  ad  annos  1734-5,  7-14. 


18  History  of  the  Theory  of  Numbers.  [Chap,  i 

Johann  Christoph  Heilbronner^^  stated  that  the  perfect  numbers  up  to 
4-10'  are  6,  28, 496,  8128, 130816,2096128.  "The  fathers  of  the  early  church 
and  many  wTiters  always  held  this  number  6  in  high  esteem.  God  com- 
pleted the  creation  in  6  days  and  since  all  things  created  by  Him  came  out 
perfect,  he  wished  the  work  of  creation  completed  according  to  the  number  6 
as  being  a  perfect  number." 

L.  Euler"  deduced  from  Fermat's  theorem,  which  he  here  proved  by 
use  of  the  binomial  theorem,  the  result*  that,  if  m  is  a  prime,  2"*  — 1,  when 
composite,  has  no  prime  factors  other  than  those  of  the  form  wn+l. 

J.  Landen^^  noted  that  196  is  the  least  number  4a;'*,  where  x  is  prime,  the 
sum  of  whose  ahquot  parts  exceeds  the  number  by  7. 

L.  Euler^^  gave  a  table  of  the  prime  factors  of  2"  — 1  for  n^37. 

C.  N.  de  Winsheim^°  noted  that  2^'^  — 1  has  the  factor  2351,  and  stated 
that  2"  — 1  is  a  prime  for  n  =  2,  3,  5,  7,  13,  17,  19,  31,  composite  for  the 
remaining  n<48,  but  was  doubtful  as  to  n  =  41,  thus  reducing  the  Hst  of 
perfect  numbers  given  by  Euler^  by  one  or  perhaps  two.  He  suspected 
that  n  =  41  leads  to  an  imperfect  number  since  it  was  excluded  by  the  acute 
Mersenne,^°  who  gave  instead  2^^(2^"  —  1)  as  the  ninth  perfect  number.  He 
remarked  that  the  basis  of  Mersenne's  assertion  is  doubtless  to  be  found  in 
the  stupendous  genius  of  Mersenne  which  perhaps  recognized  more  truths 
than  he  could  demonstrate.  He  discussed  the  error  of  Hansch^  that  2"  —  ! 
is  a  prime  if  n  is  a  prime  ^  79. 

G.  W.  Kraft^^  considered  perfect  numbers  AP,  where  P  is  a  prime  [not 
dividing  A].  Thus  a{P-\-l)=2AP,  where  a  is  the  sum  of  all  the  divisors 
of  A.  Hence  a/ {2 A— a)  equals  the  prime  P.  Let  2A— a  =  l,  a  property 
holding  for  A  =2"".  Then  P  =  2"'+^  — 1  and  the  resulting  numbers  are  of 
Euchd's  type. 

L.  Euler,^-  in  a  letter  to  Goldbach,  October  28,  1752,  stated  that  he 
knew  only  seven  perfect  numbers,  viz.,  2p~^(2^  — 1)  for  p  =  2,  3,  5,  7,  13,  17, 
19,  and  was  uncertain  whether  2^^  —  1  is  prime  or  not  (a  factor  is  necessarily 
of  the  form  64n+l  and  none  are  <2000). 

^^Historia  matheseos  universae.  Accedit  recensio  elementorum  compendiorum  et  openim  math, 
atque  historia  arithmetices  ad  nostra  tempora,  Lipsiae,  1742,  755-6.  There  is  a  63-page 
Ust  of  arithmetics  of  the  16th  century. 

«^^ovi  Comm.  Ac.  Petrop.,  1,  1747-8,  20;  Comm.  .\rith.,  I,  56,  §39. 

*We  may  simpUfy  the  proof  by  using  the  fact  that  2  belongs  to  an  e.xponent  e  modulo  p  (p  a 
prime)  such  that  e  divides  p  —  1.  For,  if  p  is  a  factor  of  2'"—  1,  m  is  a  multiple  of  e,  whence 
e  equals  the  prime  m.  Thus  p  — 1  =n7«.  If  we  take  m>2,  we  see  that  n  is  even  since 
p  is  odd  and  conclude  with  Fermat^'  that,  if  m  is  an  odd  prime,  2"»— 1  is  divisible  by  no 
primes  other  than  those  of  the  form  2km  +  l. 

"•Ladies'  Diary,  1748,  Question  305.  The  Diarian  Repository,  Collection  of  all  the  mathe- 
matical questions  from  the  Ladies'  Diary,  1704-1760,  by  a  society  of  mathematicians, 
London,  1774,  509.  Button's  The  Diarian  Miscellany  (from  Ladies'  Diarj-,  1704-1773), 
London,  1775,  vol.  2,  271.  Leyboiu-n's  Math.  Quest,  proposed  in  Ladies'  D.,  2,  1817, 
9-10. 

"Opuscula  varii  argumenti,  Berlin,  2,  1750,  25;  Comm.  Arith.,  1,  1849,  104. 

•"Novi  Comm.  Ac.  Petrop.,  2,  1751,  ad  annum  1749,  mem.,  68-99. 

"/bid.,  mem.,  112-3. 

•^Corresp.  Math.  Phys.  (ed.,  Fuss),  I,  1843,  590,  597-8. 


Chap.  I]     PERFECT,  MULTIPLY  PERFECT,  AND   AmICABLE   NUMBERS.  19 

G.  W.  Kraft^^  stated  (p.  114)  that  Euler  had  communicated  to  him  pri- 
vately in  1741  the  fact  that  2*^-1  is  divisible  by  2351.  He  stated  (p.  121) 
that  if  2^  —  1  is  composite  {p  being  prime),  it  has  a  factor  of  the  form 
2q'^p  +  l,  where  g  is  a  prime  [including  unity],  using  as  illustrations  the 
factorizations  noted  by  Euler. ^^  Of  the  numbers  2"  — 1,  n  a  prime  ^71, 
stated  to  be  prime  by  Hansch,^^  six  are  composite,  while  the  cases  53, . . . ,  71 
are  in  doubt  (p.  115). 

A.  Saverien^^  repeated  the  remarks  by  Ens^^  without  reference. 

L.  Euler^^  stated  in  a  letter  to  Bernoulli  that  he  had  verified  that  2^^  —  1 
is  a  prime  by  examining  the  primes  up  to  46339  which  are  contained  in  the 
possible  forms  248n+l  and  248n+63  of  divisors. 

L.  Euler^^  gave  a  prime  factor  of  2"=»=  1  for  various  values  of  n,  but  no 
new  cases  2^—1  with  n  a  prime. 

L.  Euler,^^  in  a  posthumous  paper,  proved  that  every  even  perfect  number 
is  of  Euclid's  type.  Let  o  =  2"6  be  perfect,  where  b  is  odd.  Let  B  denote 
the  sum  of  the  divisors  of  b.  The  sum  {2'*^^  —  l)B  of  the  divisors  of  a  must 
equal  2a.  Thus  6/5  =  (2"+^-l)/2"+\  a  fraction  in  its  lowest  terms. 
Hence  6  =  (2^+^  —  1  )c.  If  c  =  l,  6  =  2"'*'^  — 1  must  be  a  prime  since  the  sum 
of  its  divisors  is  5  =  2""^^  whence  Euclid's  formula.  If  c>l,  the  sum  B  of 
the  divisors  of  b  is  not  less  than  6+2""''^  — 1+c+l;  hence 

^^2"+nc+l)        2"+' 
6=         h         '^2"+^-l' 

contrary  to  the  earlier  equation.  The  proof  given  in  another  posthumous 
paper  by  Euler^^  is  not  complete. 

L.  Euler^^  proved  that  any  odd  perfect  number  must  be  of  the  form 
y.4x+ip2^  where  r  is  a  prime  of  the  form  4nH-l  [Frenicle®^].  Express  it  as  a 
product  ABC.  . .  of  powers  of  distinct  primes.  Denote  by  a,  b,  c, .  .  .the 
sums  of  the  divisors  oi  A,  B,C,. . .,  respectively.  Then  abc . .  .  =  2  ABC .... 
Thus  one  of  the  numbers  a,  b,  . . . ,  say  a,  is  the  double  of  an  odd  number, 
and  the  remaining  ones  are  odd.  Thus  B,  C,.  .  . are  even  powers  of  primes, 
while  A  =r*^"^^  In  particular,  no  odd  perfect  number  has  the  form  4n+3. 
Amplifications  of  this  proof  have  been  given  by  Lionnet,^^^  Stern,  ^^'^  Syl- 
vester, ^^^  Lucas. ^"     See  also  Liouville^°  in  Chapter  X. 

Montucla^^  remarked  that  Euclid's  rule  does  not  give  as  many  perfect 
numbers  as  believed  by  various  writers;  the  one  often  cited  [Paciuolo^®]  as 
the  fourteenth  perfect  number  is  imperfect;  the  rule  by  Ozanam^^  is  false 
since  511  and  2047  are  not  primes. 

"Novi  Comm.  Ac.  Petrop.,  3,  1753,  ad  annos  1750-1. 

"Dictionnaire  universel  de  math,  et  physique,  two  vols.,  Paris,  1753,  vol.  2,  p.  216. 
»^Nouv.  Mim.  Acad.  BerUn,  ann6e  1772,  hist.,  1774,  p.  35;  Euler,  Comm.  Arith.,  1,  1849,  584. 
"Opusc.  anal.,  1,  1773,  242;  Comm.  Arith.,  2,  p.  8. 

"De  numeris  amicabihbus,  Comm.  Arith.,  2,  1849,  630;  Opera  postuma,  1,  1862,  88. 
'^Tractatus  de  numerorum  doctrina,  Comm.  Arith.,  2,  514;  Opera  postuma,  1,  14-15. 
"Recreations  math,  et  physiques  par  Ozanam,  nouvelle  6d.  par  M.,  Paris,  1,  1778,  1790,  p.  33. 
Engl,  transl.  by  C.  Hutton,  London,  1803,  p.  35. 


/ 


20  History  of  the  Theory  of  Numbers.  [Chap.  I 

Johann  Philipp  Griison^^''  made  the  same  criticism  of  Ozanam"^  and 
noted  that,  if  2"x  is  perfect  and  x  is  an  odd  prime, 

1+2+ .  .  .+2''  =  2'*x-a:-2x-.  .  .-2'*-^x  =  x. 

M.  Fontana^"^  noted  that  the  theorem  that  all  perfect  numbers  are 
triangular  is  due  to  Maurolycus^^  and  not  to  T.  Maier  (cf.  Kraft^^). 

Thomas  Taylor^"-  stated  that  only  eight  perfect  numbers  have  been 
found  so  far  [the  8  listed  are  those  of  Mersenne^^j. 

J.  Struve^^^  considered  abundant  numbers  which  are  products  ahc  of 
three  distinct  primes  in  ascending  order;  thus 

ob+o+M-l  2  ^    .- 

— ; ; >C,  >C  +  1. 

ah-a-h-1      '  i_i_l_jL 

a     h     ab 

The  case  a^3  is  easily  excluded,  also  a  =  2,  6^5  [except  2-5'7].    For 
a  =  2,  6  =  3,  c  any  prime  >  3,  6c  is  abundant.     Next,  abed  is  abundant  if 

^"^'  >d+i. 


a6c  — (a6+ac+6c+a+6+c+l)' 

For  a  =  2,  6  =  3,  c  =  5  or  7,  and  for  a  =  2,  6  =  5,  c  =  7,  abed  is  abundant  for 
any  prime  d  [>c].     Of  the  numbers  ^  1000,  52  are  abundant. 

J.  Westerberg^*^  gave  the  factors  of  2"='=1  for  n  =  l,...,  32,  and  of 
10''±l,n  =  l,...,  15. 

O.  Terquem^°^  Usted  2*^-1  and  2*^-1  as  primes. 

L.  WantzeP"®  proved  the  remark  of  Kraft*^  that  if  A^i  be  the  sum  of  the 
digits  of  a  perfect  number  N>6  [of  Euclid's  type],  and  N2  the  sum  of  the 
digits  of  A^i,  etc.,  a  certain  iV,  is  unity.  Since  iV=l(mod  9),  each  Ni=l 
(mod  9),  while  the  NiS  decrease. 

V.  A.  Lebesgue^°^  stated  that  he  had  a  proof  that  there  is  no  odd  perfect 
number  with  fewer  than  four  distinct  prime  factors.  For  an  even  perfect 
number  2"?/ V . .  . , 

y'^"  ■  ■  •  +prrY = (1 +y+  ■  ■  ■  +y')  d +^+  •  •  •  +^')  ■  •  • » 

""Enthiillte  Zaubereyen  und  Geheimnisse  der  Arithmetik,  erster  Theil,  Berlin,  1796,  p.  85,  and 

Zusatz  (end  of  Theil  I). 
»<»Memorie  dell'  Istituto  Nazionale  Ital.,  mat.,  2,  pt.  1,  1808,  285-6. 
'°*The  elements  of  a  new  arithmetical  notation  and  of  a  new  arithmetic  of  infinites,  with  an 

appendix ....  of  perfect,  amicable  and  other  numbers  no  less  remarkable  than  novel, 

London,  1823,  131. 
^''Ueber  die  so  gennannten  numeri  abundantes  oder  die  Ueberfluss  mit  sich  fiihrenden  Zahlen, 

besonders  im  ersten  Tausend  unsrer  Zahlen,  Altona,  1827,  20  pp. 
*'**De  factoribus  numerorum  compositorum  dignoscendis,  Disquisitio  Acad.  CaroUna,  Lundae, 

1838.     In  the  volume,  Meditationum  Math publice  defendent  C.  J.  D.  Hill,  Pt.  II, 

1831. 
i^Nouv.  Ann.  Math.,  3,  1844.  219  (cf.  553). 
^<*Ibid.,  p.  337. 
i"76id.,  552-3. 


Chap.  I]    PERFECT,  MULTIPLY  Perfect,  and  Amicable  Numbers.  21 

the  impossibility  of  which  is  evident  when  the  exponents  j3,  7, . . .  are  other 
than  1,  0,  0, . .  .,  a  case  giving  Euclid's  solution  [cf.  Desboves^^']. 

C.  G.  Reuschle^^*  gave  in  his  table  C  the  exponent  to  which  2  belongs 
modulo  p,  for  each  prime  p<5000.  Thus  2"  — 1  has  the  factor  1399  for 
n  =  233,  the  factor  2687  forn  =  79,  and  3391  for  n  =  113  [as  stated  exphcitly 
by  Le  Lasseur^^^'^^^].  ^iso  23514513  for  n  =  47,  1433  for  w  =  179,  and  1913 
for  n  =  239.  In  the  addition  (p.  22)  to  Table  A,  he  gave  the  prime  factors  of 
2^*  —  1  for  various  n's  to  156,  37  being  the  least  n  for  which  the  decomposition 
is  not  given  completely,  while  41  is  the  least  n  for  which  no  factor  is  known. 
For  34  errata  in  Table  C,  see  Cunningham^^°  of  Ch.  VII. 

F.  Landry^^^  gave  a  new  proof  that  2^^  —  1  is  a  prime. 

Jean  Plana^^°  gave  (p.  130)  the  factorization  into  two  primes: 

2*^-1  =  13367X 164511353. 

His  statement  (p.  141)  that  2^^  — 1  has  no  factor  <  50033  was  corrected  by 
Landry^^^  (quoted  by  Lucas, "^  p.  280)  and  Gerardin."' 

Giov.  Nocco^"  showed  that  an  odd  perfect  number  has  at  least  three 
distinct  prime  factors.     For,  if  a"*6'*  is  perfect, 

2a-  =  V-T^'  6"  =  ^ -^, 

0—1  a— 1 

whence 

^       ^      Q^""^^      _(a-l)b"+l 
2(5-1)"  2(6 -l)a"»"     6"+^-l    ' 

a+fe(a6"+26"-'+2)=2+6(26"+2a6"-^). 

But  the  minunum  values  of  a,  h  are  3,  5.    Thus  6(a— 2)>2a  — 2, 

a6''-26"  =  6"-^-6(a-2)>6'*-^(2a-2),        a6'»+26"-'>26"+2a6''-\ 

contrary  to  the  earlier  equation.  In  attempting  to  prove  that  every  even 
perfect  number  2'"6Vd' ...  is  of  EucUd's  type,  he  stated  without  proof  that 

2-+16V. . .  =(2"'+^-l)J5C. . .,  B=  \     /,  C  =  - -,.. . 

0  —  1  c  — 1 

require  that  2"*+^  =  B,  6"  =  2^"+^  - 1 ,  d'  =  C, . .  .  (the  first  two  of  which  results 
yield  Euclid's  formula). 

F.  Landry^^^  stated  (p.  8)  that  he  possessed  the  complete  decomposi- 
tion of  2"±l(n^64)  except  for  2^^±1,  2«Hl,  and  gave  (pp.  10-11)  the 
factors  of  2^^-l  and  of  2"+l  for  n  =  65,  66,  69,  75,  90,  105. 

"^Mathematische  Abhandlung,  enthaltend  neue  Zahlentheoretische  Tabellen  sammt  einer 
dieselben  betreflfenden  Correspondenz  mit  dem  verewigten  C.  G.  J.  Jacobi.  Prog.,  Stutt- 
gart, 1856,  61  pp.     Described  by  Kummer,  Jour,  fur  Math.,  53,  1857,  379. 

^°'Proc6des  nouveaux  pour  demontrer  que  le  nonabre  2147483647  est  premier.  Paris,  1859. 
Reprinted  in  SpLinx-Oedipe,  Nancy,  1909,  6-9. 

""Mem.  Reale  Ac.  Sc.  Torino,  (2),  20,  1863,  dated  Nov.  20,  1859. 

'"Alcune  teorie  su'numeri  pari,  impari,  e  perfetti,  Lecce,  1863. 

"^Aux  math^maticiens  de  toutes  les  parties  du  monde:  communicatidn  but  la  decomposition  des 
nombres  en  leurs  facteurs  simples,  Paris,  1867,  12  pp. 


22  History  of  the  Theory  of  Numbers.  [Chap.  I 

F.  Landry"^  soon  published  his  table.     It  includes  the  entries  (quoted 

byLucas^2o.i22). 

2«-l=431-9719-2099863,  2*^-1=23514513.13264529, 
253-1  =  6361.69431-20394401,  2^^-1  =  179951.3203431780337, 

the  least  factors  of  the  first  two  of  which  had  been  given  by  Euler.^'  •' 
This  table  was  republished  by  Lucas^-^  (p.  239),  who  stated  that  only  three 
entries  remain  in  doubt:  2^^  — 1,  (2^^  +  l)/3,  2^*+l,  each  being  conjectured 
a  prime  by  Landry.  The  second  was  believed  to  be  prime  by  Kraitchik.^^^" 
Landr>''s  factors  of  2"+l,  for  28^n^64  were  quoted  elsewhere."^'' 

Jules  Carvallo^^*  announced  that  he  had  a  proof  that  there  exists  no  odd 
perfect  number.  Without  indication  of  proof,  he  stated  that  an  odd  per- 
fect number  must  be  a  square  and  that  the  ratio  of  the  sum  of  the  divisors 
of  an  odd  square  to  itself  cannot  be  2.  The  first  statement  was  abandoned 
in  his  published  erroneous  proof,  ^^^  while  the  second  follows  at  once  from 
the  fact  that,  when  p  is  an  odd  prime,  the  sum  of  the  2n+l  divisors,  each 
odd,  of  p^"  is  odd. 

E.  Lucas^^^  stated  that  long  calculations  of  his  indicated  that  2°^  — 1 
and  2^^ - 1  are  composite  [cf .  Cole,^"  Powers^^^].     See  Lucas-°  of  Ch.  XVII. 

E.  Lucas^^^  stated  that  2^^  — 1  and  2^^^  — 1  are  primes. 

E.  Catalan^^^  remarked  that,  if  we  admit  the  last  statement,  and  note 
that  2^  —  1,  2^  —  1,  2^  —  1  are  primes,  we  may  state  empirically  that,  up  to  a 
certain  limit,  if  2"  — 1  is  a  prime  p,  then  2^  —  1  is  a  prime  g,  2^  —  1  is  a  prime, 
etc.  [cf.  Catalan^^^]. 

G.  de  Longchamps^^'^  suggested  that  the  composition  of  2"±1  might  be 
obtained  by  continued  multiplications,  made  by  simple  displacements  from 
right  to  left,  of  the  primes  written  to  the  base  2. 

E.  Lucas^^^  verified  once  only  that  2^^^  — 1,  a  number  of  39  digits,  is  a 
prime.  The  method  will  be  given  in  Ch.  XVII,  where  are  given  various 
results  relating  indirectly  to  perfect  numbers.  He  stated  (p.  162)  that  he 
had  the  plan  of  a  mechanism  which  will  permit  one  to  decide  almost  instan- 
taneously whether  the  assertions  of  Mersenne  and  Plana  that  2"  — 1  is  a 
prime  for  n  =  53,  67,  127,  257  are  correct.  The  inclusion  of  n  =  53  is  an 
error  of  citation.     He  tabulated  prime  factors  of  2"  — 1  for  n^40. 

E.  Lucas^^^  gave  a  table  of  primes  with  12  to  16  digits  occurring  as  a 
factor  in  2"-l  for  n  =  49,  59,  65,  69,  87,  and  in  2''+l  forn  =  43,  47,  49,  53, 
69,  72,  75,  86,  94,  98,  99,  135,  and  several  even  values  of  n>100.     The 

'"Decomposition  des  nombres  2"=!=  1  en  leurs  facteurs  premiers  de  n  =  1  ^  n  =  64,  moins  quatre, 

Paris,  1869,  8  pp. 
"3<»Sphinx-0edipe,  1911,  70,  95. 
"'^L'interm^diaire  des  math.,  9,  1902,  186. 
'"Comptes  Rendus  Paris,  81,  1875,  73-75. 

"'Sur  la  th^orie  des  nombres  premiers,  Turin,  1876,  p.  11;  TWorie  des  nombres,  1891,  376. 
"«Nouv.  Corresp.  Math.,  2,  1876,  96. 
i^Comptes  Rendus  Paris,  85,  1877,  950-2. 

"sBull.  Bibl.  Storia  Sc.  Mat.  e  Fis.;  10,  1877,  152  (278-287).     Lucas"- "  of  Ch.  XVII. 
"'Atti  R.  Ac.  Sc.  Torino,  13,  1877-8,  279. 


Chap.  I]     PERFECT,  MULTIPLY  PERFECT,  AND   AmICABLE   NUMBERS.  23 

verification  of  the  primality  was  made  by  H.  Le  Lasseur.  To  the  latter 
is  attributed  (p.  283)  the  factorization  of  2'*-!  for  n  =  73,  79,  113.  These 
had  been  given  without  reference  by  Lucas. ^^° 

E.  Lucas^^^  proposed  as  a  problem  the  proof  that  if  8q+7  is  a  prime, 
24a+3_i  is  not. 

E.  Lucas^^^  stated  as  new  the  assertion  of  Euler^^  that  if  4m  — 1  and 
8m  — 1  are  primes,  the  latter  divides  A  =  2*"*~^  — 1. 

E.  Lucas^^^  proved  the  related  fact  that  if  8m  —  1  is  a  prime,  it  divides  A. 
For,  by  Fermat's  theorem,  it  divides  2^"*"^  — 1  and  hence  divides  A  or 
2^~^H-1.  That  the  prime  8m  —  1  divides  A  and  not  the  latter,  follows  from 
Euler's  criterion  that  2^^"^^''^  — 1  is  divisible  by  the  prime  p  if  2  is  a  quad- 
ratic residue  of  p,  which  is  the  case  if  p  =  8m='=l.  No  reference  was  made 
to  Euler,  who  gave  the  first  seven  primes  4m  —  1  for  which  8m  —  1  is  a  prime. 
Lucas  gave  the  new  cases  251,  359,  419,  431,  443,  491.  Lucas^^^  elsewhere 
stated  that  the  theorem  results  from  the  law  of  reciprocity  for  quadratic 
residues,  again  without  citing  Euler.  Later,  Lucas^^^  again  expressly 
claimed  the  theorem  as  his  own  discovery. 

T.  Pepin^^^  noted  that  if  p  is  a  prime  and  q  =  2^  —  1  is  a  quadratic  non- 
residue  of  a  prime  4n + 1  =  a^ + 6^,  then  qisa,  prime  if  and  only  if  (a  —  hi)  /  (o + hi) 
is  a  quadratic  non-residue  of  q. 

A.  Desboves^^^  amplified  the  proof  by  Lebesgue^^^  that  every  even  per- 
fect number  is  of  Euclid's  type  by  noting  that  the  fractional  expression  in 
Lebesgue's  equation  must  be  an  integer  which  divides  y^z'^ . . .  and  hence  is 
a  term  of  the  expansion  of  the  second  member.  Hence  this  expansion 
produces  only  the  two  terms  in  the  left  member,  so  that  (j8+l)(7+l) . .  .  =  2. 
Thus  one  of  the  exponents,  say  /3,  is  unity  and  the  others  are  zero.  The 
same  proof  has  been  given  by  Lucas^^^  (pp.  234-5)  and  Th^orie  des  Nombres, 
1891,  p.  375.  Desboves  (p.  490,  exs.  31-33)  stated  that  no  odd  perfect 
number  is  divisible  by  only  2  or  3  distinct  primes,  and  that  in  an  odd  perfect 
number  which  is  divisible  by  just  n  distinct  primes  the  least  prime  is  less 
than  2". 

F.  J.  E.  Lionnet^^*  amplified  Euler's^^  proof  about  odd  perfect  numbers. 
F.  Landry^^^  stated  that  2^^=*=  1  are  the  only  cases  in  doubt  in  his  table."' 
Moret-Blanc^^°  gave  another  proof  that  2^^  —  1  is  a  prime. 

""Assoc.  franQ.  avanc.  sc,  6,  1877,  165. 

»2iNouv.  Corresp.  Math.,  3,  1877,  433. 

i"Mess.  Math.,  7,  1877-8,  186.     AJso,  Lucas."" 

i«Amer.  Jour.  Math.,  1,  1878,  236. 

i^^BuU.  Bibl.  Storia  Sc.  Mat.  e  Fis.,  11, 1878,  792.     The  results  of  this  paper  will  be  cited  in  Ch. 

XVI. 
^Recreations  math.,  ed.  2,  1891,  1,  p.  236. 
"^Comptes  Rendus  Paris,  86,  1878,  307-310. 
"'Questions  d'algebre  616mentau-e,  ed.  2,  Paris,  1878,  487-8. 
'"Nouv.  Ann.  Math.,  (2),  18,  1879,  306. 

"sBull.  Bibl.  Storia  Sc.  Mat.,  13,  1880,  470,  letter  to  C.  Henry. 
"«Nouv.  Ann.  Math.,    (2),  20,   1881,  263.     Quoted,  with  Lucas'  proof,  Sphinx-Oedipe,  4, 

1909,  9-12. 


24  History  of  the  Theory  of  Numbers.  [Chap,  i 

H.  LeLasseur  found  after^^^  1878  and  apparently  just  before^^^  1882 
that  2'*-l  has  the  prime  factor  11447  if  n  =  97,  15193  if  n  =  211,  18121  if 
71  =  151,  18287  if  n  =  223,  and  that  there  is  no  divisor  <30000  of  2"-l  for 
the  24  prime  values  of  n,  n^257,  which  remain  in  doubt,  viz.  [cf.  Lucas^^^], 

61,    67,    71,    89,  101,  103,  107,  109,  127,  137,  139,  149, 
157,  163,  167,  173,  181,  193,  197,  199,  227,  229,  241,  257. 

J.  Carvallo^^^  attempted  again^^*  to  prove  the  non-existence  of  odd 
perfect  numbers y^'z^ . .  .u\  where y,. .  .u  are  distinct  odd  primes.  He  began 
by  noting  that  one  and  only  one  of  the  exponents  n, . .  .,  r  is  odd  [Euler^^]. 
Let  y<z<  . .  .<u,  and  call  their  number  jjl.  From  the  definition  of  a 
perfect  number, 

y  —  l      "'    u  —  1         '  y  —  l'u  —  l 

The  fractions  in  this  inequality  form  a  decreasing  series.     Hence 

fe)'>^.  ^<A'  -~i>^'  H^r 

Thus  w(2  — A;)<2.  By  a  petitio  principii  (the  division  by  2— A:,  not  known 
to  be  positive),  it  was  concluded  (p,  10)  that 

^<2i:^'       ^^<2,       y>2i/(M-i)_i- 

[This  error,  repeated  on  p.  15,  was  noted  by  P.  Mansion. ^^]  For  a 
given  n,  there  is  at  most  one  prime  between  the  two  limits  (of  difference  <  2) 
for  y.  A  superior  limit  is  found  for  2  as  a  function  of  y.  An  incomplete 
computation  is  made  to  show  that,  if  /x>8,  z  <y-\-l. 

It  is  shown  (p.  7)  that  an  odd  perfect  number  has  a  prime  factor  greater 
than  the  prime  factor  w  entering  to  an  odd  power,  since  w+l  divides  the 
sum  of  the  divisors.  In  a  table  (p.  30)  of  the  first  ten  perfect  numbers, 
2^^  — 1  and  2*^  —  1  are  entered  as  primes  [contrary  to  Euler^^  and  Plana^^°]. 

E.  Catalan^^^  stated  that  2^  —  1  is  a  prime  if  p  is  a  prime  of  the  form 
2^  —  1.     If  correct  this  would  imply  that  2^^^  — 1  is  a  prime  [cf.  Catalan^^^]. 

E.  Lucas^^^  repeated  the  remark  of  LeLasseur^^^  on  the  24  prime  values 
of  n^257  for  which  the  composition  of  2^  —  1  is  in  doubt.    According  to  a 

"iSince  these  four  values  of  n  are  included  in  the  list  by  Lucas^**  of  the  28  values  of  n  ^  257  for 
which  the  composition  of  2"—!  is  unknown.     Cf.  Lucas^^^  p.  236. 

"2Lucas,  Recreations  math.,  1,  1882,  241;  2,  1883,  230.  Later,  Lucas^^s  credited  LeLasseur 
with  these  four  cases  as  well  as  n  =  73  [Eulers^]  and  n  =  79,  113,  233  [cf.  Reuschlei"]. 
The  last  four  cases  were  given  by  Lucas"*,  while  the  last  three  do  not  occur  in  the  table 
(Lucasi24^  pp  7gg_9)  by  LeLasseur  of  the  proper  divisors  of  2"— 1  for  each  odd  n,  n<79, 
and  for  a  few  larger  composite  n's.  The  last  three  were  given  also  by  Lucas"^  (p.  236) 
without  reference. 

'"Th^orie  des  nombres  parfaits,  par  M.  Jules  Carvallo,  Paris,  1883,  32  pp. 

"<Mathesis,  6,  1886,  147. 

'"Melanges  Math.,  Bruxelles,  1,  1885,  376. 

'"Mathesis,  6,  1886,  146. 


Chap.  IJ     PERFECT,  MULTIPLY   PERFECT,  AND   AMICABLE    NUMBERS.  25 

communication  from  Pellet,  2"  — 1  is  divisible  by  6n+l  if  n  and  6n+l  are 
primes  such  that  6n+l  =4L^+27M^  [provided*  n=  1  (mod  4),  i.  e.,  L  is  odd]. 

M.  A.  Stern^"  amplified  Euler's^^  proof  concerning  odd  perfect  numbers. 

E.  Lucas^^^  repeated  the  statement  [Desboves^^^]  that  an  odd  perfect 
number  must  contain  at  least  four  distinct  primes. 

G.  Valentin^^^  gave  a  table,  computed  in  1872,  showing  factors  of 
2"— 1  for  n  =  79,  113,  233,  etc.,  but  not  the  new  cases  of  LeLasseur.^^^ 

The  primality  of  iV  =  2®^  — 1,  a  number  of  19  digits,  considered  composite 
by  Mersenne  and  prime  by  Landry,  was  established  by  J.  Pervusin"°  and 
P.  Seelhoff ^^^  independently.  The  latter  claimed  to  verify  that  there  is  no 
factor  <N^'^  of  the  form  8?i+7,  abbreviating  the  work  by  use  of  various 
numbers  of  which  iV  is  a  quadratic  residue;  thus  iV  is  a  prime  or  the  product 
of  two  primes.  Since  iV  =  2(2^°)^  — 1,  2  is  a  quadratic  residue  of  any  prime 
factor  of  N,  so  that  the  factor  is  8n=i=  1.  It  was  verified  that  3^= 1  (mod  N), 
where  i8  =  (iV  — 1)/9.  If  N=fF,  where  F  is  the  prime  factor  8n-|-l,  then 
3^^1(mod  F)  and,  by  Fermat's  theorem,  3^~^=l(mod  F).  It  is  stated 
without  proof  that  one  of  the  exponents  /S  and  F  —  1  divides  the  other. 
Cole^^^  regarded  the  proof  as  unsatisfactory. 

Seelhoff  proved  that  a  perfect  number  of  the  form  pV  is  of  Euclid's 
type  if  p  and  r  are  primes  and  p<r.    The  condition  is 


r''+\2-p)-2r''{l-p)-p 
If  p  >  2,  the  denominator  is  negative.    Hence  p  =  2  and 

^'=2K^'  2'+'  =  r+;j-j,  p  =  l,  r=2-+'-l. 

His  statements  (p.  177)  about  the  factors  of  2"  — 1,  n  =  37,  47,  53,  59, 
were  corrected  by  him  {ibid.,  p.  320)  to  accord  with  Landry.^^^ 

P.  Seelhoff ^^^  obtained  the  known  factors  of  these  2"  — 1  and  proved  that 
2^^  —  1  is  a  prime,  by  use  of  his  method  of  quadratic  residues. 

H.  Novarese^^^  proved  that  every  perfect  number  of  Euclid's  type  ends 
in  6  or  28,  and  that  each  one  >  6  is  of  the  form  9A;+1. 

Jules  Hudelot"^  verified  in  54  hours  that  2^^  —  1  is  a  prime  by  use  of  the 
test  by  Lucas,  Recreations  math.,  2,  1883,  233. 

♦Correction  by  Kraitchik,  Sphinx-Oedipe,  6,  1911,  73;  Pellet,  7,  1912,  15. 
"'Mathesis,  6,  1886,  p.  248. 
"8/6td.,  p.  250. 

""Archiv  Math.  Phys.,  (2),  4,  1886,  100-3. 
""Bull.  Acad.  Sc.  St.  Petersb.,  (3),  31,  1887,  p.  532;  Melanges  math.  astr.  ac.  St.  P^tersb.,  6, 

1881-8,  553;  communicated  Nov.  1883. 
"iZeitschr.  Math.  Phys.,  31,  1886,  174-8. 

»"Archiv  Math.  Phys.,  (2),  2,  1885,  327;  5,  1887,  221-3  (misprint  forn  =  41). 
*«Jomal  de  sciencias  math,  e  astr.,  8,  1887,  11-14.     [Servais"*.] 
"♦Mathesis,  7,  1887,  46.     Sphinx-Oedipe,  1909,  16. 


26  History  of  the  Theory  of  Numbers.  [Chap,  i 

CI.  Servais"^  republished  the  proofs  by  Novarese^"*^  and  proved  that 
a'"6'*  is  not  perfect  if  a  and  h  are  odd  primes.     For,  by  the  equations  [Nocco"^] 

a-+i_l  =  6"(a-l),  b''+^-l=2a"'(6-l), 

we  obtain,  by  subtraction, 

Thus2a'">6".  Since  a^3,  a'"+^^3a"'>a'"+6'*>a+6-l.  He  next  proved 
that,  if  an  odd  perfect  number  is  divisible  by  only  three  distinct  primes  a, 
h,  c,  two  of  them  are  3  and  5,  since  [as  by  Carvallo^^^] 


04)04)04)<l 


Taking  a  =  3,  6  =  5,  we  have  c<16,  whence  c  =  7,  11,  or  13.  He  quoted 
from  a  letter  from  Catalan  that  the  sum  of  the  reciprocals  of  the  divisors 
of  a  perfect  number  equals  2. 

E.  Cesaro^*^  proved  that  in  an  odd  perfect jiumber  containing  n  distinct 
prime  factors,  the  least  prime  factor  is  ^n\/2. 

CI.  Servais^^^  showed  that  it  does  not  exceed  n  since,  if  a<h<c<  .  . . , 

b       a+1  c       a-\-2 


6  —  1       a  c  —  \     a+1 

ah  a    a+1  a+2       a-\-n  —  \ 

2i^. . .      ^ . .  .  . > 

a—lb—1      '     a— r    a      a+1       a+n— 2 

whence  2(a  — l)<a+n  — 1,  a<n+l.     If  I  is  the  (m  — l)th  prime  factor  and 
s  is  the  772th,  and  if 

a       b 


a-l'b-l"l-l 
then 


^L<2, 


s+1  s+n—m       _.  ^L{n—m)-\-2 

>2,  s< 


s  —  l     s    '  '  '  s-\-n—m-\-l      '  2—L 

J.  J.  Sylvester ^'^^  reproduced  Euler's^^  proof  that  every  even  perfect 
number  is  of  EucUd's  type.  From  the  fact  that  ■|.|-<2,  he  concluded  that 
there  is  no  odd  perfect  number  a'"6'*.  For  the  case  of  three  prime  factors 
he  obtained  the  result  of  Servais^'*^  in  the  same  manner.  He  proved  that 
no  odd  perfect  number  is  divisible  by  105  and  stated  that  there  is  none  with 
fewer  than  six  distinct  prime  factors. 

Sylvester^^^  and  Servais^^*^  gave  complete  proofs  that  there  exists  no  odd 
perfect  number  with  only  three  distinct  prime  factors. 

i«Mathesis,  7,  1887,  228-230. 
»«/6id.,  245-6. 
"'Mathesis,  8,  1888,  92-3. 
"8Nature,  37,  Dec.  15,  1887,  152  (minor  correction,  p.  179);  Coll.  Math.  Papers,  4,  1912,  588. 
"'Comptes  Rendus  Paris,  106,  1888,  403-5  (correction,  p.  641);  reproduced  with  notes  by  P. 
Mansion,  Mathesis,  8,  1888,  57-61.  Sylvester's  Coll.  Math.  Papers,  4,  1912,  604,  615. 
""Mathesis,  8,  1888,  135. 


Chap.  I]     PERFECT,  MULTIPLY  PERFECT,  AND   AMICABLE   NUMBERS.  27 

Sylvester ^^^  proved  there  is  no  odd  perfect  number  not  divisible  by  3 
with  fewer  than  eight  distinct  prime  factors. 

Sylvester^^^  proved  there  is  no  odd  perfect  number  with  four  distinct 
prime  factors. 

Sylvester^^^  spoke  of  the  question  of  the  non-existence  of  odd  perfect 
numbers  as  a  "problem  of  the  ages  comparable  in  difficulty  to  that  which 
previously  to  the  labors  of  Hermite  and  Lindemann  environed  the  subject 
of  the  quadrature  of  the  circle."  He  gave  a  theorem  useful  for  the  investi- 
gation of  this  question:  For  r  an  integer  other  than  1  or  —1,  the  sum 
l+r+r^+ .  .  .  -\-r^~^  contains  at  least  as  many  distinct  prime  factors  as  p 
contains  divisors  >1,  with  a  possible  reduction  by  one  in  the  number  of 
prime  factors  when  r=  —  2,  p  even,  and  when  r  =  2,  p  divisible  by  6. 

E.  Catalan ^^^  proved  that  if  an  odd  perfect  number  is  not  divisible  by 
3,  5,  or  7,  it  has  at  least  26  distinct  prime  factors  and  thus  has  at  least  45 
digits.    In  fact,  the  usual  inequality  gives 

lTl3--     I    ^2  ^^^^"3   5  7   11     •     I    <2  3   5  7<^-^^^^- 

By  Legendre's  table  IX,  Theorie  des  nombres,  ed.  2,  1808;  ed.  3,  1830,  of 
the  values  of  P{w)  up  to  w;  =  1229,  we  see  that  I  ^  127.  But  127  is  the  27th 
prime  >7. 

R.  W.  D.  Christie^^^  erroneously  considered  2^^  — 1  and  2^^  — 1  as  primes. 

E.  Lucas^^®  proved  that  every  even  perfect  number,  aside  from  6  and 
496,  ends  with  16, 28,  36,  56,  or  76;  any  one  except  28  is  of  the  form  7k='=  1 ; 
any  one  except  6  has  the  remainder  1,2,  3,  or  8  when  divided  by  13,  etc. 

E.  Lucas^"  reproduced  his^^^  proofs  and  the  proof  by  Euler,^^  and  gave 
(p.  375)  a  list  of  known  factorizations  of  2"  — 1. 

Genaille^^^  stated  that  his  machine  "piano  arithm^tique "  gives  a  prac- 
tical means  of  applying  in  a  few  hours  the  test  by  Lucas  {ibid.,  5,  1876,  61) 
for  the  primality  of  2"  —  1 . 

J.  Fitz-Patrick  and  G.  ChevreP^^  stated  that  2^8(229-1)  is  perfect. 

E.  Fauquembergue^®^  found  that  2®''  — 1  is  composite  by  a  process  not 
yielding  its  factors  [cf.  Mersenne,^"  Lucas,^^^  Cole^'^^]. 

A.  Cunningham^^^  called  2^  —  1  a  Lucassian  if  p  is  a  prime  of  the  form 
4A;+3  such  that  also  2p+l  is  a  prime,  stating  that  Lucas^^^  had  proved  that 
2''  — 1  has  the  factor  2p+l.     Cunningham  listed  all  such  primes  p<2500 

i"Comptes  Rendus  Paris,  106,  1888,  448-450;  Coll.  M.  Papers,  IV,  609-610. 

^mid.,  522-6;  Coll.  M.  Papers,  IV,  611-4. 

""Nature,  37,  1888,  417-8;  Coll.  M.  Papers,  IV,  625-9. 

""Mathesis,  8,  1888,  112-3.     M6m.  soc.  sc.  Ukge,  (2),  15,  1888,  205-7  (Melanges  math.,  III). 

»*Math.  Quest.  Educat.  Times,  48,  1888,  p.  xxxvi,  183;  49,  p.  85. 

"«Mathe8is,  10,  1890,  74-76. 

"'Theorie  des  nombres,  1891,  424-5. 

"'Assoc,  frang.  avanc.  sc,  20,  I,  1891,  159. 

"'Exercices  d'Arith.,  Paris,  1893,  363. 

""L'interm^diaire  des  math.,  1,  1894,  148;  1915,  105,  for  representations  by  u*+67t;*. 

""British  Assoc.  Reports,  1894,  563. 


28  History  of  the  Theory  of  Numbers.  [Chap,  i 

and  considered  it  probable  that  primes  of  the  forms  2''±1,  2'^S  (if  not 
yielding  Lucassians)  generally  yield  prime  values  of  2^  —  1,  and  that  no 
other  primes  will.  All  known  and  conjectured  primes  2^  —  1,  with  p  prime, 
fall  under  this  rule. 

In  a  letter  to  Tannery/^-  Lucas  stated  that  Mersenne^°'^^  implied  that 
a  necessary  and  sufficient  condition  that  2^—1  be  a  prime  is  that  p  be  a 
prime  of  one  of  the  forms  2^"+l,  2^''±3,  2^"+^  — 1.  Tannery  expressed  his 
behef  that  the  theorem  was  empirical  and  due  to  Frenicle,  rather  than  to 
Fermat,  and  noted  that  the  sufficient  condition  would  be  false  if  2^'  —  1  is 
composite  [as  is  the  case,  Fauquembergue^^°]. 

Goulard  and  Tannery^^^  made  minor  remarks  on  the  subject  of  the  last 
two  papers. 

A.  Cunningham^^  found  that  2^^' - 1  has  the  factor  7487.  This  contra- 
dicts LeLasseur's^^-  statement  on  di visors <  30000  of  Mersenne's  numbers. 

A.  Cunningham^^^  found  13  new  cases  (317,  337,  547,  937, . . .)  in  which 
2^—1  is  composite,  and  stated  that  for  the  22  outstanding  primes  5^257 
[above  list^^-  except  61,  197]  2^  —  1  has  no  divisor  <  50,000  (error  as  to 
q  =  lSl,  see  Woodall^^).  The  factors  obtained  in  the  mentioned  13  cases 
were  found  after  much  labor  by  the  indirect  method  of  Bickmore,^^^  who 
gave  the  factors  1913  and  5737  of  2-^^-1. 

A.  Cunningham^"  gave  a  factor  of  2^-1  for  g  =  397,  1801,  1367,  5011 
and  for  five  larger  primes  q. 

C.  Bourlet"^  proved  that  the  sum  of  the  reciprocals  of  all  the  divisors 
di  of  a  perfect  number  n  equals  2  [Catalan ^^^],  by  noting  that  n/di  ranges 
with  di  over  the  divisors  of  n,  so  that  2n  =  'Zn/di.  The  same  proof  occurs 
in  II  Pitagora,  Palermo,  16,  1909-10,  6-7. 

M.  Stuyvaert^^^  remarked  that  an  odd  perfect  number,  if  it  exists,  is  a 
sum  of  two  squares  since  it  is  of  the  form  pk^,  where  p  is  a  prime  4n+l 
[Frenicle,^  Euler^sj 

T.  Pepin^"°  proved  that  an  odd  perfect  number  relatively  prime  to  3-7, 
3-5  or  3-5-7  contains  at  least  11,  14  or  19  distinct  prime  factors,  respectively, 
and  can  not  have  the  form  6/cH-5. 

F.  J.  Studnicka^^^  called  Ep  =  2''-\2''-l)  an  Euclidean  number  if  2^-1 
is  a  prime.  The  product  of  all  the  divisors  <Ep  of  Ep  is  E/~'^.  When 
Ep  is  written  in  the  diadic  system  (base  2),  it  has  2p  —  l  digits,  the  first  p  of 
which  are  unity  and  the  last  p  —  1  are  zero. 

"^L'interm^diaire  des  math.,  2,  1895,  317. 

i«/6Mi.,  3,  1896,  115,  188,  281. 

^"Nature,  51,  1894-5,  533;  Proc.  Lond.  Math.  Soc,  26,  1895,  261;  Math.  Quest.  Educat.  Times, 

5,  1904,  108,  last  footnote. 
i«British  Assoc.  Reports,  1895,  614. 
i«On  the  numerical  factors  of  a"-l,  Messenger  Math.,  25,  1895-6,  1-44;  26,  1896-7,  1-38. 

French  transl.  by  Fitz-Patrick,  Sphinx-Oedipe,  1912,  129-144.  155-160. 
"Troc.  London  Math.  Soc,  27,  1895-6,  111. 
"8Nouv.  Ann.  Math.,  (3),  15,  1896,  299. 
"•Mathesis,  (2),  6,  1896,  132. 

''"Memou-e  Accad.  Pont.  Nuovi  Lincei,  13,  1897,  345-420. 
"'Sitzungsber.  Bohm.  Gesell.,  Prag,  1899,  math,  nat.,  No.  30. 


Chap.  I]     PERFECT,  MULTIPLY  PERFECT,  AND   AmICABLE   NUMBERS.  29 

Mario  Lazzarini^"  attempted  to  prove  that  there  is  no  odd  perfect  num- 
ber a'^b^c'^,  but  made  the  error  of  thinking  that  a  is  relatively  prime  to 
6^+. .  .+6+1.  He  attempted  to  show  that  p  =  2"  — 1  is  a  prime  if  and 
only  if  p  divides  iV  =  3*^+1,  where  fc  =  2"~^  — 1  [false  for  a  =  2,  since  p  =  S, 
N  =  4:].  He  restricted  his  argument  to  the  case  a  odd,  whence  p  =  1  (mod  3) . 
Then,  if  p  is  a  prime,  —3  is  a  quadratic  residue  of  p,  so  that  (— 3)^^~^^^^=1 
(mod  p),  whence  p  divides  N.  Conversely,  when  this  congruence  holds,  he 
concluded  falsely  that  z^=—3  (mod  p)  has  two  and  only  two  roots,  so  that 
p  is  expressible  in  a  single  way  as  a  sum  of  a  square  and  the  triple  of  a  square 
and  hence  is  prime.  To  show  the  error,  let  p  =  ab,  where  a  =  23,  6  =  3851  are 
primes;  then 

g-l  6-1 

(-3)11  +  1=  _2a6,(-3)  2  =-l(mod5),(-3)  ^  =(-3)^^-^^^= -1  (mod a), 

whence  (—3)^^"^^^^=!  (mod  p).  Cipolla  remarked  (p.  288)  that  we  may 
deduce  from  a  result  of  Lucas^^°  that  p  is  a  prime  if  it  divides  N  without 
dividing  3*+l  for  any  divisor  8  of  p=2"~^  — 1. 

F.  N.  Cole^'^  found  that  2^"^  - 1  is  the  product  of  the  two  primes  193707721 , 
761838257287.  In  the  footnote  to  p.  136,  he  criticized  the  proof  by  Seel- 
hoff"^  of  the  primality  of  A^  =  2^^  — 1  and  stated  he  had  verified  that  N  is 
prime  by  an  actual  computation  of  a  series  of  primes  of  which  iV  is  a 
quadratic  residue. 

R.  D.  Carmichael^^^  proved  that  any  even  perfect  number  Tp2\  .  .p/" 
is  of  Euclid's  type.    Write  d  for  2'*+^  — 1.     Then,  as  usual, 

d  pf  d        \      p/ 

If  n>2.  Pi  is  less  than  d,  being  an  aliquot  divisor  of  it,  so  that  1  +  1/p, 
exceeds  the  left  member  of  the  inequality.     Hence  n  =  2,  p2  =  d. 

A.  Cunningham^^^  gave  the  residues  of  A; =2^"*,  2*,  etc.,  modulo  2^  —  1  for 
primes  g^lOl. 

A.  Turcaninov^"^  (Turtschaninov)  proved  that  an  odd  perfect  number 
has  at  least  four  distinct  prime  factors  and  exceeds  2000000. 

A.  Gerardin^^^  noted  the  error  by  Plana. ^^° 

A.  Gerardin^^^  stated  the  empirical  laws:  If  n  is  a  prime  of  the  form 
24a^+ll  and  if  2"  — 1  is  composite,  the  least  factor  is  of  the  form  24?/ +23 

"^Periodico  di  mat.  insegn.  sec,  18,  1903,  203;  criticized  by  C.  Ciamberlini,  p.  283,  and  by  M. 

Cipolla,  p.  285. 
i"Bull.  Amer.  Math.  Soc,  10,  1903-4,  134-7.     French  transl.,  Sphinx-Oedipe,  1910,  122-4. 

Cf.  Fauquembergue.^^" 
i^^Annals  of  Math.,  (2),  8,  1906-7,  149. 
I'sproc.  London  Math.  Soc,  (2),  5,  1907,  259  [250]. 
i'6  Vest,  opytn.  fiziki  (Spaczinskis  Bote),  Odessa,  1908,  No.  461  (pp.  106-113),  No.  463  (162-3), 

No.  465-6  (213-9),  No.  470  (314-8).     In  Russian.     Cf.  Bourlet.is* 
'"L'interm^diaire  des  math.,  15,  1908,  230-1. 
'"Sphinx-Oedipe,  Nancy,  3,  1908-9, 113-123;  Assoc,  frang.  avanc  sc,  1909, 145-156.     In  Wis- 

kundig  Tijdschrift,  10, 1913,  61,  he  added  that  in  the  remaining  three  cases  <257,  n  =  107, 

167,  227,  the  least   divisor   (necessarily   >1  mUlion)   is   respectively   5136   y+2783, 

8016  y+335,  10896  J/+5903. 


30  History  of  the  Theory  of  Numbers.  [Chap,  i 

{e.  g.,  n  =  ll,  59,  83,  131,  179,  251).  If  n  is  a  prime  24a:+23  and  2'*-l  is 
composite,  the  least  factor  is  of  the  form  48?/ +47  (e.  g.,  n  =  47,  ?/  =  48,  factor 
2351;  n  =  23,  71,  191,  239).  Gerardin^^^  gave  tables  of  the  possible,  but 
(unverified,  factors  of  2"  — 1,  n<257. 

A.  Cunningham^^o  gave  the  factor  150287  of  2^^^-\. 

A.  Cunningham^^^  found  the  factor  228479  of  2^^-l. 

T.  M.  Putnam^^^  proved  that  not  all  of  the  r  distinct  prime  factors  of  a 
perfect  number  exceed  1  +r/loge2  and  hence  do  not  all  equal  or  exceed  1  +3r/2. 

L.  E.  Dickson^^^  gave  an  immediate  proof  that  every  even  perfect  num- 
ber is  of  Euclid's  type.  Let  2"g  be  perfect,  where  q  is  odd  and  n  >  0.  Then 
(2"+^  —  l)s  =  2'^'^^q,  where  s  is  the  sum  of  all  the  divisors  of  q.  Thus  s  =  q-\-d, 
where  d  =  q/{2''^^  —  l).  Hence  d  is  an  integral  divisor  of  q,  so  that  q  and  d 
are  the  only  divisors  of  q.     Hence  d  =  \  and  5  is  a  prime. 

H.  J.  WoodalP^^  obtained  the  factor  43441  of  2^^^-l. 

R.  E.  Powers^^^  verified  that  2^^  — 1  is  a  prime  by  use  of  Lucas'  test  on 
the  series  4,  14,  194, ....  H.  Tarry^^^  made  an  incomplete  examination. 
E.  Fauquembergue^^^  proved  that  2^^  — 1  is  a  prime  by  writing  the  residues 
of  that  series  to  base  2. 

A.  Cunningham^^^  noted  that  2^ — 1  is  composite  for  three  primes  of  8  digits. 
On  the  proof-sheets  of  this  history,  he  noted  that  the  first  two  should  be 

g  =  67108493,     p  =  134216987;    5  =  67108913,     p  =  134217827. 

A  G^rardin^^^''  observed  that  2'^''+^-\=F^-2(?,  F  =  2"+^±l  =  2m+l, 
G  =  2"±l,  G2  =  m2+(m+l)2-(2")2. 

H.  Tarry^^^^  verified  for  the  known  composite  numbers  2^—1,  where  p 
is  a  prime,  that,  if  a  is  the  least  factor,  2"  —  1  is  composite. 

A.  Gerardin  added  empirically  that,  if  p  is  any  number  and  a  any  di- 
visor of  2^  —  1 ,  a  =  8m  =t  1  not  being  of  the  form  2"  —  1  then  2"  —  1  is  composite. 

A.  Cunningham^^^  noted  that,  if  g  is  a  prime, 

M^  =  2^-\  =  T^-2{quY={qtf-2U\ 

If  Mq  is  a  prime  it  can  be  expressed  in  the  forms  A^-]-?>B^  —  G'^-\-QH'^,  and 
in  one  or  the  other  of  the  pairs  of  forms  f^au^  {ci  =  '^,  14,  21,  42).  He 
discussed  M^  to  the  base  2. 

>'»Sphinx-Oedipe,  3,  1908-9,  118-120,  161-5,  177-182;  4,  1909,  1-5,  158,  168;  1910,  149,  166. 
""Proc.  London  Math.  Soc,  (2),  6,  1908,  p.  xxii. 

"iL'intermddiaire  des  math.,  16,  1909,  252;  Sphinx-Oedipe,  4,  1909,  4e  Trimestre.  36-7. 
"2Amer.  Math.  Monthly,  17,  1910,  167.  ^^Ibid.,  18,  1911,  109. 

'"Bull.  Amer.  Math.  Soc,  16,  1910-11,  540  (July,  1911).     Proc.  London  Math.  Soc,   (2),  9, 

1911,  p.  xvi.     Mem.  and  Proc.  Manchester  Literary  and  Phil.  Soc,  56, 1911-12,  No.  1, 5  pp. 

Sphinx-Oedipe,  1911,  92.     Verification  by  J.  Hammond,  Math.  Quest.   Solutions,   2, 

1916,  30-2. 
i«Bull.  Amer.  Math.  Soc,  18,  1911-12,  162  (report  of  meeting  Oct.,  1911).     Amer.  Math. 

Monthly,  18,  1911,  195.     Sphinx-Oedipe,  Feb.,  1912,  17-20. 
"«Sphinx-Oedipe,  Dec,  1911,   p.   192;  1912,  15.     (Proc.  London  Math.  Soc,  (2),  10,  1912, 

Records  of  Meetings,  1911-12,  p.  ii.) 
^"Ibid.,  1912,  20-22.  "^Messenger  Math.,  41,  1911,  4. 

"saBuU.  Soc.  Philomatiquesde  Paris,  (10),  3, 1911, 221.     isseSphinx-Oedipe,  6, 1911, 174  186, 192. 
""Math.  Quest.  Educ  Times,  (2),  19,  1911,  81-2;  20,  1911,  90-1,  105-6;  21,  1912,  58-9,  73. 


Chap.i]    Perfect,  Multiply  Perfect,  and  Amicable  Numbers.  31 

A.  Cunningham^^o  found  the  factor  730753  of  2^^^-l. 

V.  Ramesam^^i  verified  that  the  quotient  of  2"^-l  by  the  factor  228479 
[Cunningham^^^]  is  the  product  of  the  primes  48544121  and  212885833. 

A.  Aubry^^^  stated  erroneously  that  ''Mersenne  affirmed  that  2"  — 1  is 
a  prime,  for  n^257,  only  for  n  =  l,  2,  3,  4,  8,  10,  12,  29,  61,  67,  127,  257 
(which  has  now  been  almost  proved) ;  this  proposition  seems  to  be  due  to 
Frenicle.^'"  What  Mersenne^"  actually  stated  was  that  the  first  8  perfect 
numbers  occur  at  the  lines  marked  1,  2,  3,  4,  8,  etc.,  in  the  table  by  Bungus. 

A.  Cunningham^^^''  noted  that  M113,  M151,  M251  have  the  further  factors 
23279-65993,  55871,  54217,  respectively.     Cf.  Reuschle^o^  Lucas^^s 

A.  Gerardin^^-^  noted  that  there  is  no  divisor  <  1000000  of  the  composite 
Mersenne  numbers  not  already  factored.  Let  d  denote  the  least  divisor 
of  2«- 1,  g  a  prime  ^257.  li  q  =  60z^+43,  then  d=47  (mod  96),  except  for 
the  cases  given  by  Euler's^^  theorem  (verified  for  43,  163,  223).  If 
5  =  40w+33,  d=7  (mod  24),  verified  for  73,  113,  233.  If  5  =  30m+l,  d=l 
(mod  24),  verified  for  31,  61,  151,  181,  211. 

E.  Fauquembergue^^^''  proved  that  2^°^  — 1  is  composite  by  means  of 
Lucas'  test  with  4,  14,  194,. . .,  written  to  base  2  (Ch.  XVII). 

L.  E.  Dickson^^^  called  a  non-deficient  number  primitive  if  it  is  not  a 
multiple  of  a  smaller  non-deficient  number,  and  proved  that  there  is  only 
a  finite  number  of  primitive  non-deficient  numbers  having  a  given  number 
of  distinct  odd  prime  factors  and  a  given  number  of  factors  2.  As  a 
corollary,  there  is  not  an  infinitude  of  odd  perfect  numbers  with  any  given 
number  of  distinct  prime  factors.  There  is  no  odd  abundant  number  with 
fewer  than  three  distinct  prime  factors;  the  primitive  ones  with  three  are 

3^5-7,    32527,    325.72,    3^5211,    3^13,    3*5^13,    3*52132,    3^5^132. 

There  is  given  a  list  of  the  numerous  primitive  odd  abundant  numbers  with 
four  distinct  prime  factors  and  lists  of  even  non-deficient  numbers  of  certain 
types.  In  particular,  all  primitive  non-deficient  numbers  <  15000  are 
determined  (23  odd  and  78  even).  In  view  of  these  lists,  there  is  no  odd 
perfect  number  with  four  or  fewer  distinct  prime  factors  (cf .  Sylvester^*^"^^^) . 
A.  Cunningham^^*  gave  a  summary  of  the  known  results  on  the  composi- 
tion of  the  56  Mersenne  numbers  Mq  =  2^  —  1,  q  a  prime  ^257.  Of  these, 
12  have  been  proved  prime:  M^,  5  =  1,  2,  3,  5,  7,  13,  17,  19,  31,  61,  89,  127; 
while  29  of  them  have  been  proved  composite.     Thus  only  15  remain  in 

""British  Assoc.  Reports,  1912,  406-7.     Sphinx-Oedipe,  7,  1912,  38  (1910,  170,  that  730753 

is  a  possible  factor).     Cf.  Cunningham"*. 
i"Nature,  89,  1912,  p.  87;  Sphinx-Oedipe,  1912,  38.     Jour,  of  Indian  Math.  Soc,  Madras,  4 

1912,  56. 
"K)euvres  de  Fermat,  4,  1912,  250,  note  to  p.  67. 
"2"  Mem.  and  Proc.  Manchester  Lit.  and  Phil.  Soc,  56,  1911-2,  No.  1. 
i«*  Sphinx-Oedipe,  7,  1912,  num^ro  special,  15-16. 
"'•^  Ibid.,  Nov.,  1913, 176. 
i«Amer.  Jour.  Math.,  35,  1913,  413-26. 
"*Proc.  Fifth  International  Congress,  I,  Cambridge,  1913,  384-6.     Proc.  London  Math.  Soc, 

(2),  11,  1913,  Record  of  Meeting,  Apr.  11,  1912,  xxiv.     British  Assoc.  Reports,  1911, 

321.     Math.  Quest.  Educat.  Times,  (2),  23,  1913,  76. 


32  History  of  the  Theory  of  Numbers.  [Chap.  I 

doubt:  M„  q  =  101,  103,  107,  109,  137,  139,  149,  157,  167,  193,  199,  227, 
229,  241,  257.  The  last  has  no  factor  under  one  million,  as  verified  by 
R.  E.  Powers. ^^^^  No  one  of  the  other  14  has  a  factor  under  one  milhon,  as 
verified  t^dce  with  the  collaboration  of  A.  G^rardin.  Up  to  the  present 
three  errors  have  been  found  in  Mersenne's  assertion;  Mqj  has  been  proved 
composite  (Lucas,^^°  Cole^^^),  while  Mqi  and  Mgg  have  been  proved  prime 
(Pervusin,"°  Seelhoff,^*^  Cole,^"^  Powers^^^).  It  is  here  announced  that  M^^ 
has  the  factor  730753,  found  with  the  collaboration  of  A.  Gerardin. 

J.  ]McDonnell^^^  commented  on  a  test  by  Lucas  in  1878  for  the  primality 
of  2"-l. 

L.  E.  Dickson^^®  gave  a  table  of  the  even  abundant  numbers  <6232. 

R.  Niewiadomski^^^  noted  that  2^^^  — 1  has  the  factor  4567  and  gave 
known  factors  of  2'*  — 1.     He  gave  the  formula 

2^'"+^-l  =  (2^'"+2"*-l)^+(22"'-2'"-l)^  +  l. 

G.  Ricalde^^^  gave  relations  between  the  primes  p,  q  and  least  solutions  of 
22«+i_i  =  pg,  al-2h^  =  p,  c'-2dr  =  q. 

R.  E.  Powers^^^  proved  that  2^°"  —  1  is  a  prime  by  means  of  Lucas'^^  test 
in  Ch.  XVII. 

E.  Fauquembergue^''''  proved  that  2^  —  1  is  prime  for  p  =  107  and  127, 
composite  for  p  =  101,  103,  109. 

T.  E.  Mason-°^  described  a  mechanical  de\'ice  for  applying  Lucas'"^ 
method  for  testing  the  primality  of  2^^'*'^  — 1. 

R.  E.  Powers^°^  proved  that  2^°^  — 1  and  2^°^  — 1  are  composite  by  means 
of  Lucas'  tests  with  3,  7,  47, .  .  .and  4,  14,  194. . .  (Ch.  XVII),  respectively. 

A.  Gerardin^°^  gave  a  history  of  perfect  numbers  and  noted  that  2^—1 
can  be  factored  if  we  find  t  such  that  m  =  2pt-\-\  is  a  prime  not  dividing 
8  =  1+2^+22^+  .  .  .  +2^2'-^^^  since  2-p'-1=  (2^-1)8  (mod  m).  Or  we  may 
seek  to  express  2^—1  in  two  ways  in  the  form  x^—2y'. 

On  tables  of  exponents  to  which  2  belongs,  see  Ch.  VII,  Cunningham 
and  Woodam'^  Kraitchik.^-^ 

Additional  Papers  of  a  Merely  Expository  Character. 

E.  Catalan,  Mathesis,  (1),  6,  1886,  100-1,  178. 

W.  W.  Rouse  Ball,  Messenger  Math.,  21,  1891-2,  34-40,  121. 
-    Pontes  (on  Bovnius^"),  Mem.  Ac.  Sc.  Toulouse,  (9),  6,  1894,  155-67. 

J.  Bezdicek,  Casopis  Mat.  a  Fys.,  Prag,  25,  1896,  221-9. 

Hultsch  (on  lamblichus),  Nachr.  Kgl.  Sachs.  Gesell.,  1895-6. 

H.  Schubert,  Math.  Mussestunden,  I,  Leipzig,  1900,  100-5. 

M.  Nasso,  Revue  de  math.  (Peano),  7,  1900-1,  52-53. 

i»*'Sphinx-Oedipe,  1913,  49-50. 

i«*London  Math.  Soc,  Records  of  Meeting,  Dec,  1912,  v-vi. 

"«Quart.  Jour.  Math.,  44,  1913,  274-7. 

'"L'interm^diaire  des  math.,  20,  1913,  78,  167. 

"s/btd.,  7-8,  149-150;  cf.  140-1. 

'"Proc.  London  Math.  Soc,  (2),  13,  1914,  Records  of  meetings,  xxxi.x.     Bull.  Amer.  Math. 

Soc,  20,  1913^,  531.     Sphinx-Oedipe,  1914,  103-8. 
20<>Sphinx-Oedipe,  June,  1914,  85;  I'interm^diaire  des  math.,  24,  1917,  33. 
-"Proc  Indiana  Acad.  Science,  1914,  429-431. 

2«Proc  London  Math.  Soc,  (2),  15,  1916,  Records  of  meetings,  Feb.  10,  1916,  xxii. 
»"Sphinx-Oedipe,  1909,  1-26. 


Chap.  I]     PERFECT,  MULTIPLY    PERFECT,  AND    AMICABLE    NUMBERS.  33 

G.  Wertheim,  Anfangsgriinde  der  Zahlentheorie,  1902. 

G.  Giraud,  Periodico  di  Mat.,  21,  1906,  124-9. 

F.  Ferrari,  Suppl.  al  Periodico  di  Mat.,  11,  1908,  36-8,  53,  75-6  (Cipolla). 

P.  Bachmann,  Niedere  Zahlentheorie,  II,  1910,  97-101. 

A.  Aubry,  Assoc,  frang.  avanc.  sc,  40,  1911,  53-4;  42,  1913;  Tenseignement 

math.,  1911,  399;  1913,  215-6,  223. 
*M.  Kiseljak,  Beitrage  zur  Theorie  der  vollkommenen  Zahlen,  Progr.  Agram, 

1911. 
*J.  Vaes,  Wiskundig  Tijdschrift,  8,  1911,  31,  173;  9,  1912,  120,  187. 
J.  Fitz-Patrick,  Exercices  Math.,  ed.  3,  1914,  55-7. 

Multiply  Perfect  Numbers. 

A  multiply  perfect  or  pluperfect  number  n  is  one  the  sum  of  whose 
divisors,  including  n  and  1,  is  a  multiple  of  n.  If  the  sum  is  mn,  m  is  called 
the  multiplicity  of  n.  For  brevity,  a  multiply  perfect  number  of  multi- 
plicity m  shall  be  designated  by  P^.  Thus  an  ordinary  perfect  number  is 
a  P2.  Although  Robert  Recorde^^  in  1557  cited  120  as  an  abundant  number, 
since  the  sum  of  its  parts  is  240,  such  numbers  were  first  given  names  and 
investigated  by  French  writers  in  the  seventeenth  century.  As  a  P3  equals 
one-half  of  the  sum  of  its  aliquot  divisors  or  parts  (divisors  KPs),  it  was 
called  a  sous-double;  a  P4  equals  one-third  of  the  sum  of  its  aliquot  parts 
and  was  called  a  sous-triple;  a  P5  a  sous-quadruple;  etc. 

F.  Marin  Mersenne  proposed  to  R.  Descartes^°^  the  problem  to  find  a 
sous-double  other  than  P^^^^  =  120  =  2^3-5.  The  latter  did  not  react  on  the 
question  until  seven  years  later. 

Mersenne^"^  mentioned  (in  the  Epistre)  the  problem  to  find  a  P4,  a 
P5  or  a  P^,  a  P3  besides  120,  and  a  rule  to  find  as  many  as  one  pleases.  He 
remarked  (p.  211)  that  the  P3  120,  the  P4  240  [for  30240?]  and  all  other 
abundant  numbers  can  signify  the  most  fruitful  natures. 

Pierre  de  Fermat^"-  referred  in  1636  to  his  former  [lost]  letter  in  which  he 
gave  "the  proposition  concerning  aliquot  parts  and  the  construction  to 
find  an  infinitude  of  numbers  of  the  same  nature."  He^^^  found  the  second 
P3,  viz.,  P3<2)  =  672  =  2^3-7. 

Mersenne^°*  stated  that  Fermat  found  the  1  3  7  15 . .  . 
P3  672  and  knew  infallible  rules  and  analysis  2  4  8  16 . . . 
to  find  an  infinitude  of  such  numbers.  He^°^  3  5  9  17 .  .  . 
later  gave  [Fermat's]  method  of  finding  such  P3:  Begin  with  the  geometric 

'""Oeuvres  de  Descartes,  1,  Paris,  1897,  p.  229,  line  28,  letter  from  Descartes  to  Mersenne,  Oct 

or  Nov.,  1631. 
^"iLes  Preludes  de  I'Harmonie  Universelle  ou  Questions  Curiouses,  Utiles  aux  Predicateurs,  aux 

Theologiens,  Astrologues,  Medecins,  &  Philosophes,  Paris,  1634. 
'•"Oeuvres  de  Fermat,  2,  Paris,  1894,  p.  20,  No.  3,  letter  to  Mersenne,  June  24,  1636. 
"^Oeuvres  de  Fermat,  2,  p.  66  (French  transl.  3,  p.  288),  2,  p.  72,  letters  to  Mersenne  and 

Roberval,  Sept.,  1636. 
'"Harmonic  Universelle,  Paris,  1636,  Premiere  Preface  Generale  (preceded  by  a  preface  of  two 

pages),  imnumbered  page  9,  remark  10.     Extract  in  Oeuvres  de  Fermat,  2,  1894,  20-21. 
"'Mersenne,  Seconde  Partie  de  I'Harmonie  Universelle,  Paris,  1637.     Final  subdivision:  Nou- 

velles  Observations  Physiques  et  Math^matiques,  p.  26,  Observation  13.     Extract  in 

Oeuvres  de  Fermat,  2,  1894,  p.  21.  , 


34  History  of  the  Theory  of  Numbers.  [Chap.  I 

progression  2,  4,  8, ... .  Subtract  unity  and  place  the  remainders  above  the 
former.  Add  unity  and  place  the  sums  below.  Then  if  the  quotient  of 
the  (n+3)th  number  of  the  top  line  by  the  nth  number  of  the  bottom 
line  is  a  prime,  its  triple  multiplied  by  the  (n+2)th  number  of  the  middle 
line  is  a  P3.  Thus  if  ?i  =  l,  15/3  is  a  prune  and  3-5-8  =  120  is  a  P3.  For 
n  =  3,  63/9  is  a  prime  and  3-7-32  =  672  is  a  P3.  [This  rule  thus  states  in 
effect  that  3-2"+2p  is  a  P3  if  p  =  (2'»+3-l)/(2"+l)  is  a  prune.] 

The  third  P3,  discovered  by  Andr6  Jumeau,  Prior  of  Sainte-Croix,  is 

P3^'^  =  523776  =  29311-31. 

In  April,  1638,  he  communicated  it  to  Descartes^°^  and  asked  for  the  fourth 
P3  (the  fifth  and  last  of  St.  Croix's  challenge  problems). 

Descartes^*^^  stated  that  the  rule  ^^^  of  Fermat  furnishes  no  P3  other  than 
120  and  672  and  judged  that  Fermat  did  not  find  these  numbers  by  the 
formula,  but  accommodated  the  formula  to  them,  after  finding  them  by  trial. 

Descartes^^^  answered  the  challenge  of  St.  Croix  with  the  fourth  P3, 

P3^'^  =  1476304896  =  2^33.1143.127. 

Soon  afterwards  Descartes^  °^  announced  the  following  six  P4: 

P4^i)  =  30240  =  2^335.7, 
P4(2)  =32760  =  2^325.7.13, 
P^^^^  =  23569920  =  2^335. 1 1  -31 , 
P4(*)  =  142990848  =  2^327.11.13.31, 
P4(^>  =  66433720320  =  2^^335. 1 1 .43. 1 27, 
P4^®^  =403031236608  =  2^3327.11.13.43.127, 

and  the  sous-quadruple 

Ps^^^  =  14182439040  =  2^3^5.7-11217.19. 

He  stated  that  his  analysis  had  led  him  to  a  method  which  would  require 
time  to  explain  in  the  form  of  a  rule,  but  that  he  could  find,  for  example, 
a  sous-centuple,  necessarily  very  large. 

Fermat  apparently  responded  to  the  fifth  challenge  problem  of  St.  Croix 
on  the  fourth  P3.  Without  warrant,  Descartes^^"  suspected  that  Fermat 
had  not  found  independently  the  fourth  P3,  but  had  learned  from  some  one 
in  Paris  of  its  earlier  discovery  by  Descartes.  Fermat^^^  indicated  that  he 
possessed  an  analytic  method  by  which  he  could  solve  all  questions  con- 

'"^Oeuvres  de  Descartes,  2,  Paris,  1898,  p.  428,  p.  167  (latter  without  name  of  St.  Croix);  cf. 

Oeuvres  de  Fermat,  2,  1894,  pp.  63-64. 
"'Oeuvres  de  Descartes,  2,  1898,  p.  148,  letter  to  Mersenne,  May  27,  1638. 
^osQeuvres  de  Descartes,  2,  1898,  167,  letter  to  Mersenne,  June  3,  1638. 
'"'Oeuvres  de  Descartes,  2,  1898,  2.50-1,  letter  to  Mersenne,  July  13,  1638.     In  June,  1645, 

Descartes,  4,  1901,  p.  229,  again  mentioned  the  first  two  of  these  Pt. 
""Oeuvres  de  Descartes,  2,  1898,  273,  letter  to  Mersenne,  July  27,  1638. 
"^Oeuvres  de  Fermat,  2,  1894,  p.  165,  No.  4;  p.  176,  No.  1;  letters  to  Mersenne,  Aug.  10  and 

Dec.  26,  1638. 


Chap.  I]     PERFECT,  MULTIPLY  PERFECT,  AND   AmICABLE   NUMBERS.  35 

cerning  aliquot  parts,  apart  from  the  testing  of  the  primaUty  of  a  number  n, 
knowing  no  method  except  the  trial  of  each  number  <  \/n  as  a  divisor. 
Descartes"^  gave  the  following  rules  for  multiply  perfect  numbers: 

I.  If  n  is  a  P3  not  divisible  by  3,  then  3n  is  a  P4. 
II.  If  a  P3  is  divisible  by  3,  but  by  neither  5  nor  9,  then  45P3  is  a  P4. 

III.  If  a  P3  is  divisible  by  3,  but  not  by  7,  9  or  13,  then  3-7-13  P3  is  a  P4. 

IV.  If  n  is  divisible  by  2^  but  by  no  one  of  the  numbers  2^°,  31,  43,  127, 

then  31n  and  16-43-127n  are  proportional  to  the  sums  of  their 
aliquot  parts. 
V.  If  n  is  not  divisible  by  3  and  if  3n  is  a  P^k,  then  n  is  a  Ps^. 

By  applying  rule  II  to  P3^^\  Ps^^\  P3^*^  Descartes  obtained  his  P^^^^ 
P4<3^  P^^^K    By  applying  rule  III  to  Ps^'\  Ps^^\  Ps^''\  he  obtained  his 

p  (2)     p  (4)      p  (6) 
^4      )  ^4      5  ^4      • 

In  the  same  letter,  Descartes  expressed  to  Mersenne  a  desire  to  know 
what  Frenicle  de  Bessy  had  found  on  this  subject.  Frenicle  wrote  direct 
to  Descartes,  who  in  his  reply^^^  expressed  his  astonishment  that  Frenicle 
should  regard  as  sterile  the  above  rules  for  finding  P4,  since  Descartes  had 
deduced  by  them  six  P4  from  four  P3,  at  a  time  when  Mersenne  had  stated 
to  Descartes  that  it  was  thought  to  be  impossible  to  find  any  at  all.  Des- 
cartes stated  that,  since  one  can  find  an  infinity  of  such  rules,  one  has  the 
means  of  finding  an  infinitude  of  P^-  From  one  of  Frenicle's  P5  (com- 
municated to  Descartes  by  Mersenne), 

Ps^^)  =  30823866178560  =  2i°3^5-72l3- 19-23-89, 

Descartes  (p.  475)  derived  the  smaller  P5: 

Pg^^)  =  31998395520  =  2^3^5.72.13.17.19. 

Mersenne^^^  listed  various  P^  due  to  his  correspondents,  without  cita- 
tion of  names.  He  listed  the  above  Ps^'^  (^  =  1,  2,  3,  4)  and  remarked  that 
"un  excellent  esprit  "^^^  found  that  when 

P3^'^  =  459818240  =  2^5.7.19.37.73 

is  multiplied  by  3,  the  product  is  a  P4: 

P4(^  =  2^3.5.7.19.37.73, 

attributed  to  Lucas^^^  by  Carmichael.^^^ 

"^Oeuvres,  2,  1898,  427-9,  letter  to  Mersenne,  Nov.  15,  1638. 

"'Oeuvres  de  Descartes,  2,  1898,  471,  letter  to  Frenicle,  Jan.  9,  1639. 

'"Les  Nouvelles  Pensees  de  Galilei,  traduit  d'ltalien  en  Frangois,  Paris,  1639,  Preface,  pp.  6-7. 

Quoted  in  Oeuvres  de  Descartes,  10,  Paris,  1908,  pp.  564-6,  and  in  Oeuvres  de  Fermat,  4, 

1912,  pp.  65-66. 
'"Frenicle  de  Bessy,  according  to  the  editors  of  the  Oeuvres  de  Fermat,  2,  1894,  p.  255,  note  2; 

4,  1912,  p.  65,  note  2  (citing  Oeuvres  de  Descartes,  2,  letter  Descartes  to  Mersenne,  Nov. 

15,  1638,  pp.  419-448  [p.  429]).     It  is  clear  that  the  discoverers  Fermat,  St.  Croix,  and 

Descartes  of  the  P|^*)  (i  =  2,  3,4)  are  not  meant.     It  is  attributed  to  Legendre""  by 

Carmichael.*'* 


36  History  of  the  Theory  of  Numbers.  [Chap,  i 

There  are  listed  Descartes'  six  P^  and  P^^^^  Frenicle's  Ps^\  and  also 

P4^«)  =45532800  =  2^3^5217.31, 

P4^^^  =  43861478400  =  2^°3^5223.31.89, 

and  the  erroneous  P5  508666803200  (not  divisible  by  5^+5-|-l),  probably 
a  misprint  for  the  correct  P5  (in  the  list  by  Lehmer^'^) : 

P5^'^  =  518666803200  =  2^^3'5-72l3- 19-31 . 

A  part  of  these  Pm,  but  no  new  ones,  were  mentioned  by  Mersenne^"  in 
1644;  the  least  P3  is  stated  to  be  120.     (Oeuvres  de  Fermat,  4,  66-7.) 
In  1643  Fermat^ ^^  cited  a  few  of  the  P^  he  had  found: 

Ps^^^  =  51001 180160  =  2^*5.7.19.31.151 , 

P4^'^)  =  14942123276641920  =  2^3^5.17.23.137.547.1093, 
P5(^)  =  1802582780370364661760  =  22°335.72l32l9.31.61. 127.337, 
Ps^®^  =  87934476737668055040  =  2^^3^5.7313. 19-37.73. 127, 
Pg(i)  =  223375374113133^7231.41.^1.241.307.467.2801, 
Pg(2)  =  22735537.11.13219.29.31.43.61.113.127. 

He  stated  that  he  possessed  a  general  method  of  finding  all  P^n- 

Replying  to  Mersenne's  query  as  to  the  ratio  of 

Pg(3)  =  22^3^5^11.13219.31243.61.83.223.331.379.601.757 

X  1201.7019.823543-616318177:100895598169 

to  the  sum  of  its  aUquot  parts,  Fermat^^^  stated  that  it  is  a  Pq,  the  prime 

factors  of  the  final  factor  being  112303  and  898423  [on  the  finding  of  these 

factors,  see  Ch.  jXIV,  references  23,  92,  94,  103].     Note  that  823543  =  7^ 

Descartes^^^  constructed  P3^2)  =  572  =  21.32  by  starting  with  21  and 
noting  that  (r(21)  =32,  o-(32)  =63  =  3.21,  for  a  defined  as  on  p.  53. 

Mersenne^^  noted  that  if  a  P3  is  not  divisible  by  3,  then  3P3  is  a  P4 
[rule  I  of  Descartes^^-];  if  a  P5  is  not  divisible  by  5,  then  5P5  is  a  Pq,  etc. 
He  stated  that  there  had  been  found  34  P4, 18  P5, 10  Pg,  7  P7,  but  no  Pgso  far. 

In  1652,  J.  Broscius  (Apologia,^*  p.  162)  cited  the  P4^'^  [of  Descartes^"^]. 
The  P3  120  and  672  are  mentioned  in  the  1770  edition  of  Ozanam's'^ 
Recreations,  I,  p.  35,  and  in  Hutton's  translation  of  Montucla's^^  edition, 
I,  p.  39. 

A.  M.  Legendre^^^  determined  the  Pm  of  the  form  2"a/37 .  .  . ,  where  a, 
/3,  7, . .  .are  distinct  odd  primes,  for  7n  =  3,  n^8;  ?n=4,  n  =  3,  5;  m  =  5,  n  =  7. 
No  new  P„  were  found. 

"*Oeuvres,  2, 1894,  p.  247  (261),  letter  to  Carcavi;  Varia  opera,  p.  178;  Pr^cia  des  oeuvres  math, 
de  Fermat,  par  E.  Brassinne,  Toulouse,  1853,  p.  150. 

3'^Oeuvers  de  Fermat,  2,  1894,  255,  letter  to  Mersenne,  April  7,  1643.  The  editors  (p.  256,  note) 
explained  the  method  of  factoring  probably  used  by  Fermat.  The  sum  of  the  aliquot 
parts  of  23«  is  223iV,  where  N  =  616318177,  and  the  sum  of  the  aliquot  parts  of  N  is  2-7?  M 
iVf  =  898423.  As  M  does  not  occur  elsewhere  in  Pe.,  it  is  to  be  expected  as  a  factor  of 
the  final  factor  of  Pe. 

"8Manuscript  published  by  C.  Henry,  Bull.  Bibl.  Storia  Sc.  Mat.  e  Fis.,  12,  1879,  714. 

'^•Thor^ie  des  nombres,  3d  ed.,  vol.  2,  Paris,  1830,  146-7;  German  transl.  by  H.  Maser,  Leipzig, 
2,  1893,  141-3.     The  work  for  n?  =3  was  reproduced  by  Lucas'^"  without  reference. 


Chap.  I]    PERFECT,  MULTIPLY   PERFECT,  AND   AmICABLE   NUMBERS.  37 

E.  Lucas^^o  ga^g  a  table  of  P^  of  the  form  2''-\2''-l)N  which  includes 
only  15  of  the  26  Pm  given  above  and  no  additional  P^,  m>2,  except 
two  erroneous  P5 : 

2^°3^5-72.1l2.19.23.89,  2i^5-72l3.19237-73-127, 

attributed  elsewhere^^^  by  him  to  Fermat.  If  we  replace  7^  by  7  in  the 
former,  we  obtain  a  correct  P5  listed  by  Carmichael  'P"^ 

P5(7>  =  2'°3*5-7. 11219.23-89. 

If  in  the  second,  we  replace  5-7^  by  3^-5-7^  we  obtain  Fermat's  P^-^K 

A.  Desboves^^^  noted  that  120  and  672  are  the  only  P3  of  the  form 

2"-3-p,  where  p  is  a  prime. 

D.  N.  Lehmer^^^  gave  the  additional  P„: 

p^(i2)  ^22325.7213.19, 

P^(i3)  =  2«3272l3.19237.73-127, 

P5®  =2213^527.19.23231.79.89.137.547.683.1093, 

Pg(4)  =2193^5^7211. 13.19.23.31.41. 137.547.1093, 

Pg(5)  =22^3^5.7211.13.17.19231.43.53.127.379.601.757.1801. 

He  readily  proved  that  a  P3  contains  at  least  3  distinct  prime  factors,  a 
P4  at  least  4,  a  P5  at  least  6,  a  Pq  at  least  9,  a  P^  at  least  14. 

J.  Westlund324  proved  that  2^3.5  and  2^3.7  are  the  only  P3  of  the  form 
V\'V2Vzy  where  the  p's  are  primes  and  Pi<P2<P3.  He^^^  proved  that  the 
only  P3  =  Pi"p2P3P4,  Pi<P2<P3<P„  is  P3^'^  =293.11.31. 

A.  Cunningham^^^  considered  P^  of  the  form  2^  \2^—l)F,  where  F  is 
to  be  suitably  determined.  There  exists  at  least  one  such  P^  for  every  q 
up  to  39,  except  33,  35,  36,  and  one  for  g  =  45,  51,  62.  Of  the  85  P^  found, 
the  only  one  published  is  the  largest  one,  viz.,  for  q  =  Q2,  giving  Pq^^^  with 

F  =  3^5'72ll. 13.19223.59.71.79.127.157.379.757.43331.3033169; 

while  none  have  m>6,  and  for  m  =  3  at  most  one  has  a  given  q.  He  found 
in  1902  (but  did  not  publish)  the  two  P7  =  2^H2^^-1)P,  where 

F  =  C.192127  or  0.19^51-911, 

C  =  3i^-5^.7^.1M3.17-23.31-37-41.43.61.89.97-193.442151. 

R.  D.  CarmichaeP^^  has  shown  that  there  exists  no  odd  P^  with  only 
three  distinct  prime  factors;  that  2^3-5  and  2^3.7  are  the  only  P^  with  only 

'20Bull.  Bibl.  e  Storia  Mat.  e  Fis.,  10,  1877,  286.     In  253-5-7,  listed  as  a  Pt,  3  is  a  misprint  for  3». 
'"Lucas,  Theorie  des  Nombres,  1,  Paris,  1891,  380.     Here  the  factor  11^  IS^  of  Fermat's  P«(') 

is  given  erroneously  as  lllS^,  while  the  Pe^i^  of  Descartes  is  attributed  to  Fermat. 
'^Questions  d'Algebre,  2d  ed.,  1878,  p.  490,  Ex.  24. 
'^'Annals  of  Math.,  (2),  2,  1900-1,  103-4. 
"^Annals  of  Math.,  (2),  2,  1900-1,  172-4. 
'"Annals  of  Math.,  (2),  3,  1901-2,  161-3. 
'"British  Association  Reports,  1902,  528-9. 
'"American  Math.  Monthly,  13,  Feb.,  1906,  35-36. 

717  8  5 


38  History  of  the  Theory  of  Numbers.  [Chap,  i 

three  distinct  prime  factors  ;^^^  that  those  with  only  four  distinct  prime  fac- 
tors are^^"  the  P^^^^  of  St.  Croix^"^  and  the  P4^'^  of  Descartes  f°^  and  that  the 
even  P„  with  five^^^  distinct  prime  factors  are  P3^^\  Pi^\  Pi^^  of  Des- 
cartes^"^'  ^"^  and  P^^^^  of  Mersenne.^" 

CarmichaeP^^'*  stated  and  J.  Westlund  proved  that  if  n>4,  no  P„  has 
only  n  distinct  prime  factors. 

Carmichael's^^^  table  of  multiply  perfect  numbers  contains  the  misprint 
1  for  the  final  digit  0  of  Descartes'  Pi^\  and  the  erroneous  entry  919636480 
in  place  of  its  half,  viz.,  P^-^^  of  Mersenne.^^^    The  only  new   P^  is 

Pg(7)  =  2^^3^527211. 13-17.19-3143.257. 

All  P,;,<  10^  were  determined;  only  known  ones  were  found. 
CarmichaeP^^  gave  an  erroneous  P5  and  the  new  P^: 

p^(i4)^2"3272l3-1923M27.151, 

p^(i5)^2253^52l9-31.683.2731-8191, 

p^(i6)^225365-19-23-137-547-683-1093-2731-8191. 

Carmichael  and  T.  E.  Mason^^  gave  a  table  which  includes  the  above 
hsted  10  P2,  6  P3,  16  P4,  8  P5,  7  Pq,  together  with  204  new  multiply  perfect 
numbers  P,  (i  =  3, . .  . ,  7) .  Of  the  latter,  29  are  of  multiplicity  7,  each 
having  a  very  large  number  of  prime  factors.  No  P7  had  been  previously 
published. 

[As  a  generalization,  consider  numbers  n  the  sum  of  the  kth.  powers  of 
whose  divisors  <  n  is  a  multiple  of  n.  For  example,  n  =  2p,  where  p  is  a 
prime  8/1  ±3  and  k  is  such  that  2*^+1  is  divisible  by  p;  cases  are  p  =  3, 
k  =  l;  p  =  5,  k  =  2;  p  =  ll,  k  =  5;  p  =  13,  A;  =  6.] 

Amicable  Numbers. 

Two  numbers  are  called  amicable*  if  each  equals  the  sum  of  the  aliquot 
divisors  of  the  other. 

According  to  lamblichus^  (pp.  47-48),  "certain  men  steeped  in  mistaken 
opinion  thought  that  the  perfect  number  was  called  love  by  the  Pythago- 
reans on  account  of  the  union  of  different  elements  and  affinity  which  exists 
in  it;  for  they  call  certain  other  numbers,  on  the  contrary,  amicable  num- 
bers, adopting  virtues  and  social  quahties  to  numbers,  as  284  and  220,  for 
the  parts  of  each  have  the  power  to  generate  the  other,  according  to  the  rule 
of  friendship,  as  Pythagoras  affirmed.  WTien  asked  what  is  a  friend,  he 
replied,  'another  I,'  which  is  shown  in  these  numbers.  Aristotle  so  defined 
a  friend  in  his  Ethics." 

«»Aimalsof  Math.,  (2),  7,  1905-6,  153;  8, 1906-7,  49-56;  9, 1907-8,  180,  for  a  simpler  proof  that 

there  is  no  Pa  =  Pi^p^Vi^t  c>  1. 
""Annals  of  Math.,  (2),  8,  1906-7,  149-158. 

"'Bull.  Amer.  Math.  Soc,  15, 1908-9,  pp.  7-8.    Fr.  transl.,  Sphinx-Oedipe,  Nancy,  5, 1910, 164-5. 
wi^Amer.  Math.  Monthly,  13,  1906,  165. 

»«Bull.  Amer.  Math.  Soc,  13, 1906-7,  383-6.     Fr.  transl.,  Sphinx-Oedipe,  Nancy,  5,  1910, 161-4. 
»"Sphinx-Oedipe,  Nancy,  5,  1910,  166. 
»"Proc.  Indiana  Acad.  Sc,  1911,  257-270. 
*Amiable,  agreeable,  befreundete,  verwandte. 


Chap.  I]     PERFECT,  MULTIPLY  PERFECT,  AND   AMICABLE   NUMBERS.  36 

In  the  ninth  century  the  Arab  Thabit  ben  Korrah^°  (prop.  10)  noted  that 
2'*/i^  and  2"s  are  amicable  numbers  if 

(1)  /i  =  3-2"-l,  f=3-2"-i-l,  s  =  9'2^''-'-l 

are  primes  >  2,  literally,  if 

/l  =  2+2^      t  =  z-2''-\      3  =  H-2+...+2^      s  =  (2"+^+2'*-2)2''+^-l. 

The  term  used  for  amicable  numbers  was  se  invicem  amantes.  In  the  article 
in  which  F.  Woepcke^'^  translated  this  Arabic  manuscript  into  French,  he 
noted  that  a  definition  of  these  numbers,  called  congeneres,  occurs  in  the 
51st  treatise  (on  arithmetic)  of  Ikhovan  Algafa,  manuscript  1105,  anciens 
fonds  arabes,  p.  15,  of  the  National  Library  of  Paris. 

Among  Jacob's  presents  to  Esau  were  200  she-goats  and  20  he-goats, 
200  ewes  and  20  rams  (Genesis,  XXXII,  14).  Abraham  Azulai^^^  (1570- 
1643),  in  commenting  on  this  passage  from  the  Bible,  remarked  that  he  had 
found  written  in  the  name  of  Rau  Nachshon  (ninth  century  A.  D.):  Our 
ancestor  Jacob  prepared  his  present  in  a  wise  way.  This  number  220  (of 
goats)  is  a  hidden  secret,  being  one  of  a  pair  of  numbers  such  that  the  parts 
of  it  are  equal  to  the  other  one  284,  and  conversely.  And  Jacob  had  this  in 
mind;  this  has  been  tried  by  the  ancients  in  securing  the  love  of  kings  and 
dignatories. 

Ibn  Khaldoun^^°  related  "that  persons  who  have  concerned  themselves 
with  talismans  affirm  that  the  amicable  numbers  220  and  284  have  an 
influence  to  establish  a  union  or  close  friendship  between  two  individuals. 
To  this  end  a  theme  is  prepared  for  each  individual,  one  during  the  ascend- 
ency of  Venus,  when  that  planet  is  in  its  exaltation  and  presents  to  the 
moon  an  aspect  of  love  or  benevolence;  for  the  second  theme  the  ascendency 
should  be  in  the  seventh.  On  each  of  these  themes  is  written  one  of  the 
specified  numbers,  the  greater  (or  that  with  the  greater  sum  of  its  aliquot 
parts?)  being  attributed  to  the  person  whose  friendship  is  sought." 

The  Arab  El  Madschriti,^^!  or  el-Magriti,  (flOOT)  of  Madrid  related  that 
he  had  himself  put  to  the  test  the  erotic  effect  of  ''giving  any  one  the 
smaller  number  220  to  eat,  and  himself  eating  the  larger  number  284." 

Ibn  el-Hasan^"''  (tl320)  wrote  several  works,  including  the  "Memory 
of  Friends,"  on  the  explanation  of  amicable  numbers. 

Ben  Kalonymos^^^^  discussed  amicable  numbers  in  1320  in  a  work 
written  for  Robert  of  Anjou,  a  fragment  of  which  is  in  Munich  (Hebr.  MS. 
290,  f.  60).  A  knowledge  of  amicable  numbers  was  considered  necessary 
by  Jochanan  Allemanno  (fifteenth  century)  to  determine  whether  an 
aspect  of  the  planets  was  friendly  or  not. 

^^^Baale  Brith  Abraham  [Commentary  on  the  Bible],  Wilna,  1873,  22.     Quotation  suppUed  by 

Mr.  Ginsburg. 
'^oProlegomenes  hist.  d'Ibn  Khaldoun,  French  transl.  by  De  Slane,  Notices  et  Extraits  des 

Manuscrits  de  la  Bibl.  Imperiale,  Paris,  21,  I,  1868,  178-9. 
"^'Manuscript  Magriti;  Steinschneider,  Zur  pseudoepigraphischen   Literatur  inbesondere  der 

geheimen  Wissenschaften  des  Mittelalters,  Berlin,  1862,  p.  37  (cf.  p.  41). 
'""H.  Suter,  Abh.  Gesch.  Math.  Wiss.,  10,  1900,  159,  §  389. 
»"*Hebr.  Bibl.,  VII,  91.     Steinschneider,  Zeitschrift  der  Morgenlandischen  Ges.,  24,  1870,  369. 


5 

11 

23 

47 

2 

4 

8 

16 

6 

12 

24 

48 

71 

287 

1151 

40  History  of  the  Theory  of  Numbers.  [Chap,  i 

Alkalacadi,^"  a  Spanish  Arab  (tl486),  showed  the  method  of  finding  the 
least  amicable  numbers  220,  284. 

Nicolas  Chuquet^^  in  1484  and  de  la  Roche^^  in  1538  cited  the  amicable 
numbers 220, 284,  "de  merueilleuse  familiarite  lung  auec  laultre."  In  1553, 
Michael  StifeP^  (folios  26v-27v)  mentioned  only  this  pair  of  amicable  num- 
bers. The  same  is  true  of  Cardan, ^^  of  Peter  Bungus^-  (Mysticae  numerorum 
signif.,  1585,  105),  and  of  TartagUa.^^  Reference  may  be  made  also  to 
Schwenter." 

In  1634  Mersenne^"^  (p.  212)  remarked  that  "220  and  284  can  signify 
the  perfect  friendship  of  two  persons  since  the  sum  of  the  aliquot  parts  of 
220  is  284  and  conversely,  as  if  these  two  numbers  were  only  the  same  thing." 

According  to  Mersenne's^"^  statement  in  1636,  Fermat^^  found  the 
second  pair  of  amicable  numbers 

17296  =  2'.23.47,  18416  =  2^-1151, 

and  communicated  to  Mersenne^°^  the  general  rule:  Begin  with  the  geo- 
metric progression  2,  4,  8, ... ,  write  the  prod- 
ucts by  3  in  the  line  below;  subtract  1  from 
the  products  and  enter  in  the  top  row.  The 
bottom  row  is  6-12-1,  12-24-1,.  .  .When  a 
mmiber  of  the  last  row  is  a  prime  (as  71)  and 
the  one  (11)  above  it  in  the  top  row  is  a  prime, 

and  the  one  (5)  preceding  that  is  also  a  prime,  then  71.4  =  284,  5-11-4  =  220 
are  amicable.     Similarly  for 

1151-16  =  18416,  23-47-16  =  17296, 

and  so  to  infinity.     [The  rule  leads  to  the  pair  2"/i<,  2'*s,  where  h,  t,  s  are 
given  by  (1).] 

Descartes^^^  gave  the  rule:  Take  (2  or)  any  power  of  2  such  that  its 
triple  less  1,  its  sextuple  less  1,  and  the  18-fold  of  its  square  less  1  are  all 
primes;*  the  product  of  the  last  prime  by  the  double  of  the  assumed  power 
of  2  is  one  of  a  pair  of  amicable  numbers.  Starting  with  the  powers  2,  8,  64, 
we  get  284,  18416,  9437056,  whose  aliquot  parts  make  220,  etc.  Thus  the 
third  pair  is 

9363584  =  2^-191-383,  9437056  =  2"-73727. 

Descartes^^^  stated  that  Fermat's  rule  agrees  exactly  with  his  own. 

Although  we  saw  that  Mersenne  quoted  in  1637  the  rule  in  Fermat's 
form  and  expressly  attributed  it  to  Fermat,  curiously  enough  Mersenne^ ^^ 
gave  in  1639  the  rule  in  Descartes'  form,  attributing  it  to  "un  excellent 
Geometre"  (meaning  without  doubt  Descartes,  according  to  C.  Henry^"), 

''^Manuscript  in  Biblioth^que  Nationale  Paris,  a  commentary  on  the  arithmetic  Talkhys  of 
Ibn  Albanna  (13th  cent.).     Cf.  E.  Lucas,  L'arithm^tique  amusante,  Paris,  1895,  p.  64. 

'"Quesiti  et  Inventione,  1554,  fol.  98  v. 

'^♦Oeuvres  de  Fermat,  2,  1894,  p.  72,  letter  to  Roberval,  Sept.  22, 1636;  p.  208,  letter  to  Frenicle, 
Oct.  18,  1640. 

»^euvre8  de  Descartes,  2,  1898,  93-94,  letter  to  Mersenne,  Mar.  31, 1638. 
•Evidently  the  numbers  (1)  if  the  initial  power  of  2  be  2""^ 

"•Oeuvres  de  Descartes,  2,  1898,  148,  letter  to  Mersenne,  May  27,  1638. 

"'Bull.  Bibl.  Storia  So.  Mat.  e  Fis.,  12,  1879,  523. 


Chap.  I)     PERFECT,  MULTIPLY   PERFECT,  AND   AmICABLE   NUMBERS.  41 

and  derived  as  did  Descartes  the  first  three  pairs  of  amicable  numbers  from 
2,  8,  64.  We  shall  see  that  various  later  writers  attributed  the  rule  to 
Descartes. 

Mersenne^^  again  in  1644  gave  the  above  three  pairs  of  amicable  num- 
bers, the  misprints  in  both^^^  of  the  numbers  of  the  third  pair  being  noticed 
at  the  end  of  his  book,  and  stated  there  are  others  innumerable. 

Mersenne®^  in  1647  gave  without  citation  of  his  source  the  rule  in  the 
form  2-2%  2-2"/i<,  where  1  =  3-2'' -1,  h  =  2t+l,  s  =  ht+h-i-t  are  primes 
[as  in  (1)]. 

Frans  van  Schooten,^^^  the  younger,  showed  how  to  find  amicable 
numbers  by  indeterminate  analysis.  Consider  the  pair  4a:,  4yz  [x,  y,  z  odd 
primes];  then 

7+3a:  =  4i/0,  7-\-7y+7z+Syz  =  4x. 

Eliminating  x,  we  get  2  =  34-16/(2/  — 3).  The  case  ^  =  5  gives  2;=  11,  ^'  =  71, 
yielding  284,  220.  He  proved  that  there  are  none  of  the  type  2x,  2yz,  or 
8x,  Syz,  and  argued  that  no  pair  is  smaller  than  284,  220.  For  16.x,  16yz, 
he  found  2=15-l-256/(i/  — 15),  which  for  y  =  47  yields  the  second  known 
pair.  There  are  none  of  the  type  32a;,  S2yz,  or  type  64a:,  Myz.  For  128a:, 
1281/2,  he  got  2=  127-^16384/(1/ -127),  which  for  2/  =  191  yields  the  third 
known  pair.     Finally,  he  quoted  the  rule  of  Descartes. 

W.  Leyboum^^  stated  in  1667  that  ''there  is  a  fine  harmony  between 
these  two  numbers  220  and  284,  that  the  aliquot  parts  of  the  one  do  make 
up  the  other . .  .  and  this  harmony  is  not  to  be  found  in  many  other  numbers." 

In  1696,  Ozanam''^  gave  in  great  detail  the  derivation  of  the  three  known 
pairs  of  "amiable"  numbers  by  the  rule  as  stated  by  Descartes,  whose  name 
was  not  cited.     Nothing  was  added  in  the  later  editions.'^'  ^^ 

Paul  Halcke^^"  gave  Stifel's^^  rule,  as  expressed  by  Descartes.^^^ 

E.  Stone^^^  quoted  Descartes'  rule  in  the  incorrect  form  that  2^''pq  and 
3-2"p  are  amicable  if  p  =  3-2'*— 1  and  g  =  6-2'*— 1  are  primes. 

Leonard  Euler^^^  remarked  that  Descartes  and  van  Schooten  found  only 
three  pairs  of  amicable  numbers,  and  gave,  without  details,  a  fist  of  30  pairs, 
all  included  in  the  later  paper  by  Euler.^^^ 

G.  W.  Kraft^^^  considered  amicable  numbers  of  the  type  APQ,  AR, 
where  P,  Q,  R  are  primes  not  dividing  A.  Let  a  be  the  sum  of  all  the  divi- 
sors of  A .    Then 

R+1  =  {P+1){Q+1),  {R-{-l)a  =  APQ-^AR. 

Assuming  prime  values  of  P  and  Q  such  that  the  resulting  R  is  prime,  he 
sought  a  number  A  for  which  A  /a  has  the  derived  value.     For  P  =  3,  Q  =  1 1 , 

"'Not  noticed  in  the  correction  (left  in  doubt)  in  Oeuvres  de  Fermat,  4,  1912,  p.  250  (on  pp. 

66-7).     One  error  is  noted  in  Broscius*^,  Apologia,  1652,  p.  154. 
"'Exercitationum  mathematicarum  libri  quinque,  Ludg.  Batav.,  1657,  liber  V :  sectiones  triginta 

miscellaneas,  sect.  9,  419-425.     Quoted  by  J.  Landen.*^ 
""Deliciae  Mathematicae,  oder  Math.  Sinnen-Confect,  Hamburg,  1719,  197-9. 
*"New  Mathematical  Dictionary,  1743  (under  amicable) . 
"'De  numeria  amicabilibus.  Nova  Acta  Eruditorum,  Lipsiae,  1747,  267-9;  Comm.  Arith.  Coll., 

II,  1849,  637-8. 
-•»Novi  Comm.  Ac.  Petrop.,  2,  1751,  ad  annum  1749,  Mem.,  100-18. 


42  History  of  the  Theory  of  Numbers.  [Chap,  i 

then  A:a  =  S:5;  he  took  A  =  3B,  S^B,  3^5,  but  found  no  solution.  For 
p  =  5,  Q  =  41,  we  have  72  =  251,  38A  =  21a;  set  A  =  495,  whence  3-576  = 
38-75,  where  b  is  the  sum  of  the  divisors  of  B;  set  B  =  9C,  whence  C:c  = 
13:14,  C=13,  yielding  the  amicable  numbers  5-41A.,  251A,  where 
A  =  3^-7213  =  5733  [the  pair  VII  in  Euler's^'^^  ijgt  and  (7)  in  the  table  below]. 
Again,  to  make  A/a  =  3/8,  set  A  =  35,  whence  a  =  46  and  the  condition  is 
6  =  25,  whence  5  is  a  perfect  number  prime  to  3.  Using  5  =  28,  we  get 
A  =  84.  For  use  in  such  questions,  Kraft  gave  a  table  of  the  sum  of  the 
divisors  of  each  number^  150.     He  quoted  the  rule  of  Descartes. 

L.  Euler^*^^  obtained,  in  addition  to  two  special  pairs,  62  pairs  [including 
two  false  pairs]  of  amicable  numbers  of  the  type  am,  an,  in  which  the 
common  factor  a  is  relatively  prime  to  both  vi  and  n.  He  wrote  jm  for 
the  sum  of  all  the  divisors  of  m.    The  conditions  are  therefore 

jm=jn,  fa-jm  =  a{m-{-n). 

If  m  and  n  are  both  primes,  then  7n  =  n  and  we  have  a  repeated  perfect 
number.     Euler  treated  five  problems. 

(1)  Euler's  problem  1  is  to  find  amicable  numbers  apq,  ar,  where  p,  q,  r, 
are  distinct  primes  not  dividing  the  given  number  a.  From  the  first  con- 
dition we  have  r  =  xy  —  \,  where  x  —  p-\-\,  y  =  q-\-\.     From  the  second, 

xyi  a  =  a{2xy —x  —  y). 

Let  a/{2a—  \a)  equal  6/c,  a  fraction  in  its  lowest  terms.     Then 

y  =  bx/{cx  —  b),  {cx  —  b){cy  —  b)=b^. 

Thus  x  and  y  are  to  be  found  by  expressing  6^  as  a  product  of  two  factors, 
increasing  each  by  6,  and  dividing  the  results  by  c. 

(li)  Fu-st,  takea  =  2".  Then6  =  2^  c=l,  x,  2/  =  2"**+2^  Letri-A;  =  m. 
Then 

p  =  2"»(22*=+2*)-l,         g  =  2'"(l+2*)-l,         r  =  2^"'(2^''+^+2^''+2'')-l. 

When  these  three  are  primes,  2"*"^^'^^  and  2'"''"^  are  amicable.  Euler  noted 
that  the  rule  communicated  by  Descartes  to  van  Schooten  is  obtained  by 
taking  A:=  1,  and  stated  that  1,  3,  6  are  the  only  values  ^  8  of  m  which  yield 
amicable  numbers  (above^^^).  For  k  =  2  or  4,  Euler  remarked  that  r  is 
divisible  by  3;  for  k  =  3,  vi<Q,  and  for  k  =  5,  mS2,  p,  q,  or  r  is  composite. 

(I2)  Take  a  =  2J,  where /=2"+^+e  is  a  prune.  Then  2a-fa  =  e+l. 
If  e+1  divides  a,  we  have  c  =  l.     Set  e+ 1  =  2^,  A;^?n,  n  =  m+A:.     Then 

/=2*(2'"+^  +  l)-l,        a  =  2"'+i,         6  =  27,         b""={x-b){y-b). 

For  k  =  l,  /=2'"+2+l  is  to  be  a  prime,  whence  m+2  is  a  power  of  2. 
If  w  =  0,  6=/=5,  and  either  x  =  y,  p  =  q;  or  x,  y  =  Q,30;  p,  q  =  5,  29,  whereas 
p  and  q  are  to  be  distinct  and  prime  to  10.  If  m  =  2,  /=17,  68^  is  to  be 
resolved  into  distinct  even  factors;  in  the  four  resulting  cases,  p,  q,  r  are 

'"De  numeris  amicabilibus,  Opuscula  varii  argumenti,  2,  1750,  23-107,  Berlin;  Comm.  Arith.,  1, 
1849,  102-145.  French  transl.  in  Sphinx-Oedipe,  Nancy,  1,  1906-7,  Supplement 
I-LXXVI. 


Chap.  I]     PERFECT,  MULTIPLY  PERFECT,  AND   AMICABLE   NUMBERS.  43 

not  all  prime.  In  the  next  case  w  =  6,  /=257,  Euler  examined  only  the 
case*  x  — 6  =  2^-257,  finding  q  composite. 

For  A;  =  2,  Euler  excluded  m  =  1,  3  [m  =  4  is  easily  excluded]. 

(13)  For  k^n  in  (I2),  0  =  2*",  where  m  =  k—n.     Then 

6  =  2"+i+2"'+"-l=/ 
must  be  a  prime.     Thus  we  must  take  as  the  factors  of  6^ 

2'"x-6  =  l,  2"'2/-fe  =  62, 

whence  z  =  2" +2"+^"'",  y-=hx.     If  m  =  1,  one  of 

/=2'»+2-l,  p  =  2^+'-l 

has  the  factor  3  and  yet  must  be  a  prime;  hence  n=l,  g  =  27.  If  m  =  2, 
Euler  treated  the  cases  n^5  and  found  (for  n  =  2)  the  pair  (4)  of  the  table. 
[For  6^n^l7,  /  or  p  is  composite.]  For  m  odd  and  >1,  /  or  p  has  the 
factor  3.     For  m  =  4,  n^  17,  no  solution  results. 

(14)  For  a  =  2"(gr— l)(/i  — 1),  where  the  last  two  factors  are  prime,  set 
d  =  2a-fa.    Then 

ig  -  2^*+^)  {h  -  2^^+^)  =d-  2"+^ + 2^""+^ 
Euler  treated  the  cases  n^3,  d  =  4,  8,  16,  finding  only  the  pair  (9). 

(15)  Special  odd  values  of  a  led  (§§56-65)  to  seven  pairs  (5)-(8), 
(11)-(13).    The  cases  a  =  3^.5,  32-72-13-19  were  unfruitful. 

(2)  Euler's  problem  2  is  to  find  amicable  numbers  apq,  ars,  where  p,  q, 
r,  s  are  distinct  primes  not  dividing  the  given  number  a.  Since  jP'j2 
—  ['>'']  s,  we  may  set 

p  =  ax  —  l,  q  =  ^y  —  l,  r  =  /3x  — 1,  s  =  ay  —  l. 

We  set  fa:a  =  2b—c:h,  where  b  and  c  are  relatively  prime.  The  second 
condition  fa- fpq  =  a{pq+rs)  gives 

ca/3x2/ =  6(a+iS)  (x+2/) -26. 
Multiply  it  by  ca^.    Then 

[ca^x-h{a+^)][ca^y-b(a+^)]  =  h\a+^y-2hca^. 
Given  a,  ^,  a  and  hence  b,  c,  we  are  to  express  the  second  member  as  a  prod- 
uct of  two  factors  and  then  find  x,  y. 

For  a  =  l,  i3  =  3,  0  =  2^  Euler  obtained  the  pairs  (a),  (28).  For  a  =  2, 
^  =  3,  a  =  32-5-13,  he  got  (32);  for  a  =  l,  /3  =  4,  a  =  3^-5,  (30).  The  ratio  a:^ 
may  be  more  complex,  as  5 :21  or  1 :102,  in  (7).  As  noted  by  K.  Hunrath,^^^* 
the  numbers  (7)  are  not  amicable.  Nor  are  the  ratios  as  given,  although 
these  ratios  result  if  we  replace  8563  by  8567  =  13-659.  This  false  pair 
occurs  as  XIII  in  Euler's^^^  list. 

(3)  Problem  3  is  derived  from  problem  2  by  replacing  s  by  a  number  / 
not  necessarily  prime.  Let  h  be  the  greatest  common  divisor  of  ff=hg 
andp+l  =  ^x.    Thenr-{-l=xy,  q-{-l  =  gy.    Also 

ghxyfa=f{afr)=a(pq+fr)^a\{hx-l){gy-l)+f{xy-l)\. 

*A11  the  remaining  cases  are  readily  excluded. 
"♦"BibUotheca  Math.,  (3),  10,  1909-10,  80-81. 


44  History  of  the  Theory  of  Numbers.  [Chap,  i 

Multiply  by  6/a  and  replace  bja  by  2ab  —  ac  [see  case  (1)].     Thus 
exy  —  bhx  —  hgy  =  b{f—l),  e=bf—bgh-{-cgh. 

Thus  L^b'^gh+be{f—l)  is  to  be  expressed  as  the  product  PQ  of  two  factors 
and  they  are  to  be  equated  to  ex—bg,  ey  —  bh.  The  case  a  =  2  is  unfruitful. 
(3i)  Let  a  =  4.  Then  6  =  4,  c  =  l,  e  =  4/— 3^/i.  The  case/= 3  is  excluded 
since  it  gives  e  =  0.  For/ =5,  g  =  2,  h  =  S,  we  again  get  (a)  and  also  (j3). 
For/=5,  ^  =  1,  h  =  6,  we  get  only  the  same  two  pairs.  For  a  prime /^  7, 
no  new  solutions  are  found.     For /= 5-13,  (51)  results. 

(32)  Let  a  =  8,  whence  6  =  8,  c  =  L  The  cases /=  11,  13  are  fruitless, 
while /=  17  yields  (16).  The  least  composite/  yielding  solutions  is  11-23, 
giving  (44),  (45),  (46).  This  fruitful  case  led  Euler  to  the  more  convenient 
notations  (§88)  M  =  hP,  N  =  gQ,  L  =  PQ.  The  problem  is  now  to  resolve 
L  ff  into  two  factors,  Af ,  N,  such  that 

M+bff  N+bff 

are  integers  and  primes,  while  in  r+1  =  {p+l){q+l)/jf,  r  is  a  prime. 

(33)  Let  a  =  16.  For/=17,  we  obtain  the  pairs  (21),  (22);  for/=19, 
(23);  for/=23,  (17),  (19),  (20);  for/=47,  (18);  for/=  17-167,  (49).  Cases 
/=31,  17-151  are  fruitless  [the  last  since  129503  has  the  factor  11,  not 
noticed  by  Euler]. 

(34)  For  a  =  3^-5  or  32.7-13,  6  =  9,  c  =  2;  the  first  a  with/=7  yields  (30). 

(4)  Problem  4  relates  to  amicable  numbers  agpq,  ahr,  where  p,  q,  r  are 
primes.  Eventually  he  took  also  g  and  h  as  primes.  We  may  then  set 
g-\-l  —  km,  h-\-\  =  kn.  For  m  =  \,  n  =  3,  a  =  4  or  8,  no  amicables  are  found. 
Form  =  3,  n  =  l,  the  cases  a  =10,  A:  =  8  and  a  =  3^-5,  ^'  =  8,  yield  (38),  (55). 

(5)  Euler's  final  problem  5  is  of  a  new  type.  He  discussed  amicable 
numbers  zap,  zbq,  where  a  and  6  are  given  numbers,  p  and  q  are  unknown 
primes,  while  z  is  unknown  but  relatively  prime  to  a,  6,  p,  q.  Set 
JoM  6  =  m:n,  where  m  and  n  are  relatively  prime.  Since(p-fl)j  a  =  (54-l)j6, 
we  may  set  p-\-\=^nx,  q-{-\=mx.    The  usual  second  condition  gives 

r    r         /  K      7  /  ^  —  nxfa 

nx\a'{z  =  za{nx  —  l)-j-zb{mx  —  l),  C  —-, 1^ r* 

■^     -^  j^     {na-\-mb)x  —  a  —  b 

Let  the  latter  fraction  in  its  lowest  terms  be  r/s.  Then  z  =  kr,  jz  =  k$. 
Since  f{kr)'^kCr,  we  have  s'^ff.  Hence  we  have  the  useful  theorem: 
if  z:Cz  =  r':s',  s'<\r',  then  r'  and  s'  have  a  common  factor  >  1. 

(5i)  The  unfruitful  case  a  =  3,  6  =  1,  was  treated  like  the  next. 

(52)  Let  0  =  5,  6=1,  whence  w  =  6,  n  =  l,  2:J  z  =  6a::llx  — 6.  By  the 
theorem  in  (5),  x  must  be  divisible  by  2  or  3.  Euler  treated  the  cases 
x  =  3(3^+1),  x  =  2{2t+\).     But  this  classification  is  both  incomplete  and 


Chap.  I]     PERFECT,  MULTIPLY  PERFECT,  AND   AMICABLE   NUMBERS. 


45 


overlapping.  Since  p  =  x  —  l  is  to  be  prime,  x  is  even  (since  a;  =  3  makes  z 
divisible  by  p  =  2) .  Hence  x  =  2P,z:fz  =  QiP:llP—S.  By  the  theorem  in  (5) , 
QP  and  IIP  — 3  have  a  common  factor  2  or  3,  so  that  P  is  either  odd  or  divis- 
ible by  6.  For  P  =  Ql,  the  ratio  is  that  of  126  to  22Z  —  1 ,  which  as  before  must 
have  the  common  factor  3,  whence  l  =  3t-\-l.  Then  z:fz  =  4:{St-{-l) :22t-\-7, 
a  ratio  of  relatively  prime  numbers,  whence  22t+7'^f 4(31+1),  and 
hence  t  =  2k,  k  =  0  or  k>3.  For  A;  =  0,  we  obtain  the  pair  220,  284. 
The  next  value  >3  of  A;  for  which  p  =  x  —  l  and  q  =  Qx—l  are  primes  is 
k  =  Q,  giving  p  =  443,  g  =  2663,  numbers  much  larger  than  those  in  the 
(unnecessary)  cases  treated  by  Euler.  Then  z:jz  =  4-37:271;  set  z  =  S7''d, 
d  not  divisible  by  37;  the  cases  e  =  1,  2,  3  are  excluded  by  the  theorem  in  (5). 
For  the  remaining  case  P  odd,  P  =  2Q-\-l,  Euler  treated  those  values  ^100 
of  Q,  and  also  Q  =  244,  for  which  p  and  q  are  primes  and  obtained  the  pair 
in  (I3),  two  pairs  in  (I5),  and  (14),  (15). 

(53)  Euler  treated  in  §§112-7  various  sets  a,  b,  and  obtained  (a)  and  nine 
new  pairs  given  in  the  table. 

In  the  following  table  of  the  64  pairs  of  amicable  numbers  obtained  by 
Euler,  the  numbering  of  any  pair  is  the  same  as  in  Euler's  list,  but  the  pairs 
have  been  rearranged  so  that  it  becomes  easy  to  decide  if  any  proposed 
pair  is  one  of  Euler's.  As  noted  by  F.  Rudio,^^^''  (37)  contained  the  mis- 
print 3^  for  3^,  w^hile  (7)  and  (34)  are  erroneous,  220499  being  composite 
(311-709);  he  checked  that  all  other  entries  are  correct. 


(4)  22-23{^27^^ 

(9)  2M3.17{3||09 

(47)  23/11-29-239 
^*'^  "^  1191449 

^4»;  ^  \29-47-59 

(21)  2417'5119 

(^)2^{83lo39(f-l««) 

(18)  2^{i:?^ 

(27)  o«/79- 11087 
^^^^  ^  \383-2309 


(37)  s^-sgi'ig'' 


(15)  32.7-13-4M63< 
(35)  32.5.19{7^227 


5-977 
5867 


(38)  2-5{7: 


(a)22 


60659 
23-29-673 
5-131 
17-43 


^•'  ^  1647-719 


(43)  2' 
(60) 


11-59-173 
47-2609 
2^-19-41 
26-199 
(oo\  94/17-10303 


(2)  2^ 


[23-47 
\1151 


(19)  2^{i:t2? 

(25)  2»P-12671 
K^Q)  z  ^227-2111 

,3.  27/191-383 


(5)  32-7-13[ 


(14)  32-72-13-97 


5-17 
107 


5193 
1163 


(8)  3^-5-7{i?2S 


251 
107 


(1)  22{^jll 

i3)  2^(^32 

,45.  23(ll-23-1871 
V4^)  ^  \467-1151 

.4Q.  23/11-163-191 
^4u;  z  131.11807 

,r.,.  /23-41-467 
^^^1  \25-19-233 

(23)  2^{}9gl439 

(50)  2423-47-9767 
\tM)  z  |i583.7io3 

,17s  24/23-1367 
^^'^  "^153-607 

(26)  2^{^|JJ^f 

/90X  r,8/383-9203 
K^-6)  z  |ii5i.3067 

(7)  32-72-13{|5Y 
(10)  32-5- 19-37 (710^ 
(6)  3^-5-13gl9l9 


(51)  2^^^^ 


•131187 
-2267 
(29)  2^-ll{17:263 

,,,.  23/11-23-2543 
^**^  ^  1383-1907 

(16)  2^(M:^^ 

(49)  24/17-167-13679 
KV6)  z  I809.51071 


(36)  2^-67{|72411 

.24)  26/59-1103 
(.Z4J  L  J79.827 


{b2)Z^-l-\Z^^lll]f 


'"^Bibliotheca  Math.,  (3),  14,  1915,  351-4. 


46  History  of  the  Theory  of  Numbers.  [Chap,  i 

(31)  S^-5'13{ll:]f  (54)  3«.5^{}1:^9.179  ^g^  ^,_^_^^,^^l29.5m 

(33)  33.5.13.19^/2711      (32)  3^-5-13{^^:^J  (12)  S^.r-lVlsl^lf^^' 

(41)33.7.13.23|JJ;1%367    ^g^^  3,.-,. ^g.^gjl  1-220499    ^^^^^^ 

(30)  33.5{[7^1j  (55)  ■S^-5{',lll%^  (42)  33.5.23(11 -.J^SIT 

(11)  3^.5.1l{29g89  (56)  3^.7.11M9{g97019      (57)3^.7.11M9{^3^6959^ 

(53)  3».7M3.53{ll4211        (53)  3s.7M3.19{4™19       (59^  3^.7M3.19{53g6959^ 

Euler's  final  list  of  61  pairs  did  not  include  the  pairs  a,  jS,  7,  although  he 
had  obtained  a  four  times  in  the  body  of  his  paper,  viz.,  in  (2),  (3i),  (63); 
jS  twice  in  (3i);  7  in  (2).  Moreover,  these  three  unlisted  pairs  occur  as 
VIII,  IX,  and  XIII  among  the  30  pairs  in  Euler's^^^  earlier  list,  a  fact  noted 
on  p.  XXVI  and  p.  LVIII  of  the  Preface  by  P.  H.  Fuss  and  N.  Fuss  to 
Euler's  Comm.  Arith.  Coll.,  who  failed  to  observe  that  these  three  pairs 
occur  in  the  text  of  Euler's  present  paper.  Nor  did  these  editors  note 
that  the  fourth  mentioned  case  of  divergence  between  the  two  lists  is  due 
merely  to  the  misprint^^^''  of  57  for  47  in  (43)  of  the  present  list,  so  that 
the  correctly  printed  pair  XXVIII  of  the  list  of  30  is  really  this  (43)  and 
not  a  new  pair,  as  supposed  by  them. 

From  the  fact  that  Euler  obtained  in  his  posthumous  tract®^  on  amicable 
numbers  the  pairs  a,  jS  (once  on  p.  631  and  again  on  p.  633  and  finally  on 
p.  635),  the  editors  inferred,  p.  LXXXI  of  the  Preface,  that  the  tract  differs 
in  analysis  from  the  long  paper  just  discussed.  But  no  new  pairs  are  found, 
while  the  cases  treated  on  pp.  631-2  are  merely  problems  1  and  2  of  Euler's 
preceding  paper.  It  is  different  with  p.  634,  where  Euler  started  with  two 
numbers  like  71  and  5-11  which,  by  his  table,  have  the  same  sum,  72,  of 
divisors,  and  required  a  number  a  relatively  prime  to  them  such  that  71a 
and  55o  are  amicable.  The  single  condition  is  72J  a=(71+55)o,  whence 
ja:a  =  7A.  Thus  a  has  the  factor  4.  If  a  =  46,  where  b  is  odd,  then 
Ch  =  h  =  l,  and  the  pair  284,  220  results.  The  case  a  =  86  is  impossible.  This 
method  was  used  in  a  special  way  by  Kraft^^^  who  limited  the  numbers 
from  which  one  starts  to  a  prime  and  a  product  of  two  primes. 

In  the  Encyclopedie  Sc.  Math.,  I,  3i,  p.  59,  note  320,  it  is  stated  that  this 
posthumous  tract  contains  four  pairs  not  in  Euler's  list  of  61,  two  pairs  being 
those  of  Fermat^^^  and  Descartes.^^^  But  these  were  fisted  as  (2)  and  (3) 
by  Euler  and  were  obtained  by  him  in  case  (li)  and  attributed  to  Descartes. 

E.  Waring^^^  noted  that  2'*x,  2''yz  are  amicable  if 

2"'?/z_2"+^  +  l  2^" 


2"-l  '  i/-2"+l 

where  x^  y,  z  are  primes  and  ?/  — 2"H-1  divides  2^**.     He  cited  the  first  two 
such  pairs  of  amicable  numbers. 

3"«G.  Enestrom,  Bibliotheca  Math.,  (3),  9,  1909,  263. 
3«*Meditationes  algebraicae,  1770,  201;  ed.  3,  1782,  342-3. 


Chap.  I]     PERFECT,  MULTIPLY  PERFECT,  AND   AmICABLE   NUMBERS.  47 

The  first  three  pairs  were  given  in  an  anonymous  work.^^® 

In  1796,  J.  P.  Gruson^°°  (p.  87)  gave  the  usual  rule  (1)  leading  to  the 
three  first  known  amicable  pairs  (verwandte  Zahlen). 

A.  M.  Legendre^^^  attributed  the  rule  (1)  to  Descartes. 

G.  S.  KliigeP^^  gave  a  process  leading  to  the  choice  of  P  and  Q,  left 
arbitrary  by  Kraft.^^^  ^^  ^ave  A:a  =  R+l:PQ-\-R  =  2R-P-Q.  Thus 
P-(-Q=  \R{2A—a)—a\ /A,  while  PQ  is  given  by  Kraft's  second  equation. 
Hence  P  and  Q  are  the  roots  of  a  quadratic  equation.     For  example,  if 

A  =  4,  then  

8P,  SQ  =  R-7±VR^-Q2R-QS. 

The  positive  root  of  a;^  —  62a:  —  63  =  0  lies  between  60  and  61.  Thus  we 
try  primes  ^  61  for  R,  such  that  i^  — 7  is  divisible  by  8.  The  first  available 
R  is  71,  giving  P=ll,  Q  —  5  and  the  amicable  pair  220,  284.  In  general, 
the  quantity  a^R^+2^R-\-y  under  the  radical  sign  can  be  made  equal  to  the 
square  of  ai^+P  ip  arbitrary)  by  choice  of  R. 

John  Gough^^^  considered  amicable  numbers  ax,  ayz,  where  x,  y,  z  are 
distinct  primes  not  dividing  a.     Let  q  be  the  sum  of  the  aliquot  divisors 

of  a.    Then 

a+q-\-qx  =  ayz,  x+l  =  {y+l){z+l). 

If  q^a/i,  the  first  gives  ayz<  (l+a:)a/4,  while  2y-2z>x-\-l  by  the  second, 
Thusg'>a/4.  Let  a  =  r",  where  r  is  a  prime  >  1 .  Then  Q'=(a  — l)/(r'  — 1), 
which  with  g>a/4  implies  a(5— r)>4,  r  =  2  or  3.  He  proved  that  Tt^S. 
whence  r  =  2,  the  case  treated  by  van  Schooten.^^^ 

J.  Struve^^^  cited  his  Osterprogramm,  1815,  on  amicable  numbers. 

A.  M.  Legendre^^°  discussed  the  amicable  numbers  of  the  type  (li)  of 
Euler^^^  (with  Euler's  m,  k  replaced  by  m—iJi,,fx).  Legendre  noted  that 
r  =  2^'""''^ (2*^+1)^  —  1  is  of  the  form  s^  —  1  and  hence  composite,  if  k  is  even; 
also  that,  if  A:  =  3,  p  =  9-2'"+3-l,  g  =  9-2"'-l,  one  of  which  is  of  the  form 
s^  —  1.  He  considered  the  new  case  k  =  7  and  found  for  m  =  l  that  p  =  33023, 
q  =  257,  r  =  8520191,  stating  that  if  r  be  a  prime  we  have  the  amicable  num- 
bers 2^pq,  2V.  This  is  in  fact  the  case.^^^  For  ^  =  1,  we  have  the  ancient 
rule  (1);  he  proved  that  for  n^l5  it  gives  only  the  known  three  pairs  of 
amicable  numbers. 

Paganini^'^^,  at  age  16,  announced  the  amicable  numbers  1184  =  2^37, 
1210  =  2.5.11"^,  not  in  the  list  by  Euler^^'*,  but  gave  no  indication  of  the 
method  of  discovery. 

^^'EncyclopMie  methodique. .  .Amusemens  des  Sciences  Math,  et  Phys.,  nouv.  6d.,  Padoue, 
1793,  I,  116.     Cf.  Les  amusemens  math.,  Lille,  1749,  315. 

»"Th6orie  des  nombres,  1798,  463. 

«8Math.  Worterbuch,  1,  1803,  246-252  [5,  1831,  55]. 

«»New  Series  of  the  Math.  Repository  (ed.,  Th.  Leyboum),  vol.  2,  pt.  2,  1807,  34-39.  He  cited 
Button's  Math.  Diet.,  article  Amicable  Numbers,  taken  from  van  Schooten^^'. 

""Theorie  des  nombres,  ed.  3,  1830,  II,  §472,  p.  150.  German  transl.  by  H.  Maser,  Leipzig, 
1893,  II,  p.  145. 

"iTchebychef,  Jour,  de  Math.,  16,  1851,  275;  Werke,  1,  90.  T.  Pepin,  Atti  Ace.  Pont.  Nuovi 
Lincei,  48,  1889,  152-6.  Kraitchik,  Sphinx-Oedipe,  6,  1911,  92.  Also  by  Lehmer's  Fac- 
tor Table  or  Table  of  Primes. 

'"B.  Nicol6  I.  Paganini,  Atti  deUa  R.  Accad.  Sc.  Torino,  2,  1866-7,  362.  Cf.  Cremona's  Ital. 
transl.  of  Baltzer's  Mathematik,  pt.  III. 


48  History  of  the  Theory  of  Numbers.  (Chap,  i 

P.  Seelhoff^^^  treated  Euler's^^  problems  1  and  2  by  Euler's  methods 
(though  the  contrary  is  implied),  and  gave  about  20  pairs  of  amicable 
numbers  due  to  Euler,  with  due  credit  for  only  three  pairs.  The  only  new- 
pairs  (pp.  79,  84,  89)  are 

Q2721Q  in  oQf83-1931  ^ef  139-863 

6  i  A^-Ay'^'^|i62287  "^1167.719. 

E.  Catalan^^^  stated  empirically  that  if  ??i  is  the  sum  of  the  divisors 
<n  of  n,  and  n2  is  the  sum  of  the  divisors  <ni  of  ?ii,  etc.,  then  n,  nj,  n2, .  .  . 
have  a  limit  X,  where  X  is  unity  or  a  perfect  number. 

J.  Perrott"^  [Perott]  noted  that  there  is  no  limit  for  n  =  220,  since 

^1  =  713=  .  .  .  =284,  n2  =  n4=  .  .  .  =220. 

H.  LeLasseur^^*^  found  that  for  n<  35  the  numbers  (1)  are  all  odd  primes, 
and  hence  give  amicable  numbers,  only  when  n  =  2,  4,  7. 

Josef  Bezdicek^^^  gave  a  translation  into  Bohemian  of  Euler,^^  without 
credit  to  Euler,  and  a  table  of  65  pairs  of  amicable  numbers. 

Aug.  Haas"^  proved  that,  if  M  and  A^  are  amicable  numbers, 

l/S-+l/si=l, 
m  n 

where  m  and  n  range  over  all  divisors  of  M  and  iV,  respectively.     For, 

2w  =  Sn  =  M+iV,  so  that 

m~  M  ~     M    '  n~  N  ~     N 

If  M  =  N,  N  is  perfect  and  the  result  becomes  that  of  Catalan. ^"'^ 

A.  Cunningham^'^  considered  the  sum  s{n)  of  the  divisors  <n  of  ?i  and 
wrote  s^{n)  for  s]s(??)},  etc.  For  most  numbers,  s^(n)  =  l  when  A;  is  suffi- 
ciently large.  There  is  a  small  class  of  perfect  and  amicable  numbers,  and 
a  small  class  of  numbers  n  (even  when  n<  1000)  for  which  s''{n)  increases 
beyond  the  practical  power  of  calculation  [cf.  Catalan^'^]. 

A.  Gerardin^^"  proved  that  the  only  pairs  2^-5a;,  2-yz  of  amicable  num- 
bers, where  x,  y,  z  are  odd  primes,  are  Euler's  (a),  (^3) ;  the  only  pairs  2*-23x, 
2^yz  are  Euler's  (17),  (19),  (20).  He  cited  the  Exercices  d'arithm^tique  of 
Fitz-Patrick  and  Chevrel;  also  Dupuis'  Table  de  logarithmes,  which  gives 
24  pairs  of  amicable  numbers. 

G^rardin^^^  proved  that  the  only  pair  Sxy,  S2z  is  Euler's  (60).  He  made 
an  incomplete  examination  of  16-53a;,  IQyz,  but  found  no  new  pairs. 

3"Archiv  Math.  Phys.,  70,  1884,  75-89. 

"*Bull.  Soc.  Math.  France,  16,  1887-8,  129.     Mathesis,  8,  1888,  130. 

'"76id.,  17,  1888-9,  155-6. 

"•Lucas,  Theorie  dcs  nombres,  1,  1891,  381. 

"'Casopis  mat.  a  fys.,  Praze  (Prag),  25,  1896,  129-142,  209-221. 

"8/6id.,  349-350. 

"«Proc.  London  Math.  Soc,  35,  1902-3,  40. 

""Matheshs,  6,  1906,  41^4. 

"'Sphinx-Oedipe,  Nancy,  1906-7,  14-15,  53. 


Chap.  I]     PERFECT,  MULTIPLY   PERFECT,  AND   AmICABLE   NUMBERS.  49 

G^rardin^^^  proved  that  the  three  numbers  (1)  with  n  =  m-f  2  are  not  all 
primes  if  34<  m^  60,  the  cases  m  =  38  and  53  not  being  decided.  Replacing 
m  by  m+1  and  A;  by  2^+1  in  case  (li)  of  Euler^^'*,  we  get  the  pair  2"pg, 
2"r,  where  n  =  m+2g-]-2, 

p  =  2"*+2''+^P-l,  5  =  2"*+^P-l,  ^  =  22'"+2''+3p2_i^ 

with  P  =  2^^'*"^  +  l.  For  9^  =  0,  we  have  the  case  (1)  just  mentioned;  all 
values  m^200  are  excluded  except  m  =  38,  74,  98,  146,  149,  182,  185,  197. 
The  case  gr=  1  is  excluded  since  i/  or  2  is  a  difference  of  two  squares.  For 
g  =  2,  all  values  m  ^  60  are  excluded  except  m= 29, 34,  37, 49.  For  g  =  3,  all 
values  <100  are  excluded  except  m  =  8,  15,  23,  92. 

0.  Meissner,^^^  using  the  notation  of  Cunningham,^^^  noted  that  n  and 
s{n)  are  amicable  if  s^(n)=n  and  raised  the  question  of  the  existence  of 
numbers  n  for  which  s''{n)—n  for  k^S,  so  that  n,  s{n),. .  .,s*~^(n)  would 
give  amicable  numbers  of  higher  order.  He  asked  if  the  repetition  of  the 
operation  s,  a  finite  number  (k)  of  times  always  leads  to  a  prime,  a  perfect 
or  amicable  number;  also  if  k  increases  with  n  to  infinity.  On  these  ques- 
tions, see  Dickson^^^  and  Poulet.^^^ 

A.  Gerardin^^^  stated  that  the  only  values  n<200  for  which  the 
numbers  (1)  are  all  primes  are  the  three  known  to  Descartes. 

L.  E.  Dickson^^^  obtained  the  two  new  pairs  of  amicable  numbers 

2*-12959-50231,   2*- 17- 137-262079;   2*- 10103-735263,   2^-17-137-2990783, 

by  treating  the  type  IQpq,  16-17-137r,  where  p,  q,  r  are  distinct  odd  primes. 
These  are  amicable  if  and  only  if 

p  =  m+9935,        g  =  w+9935,        r  =  4(w+n) +88799,        wn  =  2^3*7-23-73. 

Although  Euler^^  mentioned  this  type  (33)  in  §95,  he  made  no  discussion 
of  it  since  r  always  exceeds  the  limit  100000  of  the  table  of  primes  accessible 
to  him.  An  examination  of  the  120  distinct  cases  led  only  to  the  above 
two  amicable  pairs. 

Dickson^^®  proved  that  there  exist  only  five  pairs  of  amicable  numbers 
in  which  the  smaller  number  is  <6233,  viz.,  (1),  (a),  (^),  (60)  in  Euler's^^* 
table,  and  Paganini's^^^  pair.  In  the  notation  of  Cunningham,^^^  the  chain 
n,  s{n),  s^{n), . .  .is  said  to  be  of  period  k  if  s^(n)  =n.  The  empirical  theorem 
of  Catalan'^*  is  stated  in  the  corrected  form  that  every  non-periodic 
chain  contains  a  prime  and  verified  for  a  wide  range  of  values  of  n.  In 
particular,  if  n<6233,  there  is  no  chain  of  period  3,  4,  5,  or  6.  For  k  odd 
and  >  1,  there  is  no  chain  arii,  an2, .  .  . ,  aUk  of  period  k  in  which  /ii, .  .  . ,  n^ 
have  no  common  factor  and  each  rij  is  prime  to  a>  1. 

'^^Sphinx-Oedipe,  1907-8,  49-56,  65-71 ;  some  details  are  inaccurate,  but  the  results  correct. 
'S'Archiv  Math.  Phys.,  (3),  12,  1907,  199;  Math.-Naturw.  Blatter,  4,  1907,  86  (for  k=3). 
'** Assoc,  frang.  avanc.  sc,  37,  1908,  36-48;  I'intenn^diaire  des  math.,  1909,  104. 
'8*Amer.  Math.  Monthly,  18,  1911,  109. 
"•Quart.  Jour.  Math.,  44,  1913,  264-296. 


50  History  of  the  Theory  of  Numbers.  [Chap,  i 

P.  Poulet'^'  discovered  the  chain  of  period  five, 

71  =  12496  =  24-11.71,  s(n)=  2^- 1947,  s\n)  =  2^-967,  s\n)  =2^-23-79, 

sHn)  =2^.1783, 

with  s^{n)  =n;  and  noted  that  14316  leads  a  chain  of  28  terms. 
Generalizations  of  Amicable  Numbers. 

Daniel  Schwenter^^  noted  in  1636  that  27  and  35  have  the  same  sum  of 
ahquot  parts.  Kraft^^^  noted  in  1749  that  this  is  true  of  the  pairs  45,  3-29; 
39,  55;  93,  145;  and  45,  13-19.  In  1823,  Thomas  Taylor^^^  called  two  such 
numbers  imperfectly  amicable,  citing  the  pairs  27,  35;  39,  55;  65,  77;  51, 
91;  95,  119;  69,  133;  115,  187;  87,  247.  George  Peacock^°o  used  the  same 
term. 

E.  B.  Escott^"^  asked  if  there  exist  three  or  more  numbers  such  that  each 
equals  the  sum  of  the  [aliquot]  divisors  of  the  others. 

A.  G^rardin^°^  called  numbers  with  the  same  sum  of  aliquot  parts 
nombres  associes,  citing  6  and  25;  5-19,  7-17,  and  11-13,  and  many  more  sets. 
An  equivalent  definition  is  that  the  n  numbers  be  such  that  the  product  of 
n  —  1  by  the  sum  of  the  aliquot  divisors  of  any  one  of  them  shall  equal  the 
sum  of  the  aliquot  divisors  of  the  remaining  n  —  1  numbers. 

L.  E.  Dickson^°^  defined  an  amicable  triple  to  be  three  numbers  such 
that  the  sum  of  the  aliquot  divisors  of  each  equals  the  sum  of  the  remaining 
two  numbers.  After  developing  a  theory  analogous  to  that  by  Euler^®*  for 
amicable  numbers,  Dickson  obtained  eight  sets  of  amicable  triples  in  which 
two  of  the  numbers  are  equal,  and  two  triples  of  distinct  numbers: 

293-3370,  5- 16561a,  99371o  (a  =  25.3-13), 

3-896,  11-296,  3596  (6  =  2i*.5-19-31-151). 

^L'intermddiaire  des  math.,  25.  1918,  lOO-l. 
♦""Encyclopaedia  Metropolitana,  London,  I,  1845,  422. 
«"L'interm6diaire  des  math.,  6,  1899,  152. 
♦""Sphinx-Oedipe,  1907-8,  81-83. 
♦MAmer.  Math.  Monthly,  20,  1913,  84-92. 


CHAPTER  II. 

FORMULAS   FOR  THE  NUMBER  AND  SUM  OF  DIVISORS.  PROBLEMS  OF 

FERMAT  AND  WALLIS. 

Formula  for  the  Number  of  the  Divisors  of  a  Number. 

Cardan^  stated  that  a  product  P  oi  k  distinct  primes  has  1+2+2^+  . . 
-\-2^~'^  aUquot  parts  (divisors  <P). 

Michael  StifeP  proved  this  rule  and  found^  the  number  of  divisors  of 
2*3^52p,  where  P  =  7-11.13-17-19-23-29,  by  first  noting  that  there  are 
1+2+ . .  .+64  divisors  <P  oi  P  according  to  Cardan's  rule  and  hence 
128  divisors  of  P.  The  factor  5^  gives  rise  to  128+128  more  divisors,  so 
that  we  now  have  384  divisors.  The  factor  3^  gives  3.384  more,  so  that  we 
have  1536.    Then  the  factor  2*  gives  4.1536  more. 

Mersenne*  asked  what  number  has  60  divisors;  since  60  =  2-2-3-5,  sub- 
tract unity  from  each  prime  factor  and  use  the  remainders  1,  1,  2,  4  as 
exponents;  thus  3^-2*-7-5  =  5040  (so  much  lauded  by  Plato)  has  60  divisors. 
It  is  no  more  difficult  if  a  large  number  of  aliquot  parts  is  desired. 

I.  Newton^  found  all  the  divisors  of  60  by  dividing  it  by  2,  the  quotient 
30  by  2,  and  the  new  quotient  15  by  3.  Thus  the  prime  divisors  are  1,  2,  2, 
3,  5.  Their  products  by  twos  give  4,  6,  10,  15.  The  products  by  threes 
give  12,  20,  30.  The  product  of  all  is  60.  The  commentator  J.  Castillionei, 
of  the  1761  edition,  noted  that  the  process  proves  that  the  number  of  all 
divisors  of  a'"6'*. .  .is  (m+l)(n+l) .  .  .if  a,  6, . .  .are  distinct  primes. 

Frans  van  Schooten^  devoted  pp.  373-6  to  proving  that  a  product  of  k 
distinct  primes  has  2'''— 1  aliquot  parts  and  made  a  long  problem  (p.  379) 
of  that  to  find  the  number  of  divisors  of  a  given  number.  To  find  (pp. 
380-4)  the  numbers  having  15  aliquot  parts,  he  factored  15+1  in  all  ways 
and  subtracted  unity  from  each  factor,  obtaining  abed,  a^bc,  a%^,  a^b,  a^^. 
By  comparing  the  arithmetically  least  numbers  of  these  various  types,  he 
found  (pp.  387-9)  the  least  number  having  15  aliquot  parts. 

John  Kersey'^  cited  the  long  rule  of  van  Schooten  to  find  the  number  of 
aliquot  parts  of  a  number  and  then  gave  the  simple  rule  that  Oi" . . .  a^"  has 
(6i+l) . . .  (e„+l)  divisors  in  all  if  ai, . . . ,  a„  are  distinct  primes. 

John  Wallis^  gave  the  last  rule.  To  find  a  number  with  a  prescribed 
number  of  divisors,  factor  the  latter  number  in  all  possible  ways;  if  the 

iPractica  Arith.  &  Mensurandi,  Milan,  1537;  Opera,  IV,  1663. 

*Arithmetica  Integra,  Norimbergae,  1544,  lib.  1,  fol.  101. 

'Stifel's  posthumous  manuscript,  fol.  12,  preceding  the  printed  text  of  Arith.  Integra;  cf.  E. 
Hoppe,  Mitt.  Math.  Gesell.  Hamburg,  3,  1900,  413. 

*Cogitata  Physico  Math.,  II,  Hydravhca  Pnevmatica,  Preface,  No.  14,  Paris,  1644.  (Quoted 
by  Winsheim,  Novi  Comm.  Ac.  Petrop.,  II,  ad  annum  1749,  Mem.,  68-99).  Also  letter 
from  Mersenne  to  Torricello,  June  24,  1644,  Bull.  Bibl.  Storia  Sc.  Mat.,  8,  1875,  414-5. 

•Arithmetica  UniversaUs,  ed.  1732,  p.  37;  ed.  1761,  I,  p.  61.     De  Inventione  Divisorum. 

•Exercitationum  Math.,  Lugd.  Batav.,  1657. 

^The  Elements  of  Algebra,  London,  vol.  1,  1673,  p.  199. 

•A  Treatise  of  Algebra,  London,  1685,  additional  treatise,  Ch.  III. 

61 


52  History  of  the  Theory  of  Numbers.  [Chap,  ii 

factors  are  r,  s, .  .  .,  the  required  number  is  p''~^q'~^ .  .  .,  where  p,  q,.  .  are 
any  distinct  primes.  WTien  the  number  of  divisors  is  odd,  the  number 
itself  is  a  square,  and  conversely.  The  number  of  ways  A^  =  a^b^ .  .  .  can  be 
expressed  as  a  product  of  two  factors  is  A  =  |(a+l)(j3+l) .  .  .or  \-\-k, 
according  as  N  is  not  or  is  a  square. 

Jean  Prestet^  noted  that  a  product  of  k  distinct  primes  has  2''  divisors, 
while  the  ?ith  power  of  a  prime  has  n+1  divisors.  The  divisors  of  a^h^c^ 
are  the  12  divisors  of  or}?,  their  products  by  c  and  by  c^,  the  general  rule 
not  being  stated  explicitly. 

Pierre  R^mond  de  Montmort^°  stated  in  words  that  the  number  of 
divisors  of  Oi**. .  .a„*"  is  (ci+l) . .  .(e„+l)  if  the  a's  are  distinct  primes. 

Abb^  Deidier^^  noted  that  a  product  of  k  distinct  primes  has 

^+^+(2)  +  (3)+     • 

divisors,  treating  the  problem  as  one  on  combinations  (but  did  not  sum  the 
series  and  find  2*").  To  find  the  number  of  divisors  of  2*3^5^  he  noted  that 
five  are  powers  of  2  (including  unity).  Since  there  are  three  divisors  of  3^, 
multiply  5  by  3  and  add  5,  obtaining  20.  In  view  of  the  two  divisors  of 
5^,  multiply  20  by  2  and  add  20.     The  answer  is  60. 

E.  Waring^-  proved  that  the  number  of  divisors  of  a"'?)". .  .is  (m+1) 
(n+1) .  .  .if  a,  6, .  .  are  distinct  primes,  and  that  the  number  is  a  square  if 
the  number  of  its  divisors  is  odd. 

E.  Lionnet^^  proved  that  if  a,  b,  c, . .  .are  relatively  prime  in  pairs,  the 
number  of  divisors  of  abc. .  .equals  the  product  of  the  number  of  divisors 
of  a  by  the  number  for  b,  etc.  According  as  a  number  is  a  square  or  not, 
the  number  of  its  divisors  is  odd  or  even. 

T.  L.  Pujo^^  noted  the  property  last  mentioned. 

Emil  Hain^^  derived  the  last  theorem  from  a"*  =  (<i .  .  .  t„y,  where  <i, .  .  . ,  <„ 
denote  the  divisors  of  a. 

A.  P.  Minin^^  determined  the  smallest  integer  with  a  given  number  of 
divisors. 

G.  Fontene'"  noted  that,  if  2"3^.  .  .mV  (a^/S^  .  .  .  ^n^v)  is  the  least 
number  with  a  given  number  of  di\4sors,  then  I'+l  is  a  prime,  and  /x+1  is 
a  prime  except  for  the  least  number  2^3  ha\'ing  eight  di\'isors. 

Formula  for  the  Sum  of  the  Divisors  of  a  Number.    ' 

R.  Descartes,^^  in  a  manuscript,  doubtless  of  date  1638,  noted  that,  if  p 
is  a  prim6,  the  sum  of  the  aliquot  parts  of  p"  is  (p"—  l)/(p  —  1).     If  6  is  the 

"Nouv.  Elemens  des  Math.,  Paris,  1689,  vol.  1,  p.  149. 

loEssay  d'analyse  sur  les  jeux  de  hazard,  ed.  2,  Paris,  1713,  p.  55.     Not  in  ed.  1,  1708. 
"Suite  de  I'arithm^tique  des  g^om^tres,  Paris,  1739,  p.  311. 
i^Medit.  Algebr.,  1770,  200;  ed.  3,  1782,  341. 
"Nouv.  Ann.  Math.,  (2),  7,  1868,  68-72. 
"Les  Mondes,  27,  1872,  653-4. 
"Archiv  Math.  Phys.,  55,  1873,  290-3. 
"Math.  Soc.  Moscow  (in  Russian),  11,  1883-4,  632. 
"Nouv.  Ann.  Math.,  (4),  2,  1902,  288;  proof  by  Chalde,  3,  1903,  471-3. 
*'"De  partibus  ahquotis  numerorum,"  Opuscula  Posthuma  Phys.  et  Math.,  Amstelodami, 
1701,  p.  5;  Oeuvres  de  Descartes  (ed.  Tannery  and  Adams,  1897-1909),  vol.  10,  pp.  300-2. 


23 
3 

13 


Chap.  II]  FORMULAS   FOR   NUMBER   AND    SUM    OF  DiVISORS.  53 

sum  of  the  aliquot  parts  of  a,  the  sum  of  the  ahquot  parts  of  ap  is  6p+a+6. 
If  b  is  the  sum  of  the  ahquot  parts  of  a  and  if  x  is  prime  to  a,  the  sum  of  the 
aUquot  parts  of  ax""  is 

— ^n —        [=^'+^K-j^)-^^\' 

Descartes^^  stated  a  result  which  may  be  expressed  by  the  formula 

(1)  <j{nm)=(T{n)(T{m)  (n,  m  relatively  prime), 

where  (T{n)  is  the  sum  of  the  divisors  (including  1  and  n)  of  w.     Here  he 
solved  n :  (T(n)  =  5 :  13.    Thus  n  must  be  divisible  by  5.     Enter  5 
in  column  A  and  (r(5)  =  6  in  column  B.     Then  enter  the  factor       A     B 
2  in  column  A  and  (r(2)  =3  in  column  B.     Having  two  threes 
in  column  B,  we  enter  9  in  column  A  and  cr(9)  =  13  in  B.    Every        5 
number  except  13  in  column  B  is  in  column  A.    Hence  the        2 
product  5-2-9  =  90  is  a  solution  n.    Next,  to  solve  n :  a-  (n)  =  5 :  14,         9 
we  enter  also  13  in  column  A  and  14  in  B,  and  obtain  the  solu- 
tion 90-13.     If  ?i  is  a  perfect  number,  5n:  (7(5n)  =  5: 12  and,  if  n?^6,  15n: 
(r(15n)  =  5:16. 

Descartes^^  stated  that  he  possessed  a  general  rule  [illustrated  above] 
for  finding  numbers  having  any  given  ratio  to  the  sum  of  their  aliquot  parts. 

Fermat^^  had  treated  the  same  problem.  Replying  to  Mersenne's 
remark  that  the  sum  of  the  aliquot  parts  of  360  bears  to  360  the  ratio  9  to  4, 
Fermat^^  noted  that  2016  has  the  same  property. 

John  Wallis^^  noted  that  Frenicle  knew  formula  (1). 

Wallis^^  knew  the  formula 

(2)  ^(a•6^...)  =  211^1-*^^.... 

a— 1        0—1 

Thus  these  formulae  were  known  before  1685,  the  date  set  by  Peano,^^  who 
attributed  them  to  Wallis.^^ 

G.  W.  Kraft^^  noted  that  the  method  of  Newton^  shows  that  the  sum  of 
the  divisors  of  a  product  of  distinct  primes  P, . . .,  S  is  (P+1) . .  .(S+1). 
He  gave  formula  (1)  and  also  (2),  a  formula  which  Cantor^"  stated  had 
probably  not  earlier  been  in  print.  To  find  a  number  the  sum  of  whose 
divisors  is  a  square,  Kraft  took  PA,  where  P  is  a  prime  not  dividing  A. 
If  (r(A)=a,  then  (r(PA)  =  (P-f-l)a  will  be  the  square  of  (P+l)5  if  P  = 

^^"De  la  fagon  de  trouver  le  nombres  de  parties  aliquotes  in  ratione  data,"  manuscript  Fonds- 
frangais,  nouv.  acquisitions,  No.  3280,  ff.  156-7,  Bibliothfeque  Nationale,  Paris.  Pub' 
Ushed  by  C.  Henry,  BuU.  Bibl.  Storia  Sc.  Mat.  e  Fis.,  12,  1879,  713-5. 

^Oeuvres,  2,  p.  149,  letter  to  Mersenne,  May  27,  1638. 

K)euvre8  de  Fermat,  2,  top  of  p.  73,  letter  to  Roberval,  Sept.  22,  1636. 

"Oeuvres,  2,  179,  letter  to  Mersenne,  Feb.  20,  1639. 

''^ommercium  Epistolicum,  letter  32,  April  13,  1658;  French  transl.  in  Oeuvres  de  Fermat,  3, 
553. 

"Commercium  Epist.,  letter  23,  March,  1658;  Oeuvres  de  Fermat,  3,  515-7. 

"Formulaire  Math.,  3,  Turin,  1901,  100-1. 

"Novi  Comm.  Ac.  Petrop.,  2,  1751,  ad  annum  1749,  100-109. 

"Geschichte  Math.,  3,  595;  ed.  2,  616. 


54  History  of  the  Theory  of  Numbers.  [ChapII 

a/S^  — 1;  for  A  =  14,  take  B  =  2,  whence  P  =  5.  Again,  the  sum  of  the 
aUquot  parts  of  3P~  is  (2+-P)^.  The  numbers  AP  and  BPQ  have  the  same 
sum  of  divisors  if  a(P+l)  =  6(P+1)(Q+1),  i.  e.,  if  Q  =  a/6-l;  takmg 
a  =  24,  6  =  6,  we  have  Q  =  3,  a  prime,  .4  =  14,  B  =  5  (by  his  table  of  the  sum 
of  the  divisors  of  1,. .  .,  150);  this  problem  had  been  solved  otherwise  by 
Wolff." 

L.  Euler^^  gave  a  table  of  the  prime  factors  of  o-(p),  a(p^),  and  <t(p^)  for 
each  prime  p<1000;  also  those  of  aip")  for  various  a's  for  p^23  (for 
instance,  a  ^36  when  p  =  2).  He  proved  formulas  (1)  and  (2)  here  and  in 
his^  posthumous  tract,  where  he  noted  (p.  514)  all  the  cases  in  which 
a{n)  =a-(?70  =  60. 

E.  Waring^2  proved  formula  (2).  He^  noted  that  if  P  =  arir.  .  .and 
Q  =  a'^h^ .  .  . ,  where  m  —  a,n  —  ^,...  are  large,  then  a{PQ)/a{P)  is  just  greater 
than  Q.  If  A  =  {1-1)\,  <t{IA)/(t{A)^1-^1.  If  a'br..=A  and  (x+l) 
(2/+ 1) . .  .  is  a  maximum,  then  a'"''"^  =  6""^^  =  . . .  For  a,  6, .  .  .  distinct  primes, 
a{A)  is  not  a  maximum.  He  cited  numbers  with  equal  sums  of  divisors: 
6  and  11,  10  and  17,  14  and  15  and  23. 

L.  Kronecker^^  derived  the  formulas  for  the  number  and  sum  of  the 
divisors  of  an  integer  by  use  of  infinite  series  and  products. 

E.  B.  Escott^^  listed  integers  whose  sum  of  divisors  is  a  square. 

Problems  of  Fer\la.t  and  Wallis  on  Sums  of  Divisors. 

Fermat'*^  proposed  January  3,  1657,  the  two  problems:  (i)  Find  a  cube 
which  when  increased  by  the  sum  of  its  aUquot  parts  becomes  a  square;* 
for  example,  7^ + ( 1 + 7 + 7^)  =  20^.  (ii)  Find  a  square  which  when  increased 
by  the  sum  of  its  aliquot  parts  becomes  a  cube. 

John  WalUs^^  replied  that  unity  is  a  solution  of  both  problems  and  pro- 
posed the  new  problem:  (m)  Find  two  squares,  other  than  16  and  25,  such 
that  if  each  is  increased  by  the  sum  of  its  ahquot  parts  the  resulting  sums 
are  equal. 

Brouncker*^  gave  1/n^  and  343/n^  as  solutions  (!)  of  problem  (i). 

"Elementa  Analyseos,  Cap.  2,  prob.  87. 

«Opuscula  varii  argumenti,  2,  Berlin,  1750,  p.  23;  Comm.  Arith.,  1, 102  (p.  147  for  table  to  100). 

Opera  postuma,  I,  1862,  95-100.     F.  Rudio,  Bibl.  Math.,  (3),  14,  1915,  351,  stated  that 

there  are  fully  15  errors. 
"Comm.  Arith.,  2,  512,  629.     Opera  postuma,  I,  12-13. 
«Meditationea  Algebr.,  ed.  3,  1782,  343.     (Not  in  ed.  of  1770.) 
"Vorlesungen  iiber  Zahlentheorie,  I,  1901,  265-6. 
^Amer.  Math.  Monthly,  23,  1916,  394. 

*Erroneou8ly  given  as  "cube"  in  the  French  tr.,  Oeuvres  de  Fermat,  3,  311. 
'"OeuvTes,  2,  332,  "premier  d6fi  aux  mathdmaticiens;"  also,  pp.  341-2,  Fermat  to  Digby,  June 

6,  1657,  where  7'  is  said  to  be  not  the  only  solution.     These  two  problems  by  Fermat 

were  quoted  in  a  letter  by  the  Astronomer  Jean  H6v4hus,  Nov.  1,  1657,  pubhshed  by  C. 

Henry,  Bull.  Bibl.  Storia  Sc.  Mat.  e  Fis.,  12,  1879,  683-5,  along  with  extracts  from  the 

Commercium  EpistoUcum.     Cf.  G.  Wertheim,  Abh.  Geschichte  Math.,  9,  1899,  558-561, 

570-2  (  =  Zeitschr.  Math.  Phys.,  44,  Suppl.  14). 
"Commercium  Epistohcum  de  Wallis,  Oxford,  1658;  Walhs,  Opera,  2,  1693.     Letter  II,  from 

WaUis  to  Brouncker,  Mar.  17,  1657;  letter  XVI,  Walhs  to  Digby,  Dec.  1,  1657.     Oeuvres 

de  Fermat,  3,  404,  414,  427,  482-3,  503-4,  513-5. 
**Commercium,  letter  IX,  Wallis  to  Digby;  Fermat'e  Oeuvres,  3,  419. 


Chap.  II]  PkOBLEMS    ON  SUMS   OF   DiVISORS.  55 

Frenicle^^  expressed  his  astonishment  that  experienced  mathematicians 
should  not  hesitate  to  present,  for  the  third  time,  unity  as  a  solution. 

Wains'*^  tabulated  <r{x^)  for  each  prime  a:<100  and  for  low  powers  of 
2, 3, 5,  and  then  excluded  those  primes  a:  for  which  (T(rc^)has  a  prime  factor 
not  occurring  elsewhere  in  the  table.  By  similar  eliminations  and  successive 
trials,  he  was  led  to  the  solutions^^  of  (i) : 

a=3^5-lM3-41-47,  6=2-3-5-13-41-47;     7a,  7b, 

adding  that  they  are  identical  with  the  four  numbers  given  by  Frenicle.*'' 
Note  that  o-(a)  is  the  square  of  2^3^5-7-lM3-17-29-61,  while  a{b)  is  the  square 
of  2^3V7-13-17-29.     Wallis^^  gave  the  further  solutions  of  a{x^)  =  y^: 

a:=17-3147-191,  ?/=2^°325-13-17-29-37, 

2-3-5-13-17.314M91,  2^^33527.  i3.;^7.29237^ 

3'5-lM3-17-31-4M91,  2'23^5-7-lM3-17-29237-61, 

and  the  products  of  each  x  by  7. 

Wallis^^  gave  solutions  of  his  problem  (m) : 

2^3.37,  2-19-29;  223-1M9-37,  2'7-29.67; 

29-67,  2-3-5-37;  2^7.29.67,  3-5-1M9-37. 

Frenicle*^  gave  48  solutions  of  WalUs'  problem  (m),  including  2-163 
11-37;  3-11-19,  7-107;  2-5-151,  3^-67;  also  83  sets  of  three  squares  having  the 
same  sum  of  divisors,  for  example,  the  squares  of 

2^11.37-151,  3^67-163,  5-11-37-151,  (7  =  3^7^19-31-67-1093; 

also  various  such  sets  of  n  squares  (with  prime  factors  <500)  for  n^l9, 
for  example,  the  squares  of  ac,  ad,  4:bd,  46c,  5bd,  and  56c,  where 

a  =  2-5-29-47-67-139,  6  =  13-37-191-359,  c  =  7.107,  d  =  3-ll-19. 

Frans  van  Schooten^"  made  ineffective  attempts  to  solve  problems 
(i),  (n). 

Frenicle^^  gave  the  solution 

a:  =  225-7.11-37-67-163-191-263-439-499,  t/  =  327^3- 19-31^67- 109 

of  problem  (n),  a{x^)=y^;  also  a  new  solution  of  a{x^)=y^: 
a;  =  255-7-31-73-241-243-467,  i/  =  2i23253ii. 13217.37.41. 113.193.257. 

^'Letter  XXII,  to  Digby,  Feb.  3,  1658.     Cf.  Leibnitii  et  BernouUii  Commercium  philos.  et 

math.,  I,    1795,  263,  letter  from  Johann  Bernoulli  to  Leibniz,  Apr.  3,  1697. 
"Letter  XXIII,  to  Digby,  Mar.  14,  1658. 
"The  same  tentative  process  for  finding  this  solution  a  was  given  by  E.Waring,  Meditationes 

Algebraicae,  1770,  pp.  216-7;  ed.  3,  1782,  377-8.      The  solution  6  =  751530  was  quoted 

by  Lucas,  Thiorie  des  nombres,  1891,  380,  ex.  3. 
**Solutio  duorum  problematum  circa  numeros  cubos . . .  1657,  dedicated  to  Digby  [lost  work]. 

See  Oeuvres  de  Fermat,  p.  2.  434,  Note;  WaUis." 
"Letter  XXVIII,  March  25,  1658;  WaUis,  Opera,  2,  814;  Wallia". 
««Letter  XXIX,  Mar.  29,  1658;  WaUis^^. 
"Letter  XXXI,  Apr.  11,   1658. 
"Letter  XXXIII,  Feb.  17,  1657  and  Mar.  18,  1658. 
"Letter  XLIII,  May  2,  1658. 


56  History  of  the  Theory  of  Numbers.  [Chap,  ii 

Wallis^'^  for  use  in  problem  (ii)  gave  a  table  showing  the  sum  of  the 
divisors  of  the  square  of  each  number  <  500.  Excluding  numbers  in  whose 
divisor  sum  occurs  a  prime  entering  the  table  only  once  or  twice,  there  are 
left  the  squares  of  2, 4,  8,  3,  5,  7, 11, 19,  29,  37,  67, 107, 163, 191,  263, 439, 499. 
By  a  very  long  process  of  exclusion  he  found  only  two  solutions  within  the 
limits  of  the  table,  viz.,  Frenicle's"  and 

(rj(7.11-29.163-191439)2[  =  ]3.7-13-19-31-67(^ 

Jacques  Ozanam"  stated  that  Fermat  had  proposed  the  problem  to  find 
a  square  which  with  its  aliquot  parts  makes  a  square  (giving  81  as  the  answer) 
and  the  problem  to  find  a  square  whose  aliquot  parts  make  a  square.  For 
the  latter,  Ozanam  found  9  and  2401,  whose  aliquot  parts  make  4  and 
400,  and  remarked  that  he  did  not  believe  that  Fermat  ever  solved  these 
questions,  although  he  proposed  them  as  if  he  knew  how. 

Ozanam^  noted  that  the  sum  of  961  =  31^  and  its  aliquot  parts  1  and  31 
is  993,  which  equals  the  sum  of  the  aliquot  parts  of  1 156  =  34".  As  examples 
of  two  squares  with  equal  total  sums  of  divisors  [WaUis'  problem  (m)],  he 
cited  16  and  25,  326^  and  407^,  while  others  may  be  derived  by  multiplying 
these  by  an  odd  square  not  divisible  by  5.  The  sum  of  all  the  divisors  of 
9^  is  11^  that  of  20^  is  31l  The  numbers  99  and  63  have  the  property  that 
the  sum  57  of  the  aliquot  parts  of  99  exceeds  the  sum  41  of  the  aliquot 
parts  of  63  by  the  square  16;  similarly  for  325  and  175. 

E.  Lucas^^  noted  that  the  problem  to  find  all  integral  solutions  of 

(1)  l-\-x+z^-\-x^  =  y^ 
is  equivalent  to  the  solution  of  the  system 

(2)  l+x  =  2w2,  l+x2  =  2t;2,  y  =  2uv, 

and  stated  that  the  complete  solution  is  given  by  that  of  2y^— x^  =  l. 
E.  Gerono^^  proved  that  the  only  solutions  of  (1)  are 

(x,  2/)  =  (-l,  0),         (0,  ±1),         (1,  ±2),         (7,  ±20). 

E.  Lucas^^  stated  that  there  is  an  infinitude  of  solutions  of  Fermat's 
problem  (i);  the  least  composite  solution  is  the  cube  of  2-3-5-13'41-47,  the 
sum  of  whose  divisors  is  the  square  of  2^3^5^7- 13- 17-29.  [This  solution  was 
given  by  Frenicle.^®]     For  the  case  of  a  prime,  the  problem  becomes  (1). 

A.  S.  Bang^^  gave  for  problem  (i)  the  first  of  the  three  answers  by  Wallis;*' 
for  (it),  (7(43098^)  =  1729^  for  {in),  29-67,  2-3-5-37  of  Wallis^*  and  the  first 
two  by  Frenicle;^^  all  without  references. 

"A  Treatise  of  Algebra,  1685,  additional  treatises,  Ch.  IV. 

"Letter  to  De  Billy,  Nov.  1,  1677,  published  by  C.  Henry,  Bull.  Bibl.  Storia  Sc.  Mat.  e  Fia.,  12, 

1879,  519.     Reprinted  in  Oeuvres  de  Fermat,  4,  1912,  p.  140. 
"Recreations  Math6matiques  et  Phys.,  new  ed.,  1723,  1724,  1735,  etc.,  Paris,  I,  41-43. 
"Nouv.  Corresp.  Math.,  2,  1876,  87-8. 
••Nouv.  Ann.  Math.,  (2),  16,  1877,  230-4. 
"Bull.  Bibl.  Storia  Sc.  Mat.  e  Fis.,  10,  1877,  287. 
"Nyt  Tidsskrift  for  Mat.,  1878,  107-8;  on  problems  in  1877,  180. 


Chap.  II]  PROBLEMS  ON   SUMS   OF   DiVISORS.  57 

E.  Fauquembergue,^^  after  remarking  that  (1)  is  equivalent  to  the  sys- 
tem (2),  cited  Fermat's^"  assertion  that  the  first  two  equations  (2)  hold 
only  for  a:  =  7  [aside  from  the  evident  solutions  a:  =  ±  1,  0],  which  has  been 
proved  by  Genocchi.^^ 

H.  Brocard^^  thought  that  Fermat's  assertion  that  7^  is  not  the  only 
solution  of  problem  (i)  implied  a  contradiction  with  Genocchi.^^  G.  Vacca 
{ihid.,  p.  384)  noted  the  absence  of  contradiction  as  (i)  leads  to  equation  (1) 
only  if  a:  be  a  prime. 

C.  Moreau®^  treated  the  equation,  of  type  (1), 

While  he  used  the  language  of  extracting  the  square  root  of  X  =  x*+ . . . 
written  to  the  base  x,  he  in  effect  put  X={x^-\-a)^,  0<a<x.  Then  a^  = 
x+1,  2ax^  =  x^-\-x^,  whence  2a  =  a^,  a  =  2,  x  =  3,  y  =  ll. 

E.  Lucas®^  stated  that  {x^-\-y^)/{x+y)  =^  has  the  solutions 

(3,-1,  11),  (8,  11,  101),  (123,  35,  13361),. . . 

Moret-Blanc^^  gave  also  the  solutions  (0,  1,  1),  (1,  1,  1). 
E.  Landau^^  proved  that  the  equation 

— T=y^ 

x  —  1 
is  impossible  in  integers  (aside  from  x  =  0,  ?/  =  =fc  1)  for  an  infinitude  of  values 
of  n,  viz.,  for  all  n's  divisible  by  3  such  that  the  odd  prime  factors  of  n/3,  if 
any,  are  all  of  the  form  6y— 1  (the  least  such  n  being  6).  For,  setting 
n  =  3m,  we  see  that  y^  is  the  product  of  x^+x+1  and  F  =  x^"'~^-{- . . .  +a:^+l. 
These  two  factors  are  relatively  prime  since  x^=l  gives  F=m  (mod  x^+ 
a: + 1 ) .  Hence  x^+x+lis  Si  square,  which  is  impossible  for  a;  f^  0  since  it  lies 
between  x^  and  (a:+l)^. 

Brocard^^  had  noted  the  solution  a:  =  1,  y=m,  if  n  =  mP. 

A.  Gerardin^^  obtained  six  new  solutions  of  problem  (i) : 

a:  =  2.47.193.239.701,  2/  =  2^3l5M3M7.97.149. 

x  =  2.5.23.41.83.239,  y  =  2\S\5\7.1S\29.53. 

x  =  3.13.23.47.83.239,  y  =  2^^3^517.13117.53. 

X  =  2.3.13.23.83.193.701,  y  =  2^3^5^7.13.17.53.97.149. 

a;  =  3.5.13.41.193.239.701,  2/  =  2^3l5l7.13M7.29.97.149. 

a;  =  2.5.13.43.191.239.307,  ?/  =  2i^32.5MlM7.29.37.53.113.197.241.257. 

Also  <t{N^)=S^  for  Ar  =  3-7-ll-29-37,  ^  =  3-7-13-19-67. 

"Nouv.  Ann.  Math.,  (3),  3,  1884,  538-9. 

'"Oeuvres,  2,  434,  letter  to  Carcavi,  Aug.,  1659. 

"Nouv.  Ann.  Math.,  (3),  2,  1883,  30&-10.     Cf.  Chapter  on  Diophantine  Equations  of  order  2. 

"L'intermMiaire  des  math.,  7,  1900,  31,  84. 

"Nouv.  Ann.  Math.,  (2),  14,  1875,  335. 

»Ibid.,  509. 

«76id.,  (2),  20,  1881,  150. 

"L'interm^diaire  des  math.,  8,  1901,  149-150. 

"Ibid.,  22,  1915,  111-4,  127. 


58  History  of  the  Theory  of  Numbers.  [Chap.ii 

G^rardin^^  gave  five  new  solutions  of  (i) : 

X  =  3.11.31.443.499,  i/  =  2^3.5M3.37.61.157. 

x  =  2.3^31.443.449,  2/ =  2^3.5ni. 13.37.61.157. 

a;  =  1 1 .  17.41 .43.239.307.443.499, 

2/  =  2^2  3^5'.7.11. 13^29^.37.61. 157. 
x  =  2.11. 17.23.41.211.467.577.853, 

t/  =  2^''.3^5l7.13M7.292.53.61. 113.193.197. 
x  =  3ni. 13.23.83.193.701, 

?/  =  293'537.11. 13.17.53.61.97.149, 
the  last  following  from  his*^^  fourth  pair  in  \'iew  of 

a{SnV):  a{2'S')  =  2'3.nm^:  233.52  =  2=11-612;  5=. 

A.  Cunningham  and  J.  Blaikie^^  found  solutions  of  the  form  x  =  2'p  of 
s{x)  =g^,  where  s{n)  is  the  sura  of  the  divisors  <n  of  n. 

product  of  aliquot  parts. 

Paul  Halcke'^^  noted  that  the  product  of  the  aliquot  parts  of  12,  20,  or 
45  is  the  square  of  the  number;  the  product  for  24  or  40  is  the  cube;  the 
product  for  48,  80  or  405  is  the  biquadrate. 

E.  Lionnet"^  defined  a  perfect  number  of  the  second  kind  to  be  a  number 
equal  to  the  product  of  its  aliquot  parts.  The  only  ones  are  p^  and  pq, 
where  p  and  q  are  distinct  primes. 

"L'interm^diaire  des  math.,  24,  1917,  132-3. 

"Math.  Quest.  Educ.  Times,  (2),  7,  1905,  68-9. 

"Dehciae  Math,  oder  Math.  Sinnen-Confect,  Hamburg,  1719,  197,  Exs.  150-2. 

"Nouv.  Ann.  Math.,  (2),  18,  1879,  306-8.     Lucas,  Th6orie  des  nombres,  1891,  373,  Ex.  6 


I 


CHAPTER  III. 

FERMAT'S  AND  WILSON'S  THEOREMS,  GENERALIZATIONS  AND 

CONVERSES;  SYMMETRIC  FUNCTIONS  OF 

1,2 P-\  MODULO  P. 

Fermat's  and  Wilson's  Theorems;  Immediate  Generalizations. 

The  Chinese^  seem  to  have  known  as  early  as  500  B.  C.  that  2^—2  is 
divisible  by  the  prime  p.  This  fact  was  rediscovered  by  P.  de  Fermat^ 
while  investigating  perfect  numbers.  Shortly  afterwards,  Fermat^  stated 
that  he  had  a  proof  of  the  more  general  fact  now  known  as  Fermat's  theorem: 
If  p  is  any  prime  and  x  is  any  integer  not  divisible  by  p,  then  x^~^  —  1  is 
divisible  by  p. 

G.  W.  Leibniz^  (1646-1716)  left  a  manuscript  giving  a  proof  of  Fermat's 
theorem.  Let  p  be  a  prime  and  set  x  =  a+6+c+. . ..  Then  each  multi- 
nominal  coefficient  appearing  in  the  expansion  of  x^  — 2a^  is  divisible  by  p. 
Take  a  =  6  =  c=...=l.    Thus  a;^  —  a:  is  divisible  by  p  for  every  integer  x. 

G.  Vacca^  called  attention  to  this  proof  by  Leibniz. 

Vacca^  cited  manuscripts  of  Leibniz  in  the  Hannover  Library  showing 
that  he  proved  Fermat's  theorem  before  1683  and  that  he  knew  the  theorem 
now  known  as  Wilson's^^  theorem:  If  p  is  a  prime,  l  +  (p  — 1)!  is  divisible 
by  p.  But  Vacca  did  not  explain  an  apparent  obscurity  in  Leibniz's  state- 
ment [cf.  Mahnke'^]. 

D.  Mahnke''  gave  an  extensive  account  of  those  results  in  the  manuscripts 
of  Leibniz  in  the  Hannover  Library  which  relate  to  Fermat's  and  Wilson's 
theorems.  As  early  as  January  1676  (p.  41)  Leibniz  concluded,  from  the 
expressions  for  the  ^th  triangular  and  yth.  pyramidal  numbers,  that 

(2/+l)2/=2/'-2/=0  (mod  2),  {y+2){y+l)y=y'-y=0  (mod  3), 

and  similarly  for  moduU  5  and  7,  whereas  the  corresponding  formula  for 
modulus  9  fails  for  y  =  2, — thus  forestalling  the  general  formula  by  Lagrange.^* 
On  September  12,  1680  (p.  49),  Leibniz  gave  the  formula  now  known  as 
Newton's  formula  for  the  sum  of  like  powers  and  noted  (by  incomplete 
induction)  that  all  the  coefficients  except  the  first  are  divisible  by  the 
exponent  p,  when  p  is  a  prime,  so  that 

a''+h''+c''-{- . .  .  =  {a+h+c+  . .  .Y  (mod  p). 

Taking  a  =  b=  . . .  =1,  we  obtain  Fermat's  theorem  as  above.'*  That  the 
binomial  coefficients  in  (1  +  1)^  —  1  —  1  are  divisible  by  the  prime  p  was 

^G.  Peano,  Formulaire  math.,  3,  Turin,  1901,  p.  96.     Jeans.^^" 

''Oeuvres  de  Fermat,  Paris,  2,  1894,  p.  198,  2°,  letter  to  Mersenne,  June  (?),  1640;  also  p.  203, 

2;  p.  209. 
"Oeuvres,  2,  209,  letter  to  Frenicle  de  Bessy,  Oct.  18,  1640;  Opera  Math.,  Tolosae,  1679,  163. 
*Leibnizens  Math.  Schriften,  herausgegeben  von  G.  J.  Gerhardt,  VII,  1863,  180-1,  "nova 

algebrae  promotio." 
»Bibliotheca  math.,  (2),  8,  1894,  46-8. 
•Bolletino  di  BibUografia  Storia  Sc.  Mat.,  2,  1899,  113-6. 
'Bibliotheca  math.,  (3),  13,  1912-3,  29-61. 

69 


60  History  of  the  Theory  of  Numbers.  [Chap,  hi 

proved  in  1681  (p.  50).  Mahnke  gave  reasons  (pp.  54-7)  for  believing 
that  Leibniz  rediscovered  independently  Fermat's  theorem  before  he 
became  acquainted,  about  1681-2,  with  Fermat's  Varia  opera  math,  of 
1679.  In  1682  (p.  42),  Leibniz  stated  that  (p-2)!=l  (mod  p)  if  p  is  a 
prime  [equivalent  to  Wilson's  theorem],  but  that  {p—2)l=m  (mod  p),  if 
p  is  composite,  m  ha\dng  a  factor  >  1  in  common  wdth  p. 

De  la  Hire^  stated  that  if  k-'"^^  is  divided  by  2(2r+l)  we  get  A;  as  a 
remainder,  perhaps  after  adding  a  multiple  of  the  divisor.  For  example, 
if  kr'  is  divided  by  10  we  get  the  remainder  k.  He  remarked  that  Carr^ 
had  observed  that  the  cube  of  any  number  /:<6  has  the  remainder  k  when 
divided  by  6. 

L.  Euler^  stated  Fermat's  theorem  in  the  form:  If  n+1  is  a  prime  divid- 
ing neither  a  nor  h,  then  a"  — 6"  is  divisible  by  n+1.  He  was  not  able  to 
give  a  proof  at  that  time.  He  stated  the  generaUzation :  If  e  =  p'"~Hp  — 1) 
and  if  p  is  a  prime,  the  remainder  obtained  on  dividing  a*  by  p""  is  0  or  1 
[a  special  case  of  Euler^^].  He  stated  also  that  ii  m,  n,  p,. .  .  are  distinct 
primes  not  dividing  a  and  if  A  is  the  1.  c.  m.  of  m  —  1,  n  — 1,  p  — 1, . . .,  then 
o"*  —  1  is  divisible  by  mnp . .  .  [and  a*  —  1  by  m''  n\  .  .ii  k  =  A  rrC~^n^~^ . .  .]. 

Euler^°  first  published  a  proof  of  Fermat's  theorem.     For  a  prime  p, 

2''  =  (l  +  l)^  =  l+p-h(^)H-...+p+l  =  2+mp, 
3P  =  (l+2)P  =  l+A:p+2^  3^-3- (2^-2)  =  A:p, 

(1+0)"=  1+np+aP,  (l+o)P-(l+a)-(aP-a)=np. 

Hence  if  a^—a  is  divisible  by  p,  also  (1+a)"  — (1+a)  is,  and  hence  also 
(a+2)''-(a+2),. .  .,  (a+6)P-(a+6).  For  a  =  2,  2"  -  2  was  proved  divisible 
by  p.  Hence,  wTiting  x  for  2+6,  we  conclude  that  x^—x  is  divisible  by  p 
for  any  integer  x. 

G.  W.  Kraft^^  proved  similarly  that  2"  — 2  =  7np. 

L.  Euler's^-  second  proof  is  based,  hke  his  first,  on  the  binomial  theorem. 
If  a,  6  are  integers  and  p  is  a  prime,  (a+6)"— a"  — 6"  is  divisible  by  p.  Then, 
if  a^  —  a  and  6^  —  6  are  di\'isible  by  p,  also  (a+6)"  — a  — 6  is  di\4sible  by  p. 
Take  h  =  \.  Thus  (a+1)"— a— 1  is  divisible  by  p  if  a^—a  is.  Taking 
a  =  1,  2,  3, ...  in  turn,  we  conclude  that  2''— 2,  3"— 3, .  . . ,  c^  —  c  are  divisible 
by  p. 

L.  Euler^^  preferred  his  third  proof  to  his  earlier  proofs  since  it  avoids 
the  use  of  the  binomial  theorem.     If*  p  is  a  prime  and  a  is  any  integer  not 

*Hist.  Acad.  Sc.  Paris,  annee  1704,  pp.  42-4;  ra4m.,  358-362. 

»Comm.  Ac.  Petrop.,  6,  1732-3,  106;  Coram.  Arith.,  1,  1849,  p.  2.     [Opera  postuma,  I,  1862, 

167-8  (about  1778)]. 
^••Comm.  Ac.  Petrop.,  8,  ad  annum  1736,  p.  141;  Comm.  Arith.,  1,  p.  21. 
"Novi  Comm.  Ac.  Petrop.,  3,  ad  annos  1*50-1,  121-2. 
"Novi  Comm.  Ac.  Petrop.,  1,  1747-8,  20;  Comm.  Arith.,  1,  50.     Also,  letter  to  Goldbach, 

Mar.  6,  1742,  Corresp.  Math.  Phys.  (ed.  Fuss),  I,  1843,  117.     An  extract  of  the  letter  is 

given  in  Nouv.  Ann.  Math.,  12,  1853,  47. 
"Novi  Comm.  Ac.  Petrop.,  7,  1758-9,  p.  70  (ed.  1761,  p.  49);  18, 1773,  p.  85;  Comm.  Arith.,  1, 

260-9,  518-9.     Reproduced  by  Gauss,  Disq.  Arith.,  art.  49;  Werke,  1,  1863,  p.  40. 


Chap.  Ill]  FerMAt's  AND   WiLSON's  THEOREMS.  61 

divisible  by  y,  at  most  p  —  1  of  the  positive  residues  <  p,  obtained  by  dividing 
1,  a,  a^, . . .  by  p, are  distinct.  Let,  therefore;  a"  and  a",  where ix^v,  have  the 
same  residue.  Then  a"""  — 1  is  divisible  by  p.  Let  X  be  the  least  positive 
integer  for  which  a^  —  \  is  divisible  by  p.  Then  \,a,a^,...,  a^~^  have  dis- 
tinct residues  when  divided  by  p,  so  that  X^p  — L  IfX  =  p  — 1,  Fermat's 
theorem  is  proved.  If  X<p  — 1,  there  exists  a  positive  integer  k  ik<p) 
which  is  not  the  residue  of  a  power  of  a.  Then  k,  ak,  a^k, .  .  .,  a^~^k  have 
distinct  residues,  no  one  the  residue  of  a  power  of  a.  Since  the  two  sets 
give  2X  distinct  residues,  we  have  2X^ p  —  1.  If  X<  (p  — 1)/2,  we  start  with 
a  new  residue  s  and  see  that  s,  as,  a^s, . .  .,  a^~^s  have  distinct  residues,  no  one 
the  residue  of  a  power  of  a  or  of  a^k.  Hence  X^  (p  — 1)/3.  Proceeding  in 
this  manner,  we  see  that  X  divides  p  —  L  Thus  d^~^  —  1  is  divisible  by  a''  —  1 
and  hence  by  p. 

L.  Euler^^  soon  gave  his  fundamental  generalization  of  Fermat's  theorem 
from  the  case  of  a  prime  to  any  integer  N: 

Euler's  theorem:  If  n=(f){N)  is  the  number  of  positive  integers  not 
exceeding  N  and  relatively  prime  to  N,  then  x"  — 1  is  divisible  by  N  for 
every  integer  x  relatively  prime  to  N. 

Let  V  be  the  least  positive  integer  for  which  x"  has  the  residue  1  when 
divided  by  N.  Then  the  residues  of  1,  a:,  a:^, ... ,  x"'^  are  distinct  and  prime 
to  N.  Thus  v^n.  If  v<n,  there  is  an  additional  positive  integer  a  less 
than  A''  and  prime  to  N.  Then,  when  a,  ax,  ax^, .  .  .,  ax"'^  are  divided  by  N, 
the  residues  are  distinct  from  each  other  and  from  those  of  the  powers  of  x. 
Thus,  2v^n.  Similarly,  if  2v<n,  then  Zv^n.  It  follows  in  this  manner 
that  V  divides  n. 

J.  H.  Lambert^^  gave  a  proof  of  Fermat's  theorem  differing  shghtly  from 
the  first  proof  by  Euler.^"  If  h  is  not  divisible  by  the  prime  p,  6^"^  — 1  is 
divisible  by  p.    For,  set  6  =  c+L     Then 

6^-^-1  =-l+c''-i  +  (p-l)c^-2+...+l 

= -l+c^-'-c^-2 +0^"^- . . . +1+Ap, 

where  A  is  an  integer.    The  intermediate  terms  equal 


Hence 


c+1  c+1 

-fA-/,  /=' 


P  p  '  P(c+1) 


The  theorem  will  thus  follow  by  induction  if  /  is  shown  to  be  integral. 
[Take  p>2,  so  that  p  — 1  is  even.]  Then  c^"^  — 1  is  divisible  by  c+l,  and 
by  the  hypothesis  for  the  induction,  by  p.  Since  c-\-l  =  h  is  relatively 
prime  to  p,  /  is  an  integer. 


"Novi  Comm.  Ac.  Petrop.,  8,  1760-1,  p.  74;  Comm.  Arith.,  1,  274-286;  2,  524-6. 
"Nova  Acta  Eruditorum,  Lipsiae,  1769,  109. 


62  History  of  the  Theory  of  Numbers.  [Chap,  hi 

E.  Waring^^  first  published  the  theorem  that  [Leibniz®]  l  +  (p  — 1)!  is 
divisible  by  the  prime  p,  ascribing  it  to  Sir  John  Wilson^^  (1741-1793). 
Waring  (p.  207;  ed.  3,  p.  356)  proved  that  if  a^  —  a  is  divisible  by  p,  then 
(a+1)''— a  — 1  is,  since  {a+iy  =  a^-\-pA-\-l,  "a,  property  first  invented  by 
Dom.  Beaufort  and  first  proved  by  Euler." 

J.  L.  Lagrange^^  was  the  first  to  publish  a  proof  of  Wilson's  theorem.    Let 

(x+l)(x+2) . .  .  (x+p-l)  =x''-^+AiX^-'+  .  . .  -h^p-i. 

R eplace  x  by  x + 1  and  multiply  the  resulting  equation  by  x + 1 .  Comparing 
with  the  original  equation  multiplied  by  x+p,  we  get 

{x+p){x^-'+.  .  .  +A,_,)  =  {x+ir+Ai{x+ir-'  +  . . .  +^p_i(x+l). 

Apply  the  binomial  theorem  and  equate  coefl5cients  of  like  powers  of  x. 
Thus 

Let  p  be  a  prime.  Then,  for  0<k<p,  (j)  is  an  integer  divisible  by  p. 
Hence  Ai,  2A2, .  . . ,  {p—2)Ap_2  are  divisible  by  p.    Also, 

(P-1)4,..=  (P  +  (PI})A:+(P-2)^+...  =  1+^+A,+  ...+^,.2. 

Thus  1+Ap_i  is  divisible  by  p.  By  the  original  equation,  Ap_i  =  (p  — 1)!, 
so  that  Wilson's  theorem  follows. 

Moreover,  if  x  is  any  integer,  the  proof  shows  that 

xP-^-l-(x+l)(x+2)...(x+p-l) 

is  divisible  by  the  prime  p.  If  x  is  not  divisible  by  p,  some  one  of  the 
integers x+1,. .  .,x+p  — 1  is  divisible  by  p.  Hence  x''"^  — 1  is  divisible  by 
p,  giving  Fermat's  theorem. 

Lagrange  deduced  Wilson's  theorem  from  Fermat's.  By  the  formula^* 
for  the  differences  of  order  p  — 1  of  P~\ . . .,  n^~^, 

(1)         (p-i)\=p^-'-{p-i){p-iy-'+(^p~^){p-2r-' 

-(^3^)(p-3)^-^+. .  .+(-1)^-^ 

Dividing  the  second  member  by  p,  and  applying  Fermat's  theorem,  we 
obtain  the  residue 

"Meditationes  algebraicae,  Cambridge,  1770,  218;  ed.  3,  1782,  380. 

"On  his  biography  see  Nouv.  Corresp.  Math.,  2,  187.6,  110-114;  M.  Cantor,  Bibliotheca  math., 

(3),  3,  1902,  412;  4,  1903,  91. 
"Nouv.  M6m.  Acad.  Roy.  BerUn,  2,  1773,  ann^e  1771,  p.  125;  Oeuvres,  3,  1869,  425.     Cf.  N. 

Nielsen,  Danske  Vidensk.  Selsk.  Forh.,  1915,  520. 
"Euler,  Novi  Comm.  Ac.  Petrop.,  5,  1754-5,  p.  6;  Comm.  Arith.,  1,  p.  213;  2,  p.  532;  Opera 

postuma,  Petropoli,  1,  1862,  p.  32. 


Chap.  Ill]  FeRMAt's  AND   WiLSON's  THEOREMS.  63 

Finally,  Lagrange  proved  the  converse  of  Wilson's  theorem:  If  n  divides 
l  +  (n— 1)!,  then  n  is  a  prime.  For  n  =  4m+l,  w  is  a  prime  if  (2-3. .  .2my 
has  the  remainder  —1  when  divided  by  n.  For  n  =  4m  — 1,  if  (2m  — 1)!  has 
the  remainder  =±=1. 

L.  Euler^"  also  proved  by  induction  from  a:  =  n  to  n+1  that 

(2)  xl  =  a'-x{a-ir+(^iya-2r-(^f){a-Sr+..., 

which  reduces  to  (1)  ior  x  =  p  —  l,  a  =  p;  and  more  generally, 


(3)  a^_n{a-ir+(^){a-2y-  .  . . +(-l)^Q  (a-A:r+ . .  .  =|^,  ^^ 


x<n 
x  =  n. 

D'Alembert^^  stated  that  the  theorem  that  the  difference  of  order  m  of  a*" 
is  m !  had  been  long  known  and  gave  a  proof. 

L.  Euler^^  made  use  of  a  primitive  root  a  of  the  prime  p  to  prove  Wilson's 
theorem  (though  his  proof  of  the  existence  of  a  was  defective).  When 
l,a,a^,  . . .,  oF~^  are  divided  by  p,  the  remainders  are  1, 2,  3, . . . ,  p  —  1  in  some 
order.  Hence  a(p-i)(p-2)/2  j^^s  the  same  remainder  as  (p  —  1) !.  Taking  p>2, 
we  may  set  p  =  2n+l.  Since  a"  has  the  remainder  —1,  then  a"a^"^'*~^\  and 
hence  also  (p  — 1)!,  has  the  remainder  —1. 

P.  S.  Laplace^^  proved  Fermat's  theorem  essentially  by  the  first  method 
of  Euler^°  without  citing  him:  If  a  is  an  integer  <p  not  divisible  by  the 
prime  p, 

^l  =  \a-l^lY  =  \\{a-iy+p{a-iy-'+  . . .  +l[ , 
a      a  a 

aV-^-\=-\{a-lY-\-l-a+hp{a-V)\  =^—^\{a-iy-^-l+hp] . 
a  Cv 

Hence  by  induction  a^~^  —  l  is  divisible  by  p.    For  a>p,  set  a  =  np+q  and 
use  the  theorem  for  q. 

He  gave  a  proof  of  Euler's^^  generalization  by  the  method  of  powering: 
if  n  =  p''p{\  . .,  where  p,  Pi,...  are  distinct  primes,  and  if  a  is  prime  to  n, 
then  d°  —  l  is  divisible  by  n,  where 

-"(^)(^)  ■='^' 

q  =  p''-\p-l),  r  =  pr-\Pi-l)P2''-\p2-l)..  .. 

Set  a'^  =  x.    Then  a'  — l=x'"  — 1  is  divisible  by  x  —  1.    Using  the  binomial 
theorem  and  a^~^  —  l  =  hp,  we  find  that  aj  — 1  is  divisible  by  p". 

"Novi  Comm.  Ac.  Petrop.,  13,  1768,  28-30. 

"Letter  to  Turgot,  Nov.  11,  1772,  in  unedited  papers  in  the  Biblioth^que  de  I'lnstitut  de 
France.     Cf.  BuU.  Bibl.  Storia  Sc.  Mat.  e  Fis.,  18,  1885,  531. 

"Opuscula  analytica,  St.  Petersburg,  1,  1783  [Nov.  15,  1773],  p.  329;  Comm.  Arith.,  2,  p.  44; 
letter  to  Lagrange  (Oeuvres,  14,  p.  235),  Sept.  24,  1773;  Euler's  Opera  postuma,  I,  583. 

"De  la  Place,  Th^orie  abr^g^e  des  nombres  premiers,  1776,  16-23.  His  proofs  of  Fermat's 
and  Wilson's  theorems  were  inserted  at  the  end  of  Bossut's  Algdbre,  ed.  1776,  and 
reproduced  by  S.  F.  Lacroix,  Trait6  du  Calcul  Diff.  Int.,  Paris,  ed.  2,  vol.  3,  1818,  722-4, 
on  p.  10  of  which  is  a  proof  of  (2)  for  o=a;  by  the  calculus  of  differences. 


64  History  of  the  Theory  of  Numbers.  [Chap,  ill 

From  the  (p  — l)th  order  of  differences  for  x""^  — 1, 

{x+p-ir-'-i-{p-i)\{x-\-p-2r-'-i\  +  (^p~^y,{x-{-p-sr-'-i\ 

Set  x  =  l  and  use  Fermat's  theorem.     Hence  l  +  (p  — 1)!  is  divisible  by  p. 

E.  Waring/^  1782,  380-2,  made  use  of 

x'  =  xix-l).  .  .{x-r-{-l)-\-Pxix-l).  .  .(x-r+2) 

+Qxix-1) . .  .(x-r+3)+  .  .  .  +Hx(x-l)+Ix, 

where  P  =  H-2+  •  •  .  -f  (r  — 1),  Q  =  PA^—B,  etc.,  B  denoting  the  sum  of  the 
products  of  1,  2,...,  r  — 1  two  at  a  time,  and  A^  =  l+2+ . .  . +(r— 2). 
Then 

r+2'+  .  .  .+x^  =  -^{x+l)x{x-l) .  .  .ix-r+l)+-{x+l)x.  ..{x-r+2) 
r+1  r 

+-^(x+l)x.  ..{x-r+3)+.  .  .  +^ix+l)x{x-l)-\-Ux+l)x. 

Take  r  =  x  and  let  x+1  be  a  prime.  By  Fermat's  theorem,  V,  2"^, . . .,  x' 
each  has  the  remainder  unity  when  divided  by  x+1,  so  that  their  sum  has 
the  remainder  x.     Thus  l+x\  is  divisible  by  x+1. 

Genty^^  proved  the  converse  of  Wilson's  theorem  and  noted  that  an 
equivalent  test  for  the  primahty  of  p  is  that  p  divide  (p— n)!(n  — 1)!  — 
( - 1)".     For  n  =  (p+ 1)/2,  the  latter  expression  is  \  (^zi)  !^ 2±  i  [Lagrange^^]. 

Franz  von  Schaffgotsch^^  was  led  by  induction  to  the  fact  (of  which  he 
gave  no  proof)  that,  if  n  is  a  prime,  the  numbers  2,  3, . .  . ,  n  — 2  can  be  paired 
so  that  the  product  of  the  two  in  any  pair  is  of  the  form  xn+1  and  the  two 
of  a  pair  are  distinct.  Hence,  by  multipUcation,  2-3...(n  — 2)  has  the 
remainder  unity  when  divided  by  n,  so  that  (n  — 1)!  has  the  remainder 
n  — 1.  For  example,  if  n  =  19,  the  pairs  are  2-10,  4-5,  3-13,  7-11,  6-16, 
8-12,  9-17,  14-15.  Similarly,  for  n  any  power  of  a  prime  p,  we  can  so 
pair  the  integers  <n  —  l  which  are  not  divisible  by  p.  But  for  n=15,  4 
and  4  are  paired,  also  1 1  and  1 1 .  Euler^^  had  already  used  these  associated 
residues  (residua  sociata). 

F.  T.  Schubert^^"  proved  by  induction  that  the  nth  order  of  differences 
of  r,  2",....isn!. 

A.  M.  Legendre-^  reproduced  the  second  proof  by  Euler^^  of  Fermat's 
theorem  and  used  the  theory  of  differences  to  prove  (2)  for  a  =  x.  Taking 
a;  =  p  — 1  and  using  Fermat's  theorem,  we  get  (p  — 1)!=(1  — 1)"  — 1  (mod  p). 

"Histoire  et  m6m.  de  I'acad.  roy.  sc.  insc.  de  Toulouse,  3,  1788  (read  Dec.  4,  1783),  p.  91. 

"Abhandlungen  d.  Bohmischen  Gesell.  Wiss.,  Prag,  2,  1786,  134. 

"Opusc.  anal.,  1,  1783  (1772),  64,  121;  Novi  Comm.  Ac.  Petrop.,  18,  1773,  85,  §26;  Comm. 

Arith.  1,  480,  494,  519. 
""Nova  Acta  Acad.  Petrop.,  11,  ad  annum  1793,  1798,  mem.,  174-7. 
"Th6orie  des  nombres,  1798,  181-2;  ed.  2,  1808,  166-7. 


Chap.  Ill]  FeRMAt's  AND   WiLSON's  THEOREMS.  *  65 

C.  F.  Gauss^^  proved  that,  if  n  is  a  prime,  2,  3, . . . ,  n— 2  can  be  associated 
in  pairs  such  that  the  product  of  the  two  of  a  pair  is  of  the  form  xn+1. 
This  step  completes  Schaffgotsch's^^  proof  of  Wilson's  theorem. 

Gauss^^  proved  Fermat's  theorem  by  the  method  now  known  to  be 
that  used  by  Leibniz^  and  mentioned  the  fact  that  the  reputed  proof  by 
Leibniz  had  not  then  been  published. 

Gauss^*^  proved  that  if  a  belongs  to  the  exponent  t  modulo  p,  a  prime, 
then  a-a^-a^  .  .  .a^^i  —  lY'^^  (mod  p).  In  fact,  a  primitive  root  p  of  p 
may  be  chosen  so  that  a=p^^~^^'\  Thus  the  above  product  is  congruent 
to  p*,  where 

Thus  p*=(p~2~}''^^  =  (  — 1)'"*"^  (mod  p).  When  a  is  a  primitive  root,  a, 
a^,. . .,  aF~^  are  congruent  to  1,  2, . . . ,  p  —  1  in  some  order.  Hence  (p  —  1) != 
(  —  1)^.  This  method  of  proving  Wilson's  theorem  is  essentially  that  of 
Euler.22 

Gauss^^  stated  the  generalization  of  Wilson's  theorem:  The  product  of 
the  positive  integers  <  A  and  prime  to  A  is  congruent  modulo  ^  to  —  1  if 
A  =  4,  p"*  or  2p^,  where  p  is  an  odd  prime,  but  to  + 1  if  ^  is  not  of  one  of 
these  three  forms.  He  remarked  that  a  proof  could  be  made  by  use  of 
associated  numbers^^  with  the  difference  that  a;^=l  (mod  A)  may  now 
have  roots  other  than  ±  1 ;  also  by  use  of  indices  and  primitive  roots^°  of  a 
composite  modulus. 

S.  F.  Lacroix^^  reproduced  Euler's^^  third  proof  of  Fermat's  theorem 
without  giving  a  reference. 

James  Ivory^^  obtained  Fermat's  theorem  by  a  proof  later  rediscovered 
by  Dirichlet.^"  Let  N  be  any  integer  not  divisible  by  the  prime  p.  When 
the  multiples  N,  2N,  SN, . . .,  {p  —  l)N  are  divided  by  p,  there  result  p  dis- 
tinct positive  remainders  <p,  so  that  these  remainders  are  1,  2, . . .,  p  — 1 
in  some  order .^^  By  multiplication,  N^~^Q  =  Q-\-mp,  where  Q  =  (p  — 1)!. 
Hence  p  divides  iV^~^  —  1  since  it  does  not  divide  Q. 

Gauss^^  used  the  last  method  in  his  proof  of  the  lemma  (employed  in  his 
third  proof  of  the  quadratic  reciprocity  law):  If  k  is  not  divisible  by  the 
odd  prime  p,  and  if  exactly  /x  of  the  least  positive  residues  of  k,  2k,. . ., 
l{p-l)k modulo p  exceed p/2,  then k^p-'^^^^=  ( - 1)" (mod p) .   [Cf . Grunert.^^] 

''^Disquisitiones  Arith.,  1801,  arts.  24,  77;  Werke,  1,  1863, 19,  61. 

2*Disq.  Arith.,  art.  51,  footnote  to  art.  50. 

soDisq.  Arith.,  art.  75. 

^iDisq.  Arith.,  art.  78. 

32Compl4ment  des  416mens  d'alglbre,  Paris,  ed.  3,  1804,  298-303;  ed.  4,  1817,  313-7, 

"New  Series  of  the  Math.  Repository  (ed.  Th.  Leybourn),  vol.  1,  pt.  2,  1806,  6-8. 

"A  fact  known  to  Euler,  Novi  Comm.  Acad.  Petrop.,  8,  1760-1,  75;  Comm.  Arith.,  1,  275; 

and  to  Gauss,  Disq.  Arith.,  art.  23.      Cf.  G.  Tarry,   Nouv.  Ann.  Math.,  18,  1899, 

149,  292. 
^''Comm.  soc.  reg.  so.  Gottingensis,  16,  1808;  Werke,  2,  1-8.     Gauss'  Hohere  Arith.,  German 

transl.  by  H.  Maser,  Berhn,  1889,  p.  458. 


66  History  of  the  Theory  of  Numbers.  [Chap,  hi 

J.  A.  Gninert^®  considered  the  series 

K  n]  =  n-- (^)  (71-1)-+ Q  (n-2r- .  .  ., 

to  which  Euler's  (3)  reduces  for  a  =  n,  x  =  m,  and  proved  that 

[m,  n]=n\[m  —  l,  n  — l]  +  [w  — 1,  n]\ . 
This  recursion  formula  gives 

[m,n]  =  0  (m  =  0,  l,...,n-l);  [72,  n]=n\  [cf.  (2)], 

Any  [m,  n]  is  di\isible  by  n\.  As  by  the  proof  of  Lagrange,^^  [m,  n]  +  (  — 1)" 
is  di\isible  by  w  +  1  if  the  latter  is  a  prime  >n.     Again, 

which  for  x  =  0,  h=l,  gives  [m,  m]=ml. 

W.  G.  Horner^"  proved  Euler's  theorem  by  generaUzing  Ivory's^^  method. 
If  ri, .  . . ,  r^  are  the  integers  <m  and  prime  to  m,  then  riN, . . . ,  r^N have  the 
r's  as  their  residues  modulo  m. 

P.  F.  Verhulst^^  gave  Euler's  proof^^  in  a  sUghtly  different  form. 

F.  T.  Poselger^^  gave  essentially  Euler's^°  first  proof. 

G.  L.  Dirichlet^°  derived  Fermat's  and  Wilson's  theorems  from  a  com- 
mon source.  Call  m  and  n  corresponding  numbers  if  each  is  less  than  the 
prime  p  and  if  mn=a  (mod  p),  where  a  is  a  fixed  integer  not  di\dsible  by  p 
(thus  generahzing  Euler's-^  associated  numbers).  Each  number  1,  2, . .  ., 
p  — 1  has  (5ne  and  but  one  corresponding  number.  If  a:"=a  (mod  p)  has  no 
integral  solution,  corresponding  numbers  are  distinct  and 

(p-l)!=a^-^)''2  (modp). 

But  if  A;  is  a  positive  integer  <p  such  that  ^'^=a  (mod  p),  the  second  root 
is  p  — A',  and  the  product  of  the  numbers  1, .  .  .,  p  — 1,  other  than  k  and  p—k, 
has  the  same  residue  as  a^^~^^^^,  whence 

(p-l)!=-a^-i^/2(j^Q^p) 

The  case  a  =  1  leads  to  Wilson's  theorem.    By  the  latter,  we  have 

a(p-i)/2=±i  (modp), 

"Math.  Abhandlungen,  Erste  Sammlung,  Altona,  1822,  67-93.  Some  of  the  results  were 
quoted  by  Gnmert,  Archiv  Math.  Phys.,  32,  1859,  115-8.  For  an  interpretation  in 
factoring  of  [m,  n],  see  Minetola'"  of  Ch.  X. 

"Annals  of  Phil.  (Mag.  Chem.   .    .    .),  new  series,  11,  1826,  81. 

>8Corresp.  Math.  Phys.  (ed.  Qu^telet),  3,  1827,  71. 

"Abhand.  Ak.  Wiss.  Berhn  (Math.),  1827,  21. 

"Jour,  fiir  Math.,  3,  1828,  390;  Werke,  1,  1889,  105.     Dirichlet,"  §34. 


Chap.  Ill]  FeRMAT's  AND   WiLSON's  THEOREMS.  67 

the  sign  being  +  or  —  according  as  A;^=a  (mod  p)  has  or  has  not  integral 
solutions    (Euler's   criterion).     Squaring,    we   obtain   Fermat's   theorem. 
Finally,  Dirichlet  rediscovered  the  proof  by  Ivory .^^     [Cf.  Moreau.-^^^] 
J.  Binet^^  also  rediscovered  the  proof  by  Ivory .^^ 
A.  Cauchy^^  gave  a  proof  analogous  to  that  by  Euler.^" 
An  anonymous  writer^^  proved  that  if  n  is  a  prime  the  binomial  coeffi- 
cient (n  —  l)k  has  the  residue  (  —  1)*'  modulo  n,  so  that 

{l+xr-'-l=-x+x^-  . .  .+x''-\     {l+x)\{l+xy-^-l\^x{x''-^-l), 

modulo  n.    Thus  Fermat's  theorem  follows  by  induction  on  x  as  in  the 
proof  by  Euler.^^ 

V.  Bouniakowsky^  gave  a  proof  of  Euler's  theorem  similar  to  that  by 
Laplace. ^^  If  a^h  is  divisible  by  a  prime  p,  aP^-'ifo^""'  is  divisible  by  p", 
provided  p>2  when  the  sign  is  plus.  Hence  if  p,  p' ,. . .  are  distinct  primes, 
o'±6'  is  divisible  by  iV  =  p"p"*'. . . ,  where  t  =  p''~^p"''~^ . . . ,  if  a=*=6  is  divisible 
by  pp' . . . ,  provided  the  p's  are  >  2  if  the  sign  is  plus.  Replace  a  by  its 
(p  — l)th  power  and  6  by  1  and  use  Fermat's  theorem;  we  see  that  a'  —  l  is 
divisible  by  N  if  e=(f}{N).  The  same  result  gives  a  generalization  of 
Wilson's  theorem^ 

U?)-l)!t^'*"+l=0(modp"). 

He  gave  {ibid.,  563-4)  Gauss'^"  proof  of  Wilson's  theorem. 

J.  A.  Grunert^^  used  the  known  fact  that,  if  0<k<p,  then  k,  2k,.  . ., 
(p  —  l)k  are  congruent  to  1,  2, . . .,  p  — 1  in  some  order  modulo  p,  a  prime, 
to  show  that  kx=l  (mod  p)  has  a  unique  root  x.  Wilson's  theorem  then 
follows  as  by  Gauss.  If  {ibid.,  p.  1095)  we  square  Gauss'  formula,^^  we  get 
Fermat's  theorem. 

Giovanni  de  Paoli^®  proved  Fermat's  and  Euler's  theorems.     In 

(x+iy=x^-\-i-{-pS,, 

where  p  is  a  prime,  S^  is  an  integer.     Change  x  to  x  —  1, . . . ,  2,  1  and  add  the 
resulting  equations.    Thus 

x-l 

x^-x^p^S,. 

Replace  x  by  a:"*,  divide  by  x"^  and  set  y  =  x^~'^.    Thus 

r  - 1  =  pXm,  X^=XSz"'/x''  =  integer. 

Replace  m  by   2m,...,   {p  —  l)m,  add  the  resulting  equations,  and  set 

Y„=l  +Xrn+X2m-\-  ■  •  .  +X(,^i),n.       Thus 

r"-l=p(y"-l)F^  =  p^X^7^. 

"Jour,  de  I'^cole  polytechnique,  20,  1831,  291  (read  1827).     Cauchy,  Comptes  Rendus  Paris, 

12,  1841,  813,  ascribed  the  proof  to  Binet. 
«Exer.  de  math.,  4,  1829,  221;  Oeuvres,  (2),  9,  263.     R^sumg  analyt.,  Turin,  1,  1833,  10. 
«Jour.  fiir  Math.,  6,  1830,  100-6. 

"M6m.  Ac.  Sc.  St.  P^tersbourg,  Sc.  Math.  Phys.  et  Nat.,  (6),  1,  1831, 139  (read  Apr.  1, 1829). 
"Kliigel's  Math.  Worterbuch,  5,  1831,  1076-9. 
"Opuscoli  Matematici  e  Fisici  di  Diversi  Autori,  Milano,  1,  1832,  262-272. 


68  History  of  the  Theory  of  Numbers.  [Chap,  hi 

Change  m  to  mp, . . . ,  T/zp""^.     Thus 

2/-^  - 1  =  p(^-^ - 1) F^,  =  p^X^y^F^,, 

Hence  x^^^  — 1  is  divisible  by  N  for  iV"  =  p"  and  so  for  any  N. 

For  X  odd,  x^  —  1  is  divisible  by  8,  and  x*'"  —  1  by  2(x^*"— 1).  As  above, 
he  found  that  x' - 1  is  divisible  by  2*  for  t  =  m-2'-^,  i>  2.  Thus,  if  iV  =  2'n, 
n  odd,  X*— 1  is  divisible  by  N  for  A:  =  2*"^0(n). 

A.  L.  Crelle^^  employed  a  fixed  quadratic  non-residue  v  of  the  prime  p, 
and  set  j^=ry,  vf=Vj  (mod  p).     By  multipUcation  of 

ip-jf=rj,  vf=v^  (mod  p)  (i  =  1,  •  •  •  »^^) 

and  use  of  v^^~'^^'^=  —  \,  we  get 

-](p-l)!f2=nr,v,=  (p-l)!  (modp). 

F.  Minding^*  proved  the  generaUzed  Wilson  theorem.  Let  P  be  the 
product  of  the  tt  integers  a,  /3, . . .,  <A  and  relatively  prime  to  A.  Let 
A  =  2''p"'g"r* . , . ,  where  p,  q,r,. . .  are  distinct  odd  primes,  and  m>  0.  Take 
a  quadratic  non-residue  t  of  p  and  determine  a  so  that  a=t  (mod  p),  o=l 
(mod  2qr. . .).  Then  a  is  an  odd  quadratic  non-residue  of  A.  Let  ax=a 
(mod  A).  For  ^9^x,  a,  let  i3?/=a  (mod  A).  Then  y^^a,  x,  jS.  In  this  way 
the  TT  numbers  a,  j8, . . .  can  be  paired  so  that  the  product  of  the  two  in  any 
pair  is  =a  (mod  A),  whence  P=a''^^  (mod  A). 

First,  let  A  =  2''p^  Then  a'=-l  (mod  p"^),  s  =  p"*-^(p-l)/2,  whence 
P=-l  (mod^)  if  M  =  Oor  L     But,  if  m>1, 

a^={-iy     =l(modp'"),         a^=/    =l(mod2''),         P=  +  l(modA). 

Next,  let  m>l,  n>l,  in  A.  Raising  the  above  a*=— 1  to  the  power 
2''"V~^(?  — !)•  •  •>  we  get  a'^^=-\-l  (mod  p").  A  like  congruence  holds 
moduli  g",  r^ . . .,  and  2",  whence  P=4-l  (mod  A). 

Finally,  let  A  =  2",  /x>L  Then  a=— 1  is  a  quadratic  non-residue  of 
2"  and,  as  above,  P=  ( —  1)^  (mod  A),l  =  2""^.  The  proof  of  Fermat's  theorem 
due  to  Ivory^^  is  given  by  Minding  on  p.  32. 

J.  A.  Grunert*^  gave  Horner's^^  proof  of  Euler's  theorem,  attributing 
the  case  of  a  prime  to  Dirichlet  instead  of  Ivory .^^  A  part  of  the  generalized 
Wilson  theorem  was  proved  as  follows:  Let  ri,. . .,  r,  denote  the  positive 
integers  <p  and  prime  to  p.     Let  a  be  prime  to  p.     In  the  table 


riflVg,  r2a^rg, . . . ,  rqa\ 


«'Abh.  Ak.  Wiss.  Berlin  (Math.),  1832,  66.     Reprinted." 

"Anfangsgriinde  der  Hoheren  Arith.,  1832,  75-78. 

"Math.  Worterbuch,  1831,  pp.  1072-3;  Jour,  fiir  Math.  8,  1832,  187. 


Chap.  Ill]  FeKMAT's  AND   WiLSON's  THEOREMS.  69 

a  single  term  of  a  row  is  =1  (mod  p).  If  this  term  be  TkO^rk,  replace  it 
by  (p—rk)a\^-l.  Next,  if  r^^a^^  =f1,  r„aVi=±l,  then  rk-^ri=p  and 
one  of  the  r„  is  replaced  by  p—r^.  Hence  we  may  separate  riO, . . .,  r^a 
into  q/2  pairs  such  that  the  product  of  the  two  of  a  pair  is  =  ±  1  (mod  p) . 
Taking  a  =  1,  we  get  ri . .  .rg=  ±  1  (mod  p).  The  sign  was  determined  only 
for  the  case  p  a  prime  (by  Gauss'  method). 

A.  Cauchy^*^  derived  Wilson's  theorem  from  (1),  page  62  above. 
*Caraffa^^  gave  a  proof  of  Fermat's  theorem. 

E.  Midy^^  gave  Ivory's^^  proof  of  Fermat's  theorem, 

W.  G.  Horner^^  gave  Euler's^^  proof  of  his  theorem. 

G.  Libri^^  reproduced  Euler's  proof^^  without  a  reference. 

Sylvester^^  gave  the  generalized  Wilson  theorem  in  the  incomplete  form 
that  the  residue  is  ±  1. 

Th.  Schonemann^^  proved  by  use  of  symmetric  functions  of  the  roots 
that  if  s"+6i2;""^+ ...  =0  is  the  equation  for  the  pth  powers  of  the  roots 
of  x^+aiX^~^-{- ...  =0,  where  the  a's  are  integers  and  p  is  a  prime,  then 
hi=af  (mod  p).  If  the  latter  equation  is  (x  — 1)''  =  0,  the  former  is 
2'*-(nP+pQ)2"-^+.  ..=0,  and  yet  is  evidently  (2:-l)'*  =  0.  Hence 
71^=71  (mod  p). 

W.  Brennecke^^  elaborated  one  of  Gauss'^^  suggestions  for  a  proof  of  the 
generalized  Wilson  theorem.  For  a>2,  x^=l  (mod  2°)  has  exactly  four 
incongruent  roots,  =•=  1,  ='=  (l+2"~^),  since  one  of  the  factors  x=^l,  of  differ- 
ence 2,  must  be  divisible  by  2  and  the  other  by  2""^.  For  p  an  odd  prime, 
let  ri, . . .,  r^  be  the  positive  integers  <p"  and  prime  to  p",  taking  ri  =  l, 
r^  =  p"— 1.  For  2^s^)u  — 1,  the  root  x  of  r^x^l  (mod  p")  is  distinct  from 
Ti,  r^,  r^.  Thus  7-2, ... ,  r^_i  may  be  paired  so  that  the  product  of  the  two 
of  a  pair  is  =1  (mod  p").  Hence  ri . .  .r^=  —  1  (mod  p").  This  holds  also 
for  modulus  2p".     For  a  >  2, 

(2-i-i)(2»-i+l)=-l,  ri. .  .r^=-\-l  (mod  2"). 

Finally,  let  N=p''M,  where  M  is  divisible  by  an  odd  prime,  but  not  by  p. 
Then  m=(f>{M)  is  even.    The  integers  <N  and  prime  to  p  are 

rj>rj+p'^,  rj+2p%.. .,  r,.+(M-l)p»  (i  =  l,. .  .,  m). 

For  a  fixed  j,  we  obtain  m  integers  <iV  and  prime  to  N.  Hence  if  \N\ 
denotes  the  product  of  all  the  integers  <  N  and  prime  to  iV, 

\N\^{n. .  .r^)'^=l  (mod  p"). 
ForiV  =  pV.-.,  \N\=1  (mod  O,--,  whence  jiVt^l  (modiV). 


"R6suin6  analyt.,  Turin,  1,  1833,  35. 

"Elem.  di  mat.  commentati  da  Volpicelli,  Rome,  1836,  I,  89. 

"De  quelques  propri^t^s  des  nombres,  Nantes,  1836. 

"London  and  Edinb.  Phil.  Mag.,  11,  1837,  456. 

"M6m.  divers  savants  ac.  sc.  Institut  de  France  (math.),  5,  1838,  19. 

"Phil.  Mag.,  13,  1838,  454  (14,  1839,  47);  Coll.  Math.  Papers,  1,  1904,  39. 

"Jour,  fiir  Math.,  19, 1839,  290;  31, 1846,  288.    Cf.  J.  J.  Sylvester,  Phil.  Mag.,  (4),  18, 1859,  281. 

"Jour,  fiir  Math.,  19,  1839,  319. 


70 


History  of  the  Theory  of  Numbers. 


[Chap.  HI 


A.  L.  Crelle^^  proved  the  generalized  Wilson  theorem.  By  pairing  each 
root  <T  of  x-=l  (mod  s)  with  the  root  s—a,  and  each  integer  a<s,  prime  to 
s  and  not  a  root,  with  its  associated  number  a',  where  aa'=\  (mod  s),  we 
see  that  the  product  of  all  the  integers  <s  and  prime  to  s  is  =  +  1  or  —1 
(mod  s)  according  as  the  number  n  of  pairs  of  roots  o-,  s—o-  is  even  or  odd. 
To  find  n,  express  s  in  every  way  as  a  product  of  two  factors  u,  v,  whose 
g.  c.  d.  is  1  or  2;  in  the  respective  cases,  each  factor  pair  gives  a  single  root 
(T  or  two  roots.  Treating  four  subcases  at  length  it  is  shown  that  the  num- 
ber of  factor  pairs  is  2^"  in  each  case,  where  k  is  the  number  of  distinct  odd 
primes  dividing  s ;  and  then  that  n  is  odd  if  s  =  4,  p"*  or  2p",  but  even  if  n 
is  not  of  one  of  these  three  forms. 

A.  Cauchy^^"  proved  Fermat's  theorem  as  had  Leibniz.'* 

V^^  (S.  Earnshaw?)  proved  Wilson's  theorem  by  Lagrange's  method  and 
noted  that,  if  Sr  is  the  sum  of  the  products  of  the  roots  of  AqX"'+Aix"'~^-\-  . .  . 
=  0  (mod  p)  taken  r  at  a  time,  then  AoSi  —  {  —  iyAi  is  divisible  by  p. 

Paolo  Gorini^"  proved  Euler's  theorem  6'=1  (mod  A),  where  t=(f>(A), 
by  arranging  in  order  of  magnitude  the  integers  (A)  p',  p", .  . . ,  p^'^  which 
are  less  than  A  and  prime  to  A.  After  omitting  the  numbers  in  (A)  which 
are  di\'isible  by  h,  we  obtain  a  set  (B)  q',.  .  .,  q^^\  Let  5^"^  be  the  least  of 
the  latter  which  when  increased  by  A  gives  a  multiple  of  h : 


(C) 


g(-)+A  =  p^'^^6. 


V 


(a-1) 


i(0 


The  numbers*  (A)  coincide  with  those  in  sets  (B)  and  (D) : 

(D)  p%p"b,...,p^-%. 

Hence  by  multiplication  and  cancellation  of  p', 

(F)  q'...qH''-^  =  p^''\..p^ 

To  each  number  (B)  add  the  least  multiple  of  A  which  gives  a  sum  divisible 
by  b,  say  (G)  q'+g'A,...,  q^^+g^^A.  The  least  of  these  is  q^''^-\-A  = 
p^^^h,  by  (C).  Each  number  (G)  is  <6A  and  all  are  distinct.  The  quo- 
tients obtained  by  di\'iding  the  numbers  (G)  by  h  are  prime  to  A  and  hence 
included  among  the  p^^V-j  P^'\  whose  number  is  t—a-\-l=l,  so  that 
each  arises  as  a  quotient.     Hence 

(H)  n(g«+^«A)=PA+g'. .  .g^'^  =  p' 

Combine  this  with  (F)  to  eliminate  the  p's. 


pW 


-a+l 


q'. .  .g«6''-W-"+i  =  PA+5'.  .  .q 


(0 


■Q 


We  get 


6'-l  =  QA. 


"Jour,  fur  Math.,  20,  1840,  29-56.     Abstract  in  Bericht  Akad.  Wiss.  Berlin,  1839,  133-5. 

»8aM6m.  Ac.  Sc.  Paris,  17,  1840,  436;  Oeuvres,  (1),  3,  163-4. 

"Cambr.  Math.  Jour.,  2,  1841,  79-81. 

"Annali  di  Fisica,  Chimica  e  Mat.  (ed.,  G.  A.  Majocchi),  Milano,  1,  1841,  255-7. 

*To  follow  the  author's  steps,  take  A  =  15,  6  =  2,  whence  « =  8,  i  =  4,  (A)  1,  2,  4,  7,  8,  11,  13,  14; 

(B)  1,  7, 11, 13;  (C)  1  +  15  =  8-2,  ?(<>)  =8,  a  =  5;  (D)  2,  4,  8,  14;  (F)  171113  2«  =  8111314; 

(G)  1  +  15,7  +  15,  11+15;13  +  15,  each  g  =  l;  the  quotients  of  the  latter  by  2  are  8,  11, 13, 

14,  viz.,  last  four  in  (A);  (H)  P.15  +  1.7.11.13=8.11.13.14.2«;  the  second  member  ifl 

1-71113  2"  by  (F).     Hence  171113  (28-l)  =  15P. 


Chap.  Ill]  FeRMAT's  AND   WiLSON's  THEOREMS.  71 

E.  Lionnet^^  proved  that,  if  p  is  an  odd  prime,  the  sum  of  the  mth  powers 
of  1,. . .,  p  —  1  is  divisible  by  p  for  0<m<p  —  l.  Hence  the  sum  P^  of 
the  products  of  1, ...  ,p  —  1  taken  w  at  a  time  is  divisible  by  p  [Lagrange^^]. 
Since 

(l  +  l)(l+2)...(H-p-l)  =  l+Pi+P2+...H-Pp-2+(p-l)!, 

l  +  (p  — 1)!  is  divisible  by  p. 

E.  Catalan^^  gave  the  proofs  by  Ivory^^  and  Horner.^^  C.  F.  Arndt^^  gave 
Horner's  proof;  and  proved  the  generalized  Wilson  theorem  by  associated 
numbers.    O.  Terquem^^  gave  the  proofs  by  Gauss^^  and  Dirichlet.^° 

A.  L.  Crelle^^  republished  his  proof'*''  of  Wilson's  theorem,  as  well  as 
that  by  Gauss^°  and  Dirichlet.^°  Crelle^^  gave  two  proofs  of  the  generalized 
Wilson  theorem,  essentially  that  by  Minding^^  and  that  given  by  himself.^^ 
If  fj,  is  the  number  of  distinct  odd  prime  factors  of  z,  and  2^"  is  the  highest 
power  of  2  dividing  z,  and  r  is  a  quadratic  residue  of  z,  then  (p.  150)  the 
number  n  of  pairs  of  roots  ±x  of  x^=r  (mod  z)  is  2""^  if  m  =  0  or  1,  2"  if 
m  =  2,  2"'^^  if  w>2.  Using  the  fact  (p.  122)  that  the  quadratic  residues  of 
z  are  the  e=(f){z)/(2n)  roots  of  r*=l  (mod  z),  it  is  shown  (p.  173)  that,  if  v 
is  any  integer  prime  to  z,  y*'^^^'''*=l  (mod  z),  "a,  perfection  of  the  Euler- 
Fermat  theorem." 

L.  Poinsot^^  failed  in  his  attempt  to  prove  the  generalized  Wilson 
theorem.  He  began  as  had  Crelle.^^  But  he  stated  incorrectly  that  the 
number  n  of  pairs  of  roots  =^x  of  x^^l  (mod  s)  equals  the  number  v  of 
ways  of  expressing  s  as  a  product  of  two  factors  P,  Q  whose  g.  c.  d.  is  1  or  2. 
For  each  pair  =^x,  it  is  implied  that  x—1  and  x+1  uniquely  determine  P,  Q. 
For  s  =  24,  n  =  y  =  4;  but  for  the  root  x  =  7  (or  for  x  =  17),  a:  ±  1  yield 
P,  Q  =  3,  8,  and  6, 4.  To  correct  another  error  by  Poinsot,  let  n  be  the  number 
of  distinct  odd  prime  factors  of  s  and  let  2"*  be  the  highest  power  of  2  dividing 
s;  then  y  =  2''-^  2",  3-2''-^  or  2"+^  according  as  w  =  0,  1,  2,  or  ^3,  whereas 
[Crelle*^^]  n  =  2''-\  2''-\  2^  2"+^  No  difficulty  is  met  (pp.  53-5)  in  case  the 
modulus  is  a  power  of  a  prime.  He  noted  (p.  33)  that  if  Vi,  r2, . . .  are  the 
integers  <N  and  prime  to  N,  and  tt  is  their  product,  they  are  congruent 
modulo  N  to  tt/ti,  Tr/ra, ...,  whence  T=Tr''~^  (mod  N),  where  v=(f){N). 
Thus,  by  Euler's  theorem,  7r^= 1.  This  does  not  imply  that  7r=  =*=  1  as  cited 
by  Aubry,!"  pp.  30O-I. 

Poinsot  (p.  51)  proved  Euler's  theorem  by  considering  a  regular  polygon 
of  N  sides.  Let  x  be  prime  to  N  and  <  N.  Join  any  vertex  with  the  xth  ver- 
tex following  it,  the  new  vertex  with  the  a:th  vertex  following  it,  etc.,  thus 
defining  a  regular  (star)  polygon  of  N  sides.     With  the  same  x,  derive 

"Nouv.  Ann.  Math.,  1,  1842,  175-6. 
«/6td.,  462-4. 

«Archiv  Math.  Phys.,  2,  1842,  7,  22,  23. 
"Nouv.  Ann.  Math.,  2,  1843,  193;  4,  1845,  379. 
«Jour.  fur  Math.,  28,  1844,  176-8. 
«8/bid.,  29,  1845,  103-176. 

«^Jour.  de  Math.,  10, 1845,  25-30.    German  exposition  by  J.  A.  Grunert,  Archiv  Math.  Phys., 
7,  1846,  168,  367. 


72  History  of  the  Theory  of  Numbers.  [Chap.  hi 

similarly  a  new  A^-gon,  etc.,  until  the  initial  polygon  is  reached.®^  The 
number  )U  of  distinct  polygons  thus  obtained  is  seen  to  be  a  divisor  of  <t>{N), 
the  number  of  polygons  corresponding  to  the  various  a:'s.  If  in  the  initial 
polygon  we  take  the  x^th  vertex  following  any  one,  etc.,  we  obtain  the 
initial  polygon.  Hence  of  and  thus  also  x"^^^  has  the  remainder  unity  when 
divided  by  N.  [When  completed  this  proof  differs  only  shghtly  from  that 
by  Euler."] 

E.  Prouhet^^  modified  Poinsot's  method  and  obtained  a  correct  proof 
of  the  generalized  Wilson  theorem.  Let  r  be  the  number  of  roots  of  x^=l 
(mod  N),  and  w  the  number  of  ways  of  expressing  iV  as  a  product  of  two 
relatively  prime  factors.  If  AT  =  2'"pi" . . .  p/",  where  the  p's  are  distinct 
odd  primes,  evidently  w;  =  2*'  if  m>0,  1^  =  2""^  if  m  =  0.  By  considering 
divisors  of  a:  =*=  1,  it  is  proved  that  r  =  2u'  if  ttz  =  0  or  2,  r  =  w;  if  w  =  1,  r  =  4iy  if 
m>2.  Hence  r  =  2"  if  m  =  0  or  1,  2"+^  if  m  =  2,  2"+^  if  m> 2.  By  Crelle,^^ 
the  product  P  of  the  integers  <A''  and  prime  to  N  is  =(  —  1)''''^  (mod  N). 
Thus  for  jLt>0,  P=  +  l  unless  m  =  0  or  l,/i  =  l,  viz.,  N  =  p^  or  2p';  while,  for 
^  =  0,  N  =  2"',  m>2,  we  have  r  =  4,  P=-\-l. 

Friderico  Arndt'^°  elaborated  Gauss'^^  second  suggestion  for  a  proof  of 
the  generalized  Wilson  theorem.  Let  gf  be  a  primitive  root  of  the  modulus 
p"  or  2p",  where  p  is  an  odd  prime.  Set  y=0(p").  Then  g,  g^,. . .,  g"  are 
congruent  to  the  numbers  less  than  the  modulus  and  prime  to  it.  If  P  is 
the  product  of  the  latter,  P^g''-'^^^'^  But  g"^=-l.  Hence  P=-l. 
Next,  if  n>2,  the  product  of  the  incongruent  numbers  belonging  to  an 
exponent  2"""*  is  =1  (mod  2").  Next,  consider  the  modulus  M  =  AB, 
where  A  and  B  are  relatively  prime.  The  positive  integers  <  M  and  prime 
to  M  are  congruent  modulo  M  to  Ayi-\-Bxj,  where  the  0:^  are  <A  and  prime 
to  A,  the  yi  are  <B  and  prime  to  B.    But,  if  a=0(A), 

a 

7ri  =  'n.{Ayi+BXj)=B''xi.  .  .Xa=Xi. .  .x^  (mod  A), 

3  =  1 

P=riTr2.  ..^{x^..  .xJ'^^^Hniod  A). 

By  resolving  M  into  a  product  of  powers  of  primes  and  applying  the  above 
results,  we  determine  the  sign  in  P=±l  (mod  M). 

J.  A.  Grunert^^  proved  that  if  a  prime  n+l>2  divides  no  one  of  the 
integers  ai, . . .,  a„,  nor  any  of  their  differences,  it  divides  aia2.  .  .a„+l,  and 
stated  that  this  result  is  much  more  general  than  Wilson's  theorem  (the 
case  aj=j).  But  the  generalization  is  only  superficial  since  ai,. . .,  a„  are 
congruent  modulo  n+1  to  1,...,  n  in  some  order.  His  proof  employed 
Fermat's  theorem  and  certain  complex  equations  involving  products  of 
differences  of  n  numbers  and  sums  of  products  of  n  numbers  taken  m  at 
a  time. 

J.  F.  Heather^^  gave  without  reference  the  first  results  of  Grunert.^^ 

osCf.  P.  Bachmann,  Die  Elemente  der  Zahlentheorie,  1892,  19-23. 

«»Nouv.  Ann.  Math.,  4,  1845,  273-8. 

"Jour,  fvir  Math.,  31,  1846,  329-332. 

"Archiv  Math.  Phys.,  10,  1847,  312. 

"The  Mathematician,  London,  2,  1847,  296. 


Chap.  Ill]  FeRMAT's  AND   WiLSON's  THEOREMS.  73 

A.  Lista'^^  gave  Lagrange's  proof  of  Wilson's  theorem. 

V.  Bouniakowsky'^^  gave  Euler's^^  proof. 

P.  L.  Tchebychef^^  concluded  from  Fermat's  theorem  that 

(a:-l)(x-2). .  .(x-p+l)-a;^-i+l=0  (mod  p) 

is  an  identity  if  p  is  a  prime.  Hence  if  Sj  is  the  sum  of  the  products  of  1, . . . , 
p  —  1  taken  j  at  a  time,  Sj=0  (i<p  — 1),  Sp_i=  — 1  (mod  p),  the  last  being 
Wilson's  theorem. 

Sir  F.  PoUock'^^  gave  an  incomplete  statement  and  proof  of  the  general- 
ized Wilson  theorem  by  use  of  associated  numbers.  Likewise  futile  was 
his  attempt  to  extend  Dirichlet's^"  method  [not  cited]  of  association  into 
pairs  with  the  product = a  (mod  m)  to  the  case  of  a  composite  m. 

E.  Desmarest"  gave  Euler's^^  proof  of  Fermat's  theorem. 

0.  Schlomilch^^''  considered  the  quotient 

{„p_  (»)  („_i)p+  («)  („_2)p-  . . .  f/n!. 

J.  J.  Sylvester'^*  took  x  =  l,  2,. . .,  p  —  1  in  turn  in 

{x-l){x-2) . . . (x-p+l)  =x^-'+A,x^-^+  . . .  +A,_i, 

where  p  is  a  prime.  Since  x^~^=l  (mod  p),  there  result  p  —  1  congruences 
linear  and  homogeneous  in  Ai, . . . ,  Ap_2,  Ap-i+1,  the  determinant  of  whose 
coefficients  is  the  product  of  the  differences  of  1,  2, . . . ,  p  — 1  and  hence  not 
divisible  by  p.  Thus  Ai=0,...,  Ap_i+1=0,  the  last  giving  Wilson's 
theorem. 

W.  Brennecke'^^  proved  Euler's  theorem  by  the  methods  of  Horner^^ 
and  Laplace,  ^^  noting  that 

{a^-y=l  (mod  p^),  (a^-i)^'=l  (mod  p^), .... 

He  gave  the  proof  by  Tchebychef ^^  and  his  own  proof." 

J.  T.  Graves^"  employed  nx=n+l  (mod  p),  where  p  is  a  prime,  and 
stated  that,  for  n  =  l,...,  p  —  1,  then  x=2,...,  p  in  some  order.  Also 
x=p  ior  n  =  p  —  l.    Hence  2-3. .  .(p  — l)=p  — 1  (mod  p). 

H.  Durege^^  obtained  (2)  for  a  =  x  and  Grunert's^^  results  on  the  series 
[m,  n]  by  use  of  partial  fractions  for  the  reciprocal  o(  x(x  —  l) .  .  .{x  —  n). 

E.  Lottner^^  employed  for  the  same  purpose  infinite  trigonometric  and 
algebraic  series,  obtaining  recursion  formulae  for  the  coefficients. 

"Periodico  Mensual  Cienciaa  Mat.  y  Fis.,  Cadiz,  1,  1848,  63. 

T*BuU.  Ac.  Sc.  St.  P6tersbourg,  6,  1848,  205. 

"Theorie  der  Congruenzen,  1849  (Russian);  in  German,  1889,  §19.     Same  proof  by  J.  A. 

Serret,  Cours  d'algebre  sup^riem-e,  ed.  2,  1854,  324. 
^«Proc.  Roy.  Soc.  London,  5,  1851,  664. 
"TMorie  des  nombres,  Paris,  1852,  223-5. 
""Jour,  fur  Math.,  44,  1852,  348. 

"Cambridge  and  Dublin  Math.  Jour.,  9,  1854,  84;  Coll.  Math.  Papers,  2,  1908,  10. 
"Einige  Satze  aus  den  Anfangsgriinden  der  Zahlenlehre,  Progr.  Realschule  Posen,  1855. 
soBritish  Assoc.  Report,  1856,  1-3. 
"Archiv  Math.  Phys.,  30,  1858,  163-6. 
"/bid.,  32,  1859,  111-5. 


74  History  of  the  Theory  of  Numbers.  [Chap,  hi 

J.  Toeplitz^  gave  Lagrange's  proof  of  Wilson's  theorem. 

M.  A.  Stern^  made  use  of  the  series  for  log  (1  —  x)  to  show  that 

1+x+xH.  . .  =-^  =  e'+*'/2+^/3+...^ 
l—x 

Multiply  together  the  series  for  e',  e'*^^,  etc.     By  the  coeflBcient  of  x^. 


p!  P'  (p-2)!'  ••• 

Take  p  a  prime.     No  term  of  s  has  a  factor  p  in  the  denominator.     Hence 

(1-s)  •  (p-l)!  =  ^-tfcli^  =  integer. 
P 

V.  A.  Lebesgue^^  obtained  Wilson's  theorem  by  taking  x  =  p  —  l  in 

p  X  Hk+l) . . .  {k-\-p-2)  =x(x+l) . . .  (x+p-1). 
k=i 

If  P  is  a  composite  number  ?^4,  (P  — 1)!  is  di\'isible  by  P.  He  (p.  74) 
attributed  Ivory's^^  proof  of  Fermat's  theorem  to  Gauss,  without  reference. 

G.  L.  Dirichlet^^  gave  Horner's"  and  Euler's^^  proof  of  Euler's  theorem 
and  derived  it  from  Fermat's  by  the  method  of  powering.  His  proof  (§38) 
of  the  generalized  Wilson  theorem  is  by  associated  numbers,  but  is  some- 
what simpler  than  the  analogous  proofs. 

Jean  Plana"  used  the  method  of  powering.  Let  N  =  p^pi' ....  For  M 
prime  to  N,  M^~'^  =  1  +pQ.    Hence 

Thus  for  e  =  (p{p^pi'),  M"  —  !  is  divisible  bj'  p''  and  p/'  and  hence  by  their 
product,  etc.  Plana  gave  also  a  modification  of  Lagrange's  proof  of  Wilson's 
theorem  by  use  of  (2) ;  take  x=a  =  p  —  l,  subtract  the  expansion  of  (1  —  1)""^ 
and  write  the  resulting  series  in  reverse  order: 

(p-l)!+l  =  (^2^)(2^-^-l)-(V)(3^-^-l)+... 

-(^:D](p-2)^-^-lt  +  1(p-i)''-'-if- 
H.  F.  Talbot^^  gave  Euler's^^  proof  of  Fermat's  theorem. 
J.  Blissard^^"  proved  the  last  statement  of  Euler.^ 
C.  Sardi^^  gave  Lagrange's  proof  of  Wilson's  theorem. 
P.  A.  Fontebasso^*^  proved  (2)  for  x  =  a  by  finding  the  first  term  of  the 
ath  order  of  differences  ofy'',{y+hy,{y-\-2hy,. . .  and  then  setting y  =  0,h  =  l. 

^'Archiv  Math.  Phys.,  32,  1859,  104. 

"Lehrbuch  der  Algebraischen  Analysis,  Leipzig,  1860,  391. 

"Introd.  thdorie  des  nombres,  Paris,  1862,  80,  17. 

"Zahlentheorie  (ed.  Dedekind),  §§19,  20,  127,  1863;  ed.  2,  1871;  ed.  3,  1879,  ed.  4,  1894. 

8'Mem.  Acad.  Turin,  (2),  20,  1863,  148-150. 

"Trans.  Roy.  Soc.  Edinburgh,  23,  1864,  45-52. 

ss^'Math.  Quest.  Educ.  Times,  6,  1866,  26-7. 

"Giomale  di  Mat.,  5,  1867,  371-6. 

•"Saggio  di  una  introd.  arit.  trascendente,  Treviso,  1867,  77-81. 


Chap.  Ill]  FeRMAT's  AND  WilSON's  THEOREMS.  75 

C.  A.  Laisant  and  E.  Beaujeux^^  used  the  period  ai . .  .a„  of  the  periodic 
fraction  to  base  B  for  the  irreducible  fraction  pi/q,  where  q  is  prime  to  B. 
li  P2,. . .,  Pn  are  the  successive  remainders, 

Bpi  =  aiq+p2,  Bp2  =  a2q+P3,.  .  .,  Bpr,  =  anq+Pi. 

Starting  with  the  second  equation,  we  obtain  the  period  a2.  .  .a„ai  for  P2/q- 
Similarly  for  ps/q,.  .  .,  Pn/q-  Thus  the  f=(p{q)  irreducible  fractions  with 
denominator  q  separate  into  sets  of  n  each.  Hence /=A;n.  Since  5'*=!, 
B^=l  (modg). 

L.  Ottinger^-  employed  differential  calculus  to  show  that,  in 

P={a+d){a+2d).. .  \a+ip-l)d\  =aP-i+Ci^~V-2d+C2^-V-3d2_|__  ^ 

3=1  q~ri- 

Cr  being  the  sum  of  the  products  of  1,  2, . . . ,  A;  taken  r  at  a  time.  Hence,  if 
p  is  a  prime,  C?~^  (r  =  1, . . .,  p  — 2)  is  divisible  by  p,  and 

P=aP-i+c^-2d.  ..{p-l)d  (mod  p). 

For  a  =  d  =  l,  this  gives  0=l  +  (p  — 1)!  (mod  p). 

H.  Anton^^  gave  Gauss' ^^  proof  of  Wilson's  theorem. 

J.  Petersen^^  proved  Wilson's  theorem  by  dividing  the  circumference  of 
a  circle  into  p  equal  parts,  where  p  is  a  prime,  and  marking  the  points 
1, .  . .,  p.  Designate  by  12.  .  .p  the  polygon  obtained  by  joining  1  with  2, 
2  with  3,.  .  .,  p  with  1.  Rearranging  these  numbers  we  obtain  new  poly- 
gons, not  all  convex.  While  there  are  p!  rearrangements,  each  polygon  can 
be  designated  in  2p  ways  [beginning  with  any  one  of  the  p  numbers  as  first 
and  reading  forward  or  backward],  so  that  we  get  (p  — 1)!/2  figures.  Of 
these  ^(p  — 1)  are  regular.  The  others  are  congruent  in  sets  of  p,  since  by 
rotation  any  one  of  them  assumes  p  positions.  Hence  p  divides  (p  — 1)!/2 
-(p-l)/2  and  hence  (p-2)!-l.     Cf.  Cayley^o^. 

To  prove  Fermat's  theorem,  take  p  elements  from  q  with  repetitions  in 
all  ways,  that  is,  in  q^  ways.  The  q  sets  with  elements  all  alike  are  not 
changed  by  a  cyclic  permutation  of  the  elements,  while  the  remaining  q^  —  q 
sets  are  permuted  in  sets  of  p.  Hence  p  divides  q^—q.  [Cf.  Perott,^^® 
Bricard.i"] 

F.  Unferdinger^^  proved  by  use  of  series  of  exponentials  that 

2''-(';')(^-ir+(2)(^-2r-...+(-ir(^)(2-mr 

'  "Nouv.  Ann.  Math.,  (2),  7,  1868,  292-3. 
"Archiv  Math.  Phys.,  48,  1868,  159-185. 
''Ibid.,  49,  1869,  297-8. 

"Tidsskrift  for  Mathematik,  (3),  2,  1872,  64-65  (Danish). 
"^Sitzungsberichte  Ak.  Wisa.  Wien,  67,  1873,  II,  363. 


76  History  of  the  Theory  of  Numbers.  [Chap,  hi 

is  zero  ii  n<m,  but,  if  ti^tt^,  equals 

where 

For  n  =  m,  the  initial  sum  equals  Em  =  rn\. 

P.  Mansion^®  noted  that  Euler's  theorem  may  be  identified  with  a 
property  of  periodic  fractions  [cf.  Laisant^^].  Let  N  be  prime  to  R.  Taking 
R  as  the  base  of  a  scale  of  notation,  divide  100.  .  .by  A^  and  let  gi . .  .g„  be 
the  repetend.  Then  (72"  — l)/iV  =  Q'i. .  .5„.  Unless  the  n  remainders  r^ 
exhaust  the  integers  <N  and  prime  to  A^,  we  divide  r/  00. .  .by  A^,  where 
r/  is  one  of  the  integers  distinct  from  the  r,-,  and  obtain  n  new  remainders  r/. 
In  this  way  it  is  seen  that  n  divides  (p{N),  so  that  N  divides  R'^'-^  —  l.  [At 
bottom  this  is  Euler's^*  proof.] 

P.  Mansion^^  reproduced  this  proof,  made  historical  remarks  on  the 
theorem  and  indicated  an  error  by  Poinsot.^^ 

Franz  Jorcke^^  reproduced  Euler's^^  proof  of  Wilson's  theorem. 

G.  L.  P.  V.  Schaewen^^  proved  (2)  with  a  changed  to  —p,  by  expanding 
the  binomials. 

Chr.  ZeUer^o"  proved  that,  for  n  ?^  4, 

is  divisible  by  n  unless  n  is  a  prime  such  that  n  —  1  divides  x,  in  which  case 
the  expression  is  =  —  1  (mod  n) . 

A.  Cayley^°^  proved  Wilson's  theorem  as  had  Petersen.^^ 
E.  Schering^^^  took  a  prime  to  m  =  2'pi''\  .  .p'",  where  the  p's  are  dis- 
tinct odd  primes  and  proved  that  x^=a  (mod  m)  has  roots  if  and  only  if 
a  is  a  quadratic  residue  of  each  Pi  and  if  a  =  l  (mod  4)  when  7r  =  2,  a=l 
(mod  8)  when  7r>2,  and  then  has  \l/{m)  roots,  where  \p{m)=2'',  2""^^  or 
2"''"^,  according  as  7r<2, 7r  =  2,  or  7r>2.  Let  a  be  a  fixed  quadratic  residue 
of  m  and  denote  the  roots  by  ^aj  (j  =  l,.  .  .,  4'/2).  Set  a.- =m—aj.  The 
<}>{m)—\f/{ni)  integers  <m  and  prime  to  m,  other  than  the  ay,  a/,  may  be 
denoted  by  aj,  a/  (i  =  |iA+lj-  •  •?  20)j  where  aja'j=a  (mod  m).  From  the 
latter  and  —aja/=a  (i  =  l, . . .,  ^/2),  we  obtain,  by  multiplication, 
^i^(m)  =  (_j)}*(m)^^     .r^  (mod  w), 

••Messenger  Math.,  5,  1876,  33  (140);  Xouv.  Corresp.  Math.,  4,  1878,  72-6. 

"Th6orie  des  nombres,  1878,  Gand  (tract). 

•*Uber  Zahlenkongruenzen,  Progr.  Fraustadt,  1878,  p.  31. 

"Die  Binomial  Coefficienten,  Progr.  Saarbriicken,  1881,  p.  20. 
looBull.  des  sc.  math,  astr.,  (2),  5,  1881,  211^. 

"'Messenger  of  Math.,  12,  1882-3,  41;  CoU.  Math.  Papers,  12,  p.  45. 
iwActa  Math.,  1,  1882,  153-170;  Werke,  2,  1909,  69-86. 


Chap.  Ill]  FeRMAt's  AND   WiLSON's  THEOREMS.  77 

where  the  Vj  are  the  integers  <m  and  prime  to  m.  Taking  a=l,  we  have 
the  generahzed  Wilson  theorem.  Applying  a  like  argument  when  a  is  a 
quadratic  non-residue  of  m  [Minding^^],  we  get 

^J^(m)^^^_  .r^=(_l)^'^('»)  (mod  m). 

This  investigation  is  a  generaUzation  of  that  by  Dirichlet.*" 

E.  Lucas^"^  wrote  Xp  for  x{x-\-l) . .  .{x-\-p  —  l),  and  F^   for  the  sum  of 
the  products  of  1, . .  . ,  p  taken  g  at  a  time.    Thus 

^  ~rA  p-i^      ~r  •  •  •  ~rA  p-i^  —  Xp. 

Replacing  p  by  1, . . .,  n  in  turn  and  solving,  we  get 


where 


(-l)"-^+'A„-p+i  = 


•plp2  pn— p+1 

1  r^     Y'*~p 


0  o...iri 


the  subscript  p  —  1  on  the  F's  being  dropped.  After  repeating  the  argument 
by  Tchebychef^^,  Lucas  noted  that,  if  p  is  an  odd  prime,  A„_p+i=l  or  0 
(mod  p),  according  as  p  — 1  is  or  is  not  a  divisor  of  n. 

G.  Wertheim^°^  gave  Dirichlet's^^  proof  of  the  generalized  Wilson 
theorem;  also  the  first  step  in  the  proof  by  Arndt.^° 

W.  E.  HeaP''^  gave  without  reference  Euler's^'*  proof. 

E.  Catalan^°^  noted  that  if  2n+l  is  composite,  but  not  the  square  of  a 
prime,  n\  is  divisible  by  2n+l;  if  2n+l  is  the  square  of  a  prime,  (n!)^  is 
divisible  by  2n+l. 

C.  Garibaldi^"^  proved  Fermat's  theorem  by  considering  the  number  N 
of  combinations  of  ap  elements  p  at  a  time,  a  single  element  being  selected 
from  each  row  of  the  table 

en  ^12.  .  .eia 


^pl  ^p2 •  •  • ^pa. 

From  all  possible  combinations  are  to  be  omitted  those  containing  elements 
from  exactly  n  rows,  for  n  =  l,  . . .,  p  —  1.  Let  An  denote  the  number  of 
combinations  p  at  a  time  of  an  elements  forming  n  rows,  such  that  in  each 
combination  occur  elements  from  each  row.    Then 


--iV-tO^- 


"'BuU.  Soc.  Math.  France,  11,  1882-3,  69-71;  Mathesis,  3,  1883,  25-8. 

i^Elemente  der  Zahlentheorie,   1887,   186-7;  Anfangsgriinde  der  Zahlenlehre,   1902,  343-5 

(331-2). 
"'Annals  of  Math.,  3,  1887,  97-98. 

»««M6m.  soc.  roy.  so.  LiSge,  (2),  15,  1888  (Melanges  Math.,  Ill,  1887,  139). 
"^Giornale  di  Mat.,  26,  1888,  197. 


78  History  of  the  Theory  of  Numbers.  [Chap,  hi 

Take  each  e^  =  1 ;  then  N  =  a^  smce  the  number  of  the  specified  combina- 
tions becomes  the  sum  of  all  products  of  p  factors  unity,  one  from  each  row 
of  the  table.     Thus 


0"=  (  ^j=a  (mod  p). 


p-i 


R.  W.  Genese^°^  proved  Euler's  theorem  essentially  as  did  Laisant.'^ 

M.  F.  Daniels^^^  proved  the  generahzed  Wilson  theorem.  If  ^(n) 
denotes  the  product  of  the  integers  <n  and  prime  to  n,  he  proved  by  induc- 
tion that  ^(p')=  —  1  (mod  p')  for  p  an  odd  prune.  For,  if  pi, .  . .,  p„  are 
the  integers  <  p'  and  prime  to  it,  then  pi  +jp',  . .  .,  Pn  +ip'  (i  =  0, 1 , . .  . ,  p  —  1) 
are  the  integers  <  p'"^^  and  prime  to  it.  He  proved  similarly  by  induction 
that  1/^(2')  =  +  !  (mod  2')  if  7r>2.  Evidently  iA(2)  =  l  (mod  2),  »//(4)=-l 
(mod  4).  If  m  =  a''b^ . . .  and  n  =  l^,  where  I  is  a  new  prime,  then  \p(m)=e 
(mod  m),  \l/{n)  =  r)  (mod  n)  lead  by  the  preceding  method  to  \l/{mn)  =  e'^^"^ 
(mod  m),  viz.,  1,  unless  n  =  2.     The  theorem  now  follows  easily. 

E.  Lucas^^^  noted  that,  if  x  is  prime  to  n  =  AB . . .,  where  A,  B,. . .  are 
powers  of  distinct  primes,  and  if  0  is  the  1.  c.  m.  of  (f){A),  (l>{B),.  .  .,  then 
x'^=  1  (mod  n).  In  case  A  =  2^',  A:>  2,  we  may  replace  (j){A)  by  its  half.  To 
get  a  congruence  holding  whether  or  not  x  is  prime  to  n,  multiply  the  former 
congruence  by  x",  where  a  is  the  greatest  exponent  of  the  prime  factors  of  n. 
Note  that  <}>-\-<T<n  [Bachmann^^^'  "^].     CarmichaeP^^  wrote  X(n)  for  0. 

E.  Lucas^^^  found  A^~^x^~^  in  two  ways  by  the  theory  of  differences. 
Equating  the  two  results,  we  have 

(p-i)!=(p-ir^-(^7^)(p-2ri+...-(^:^)i 

Each  power  on  the  right  is  =  1  (mod  p) .    Thus 

(p-l)!=(l-l)p-i-l=-l  (modp). 

P.  A.  MacMahon^^^  proved  Fermat's  theorem  by  showing  that  the 
number  of  circular  permutations  of  p  distinct  things  n  at  a  time,  repetitions 
allowed,  is 

h<l>(d)p^^', 

ft 

where  d  ranges  over  the  divisors  of  n.    For  n  a  prime,  this  gives 

p"+(n  — l)p=0,  p"=p  (mod  n).  ^^ 

Another  specialization  led  to  Euler's  generalization. 

E.  Maillet^^^  applied  Sylow's  theorem  on  subgroups  whose  order  is 
the  highest  power  p''  of  a  prime  p  dividing  the  order  m  of  a  group,  viz., 

"^British  Association  Report,  1888,  580-1. 

""Lineaire  Congruenties,  Diss.  Amsterdam,  1890,  104-114. 

""BuU.  Ac.  Sc.  St.  P6tersbourg,  33,  1890,  496. 

">Mathesis,  (2),  1,  1891,  11;  Th^orie  des  nombres,  1891,  432. 

""Proc.  London  Math.  Soc,  23,  1891-2,  305-313. 

"'Recherches  sur  les  substitutions,  Th§se,  Paris,  1892,  115. 


Chap.  Ill]  FeRMAT's  AND   WiLSON's  THEOREMS.  79 

'm  =  pN{l+np),  when  h  =  l.  For  the  sjonmetric  group  on  p  letters,  m  =  p\ 
and  N  =  p  —  \,  so  that  (p  — 1)!=  — 1  (mod  p).  There  is  exhibited  a  special 
group  for  which  m  =  pa^,  N  =  a,  whence  a^=a  (mod  p). 

G.  Levi"*  failed  in  his  attempt  to  prove  Wilson's  theorem.  Let  b  and 
a={p  —  l)h  have  the  least  positive  residues  ri  and  r  when  divided  by  p. 
Then  r+ri  =  p.  Multiply  h/p  =  q+ri/p  by  p  —  1.  Thus  ri(p  — 1)  has  the 
same  residue  as  a,  so  that 

Ci  T 

ri(p-l)=r+mp,  -  =  q{p-l)+m+- 

He  concluded  that  ri(p  — l)=r,  falsely,  as  the  example  p  =  5,  6  =  7,  shows. 
He  added  the  last  equation  to  r+ri  =  p  and  concluded  that  ri  =  l,  r  =  p  —  l, 
so  that  (a+l)/p  is  an  integer.  The  fact  that  this  argument  is  independent 
of  Levi's  initial  choice  that  6  =  (p  — 2) !  and  his  assumption  that  p  is  a  prime 
shows  that  the  proof  is  fallacious. 

Axel  Thue"^  obtained  Fermat's  theorem  by  adding 

a''-{a-iy  =  l+kp,     {a-iy-{a-2y  =  l+hp,     ...,     P-0^=1 

[PaoU*^].  Then  the  differences  A^i^(j)  of  the  first  order  of  F{x)=x^~'^  are 
divisible  by  p  for  J  =  1,. . .,  p-2;  likewise  A^/^(l),. .  .,^''-^F{l),   By  adding 

A^+i/r(o)=A^F(l)-A^(0)     (i  =  l,. . .,  p-2), 

we  get 

-A^-^/?'(0)  =  1+AV(1)-A2/?'(1)+.  .  .+A^-2F(1),     (p-l)!+l=0(modp). 

N.  M.  Ferrers"®  repeated  Sylvester's"^^  proof  of  Wilson's  theorem. 
M.  d'Ocagne"^  proved  the  identity  in  r: 

(r+l)^+i+^^i^SPfcl^PV(^+l)'^'"''(-0*'=r'+'  +  l, 

where  g  =  [(A;+l)/2]  and  P^"^  is  the  product  of  n  consecutive  integers  of 
which  m  is  the  largest,  while  P^  =  L  Hence  if  /c+1  is  a  prime,  it  divides 
(j,_[_j)A:+i_^fc+i_2^  and  Fermat's  theorem  follows.  The  case  k  =  p  —  l 
shows  that  if  p  is  a  prime,  q={p  —  l)/2,  and  r  is  any  integer, 

S  P%-.^i  PV(^+l)''"''(-^)*=0  (mod  g!). 
t=i 

T.  del  Beccaro"^  used  products  of  linear  functions  to  obtain  a  very  com- 
pUcated  proof  of  the  generalized  Wilson  theorem. 

A.  Schmidt"^  regarded  two  permutations  of  1,  2, . . .,  p  as  identical  if 
one  is  derived  from  the  other  by  a  cyclic  substitution  of  its  elements.  From 
one  of  the  (p  —  1)!  distinct  permutations  he  derived  a  second  by  adding 

"*Atti  del  R.  Istituto  Veneto  di  Sc,  (7),  4,  1892-3,  pp.  1816-42. 

"'Archiv  Math,  og  Natur.,  Kristiania,  16,  1893,  255-265. 

'"Messenger  Math.,  23,  1893-4,  56. 

"^Jour.  de  I'^cole  polyt.,  64,  1894,  200-1. 

"8Atti  R.  Ac.  Lincei  (Fis.  Mat.),  1,  1894,  344-371. 

""Zeitschrift  Math.  Phys.,  40,  1895,  124. 


80  History  of  the  Theory  of  Numbers.  [Chap,  hi 

unity  to  each  element  and  replacing  p+1  by  1.     Let  m  be  the  least  number 
of  repetitions  of  this  process  which  will  yield  the  initial  permutation.    For 
p  a  prime,  m  =  l  or  p.     There  are  p  — 1  cases  in  which  m  =  \.     Hence 
(p  —  1) !  —  (p  —  1)  is  divisible  by  p.     Cf .  Petersen.^^ 
Many  proofs  of  (3),  p.  63,  have  been  given. ^^° 

D.  von  Sterneck^-^  gave  Legendre's  proof  of  Wilson's  theorem. 

L.  E.  Dickson^-^  noted  that,  if  p  is  a  prime,  p(p  — 1)  of  the  p!  substitu- 
tions on  p  letters  have  a  linear  representation  x'=ax-\-h,  a^O  (mod  p), 
while  the  remaining  ones  are  represented  analytically  by  functions  of  degree 
>  1  which  fall  into  sets  of  p^(p  — 1)  each,  viz.,  aJ{x-^h)-\-c,  where  a  is  prime 
top.  Hencep!—p(p  —  l)  is  a  multiple  of  p^(p  —  l),  and  therefore  (p  — 1)!+1 
is  a  multiple  of  p. 

C.  Moreau^-^  gave  without  references  Schering's^°^  extension  to  any 
modulus  of  Dirichlet's^°  proof  of  the  theorems  of  Fermat  and  Wilson. 

H.  Weber^'^  deduced  Euler's  theorem  from  the  fact  that  the  integers 
Km  and  prime  to  m  form  a  group  under  multiplication,  whence  every 
integer  belongs  to  an  exponent  dividing  the  order  0(m)  of  the  group. 

E.  Cahen^-^  proved  that  the  elementary  symmetric  functions  of  1,. . ., 
p  — 1  of  order  <p  — 1  are  divisible  by  the  prime  p.     Hence 

(a:-l)(a:-2). .  .(a:-p+l)=xP-^+(p-l)!  (modp), 

identically  in  x.  The  case  x  =  l  gives  Wilson's  theorem,  so  that  also  Fer- 
mat's  theorem  follows. 

J.  Perott^-^  gave  Petersen's^^  proof  of  Fermat's  theorem,  using  q^  ''con- 
figurations" obtained  by  placing  the  numbers  1,  2,...,  q  into  p  cases, 
arranged  in  a  line.  It  is  noted  that  the  proof  is  not  vaUd  for  p  composite; 
for  example,  if  p  =  4,  g  =  2,  the  set  of  configurations  derived  from  1212  by 
cyclic  permutations  contains  but  one  additional  configuration  2121. 

L.  Kronecker^^^  proved  the  generalized  Wilson  theorem  essentially  as 
had  Brennecke.^^ 

G.  Candido^^^  made  use  of  the  identity 

aP+6P=  (a+6)P-pa6(a+6)P-2+  . . . 

^^_^).p(p-2r+10...(p-r-l)  .,,_^    ^^^ 

1-2. . .r 

Take  p  a  prime  and  6= —1.     Thus  a^  — a=(a  — 1)^— (a  — 1)  (mod  p). 

"«L'mterm6diaire  des  math.,  3,  1896,  2&-28,  229-231;  7,  1900,  22-30;  8,  1901, 164.     A.  Capelli 

Giornale  di  Mat.,  31,  1893,  310.     S.  Pincherle,  ibid.,  40,  1902,  180-3. 
"iMonatshefte  Math.  Phys.,  7,  1896,  145. 
»»Annals  of  Math.,  (1),  11,  1896-7,  120. 
i"Nouv.  Ann.  Math.,  (3),  17,  1898,  296-302. 
i^Lehrbuch  der  Algebra,  II,  1896,  55;  ed.  2,  1899,  61. 
^"£l6ment3  de  la  throne  des  nombres,  1900,  111-2. 
>»'Bull.  des  Sc.  Math.,  24,  I,  1900,  175. 
i"Vorlesungen  uber  Zahlentheorie,  1901,  I,  127-130. 
"'Giomale  di  Mat.,  40,  1902,  223. 


Chap.  Ill]  FeRMAT's  AND   WiLSON's  THEOREMS.  81 

P.  Bachmann^^^  proved  the  first  statement  of  Lucas."*'  He  gave  as  a 
"new"  proof  of  Euler's  theorem  (p.  320)  the  proof  by  Euler,^^  and  of  the 
generaUzed  Wilson  theorem  (p.  336)  essentially  the  proof  by  Arndt.^° 

J.  W.  Nicholson^^^  proved  the  last  formula  of  Grunert.^^ 

Bricard^^^  changed  the  wording  of  Petersen's^^  proof  of  Fermat's  theorem. 
Of  the  q^  numbers  with  p  digits  written  to  the  base  q,  omit  the  q  numbers 
with  a  single  repeated  digit.  The  remaining  q^—q  numbers  fall  into  sets 
each  of  p  distinct  numbers  which  are  derived  from  one  another  by  cyclic 
permutations  of  the  digits. 

G.  A.  Miller^^^  proved  the  generalized  Wilson  theorem  by  group  theory. 
The  integers  relatively  prime  to  g  taken  modulo  g  form  under  multiplica- 
tion an  abelian  group  of  order  (f){g)  which  is  the  group  of  isomorphisms  of  a 
cyclic  group  of  order  g.  But  in  an  abelian  group  the  product  of  all  the  ele- 
ments is  the  identity  if  and  only  if  there  is  a  single  element  of  period  2. 
It  is  shown  that  a  cyclic  group  is  of  order  p",  2p*  or  4  if  its  group  of  isomor- 
phisms contains  a  single  element  of  period  2. 

V.  d'Escamard^^^  reproduced  Sylvester's''^  proof  of  Wilson's  theorem. 

K.  Petr^^*  gave  Petersen's®^  proof  of  Wilson's  theorem. 

Prompt^^®  gave  an  obscure  proof  that  2^~^  —  1  is  divisible  by  the  prime  p. 

G.  Arnoux^^®  proved  Euler's  theorem.  Let  X  be  any  one  of  the 
v=4>{m)  integers  a,  /3,  7, . . .,  prime  to  m  and  <m.  We  can  solve  the  con- 
gruences 

aa'=/3/3'=77'=  . .  .  =\  (mod  m). 
Here  a',  jS', . .  .form  a  permutation  of  a,  |S, . . . .    Thus 

In  particular,  for  X  =  l,  we  get  (a/3. .  .)^=1.  Hence  for  any  X  prime  to  m, 
V=\  (mod  m).     [Of.  Dirichlet/"  Schering,i°2  C.  Moreau.i^^] 

R.  A.  Harris^^^"  proved  that  (aj8 . .  .)^  =  1  as  did  Arnoux^^^,  but  inferred 
falsely  that  a./3 . . .  =  ±  1. 

A.  Aubry^^'^  started,  as  had  Waring  in  1782,  with 

where  yp  =  a:(a;  — 1). .  .(x— p+1).    Then 

x'»+i-a;'^=  F„+i+A7„+  . . .  -I-MF3+F2. 
Summing  for  x  =  1, . . . ,  p  —  1  and  setting  Sk  =  l*+2*+ . . .  +  (p  —  1)*,  we  get 

_\n±l\_        \n\    ,         ,MJ3J  ,  \2\ 


n+2         w+1 


«»Niedere  Zahlentheorie,  I,  1902,  157-8.  ""Amer.  Math.  Monthly,  9,  1902,  187,  211. 

"iNouv.  Ann.  Math.,  (4),  3,  1903,  340-2. 

"2Annals  of  Math.,  (2),  4,  1903,  188-190.     Cf.  V.  d'Escamard,  Giornale  di  Mat.,  41,  1903, 

203-4;  U.  Scarpis,  ihid.,  43,  1905,  323-8. 
»"Giomale  di  Mat.,  43,  1905,  379-380.  i=^Casopis,  Prag,  34,  1905,  164. 

^"Remarques  sur  le  theorSme  de  Fennat,  Grenoble,  1905,  32  pp. 
"'Arithm^tique  Graphique;  Fonctions  Arith.,  1906,  24. 
i»««Math.  Magazine,  2,  1904,  272.  "'L'enseignement  math.,  9,  1907,  434-5,  440. 


82  History  of  the  Theory  of  Numbers.  [chap.  hi 

where  \k\  =p{p  —  l) . .  .(p  —  k).  Hence,  if  p  is  a  prime  and  n<p  — 1, 
Sn+i—Sn—0.  But  Si^O.  Hence  s„=0(n<p  — 1),  Sp_i=  —  (p  — 1)!.  Thus 
Wilson's  theorem  follows  from  Fermat's. 

Without  giving  references,  Aubry  (p.  298)  attributed  Horner's"  proof 
of  Euler's  theorem  to  Gauss;  the  proof  (pp.  439-440)  by  Paoli^^  (and 
Thue^^^)  of  Fermat's  theorem  to  Euler^^;  the  proof  (p.  458)  by  Laplace^^  of 
Euler's  theorem  by  powering  to  Euler. 

R.  D.  CarmichaeP^^  noted  that,  if  L  is  the  1.  c.  m.  of  all  the  roots  z  of 
0(2)  =  a,  and  if  a:  is  prime  to  L,  then  a:°=  1  (mod  L).  Hence  except  when  n 
and  n/2  are  the  only  numbers  whose  </)-function  is  the  same  as  that  of  n, 
^•pM  =  ^  holds  for  a  modulus  M  which  is  some  multiple  of  n.  A  practical 
method  of  finding  M  is  given. 

R.  D.  CarmichaeP^^  proved  the  first  result  by  Lucas.^^° 

J.  A.  Donaldson^^°  deduced  Fermat's  theorem  from  the  theory  of 
periodic  fractions. 

W.  A.  Lindsay"^  proved  Fermat's  theorem  by  use  of  the  binomial 
theorem. 

J.  I.  Tschistjakov"^  extended  Euler's  theorem  as  had  Lucas. ^^° 

P.  Bachmann^^^  proved  the  remarks  by  Lucas,^^°  but  replaced  <f>+(r<n 
by  71^0 +(7,  stating  that  the  sign  is  >  if  n  is  divisible  by  at  least  two  distinct 
primes. 

A.  Thue^^  noted  that  a  different  kinds  of  objects  can  be  placed  into  n 
given  places  in  o"  ways.  Of  these  let  11'^  be  the  number  of  placings  such 
that  each  is  converted  into  itself  by  not  fewer  than  n  applications  of  the 
operation  which  replaces  each  by  the  next  and  the  last  by  the  first.  Then 
U2  is  divisible  by  n.  If  n  is  a  prime,  1/1  =  a""— a  and  we  have  Fermat's 
theorem.  Next,  a''=2[/a,  where  d  ranges  over  the  divisors  of  n.  Finally, 
if  p,  5, . . . ,  r  are  the  distinct  prime  factors  of  n, 

C/^=S(-l)?a"/^=0  (mod  n), 

where  D  ranges  over  the  distinct  divisors  oi  pq. .  .r,  while  0  is  the  number 
of  prime  factors  of  D.    Euler's  theorem  is  deduced  from  this. 

H.  C.  PockUngton^^^  repeated  Bricard's^^^  proof. 

U.  Scarpis^^^  proved  the  generalized  Wilson  theorem  by  a  method  similar 
to  Arndt's.^°  The  case  of  modulus  2^"  (X>2)  is  treated  by  induction. 
Assume  that  Ilr=l  (mod  2^),  where  ri, .  . .,  r„  are  the  v=4>{2^)  odd  integers 
<2^.  Then  rj, .  . .,  r„  ri+2^, .  . .,  r„+2''  are  the  residues  modulo  2^"+^  and 
their  product  is  seen  to  be  =1  (mod  2''"''^).     Next,  let  the  modulus  be 

"«BuU.  Amer.  Math.  Soc,  15,  1908-9,  221-2. 

"'/bid.,  16,  1909-10,  232-3. 

""Edinburgh  Math.  Soc.  Notes,  1909-11,  79-84. 

"i/bwf.,  78-79. 

"^Tagbl.  XII  Vers.  Russ.  Nat.,  124,  1910  (Russian). 

i«Niedere  Zahlentheorie,  II,  1910,  43-44. 

'*^Skrifter  Videnskaba-Selskabet,  Christiania,  1910,  No.  3,  7  pp. 

>«Nature,  84,  1910,  531. 

i«Periodico  di  Mat.,  27,  1912,  231-3. 


Chap.  Ill]  FeEMAT's  AND   WiLSON's  ThEOKEMS.  83 

n  =  pi'  . .  .p^'*  {h>2),  n9^2p^.    Then  a  system  of  residues  modulo  n,  each 

h 

prime  to  n,  is  given  by  S  A,r„  with 


i=l 


A 


'  w) 


n  \</>(pi°0 


} 


where  r^  ranges  over  a  system  of  residues  modulo  pi"',  each  prime  to  p,. 
Let  P  be  the  product  of  these  SA,rj.     Since  AiAj  is  divisible  by  n  if  iT^j, 

h  <p(n/pi°-i\ 

p_2Ai^W(nr,)^       ^  (modn). 
t=i 

Thus  P— 1  is  divisible  by  each  p°^  and  hence  by  n. 

*Illgner^^^  proved  Fermat's  theorem. 

A.  Bottari^^^  proved  Wilson's  theorem  by  use  of  a  primitive  root  [Gauss^°]. 

J.  Schumacher^^^  reproduced  Cayley's^°^  proof  of  Wilson's  theorem. 

A.  Arevalo^^°  employed  the  sum  /S„  of  the  products  taken  n  at  a  time  of 
1,  2, . . . ,  p  —  1.    By  the  known  formula 

it  follows  by  induction  that  Sn  is  divisible  by  the  prime  p  if  n<p  — 1.  In 
the  notation  of  Wronski,  write  a^^*"  for 

a{a+r).. .  \a+ip-l)r\  =a''-\-Sia''-''r+ . .  .+Sp_iar^-\ 

For  a  =  r  =  l,  we  have  p!  =  l+*Si+. .  .-\-Sp^i,  whence  >Sp_i=  — 1  (mod  p), 
giving  Wilson's  theorem.  Also,  a^^''=a^—a'r^~^.  Dividing  by  a  and  taking 
r  =  l,  we  have 

(a+l)(^-^^/^=a^-i-l  (modp). 

The  left  member  is  divisible  by  p  if  o  is  not.  Hence  we  have  Fermat's 
theorem.    Another  proof  follows  from  Vandermonde's  formula 

(x+ay^'=  S  (J)x^p-''^^'a^^'=x''^'+a''^'  (mod  p), 
(xi  +  . . .  +xy'-=x,^/'+ . . .  +a^/^  a^/''=a-P^ 

Remove  the  factor  a  and  set  r  =  0;  we  obtain  Fermat's  theorem. 

Prompt^^^  gave  Euler's^^  proof  of  his  theorem  and  two  proofs  of  the  type 
sketched  by  Gauss  of  his  generalization  of  Wilson's  theorem;  but  obscured 
the  proofs  by  lengthy  numerical  computations  and  the  use  of  unconven- 
tional notations. 

F.  Schuh^^^  proved  Euler's  theorem,  the  generalized  Wilson  theorem, 
and  discussed  the  symmetric  functions  of  the  roots  of  a  congruence  for  a 
prime  modulus. 

"^Lehrsatz  uber  x"— x,  Uaterrichts  Blatter  fiir  Math.  u.  Naturwisa.,  Berlin,  18,  1912,  15. 

'"II  Boll.  Matematica  Gior.  Sc.-Didat.,  11,  1912,  289. 

'"Zeitschrift  Math.-naturwiss.  Unterricht,  44,  1913,  263-4. 

""Revista  de  la  Sociedad  Mat.  Espafiola,  2,  1913,  123-131. 

"'Demonstrations  nouvelles  des  th^orlmes  de  Fermat  et  de  Wilson,  Paris,  Gauthier-Villars, 

1913,  18  pp.     Reprinted  in  Tinterm^diaire  des  math.,  20,  1913,  end. 
"^Suppl.  de  Vriend  der  Wiskunde,  25,  1913,  33-59,  143-159,  228-259. 


84  History  of  the  Theory  of  Numbers.  [Chap,  hi 

G.  Frattini^^  noted  that,  if  F{a,  j3, . .  . )  is  a  homogeneous  symmetric 
poljTiomial,  of  degree  g  with  integral  coefficients,  in  the  integers  a,  /3, . .  . 
less  than  m  and  prime  to  m,  and  if  F  is  prime  to  7??,  then  k°=  1  (mod  m)  for 
ever>'  integer  A-  prime  to  m.     In  fact, 

F(a,  jS,.  .  .)  =  F{ka,  k^,..  .)  =  k^F{a,  0,...)  (mod  m). 

Taking  F  to  be  the  product  a^.  .  .,  we  have  Euler's  theorem.  Another 
corollary  is 

u\l+j)=l  +  (p-l)l  (modp), 

for  p  a  prime,  which  implies  Wilson's  theorem. 

*J.  L.  Wildschlitz-Jessen^^^  gave  an  historical  account  of  Fermat's  and 
Wilson's  theorems. 

E.  Piccioh^"  repeated  the  work  of  Dirichlet.'*° 

The  Generalization  F{a,N)  =  0  (mod  N)  of  Fermat's  Theorem. 
C.  F.  Gauss^^°  noted  that,  if  N=pi^ . .  .p/*  (p's  distinct  primes), 

»=1  »■<;■  x<i<k 

is  divisible  by  N  when  a  is  a  prime,  the  quotient  being  the  number  of  irre- 
ducible congruences  modulo  a  of  degree  N  and  highest  coefficient  unity. 
He  proved  that 
(1)  a^=2F(a,  d),  F{a,\)=a, 

where  d  ranges  over  all  the  divisors  of  N,  and  stated  that  this  relation  read- 
ily leads  to  the  above  expression  for  F  (a,  N).     [See  Ch.  XIX  on  inversion.] 

Th.  Schonemann^^^  gave  the  generalization  that  if  a  is  a  power  p"  of  a 
prime,  the  number  of  congruences  of  degree  A^  irreducible  in  the  Galois  field 
of  order  a  is  N~'^F{a,  N). 

An  account  of  the  last  two  papers  and  later  ones  on  irreducible  con- 
gruences will  be  given  in  Ch.  VIII. 

J.  A.  Serret^^^  stated  that,  for  any  integers  a  and  iV,  F{a,  N)  is  divisible 
by  N.    For  N=p\  p  a  prime,  this  implies  that 

a</>(pO  =  l  (modpO, 

when  a  is  prime  to  p,  a  case  of  Euler's  theorem. 

S.  Kantor^^^  showed  that  the  number  of  cycHc  groups  of  order  N  in  any 
birational  transformation  of  order  a  in  the  plane  is  N~^F{a,  N) .  He  obtained 
(1)  and  then  the  expression  for  F(a,  N)  by  a  lengthy  method  completed  for 
special  cases. 

iwPeriodico  di  Mat.,  29,  1913,  49-53. 

i^Nyt  Tidsskrift  for  Mat.,  25,  A,  1914,  1-24,  49-68  (Danish). 

i"Periodico  di  Mat.,  32,  1917,  132-4. 

"oPosthumous  paper,  Werke,  2,  1863,  222;  Gauss-Maser,  611. 

"iJour.  fiir  Math.,  31,  1846,  269-325.     Progr.  Brandenburg,  1844. 

i"Nouv.  Ann.  Math.,  14,  1855,  261-2. 

"'AnnaU  di  Mat.,  (2),  10,  1880,  64-73.    Comptes  Rendus  Paris,  96,  1883,  1423. 


Chap.  Ill]  GENERALIZATIONS    OF   FeRMAT's   THEOREM.  85 

Ed.  Wey^^^^  E.  Lucas^^^  and  Pellet^^^  gave  direct  proofs  that  F{a,  N)  is 
divisible  by  N  for  any  integers  a,  N. 

H.  Picquet^^^  noted  the  divisibility  of  F{Zm  —  l,  N)  by  iV  in  an  enumera- 
tion of  certain  curvilinear  polygons  of  N  sides,  at  the  same  time  inscribed 
and  circumscribed  in  a  given  cubic  curve.  He.  gave  a  proof  of  the  divisi- 
bility of  F{a,  N)  by  N,  requiring  various  subcases.  He  stated  that  the 
function  F{a,  N)  is  characterized  by  the  two  relations 

(2)         F{a,  np") ^F{a''\  n) -F{a''"\  n),  F^a,  p')=a^'-a''"\ 

where  a  is  any  integer,  n  an  integer  not  divisible  by  the  prime  p. 
A.  Grandi^®^  proved  that  F{a,  N)  is  divisible  by  N  by  writing  it  as 

^N  _  ^N/p,  _  j  ^^N/p,  _  ^Nlp,p,-^  j^  ^^N/p,  _  ^N/p,v,^  +  .  .  .  [ 

+  \  {a^/p^P'-a^/PiP^p>)  -f- ...  J  -I- ... . 
Each  of  these  binomials  is  divisible  by  pi'  since 

G.  Koenigs^^*  considered  a  uniform  substitution  z' =4>{z)  and  its  nth 
power  z" =4)n{z).  Those  roots  of  2— 0^(2)  =0  which  satisfy  no  like  equation 
of  lower  index  are  said  to  belong  to  the  index  n.  If  x  belongs  to  the  index 
n,  so  do  also  (^i{x)  for  i—\,...,  n  —  1.  Thus  the  roots  belonging  to  the 
index  n  are  distributed  into  sets  of  n.  If  a  is  the  degree  of  the  polynomials 
in  the  numerator  and  denominator  of  0(s),  the  number  of  roots  belonging 
to  the  index  n  is  F{a,  n),  which  is  therefore  divisible  by  n. 

MacMahon's^^^  paper  contains  in  a  disguised  form  the  fact  that  F(a,  N) 
is  divisible  by  N.  Proofs  were  given  by  E.  Maillet"^  by  substitution 
groups,  and  by  G.  Cordone.^^^ 

Borel  and  Drach^'^"  made  use  of  Gauss'  result  that  F{p,  N)  is  divisible 
by  N  for  every  prime  p  and  integer  N,  and  Dirichlet's  theorem  that  there 
exist  an  infinitude  of  primes  p  congruent  modulo  N  to  any  given  integer  a 
prime  to  N,  to  conclude  that  F(a,  N)  is  divisible  by  N. 

L.  E.  Dickson^^^  proved  by  induction  (from  k  to  k-{-l  primes)  that 
F{a,  N)  is  characterized  by  properties  (2)  and  concluded  by  induction  that 
F{a,  N)  is  divisible  by  N.    A  like  conclusion  was  drawn  from 

\F{a,  N)\'-F{a,  N)=Fia,  qN)  (mod  q), 
where  g  is  a  prime.    He  gave  the  relations 

F{a,  nN)  =  Fia"",  n)  -  i  F(a^/^-,  n)  +  S  Fia"^^"'"',  n)-... 

+  {-TyF{a^^'''-'",  n)l 
F{a,N)=i:<l>id), 

"Casopis,  Prag,  11,  1882,  39. 

""Comptes  Rendus  Paris,  96,  1883,  1300-2. 

»«/6td.,  p.  1136,  1424.     Jour,  de  I'^cole  polyt.,  cah.  54,  1884,  61,  85-91. 

i"Atti  R.  Istituto  Veneto  di  Sc,  (6),  1,  1882-3,  809. 

i"Bull.  des  sciences  math.,  (2),  8,  1884,  286. 

"•Rivista  di  Mat.,  Torino,  5,  1895,  25. 

""Introd.  th^orie  dea  nombres,  1895,  50. 

"^Annals  of  Math.,  (2),  1,  1899,  35.     Abstr.  in  Comptes  Rendus  Paris,  128,  1899,  1083-6. 


86  History  of  the  Theory  of  Numbers.  [Chap,  hi 

where  d  ranges  over  those  divisors  of  a^  —  1  which  do  not  divide  d°  —  l  for 
0<t;<iV;  while,  in  the  former,  Pi,. .  .,  Pa  are  the  distinct  prime  factors  of 
N,  and  n  is  prime  to  N. 

L.  Gegenbauer^^-  wrote  F(o,  n)  in  the  form  S/i(c^)a"'''',  where  d  ranges 
over  the  divisors  of  n,  and  nid)  is  the  function  discussed  in  Chapter  XIX 
on  Inversion.  As  there  shown,  2ju(d)  =0  if  n>  1.  This  case  fix)  =ijl{x)  is 
used  to  prove  the  generaUzation :  If  the  function /(x)  has  the  property  that 
2/((i)  is  divisible  by  n,  then  for  every  integer  a  the  function  l!>f{d)a'''^  is 
divisible  by  n,  where  in  each  sum  d  ranges  over  the  divisors  of  n.  Another 
special  case,  f{x)  =4>{x),  was  noted  by  MacMahon.^^^ 

J.  Westlund^^^  considered  any  ideal  Am.  o.  given  algebraic  number  field, 
the  distinct  prime  factors  Pi, .  .  . ,  P^  of  ^,  the  norm  n(^)  of  ^,  and  proved 
that  if  a  is  any  algebraic  integer, 

^nU)  _^gn{A)ln{Pi)  _|_2^n(^)/n(PiPj)  _  4-  (  —  n'^'>(^)/«(Pi.  .  -Pi) 

is  always  divisible  by  A. 

J.  Vdlyi^^^  noted  that  the  number  of  triangles  similar  to  their  nth  pedal 
but  not  to  the  dih.  pedal  {d<n)  is 

Xin)  =^P{n) -^^(^)  +2'A(-^)  -  ■  ■ ., 

Vp/  ^PlP2^ 

if  Pi>  P2,  •  •  •  are  the  distinct  prime  factors  of  n,  and  yp{k)=2^{2^  —  \).  He 
proved  that  x(^)  is  divisible  by  n,  since  if  the  nth  pedal  to  ABC  is  the  first 
one  similar  to  ABC,  a  like  property  is  true  of  the  first  pedal, . . .,  (n  — l)th 
pedal,  so  that  the  x(^)  triangles  fall  into  sets  of  n  each  of  period  n.  [Note 
that  x(n)=P(4,  n)-F(2,  n).] 

A.  Axer^^^  proved  the  following  generalization  of  Gegenbauer's"^  theorem: 
If  G(ri, . . . ,  r/i)  is  any  polynomial  with  integral  coefficients,  and  if,  when  d 
ranges  over  all  the  divisors  of  n, 

2/(rf)G(ri"^. .  .,  rC'^)=Q  (mod  n) 

for  a  particular  function  G  =  Gq  and  a  particular  set  of  values  Txq,  . . . ,  r/,o, 
not  a  set  of  solutions  of  Gq,  and  for  which  Go  is  prime  to  n,  then  it  holds  for 
every  G  and  every  set  ri, . . . ,  r^. 

Further  Generalizations  of  Fermat's  Theorem. 

For  the  generalization  to  Galois  imaginaries,  see  Ch.  VIII. 

For  the  generalization  by  Lucas,  see  Ch.  XVII,  Lucas,^^  Carmichael.^' 

On  :x^=  1  (mod  n)  for  x  prime  to  n,  see  Cauchy,^^  Moreau,^^  Epstein,"' 
of  Ch.  VII. 

0.  H.  MitchelP'^  considered  the  2*  products  s  of  distinct  primes  dividing 
k  =  pi...pf  and  denoted  by  r/A;)  the  number  of  positive  integers 
X,<k  which  are  divisible  by  s  but  by  no  prime  factor  of  k  not  dividing  s. 

i^Monatshefte  Math.  Phys.,  11,  1900,  287-8. 

i"Proc.  Indiana  Ac.  Sc,  1902,  78-79. 

"«Monatshefte  Math.  Phys.,  14,  1903,  243-2.53. 

»"MonatBhefte  Math.  Phys.,  22,  1911,  187-194. 

"»Amer.  Jour.  Math.,  3,  1880,  300;  Johns  Hopkins  Univ.  Circular,  1,  1880-1,  67,  97. 


Chap.  Ill]  GENERALIZATIONS  OF  FeRMAt's  THEOREM.  87 

The  products  of  the  various  X^  by  any  one  of  them  are  congruent  modulo  k 
to  the  Xg  in  some  order.     Hence 

X/^w=ie^     (modifc), 

where  R^  is  the  corresponding  one  of  the  2'  roots  of  x^^x  (mod  k).  The 
analogous  extension  of  Wilson's  theorem  is  HXs^^Rs  (mod  k),  the  sign 
being  minus  only  when  k/a  =  p'',  2p'  or  4  and  at  the  same  time  a/s  is  odd. 
Here  <r  =  np/^  if  s  =  np,.     Cf.  Mitchell,^"  Ch.  V. 

F.  RogeP^^  proved  that,  if  p  is  a  prime  not  dividing  n, 

n-i  =  l+(f)(7i-l)  +  (|)(n-l)2+...  +  (f)(n-l)Hp,  A:  =  ^, 

where  p  is  divisible  by  every  prime  lying  between  k  and  p+l. 

Borel  and  Drach^^°  investigated  the  most  general  polynominal  in  x  divis- 
ible by  m  for  all  integral  values  of  x,  but  not  having  all  its  coefficients 
divisible  by  m.  If  m  =  p''q^, .  .  . ,  where  p,  q,. .  .are  distinct  primes,  and  if 
P{x),  Q{x),. . .  are  the  most  general  polynomials  divisible  by  p",  q^,. . ., 
respectively,  that  for  m  is  evidently 

{P{x)+p'^f{x)\\Q{x)+q'g{x)\.... 

For  a<p+l,  the  most  general  P{x)  is  proved  to  be 

iMx)Mx),         Mx)  =p''-\x^-x)\ 

where  the/'s  are  arbitrary  polynomials.  For  a<2(p+l),  the  most  general 
P{x)  is 

s/,<A,+  ste,        rp,=ct>{x){x^-xy-Y-''-\ 

k=l  k=l 

where  4>{x)  =  {x^—xy—p^~^{x^—x),  and  the/'s,  g's   are   arbitrary   poly- 
nomials.    Note  that  ^^(x)  -p'''-'^<f>lx)  is  divisible  by  p^'+^+K    Cf.  Nielsen.^^^ 
E.  H.  Moore^^^  proved  the  generalization  of  Fermat's  theorem: 


Xi''"*-^     X  p*""^ 


XiP 
Xi 


m     p— 1  p— 1 

=  n    H      .  .  .     H  {Xk+Ck+iXk+i+  . . .  +c^xj  (mod  p). 


F.  Gruber^^^  showed  that,  if  n  is  composite  and  ai, . . . ,  a<  are  the  ^=0(n) 
integers  <  n  and  prime  to  n,  the  congruence 

(1)  x'  — l  =  (x— fli). .  .(x— a«)     (mod  n) 

is  an  identity  in  x  if  and  only  if  n  =  4  or  2p,  where  p  is  a  prime  2*+l. 

"•Archiv  Math.  Phys.,  (2),  10,  1891,  84-94  (210). 

""Introduction  th^orie  des  nombres,  1895,  339-342. 

">BuU.  Amer.  Math.  Soc,  2,  1896,  189;  cf.  13,  1906-7,  280. 

"»Math.  Nat.  Berichte  aus  Ungarn,  13,  1896,  413-7;  Math,  term^s  ertesito,  14,  1896,  22-25. 


88 


History  of  the  Theory  of  Numbers. 


[Chap.  Ill 


E.  Malo^^  employed  integers  -4/  and  set  u  =  afz, 


Since  f^  d='Eu''/k  {k  =  n,  m+n,  2m-\-n, 


e= 


w 


l-u" 


=ScOpX''-Ma:. 


2^x^=2^^'^' 


P 


k 


p        k 


p-ltkt 


where  k  takes  the  values  n,  w+n, .  .  .which  are  ^p/fi.  If  no  prime  factor 
of  such  a  k  occurs  in  the  denominator  of  the  expansion  of  cop/p,  the  latter 
is  an  integer;  this  is  the  case  if  p  is  a  prime  and  /x= 2.     For  w  =  n  =  l,  ix  =  2, 


^(»-)-(2)-(3)- 


+ 


■ax 


a-3. 


.0-2 


we  get  o3p  =  a^—a  and  hence  Fermat's  theorem. 

L.  Kronecker^^  generalized  Fermat's  and  Wilson's  theorems  to  modular 
systems. 

R.  Le  Vavasseur^^  obtained  a  result  evidently  equivalent  to  that  by 
Moore^^^  for  the  non-homogeneous  case  Xm  =  l' 

M.  Bauer^^^  proved  that  if  n  =  p'm,  where  m  is  not  divisible  by  the  odd 
prime  p,  and  Oi, . . .,  a<  are  the  t=<l>{n)  integers  <n  and  prime  to  n, 

{x-ai) . .  .(a:-a,)  =  (xP-i-l)'/<P-'Hmod  p'), 

identically  in  a:.  If  p  =  2  and  7r>  1,  the  product  is  identically  congruent  to 
{x'^  —  iy^^.  Hence  he  found  the  values  of  d,  n  for  which  (1)  holds  modulo  d, 
when  d  is  a  divisor  of  n.  If  p  denotes  an  odd  prime  and  q  a  prime  2*+ 1,  the 
values  are 


d 

2q 

4 

P 

2 

n 

2q 

4 

p-,  2p» 

2^2"5lg2... 

M.  Bauer^*''  determined  how  n  and  N  must  be  chosen  so  that  x"  — 1 
shall  be  congruent  modulo  A^  to  a  product  of  linear  functions.  We  may 
restrict  N  to  the  case  of  a  power  of  a  prime.  If  p  is  an  odd  prime,  a;"  —  ! 
is  congruent  modulo  p°  to  a  product  of  linear  functions  only  when  p=l 
(mod  n),  a  arbitrary,  or  when  n  =  p'm,  a  =  l,  p  =  l  (mod  m).  For  p  =  2, 
only  when  n  =  2^,  a  =  l,  or  n  =  2,  a  arbitrary.  For  the  case  n  a  prime,  the 
problem  was  treated  otherwise  by  Perott.^^^ 

M.  Bauer^^^  noted  that,  if  n  =  'p'm,  where  m  is  not  divisible  by  the  odd 
prime  p, 

n(a:-i)  =  (xP-x)"/P  (mod  p'). 


t=i 


>8»L'interm6diaire  des  math.,  7,  1900,  281,  312. 
^"Vorlesungen  uber  Zahlentheorie,  I,  1901,  167,  192,  220-2. 

'"Comptes  Rendus  Paris,  135,  1902,  949;  Mdm.  Ac.  Sc.  Toulouse,  (10),  3,  1903,  39-48. 
•"Nouv.  Ann.  Math.,  (4),  2,  1902,  256-264. 

"'Math.  Nat.  Berichte  aus  Ungarn,  20,  1902,  34-38;  Math.  6s  Phys.  Lapok,  10,  1901,  274-8 
(pp.  145-152  relate  to  the  "theory  of  Fermat's  congruence";  no  report  is  available). 
"8Amer.  Jour.  Math.,  11,  1888;  13,  1891. 
'"Math.  68  Phys.  Lapok,  12,  1903,  159-160. 


Chap.  Ill]  GENERALIZATIONS   OF  FeRMAT's  ThEOREM.  89 

Richard  Sauer^^'^  proved  that,  \i  a,h,  a  —  h  are  prime  to  k, 

a^+a^-^6+a^-V+  •  •  •  +?)*'=1  (mod  k),  <P  =  (p{k), 

since  a*""^^— 6*"^^=a— 6.  Changing  alternate  signs  to  minus,  we  have  a 
congruence  valid  if  a,  6  are  prime  to  k,  and  if  0+6  is  not  divisible  by  k. 
If  p  is  an  odd  prime  dividing  a=F6, 

is  divisible  by  p,  but  not  by  p^. 

A.  Capelli^^^  showed  that,  if  a,  6  are  relatively  prime. 


ah 


=[V]+[V]+i' 


where  [x]  is  the  greatest  integer  ^  x. 

M.  Bauer^^^  proved  that,  if  p  is  an  odd  prime  and  m  =  p"  or  2p^,  every 
integer  x  relatively  prime  to  m  satisfies  the  congruence 

(a;p-i  _  l)p"- =  (x+A^i) . . .  {x-\-ki)  (mod  m), 

where  /bi, . . .,  ki  denote  the  l=<l){m)  integers  <m  and  prime  to  m>2.  If 
m  is  not  4,  p"  or  2p",  every  integer  a:  prime  to  m  satisfies  the  congruence 

(x^^-^y^-iy^(x+ki) . . .  (x+ki)  (mod  m). 

L.  E.  Dickson^^^  proved  Moore's^^^  theorem  by  invariantive  theory. 

N.  Nielsen^^^  proved  that,  if  ^{x)  is  a  polynomial  with  integral  coeffi- 
cients not  having  a  common  factor  >  1,  and  if  for  every  integral  value  of  x 
the  value  of  ^{x)  is  divisible  by  the  positive  integer  m,  then 

p-i 
^{x)  =  (f>{x)  o)p{x)+  S  rrip-s  A,  cos(x),    o)n{x)=x{x-\-l) .  .{x+n-1), 

8=  1 

where  4>(x)  is  a  polynomial  with  integral  coefficients,  the  Ag  are  integers, 
p  is  the  least  positive  integer  for  which  p !  is  divisible  by  m,  and  mp_s  is 
the  least  positive  integer  I  for  which  s\l  is  divisible  by  m.  Cf.  Borel  and 
Drach.i8° 

H.  S.  Vandiver^^^  proved  that,  if  V  ranges  over  a  complete  set  of  incon- 
gruent  residues  modulo  m  =  pi  . .  .pl^,  while  U  ranges  over  those  F's 
which  are  prime  to  m, 

A; 

ll{x-V)^^tXx''^-xT"'%    n(a;-C7)=Si,(a;P«-'-l)*'('")/(p»-^>, 

modulo  w,  where  t^  =  {mlpg^^y,  e = </)(p/») .  For  w  =  p",  the  second  congruence 
is  due  to  Bauer.^^^'  ^^^ 

""Eine  polynomische  Verallgemeinerung  des  Fermatschen  Satzes,  Diss.,  Giessen,  1905. 

»"Dritter  Internat.  Math.  Kongress,  Leipzig,  1905,  148-150. 

»»Archiv  Math.  Phys.,  (3),  17,  1910,  252-3.     Cf.  Bouniakowsky^s  of  Ch.  XI. 

"'Trans.  Amer.  Math.  Soc,  12,  1911,  76;  Madison  Colloquium  of  the  Amer.  Math.  Soc,  1914, 

39-40. 
"<Nieuw  Archief  voor  Wiskunde,  (2),  10,  1913,  100-6. 
i»Annals  of  Math.,  (2),  18,  1917,  119. 


90  History  of  the  Theory  of  Numbers.  [Chap,  hi 

Further  Generalizations  of  Wilson's  Theorem;  Related  Problems. 

J.  Steiner-°°  proved  that,  if  Ak  is  the  sum  of  all  products  of  powers  of 
Gi,  02,...,  Op-it  of  degree  k,  and  the  o's  have  incongruent  residues  p^O 
modulo  p,  a  prime,  then  A^,.  .  .,  Ap_2  are  divisible  by  p. 

He  first  showed  by  induction  that 

z      =2i.p^i-\-AiA.p^2~T'  ■  ■  ■  \Ap-2-^i\Ap^i, 
Xk={x-ai). .  .{x-ttk),  ^i  =  Oi+.  .  .+ap_i, 

^2  =  01^  +  0102+  .  .  .  +0iOp_2  +  O2^+O2O3+  .  .  .  +oJ_2, 

For  example,  to  obtain  x^  he  multipUed  the  respective  tenns  of 

V=(X  — Oi)(x  —  02) +  (01+02)  (x  —  0i)+0i^ 

by  X,  (a:-03)+a3,  (x— 02)+02,  (x  — oO+Oi.  Let  Oi,...,  Op_i  have  the 
residues  1,. . .,  p  —  l  in  some  order,  modulo  p.  For  x  — 02  divisible  by  p, 
x^~^=Ap_i  =  a{~^  (mod  p),  so  that  Ap_2Xi  and  hence  also  Ap_2  is  divisible 
by  p.  Then  for  x=a3,  ^p_3X2  and  ^p_3  are  divisible  by  p.  For  x  =  0, 
ai  =  l,  the  initial  equation  yields  Wilson's  theorem. 

C.  G.  J.  Jacobi"''^  proved  the  generahzation :  If  Oi,. . .,  a„  have  distinct 
residues  f^  0,  modulo  p,  a  prime,  and  Pr^m  is  the  sum  of  their  multipUcative 
combinations  with  repetitions  m  at  a  time,  Pnm  is  divisible  by  p  for  w  =  p— n, 
p-n+1,...,  p-2. 

Note  that  Steiner's  Ah  is  Pp-k.k-    We  have 


][ J^ I    Pfil      ■    Pn2 

(x-oi) . . .  (x-o„)  "x""^x"+i^x"+2"^  •  •  •'  '  """,=; 


(1)  7:r^7v^^^^.=i+9h+^2+--:         P..=  :^a;^-'/Dj, 


Dj  =  {cLj - Oi) . . .  (Oj - a,_i) (oy - Oy+i) . . . (o, - 0 J ,  0=2  Oy /D,  {k<n-l). 

j=i 

Let  n+m-l  =  k+^{p-\).  Then  o^+^-^^o/  (mod  p).  Hence  if 
A;<Cn  — 1 

'    Di . .  .Dr,Pn,„=D, . .  .Dj:a)/D„  P^^^O  (mod  p). 

The  theorem  follows  by  taking  /3  =  1  and  ^'  =  0,  1, . .  .,  n— 2  in  turn. 

H.  F.  Scherk^°-  gave  two  generaUzations  of  Wilson's  theorem.     Let  p  be 
a  prime.     By  use  of  Wilson's  theorem  it  is  easily  proved  that 

n! 

where  x  is  an  integer  such  that  px=l  (mod  n!).  Next,  let  C/  denote  the 
sum  of  the  products  of  1,  2, . .  . ,  ^  taken  r  at  a  time  with  repetitions.  By  use 
of  partial  fractions  it  is  proved  that 

(p-r-l)!C;_,_i+(-l)'-=0(modp)  (r<p-l). 

It  is  stated  that 

""Jour,  fiir  Math.,  13,  1834,  356;  Werke  2,  p.  9. 
""/bid.,  14,  1835,  64-5;  Werke  6,  252-3. 

'"Bericht  iiber  die  24.  Versammlung    Deutscher    Naturforscher   und   Aerzte  in  1846,   Kiel, 
1847,  204-208. 


(p-n-l)!^(-ir^^(modp), 


Chap.  Ill]  Genekalizations  OP  Wilson's  Theorem.  91 

C;.r-iCr-'  +  {-iy^O,  C:,-m\^0  (modp),  ^  =  ^- 

H.  F.  Scherk^''^  proved  Jacobi's  theorem  and  the  following:  Form  the 
sum  Pnh  of  the  multiplicative  combinations  with  repetitions  of  the  hth  class 
of  any  n  numbers  less  than  the  prime  p,  and  the  sum  of  the  combinations 
without  repetitions  out  of  the  remaining  p—n  —  1  numbers  <p;  then  the 
sum  or  the  difference  of  the  two  is  divisible  by  p  according  as  h  is  odd  or  even. 

Let  Cl  denote  the  sum  of  the  combinations  with  repetitions  of  the  hth 
class  oi  1,  2,. .  .,  k;  Al  the  sum  without  repetitions.     If  0<h<p  —  l, 

Ci^O  (mod  p),  j  =  p-k,.. .,  p-2;  Cl+,^Cl 

For  h  =  p-l,  Ci;ik=n+1  for  k  =  l,...,  p.  For  h  =  m{p-l)+t,  Cl=Ci 
when  k<p+l.  For  l<h<k,  the  sum  of  Cl  and  A^  is  divisible  by 
A;^(fc+1)^;  likewise,  each  C and  A  if /i  is  odd.  For  h<2k,  Cl—Al  is  divisible 
by  2A:H-1.     The  sum  of  the  2nth  powers  of  1, . . . ,  /c  is  divisible  by  2k-\-l. 

K.  HenseP'^  has  given  the  further  generalization:  If  ai, . . . ,  a,^,  61, . . . ,  6, 
are  n-\-v  =  p  —  l  integers  congruent  modulo  p  to  1,  2, . . . ,  p  —  1  in  some  order, 
and 

^l/{x)  =  (x-b,)...  (rr-6J  =x'-B,x'-'+  . .  .=^B„ 

then,  for  any  j,  Pnj^i  —  iy'Bj^  (mod  p),  where  jo  is  the  least  residue  of  j 
mod  p  —  1  and  Bh  =  0  {k>v). 

For  Steiner's  Z„,  Z„^(a;)=a:^-i-l  (mod  p).  Multiply  (1)  by 
a:"(x^-^-l).     Thus 

X''rP{x)^X'-'+PnlX^-'+ .  .  .+Pnp-2X  +  Pnp-l-l  +  ^"^~^"^ 

X 

+^"^^^"^"'+...  (modp). 

X 

Replace  \f/{x)  by  its  initial  expression  and  compare  coefficients.    Hence 

p^j^i-iyBj{j=i,...,v). 

Taking  v=j  =  p  —  2  and  choosing  2,...,  p  —  1  for  61,...,  6„  we  get 
1=  — (p  — 1)!  (mod  p). 

Converse  of  Fermat's  Theorem. 

In  a  Chinese  manuscript  dating  from  the  time  of  Confucius  it  is  stated 
erroneously  that  2""^  — 1  is  not  divisible  by  n  if  n  is  not  prime  (Jeans^^^). 

Leibniz  in  September  1680  and  December  1681  (Mahnke,^  49-51)  stated 
incorrectly  that  2'*— 2  is  not  divisible  by  n  if  n  is  not  a  prime.  If  n  =  rs, 
where  r  is  the  least  prime  factor  of  n,  the  binomial  coefficient  (")  was  shown 
to  be  not  divisible  by  n,  since  n  —  1,...,  n— r+1  are  not  divisible  by  r, 
whence  not  all  the  separate  terms   in  the  expansion  of  (1  +  1)"  — 2  are 

^o^Ueber  die  Theilbarkeit  der  Combinationssummen  aus  den  natiirlichen  Zahlen  durch  Prim- 

zahlen,  Progr.,  Bremen,  1864,  20  pp. 
^"Archiv  Math.  Phys.,  (3),  1,  1901,  319;  Kronecker's  Zahlentheorie  1,  1901,  503. 


92  History  of  the  Theory  of  Numbers.  [Chap,  hi 

divisible  by  n.  From  this  fact  Leibniz  concluded  erroneously  that  the 
expression  itself  is  not  divisible  by  n. 

Chr.  Goldbach^^°  stated  that  {a-\-by  —  a^  —  b^  is  divisible  by  p  also  when 
p  is  any  composite  number.  Euler  (p.  124)  points  out  the  error  by  noting 
that  2^^  —  2  is  divisible  by  neither  5  nor  7. 

In  1769  J.  H.  Lambert^'*  (p.  112)  proved  that,  if  ^"•-l  is  divisible  by  a, 
and  d"  —  1  by  6,  where  a,  b  are  relatively  prime,  then  d'  —  l  is  divisible  by 
ab  if  c  is  the  1.  c.  m.  of  m,  n  (since  divisible  by  d"*  —  1  and  hence  by  a) .  This 
was  used  to  prove  that  if  g  is  odd  [and  prime  to  5]  and  if  the  decimal  fraction 
for  l/g  has  a  period  oi  g  —  1  terms,  then  ^  is  a  prime.  For,  if  ^  =  a6  [where 
a,  b  are  relatively  prime  integers  >  1],  1/a  has  a  period  of  m  terms,  m^a  —  \, 
and  1/6  a  period  of  n  terms,  n^b  —  l,  so  that  the  number  of  terms  in  the 
period  for  1/^  is  ^  (a  — 1)(6  — 1)/2<(7  — 1.  Thus  Lambert  knew  at  least 
the  case  /c  =  10  of  the  converse  of  Fermat's  theorem  (Lucas^"'  ^^'). 

An  anonymous  writer  ^^^  stated  that  2n+l  is  or  is  not  a  prime  according 
as  one  of  the  numbers  2"=*=  1  is  or  is  not  divisible  by  n.  F.  Sarrus^^^  noted 
the  falsity  of  this  assertion  since  2^'^°  —  1  is  divisible  by  the  composite  num- 
ber 341. 

In  1830  an  anonymous  writer^  noted  that  a"~^  —  1  may  be  divisible  by  n 
when  n  is  composite.  In  a^~^  =  /cp+1,  where  p  is  a  prime,  set  k  =  \q.  Then 
^(P-i)«=l  (jjjojj  pqy  Ti^us  oP«-i  =  l  if  a«-i  =  l  (mod  pq),  and  the  last  will 
hold  if  g  —  1  is  a  multiple  of  p  —  1 ;  for  example,  ifp  =  ll,g'  =  31,a  =  2,  whence 
2340=1  (mod  341). 

V.  Bouniakowsky^^3  proved  that  if  A^  is  a  product  of  two  primes  and  if 
iV— 1  is  divisible  by  the  least  positive  integer  a  for  which  2"=1,  whence 
2^~^=1  (mod  N),  then  each  of  the  two  primes  decreased  by  unity  is  divisible 
by  a.    He  noted  that  3^=1  (mod  91  =  7-13). 

E.  Lucas^^^  noted  that  2''~^=1  (mod  n)  for  71  =  37-73  and  stated  the  true 
converse  to  Fermat's  theorem:  If  a""  —  !  is  divisible  by  p  for  x  =  p  —  l,  but 
not  for  x<p  —  lf  then  p  is  a  prime. 

F.  Proth^^*  stated  that,  when  a  is  prime  to  n,  n  is  a  prime  if  a*=  1  (mod  n) 
for  a:=  (n  — 1)/2,  but  for  no  other  divisor  of  (n  — 1)/2;  also,  if  a''=l  (mod  n) 
for  x  =  n  —  l,  but  for  no  divisor  <\/n  of  n  —  1.  If  71  =  7^-2*^+1,  where  m 
is  odd  and  <  2*",  and  if  a  is  a  quadratic  non-residue  of  n,  then  n  is  a  prime 
if  and  only  if  a^"~^^/^=  —  1  (mod  n).  If  p  is  a  prime  >^\/n,  n  =  mp+l  is  a 
prime  if  a"~^  — 1  is  divisible  by  n,  but  a^'^l  is  not. 

*F.  Thaarup^^'  showed  how  to  use  a"~^=l  (mod  ti)  to  tell  if  n  is  prime. 
E.  Lucas^^^  proved  the  converse  of  Fermat's  theorem:  If  a^=l  (mod  ti) 
for  a:  =  71  —  1,  but  not  for  x  a  proper  divisor  oi  n  —  1,  then  n  is  a  prime. 

""Corresp.  Math.  Phys.  (ed.  Fuss),  I,  1843,  122,  letter  to  Euler,  Apr.  12,  1742. 

"'Annales  de  Math.  (ed.  Gergonne),  9,  1818-9,  320. 

^"Ibid.,  10,  1819-20,  184-7. 

"»M6m.  Ac.  Sc.  St.  P6tersbourg  (math.),  (6),  2, 1841  (1839),  447-69;  extract  in  Bulletin,  6,  97-8. 

"*Assoc.  frang.  avanc.  sc,  5,  1876,  61;  6,  1877,  161-2;  Amer.  Jour.  Math.,  1,  1878,  302. 

"'Comptes  Rendus  Paris,  87,  1878,  926. 

"•Nyt  Tidsskr.  for  Mat.,  2A,  1891,  49-52. 

*"Th6orie  des  nombres,  1891,  423,  441. 


Chap.  Ill]  CONVERSE   OF  FeRMAt's  THEOREM.  93 

G.  Levi^^*  was  of  the  erroneous  opinion  that  P  is  prime  or  composite 
according  as  it  is  or  is  not  a  divisor  of  10^"^  — 1  [criticized  by  Cipolla,^^^ 
p.  142]. 

K.  Zsigmondy^^^  noted  that,  if  g  is  a  prime  =1  or  3  (mod  4),  then  2q+l 
is  a  prime  if  and  only  if  it  divides  (2*+l)/3  or  2^  —  1,  respectively;  4g+l  is 
a  prime  if  and  only  if  it  divides  (2^*+l)/5. 

E.  B.  Escott^^^  noted  that  Lucas'^^^  condition  is  sufficient  but  not 
necessary. 

J,  H.  Jeans^^®  noted  that  if  p,  q  are  distinct  primes  such  that  2^=2 
(mod  g),  2^=2  (mod  p),  then  2^^ =2  (mod  pq),  and  found  this  to  be  the  case 
for  pg  =  11-31,  19-73,  17-257,  31-151,  31-331.  He  ascribed  to  Kossett  the 
result  2"-^=l  (mod  n)  for  n  =  645. 

A.  Korselt^^^  noted  this  case  645  and  stated  that  a^=a  (mod  p)  if  and 
only  if  p  has  no  square  factor  and  p  —  1  is  divisible  by  the  1.  c.  m.  of  pi  —  1, .  .  . , 
p„  — 1,  where  pi, . . .,  p„  are  the  prime  factors  of  p. 

J.  FraneP^^  noted  that  2^^ =2  (mod  pq),  where  p,  q  are  distinct  primes, 
requires  that  p  —  1  and  g  — 1  be  divisible  by  the  least  integer  a  for  which 
2''=1  (mod  pq).     [Cf.  Bouniakowsky.^^^] 

L.  Gegenbauer222«  noted  that  2^''-^=l  (mod  pq)  if  p  =  2'-l  =  /cpr+l 
and  q  =  KT-\-l  are  primes,  as  for  p  =  Sl,  q  =  ll. 

T.  Hayashi^^^  noted  that  2''— 2  is  divisible  by  n  =  11-31.  If  odd  primes 
p  and  q  can  be  found  such  that  2^=2,  2^=2  (mod  pq),  then  2^'— 2  is  divisible 
by  pq.  This  is  the  case  if  p  —  1  and  q  —  1  have  a  common  factor  p'  for  which 
2"''=!  (mod  pq),  as  for  p  =  23,  g  =  89,  p'  =  ll. 

Ph.  Jolivald224  asked  whether  2^-^=1  (mod  N)  if  N  =  2''-l  and  p  is  a 
prime,  noting  that  this  is  true  if  p  =  ll,  whence  iV  =  2047,  not  a  prime. 
E.  Malo^^^  proved  this  as  follows: 

AT-l  =2(2^-1-1)  =2pw,  2^-^  =  (2^)2-  =  (Ar+l)2-=i  (modiV). 

G.  Ricalde^^^  noted  that  a  similar  proof  gives  a^~''+^=l  (mod  N)  if 
N  =  a^—1,  and  a  is  not  divisible  by  the  prime  p. 

H.  S.  Vandiver^^^  proved  the  conditions  of  J.  FraneP^^  and  noted  that 
they  are  not  satisfied  if  a<  10.  Solutions  for  a  =  10  and  a  =  11  are  ^3  =  11-31 
and  23-89,  respectively. 

H.  Schapira^^^  noted  that  the  test  for  the  primality  of  N  that  a^=l 

»8Monat8hefte  Math.  Phys.,  4,  1893,  79. 

*i«L'mtenn4diaire  des  math.,  4,  1897,  270. 

220Messenger  Math.,  27,  1897-8,  174. 

''"L'interm^diaire  des  math.,  6,  1899,  143. 

^Ibid.,  p.  142. 

=«""Monatshefte  Math.  Phys.,  10,  1899,  373. 

'^'Jour.  of  the  Physics  School  in  Tokio,  9, 1900, 143-4.     Reprinted  in  Abhand.  Geschichte  Math. 

Wiss.,  28,  1910,  25-26. 
*"L'interm4diaire  des  math.,  9,  1902,  258. 
^lUd.,  10,  1903,  88. 
*^Ibid.,  p.  186. 

*"Amer.  Math.  Monthly,  9,  1902,  34-36. 
»»Tchebychef's  Theorie  der  Congruenzen,  ed.  2,  1902,  306. 


94 


History  of  the  Theory  of  Numbers. 


[Chap.  Ill 


(mod  N)  ioT  q  =  N—l,  but  for  no  smaller  q,  is  practical  only  if  it  be  known 
that  a  small  number  a  is  a  primitive  root  of  N. 

G.  Arnoux--^"  gave  numerical  instances  of  the  converse  of  Fermat's 
theorem. 

M.  Cipolla^^^  stated  that  the  theorem  of  Lucas^^'  impUes  that,  if  p  is 
a  prime  and  A:  =  2,  4,  6,  or  10,  then  kp-\-l  is  a  prime  if  and  only  if  2^'^=1 
(mod  kp-\-l).  He  treated  at  length  the  problem  to  find  a  for  which  a^~^  =  1 
(mod  P),  given  a  composite  P;  and  the  problem  to  find  P,  given  a.  In 
particular,  we  may  take  P  to  be  any  odd  factor  of  (a^"  — l)/(a^  — 1)  if  p 
is  an  odd  prime  not  dividing  a^  —  1.  Again,  2^~^= 1  (mod  P)  for  P  =  F^F^  ■  .  . 
F„  m>n>  .  .  .>s,  if  and  only  if  2'>m,  where  P,  =  2^''4-l  is  a  prime.  If 
p  and  q  =  2p  —  \  are  primes  and  a  is  any  quadratic  residue  of  q,  then  a^«~^  =  1 
(mod  pq) ;  we  may  take  a  =  3  if  p  =  4n+3 ;  a  =  2\i  p  =  4n+ 1 ;  both  a  =  2  and 
a  =  3if  p  =  12A:+l;etc. 

E.  B.  Escott^^°  noted  that  e"~^  =  l  (mod  n)  if  e''  — 1  contains  two  or  more 
primes  whose  product  n  is  =1  (mod  a),  and  gave  a  list  of  54  such  n's. 

A.  Cunningham^^^  noted  the  solutions  n  =  FsF^F^QF7,  n  =  F^. .  .P15,  etc. 
[cf.  CipoUa],  and  stated  that  there  exist  solutions  in  which  n  has  more  than 
12  prime  factors.     One  with  12  factors  is  here  given  by  Escott. 

T.  Banachiewicz^^^  verified  that  2^—2  is  divisible  by  N  for  N  composite 
and  <  2000  only  when  N  is 

341  =  11-31,    561=3-1M7,    1387  =  19-73,    1729  =  7-13-19,    1905  =  3-5-127. 

Since  2^—2  is  evidently  divisible  by  N  for  every  N  =  Fk  =  2^  +1,  perhaps 
Fermat  was  thus  led  to  his  false  conjecture  that  every  Fk  is  a  prime. 

R.  D.  CarmichaeP^^  proved  that  there  are  composite  values  of  n  (a 
product  of  three  or  more  distinct  odd  primes)  for  which  e"~^=l  (mod  n) 
holds  for  every  e  prime  to  n. 

J.  C.  Morehead^^  and  A.  E.  Western  proved  the  converse  of  Fermat's 
theorem. 

D.  Mahnke^  (pp.  51-2)  discussed  Leibniz'  converse  of  Fermat's  theorem 
in  the  form  that  n  is  a  prime  if  a:"~^=l  (mod  n)  for  all  integers  x  prime  to  n 
and  noted  that  this  is  false  when  n  is  the  square  or  higher  power  of  a  prime 
or  the  product  of  two  distinct  primes,  but  is  true  for  certain  products  of 
three  or  more  primes,  as  3-11-17,  5-13-17,  5-17-29,  5-29-73,  7-13-19. 

R.  D.  CarmichaeP^^  used  the  result  of  Lucas^^"  to  prove  that  a^~^  =  l 
(mod  P)  holds  for  every  a  prime  to  P  if  and  only  if  P  — 1  is  divisible  by 
X(P).  The  latter  condition  requires  that,  if  P  is  composite,  it  be  a  product 
of  three  or  more  distinct  odd  primes.     There  are  found  14  products  P  of 

«8« Assoc,  frang.,  32,  1903,  II,  113-4. 

"•Annali  di  Mat.,  (3),  9,  1903-4,  138-160. 

""Messenger  Math.,  36,  1907,  175-6;  French  transl.,  Sphinx-Oedipe,  1907-8,  146-8. 

"'Math.  Quest.  Educat.  Times,  (2),  1^,  1908,  22-23;  6,  1904,  26-7,55-6. 

»«Spraw.  Tow.  Nauk,  Warsaw,  2,  1909,  7-10. 

"'Bull.  Amer.  Math.  Soc,  16,  1909-10,  237-8. 

^Ibid.,  p.  2. 

"»Amer.  Math.  Monthly,  19,  1912,  22-7. 


Chap.  Ill]  SYMMETRIC   FUNCTIONS   MoDULO   p.  95 

three  primes,  as  well  as  P  =  13-37-73457,  for  each  of  which  the  congruence 
holds  for  every  a  prime  to  P. 

Welsch^se  stated  that  ifk  =  4n-\-l  is  composite  and  <  1000,  2^-^  =  1  (mod 
k)  only  for  A;  =  561  and  645;  hence  71**=  1  (mod  k)  for  these  two  k's. 

P.  Bachmann^^^  proved  that  x^'^~^'=  1  (mod  pq)  is  never  satisfied  by  all 
integers  prime  to  pq  if  p  and  q  are  distinct  odd  primes  [Carmichael^^^]. 

Symmetric  Functions  of  1,  2,.  .  .p— 1  Modulo  p. 

Report  has  been  made  above  of  the  work  on  this  topic  by  Lagrange,  ^^ 
Lionnet,"  Tchebychef,^^  Sylvester,'^^  ottmger,^^  Lucas,!"^  Cahen,!^^  Aubry,^" 
Arevalo,^^^  Schuh,^^^  Frattini,!^^  steiner,^^^  Jacobi,^^!  Hensel.^o^ 

We  shall  denote  l"+2"+ .  . .  +(p  — I)'*  by  s^,  and  take  p  to  be  a  prime. 

E.  Waring^^°  wrote  a,  j3, . .  .for  1,  2, . . . ,  x,  and  considered 

s  =  a^^^y' .  .  .  ^a^^^y' .  .  .  +a"/3^7'' .... 

If  <  =  a+6+c+  . .  .is  odd  and  <a:,  andx+1  is  prime,  s  is  divisible  by  (x+l)^. 
If  t<2x  and  a,  6, . .  .are  all  even  and  prime  to  2a; +1,  s  is  divisible  by  2a;-f  1. 

V.  Bouniakowsky^^^  noted  that  s^  is  divisible  by  p^,  if  p>  2  and  m  is  odd 
and  not  =1  (mod  p  — 1);  also  if  both  w=l  (mod  p  —  1)  and  m=0  (mod  p). 

C.  von  Staudt252  proved  that,  if  S,Xx)  =  1+2"+  •  •  •  +a;% 

S^{ah)=bSM+naSn-iia)Si{b-l)     (mod  a"), 
2S2n+i{a)^{2n+l)aS2M     (mod  a^). 

If  a,h,. . .,  I  are  relatively  prime  in  pairs, 

S,Xah...l)     SM  S^il) 


ah. .  .1  a        ' ' '        I 


=  integer. 


A.  Cauchy253  proved  that  1  +  1/2+  . . .  +l/(p-l)=0  (mod  p). 
G.  Eisenstein^^^  noted  that  s^=  — 1  or  0  (mod  p)  according  as  m  is  or 
is  not  divisible  by  p  —  1.     If  w,  n  are  positive  integers  <p  — 1, 

'iyia+iy^O  or  -  (p_f_  J'    (mod  p), 

according  as  m+n<or^p  — 1. 

L.  Poinsot^^^  noted  that,  when  a  takes  the  values  1, . . . ,  p  —  1,  then  (ax)" 
has  the  same  residues  modulo  p  as  a",  order  apart.  By  addition,  SnX'*=Sn 
(mod  p).  Take  x  to  be  one  of  the  numbers  not  a  root  of  a;"=l.  Hence 
s„=0  (mod  p)  if  n  is  not  divisible  by  p  —  1. 

^''L'mtermMiaire  des  math.,  20,  1913,  94. 

*»'Archiv  Math.  Phys.,  (3),  21,  1913,  185-7. 

""Meditationes  algebraicae,  ed.  3,  1782,  382. 

">BuIl.  Ac.  Sc.  St.  P^tersbourg,  4,  1838,  65-9. 

"'Jour,  fiir  Math.,  21,  1840,  372-4. 

"'M^m.  Ac.  Sc.  de  I'lnstitut  de  France,  17,  1840,  340-1,  footnote;  Oeuvres,  (1),  3,  81-2. 

«"Jour.  fiir  Math.,  27,  1844,  292-3;  28,  1844,  232. 

»"Jour.  de  Math.,  10,  1845,  33-4. 


96  History  of  the  Theory  of  Numbers.  [Chap,  hi 

J.  A.  Serret^^^  concluded  by  applying  Newton's  identities  to  (x— 1) . . . 
(x— p+l)=0  that  Sn=0  (mod  p)  unless  n  is  divisible  by  p  — 1. 
J.  Wolstenholme^"  proved  that  the  numerators  of 

1+UU...+  1  1+1  +  .    ■    1 


2  '  3  '  •■•  '  p-r  -  .  22  '  •••  '  (p-l)2 

are  divisible  by  p^  and  p  respectively,  if  p  is  a  prime  >3.  Proofs  have  also 
been  given  by  C.  Leudesdorf^^s,  A.  Rieke,^^^  E.  Allardice,^^"  G.  Osborn,^" 
L.  Birkenmajer,^"  P.  Niewenglowski,^^  N.  Nielsen,^"  H.  Valentiner,^®^ 
and  others.^^^ 

V.  A.  Lebesgue^®'  proved  that  s^  is  divisible  by  p  if  w  is  not  divisible 
by  p  —  1  by  use  of  the  identities 

(n+1)  S  k{k+l) . .  .ik-\-n-l)=xix+l) . .  .(x+n)  (n  =  l,. . .,  p-1). 

k=l 

P.  Frost^^^  proved  that,  if  p  is  a  prime  not  dividing  2^''  — 1,  the  numera- 
tors of  a2r,  (T2r-i,  p(2r  —  l)o-2r+2(T2r-i  are  divisible  by  p,  p^,  p^,  respectively, 
where 

1J_1_L  1 


2*  '  •"  '  (p-1)' 

The  numerator  of  the  sum  of  the  first  half  of  the  terms  of  0*2,  is  divisible  by 
p;  likewise  that  of  the  sum  of  the  odd  terms. 

J.  J.  Sylvester^^^  stated  that  the  sum  S^,  m  of  all  products  of  n  distinct 
numbers  chosen  from  1,. . .,  m  is  the  coefficient  of  T  in  the  expansion  of 
{l+t){l-\-2t) . . .  (1+wO  and  is  divisible  by  each  prime  >n+l  contained  in 
any  term  of  the  set  m— n+l,. . .,  m,  m+1. 

E.  Fergola""  stated  that,  if  (a,  6, . . . ,  ly  represents  the  expression 
obtained  from  the  expansion  of  (a+6H-  . . .  +0"  by  replacing  each  numerical 
coefficient  by  unity,  then 

(X,  x+1,. . .,  x+rr=  i  CY)^^'  2,. . .,  rr-^x^. 

*"Coiirs  d'algSbre  sup^rieure,  ed.  2,  1854,  324. 

«'Quar.  Jour.  Math.,  5,  1862,  35-39. 

"sProc.  London  Math.  Soc,  20,  1889,  207. 

«»Zeit8chrift  Math.  Phys.,  34,  1889,  190-1. 

««oProc.  Edinburgh  Math.  Soc,  8,  1890,  16-19. 

"'Messenger  Math.,  22,  1892-3,  51-2;  23,  1893-4,  58. 

"^Prace  Mat.  Fiz.,  Warsaw,  7,  1896,  12-14  (Polish). 

M'Nouv.  Ann.  Math.,  (4),  5,  1905,  103. 

««Nyt  Tidsskrift  for  Mat.,  21,  B,  1909-10,  8-10. 

^Ibid.,  p.  36-7. 

^^Math.  Quest,  Educat.  Times,  48,  1888, 115;  (2),  22, 1912,  99;  Amer.  Math.  Monthly,  22, 1915, 

103,  138,  170. 
«^Introd.  k  la  thdorie  des  nombres,  1862,  79-80,  17. 
»8Quar.  Jour.  Math.,  7,  1866,  370-2. 
M»Giomale  di  Mat.,  4,  1866,  344.     Proof  by  Sharp,  Math.  Ques.  Educ.  Times,  47, 1887, 145-6; 

63,  1895,  38. 
"oibid.,  318-9.    Cf.  Wronski"!  of  Ch.  VIII. 


Chap.  Ill] 


Symmetric  Functions  Modulo  p. 


97 


The  number  (1,  2, . . .,  r)'*  is  divisible  by  every  prime  >r  which  occurs  in 
the  series  n+2,  n+3, . . .,  n+r. 
G.  ToreUi"!  proved  that 

(ai, . . . ,  GnY  =  (ai, . . . ,  a„_iy+a„{ai, .  . . ,  a^y~\ 
(fli,. . .,  o„,  ^y-Cfli,. . .,  fln,  c)'"=(6-c)(ai,. . .,  a„,  6,  cy-\ 

which  becomes  Fergola's  for  ai  =  i  (i  =  0, . . .,  n).    Proof  is  given  of  Syl- 
vester's^^^  theorem  and  the  generahzation  that  >Sy,i  is  divisible  by  (}+!). 

Torelli^^^  proved  that  the  sum  o-„, «  of  all  products  of  n  equal  or  distinct 
numbers  chosen  from  1,  2, . . .,  m  is  divisible  by  (n+T),  and  gave  recursion 
formulas  for  o-n,  m- 

C.  Sardi^'^^  deduced  Sylvester's  theorem  from  the  equations  Ai  —  (|),, . . 
used  by  Lagrange. ^^    Solving  them  for  Ap  =  Sp,n,  we  get 


pi{-iy^%,,= 


-1 

G) 

(a) 
C) 


0 
-2 


i;--i) 


0 

V  2  ; 

0 

\  3  ; 

0 

("to 

( 


n-p+2\      /n+1 
2      y     Vp+^ 


D 


If  n+l  is  a  prime  we  see  by  the  last  column  that  >S„_i.„  is  divisible  by  n+1. 
When  p  =  n  —  l,  denote  the  determinant  by  D.  Then  if  n+l  is  a  prime, 
D  is  evidently  divisible  by  n+l.  Conversely,  if  D  is  divisible  by  n+l  and 
the  quotient  by  (n  — 1) !,  then  n+l  is  a  prime.     It  is  shown  that 


p=l 


r„  =  F+...+n^ 


Using  this  for  m  =  1, . . . ,  n,  we  see  that  Vp  is  divisible  by  any  integer  prime 
to  2,  3, . .  .,  p+1  which  occurs  in  n+l  or  n.  Hence  if  n+l  is  a  prime,  it 
divides  ri, . . .,  r„_i,  while  rn=n  (mod  n+l).  If  n+l  divides  r„_i  it  is  a 
prime. 

Sardi"^  proved  Sylvester's  theorem  and  the  formula 


S  (  —  l)''>Si,,  n+r-lO'k-r,  n+r  =  0, 
r-0 


stated  by  Fergola."^ 


"iQiornale  di  Mat.,  5,  1867,  110-120. 
"276id.,  250-3. 
"mid.,  371-6. 
"*Ibid.,  169-174. 
"Ubid.,  4,  1866,  380. 


98  History  of  the  Theory  of  Numbers.  [Chap,  hi 

Sylvester"^  stated  that,  if  pi,  p2, . .  .are  the  successive  primes  2,  3,  5, . .  ., 

^'^^ ^T^^A^ ^'-'^''^' 

where  Fk{n)  is  a  polynomial  of  degree  k  with  integral  coefficients,  and  the 
exponent  e  of  the  prime  p  is  given  by 

E.  Ces^ro^"  stated  Sylvester's^®^  theorem  and  remarked  that  <S„.„— n! 
is  divisible  by  w  —n  if  m—n  is  a  prime. 

E.  Ces^ro"^  stated  that  the  prime  p  divides  /S^.p-2  — 1,  *Sp_i.p+l,  and, 
except  when  m  =  p  —  l,  S^.p-i-  Also  (p.  401),  each  prime  p>{n+l)/2 
divides  *Sp_i.„H-l,  while  a  prime  p  =  (n+l)/2  or  n/2  divides  »Sp_i.„+2. 

O.  H.  Mitchell"^  discussed  the  residues  modulo  k  (any  integer)  of  the 
symmetric  functions  of  0,  1, .  . . ,  /c  —  1.  To  this  end  he  evaluated  the  residue 
of  (x— a)(x— /3) . .  .,  where  a,  /3, .  .  .are  the  s-totitives  of  k  (numbers<A:  which 
contain  s  but  no  prime  factor  of  k  not  found  in  s) .  The  results  are  extended 
to  the  case  of  moduli  p,  f{x),  where  p  is  a  prime  [see  Ch.  VIII]. 

F.  J.  E.  Lionnet^^°  stated  and  Moret-Blanc  proved  that,  if  p  =  2n+l  is 
a  prime>  3,  the  sum  of  the  powers  with  exponent  2a  (between  zero  and  2n) 
of  1,  2, . .  . ,  n,  and  the  like  sum  for  n-\-l,  n+2, .  . . ,  2n,  are  divisible  by  p. 

M.  d'Ocagne^^^  proved  the  first  relation  of  Torelli.^^^ 

E.  Catalan^^^  stated  and  later  proved^^^  that  s^  is  divisible  by  the  prime 
p>k-\-l.  If  p  is  an  odd  prime  and  p  —  1  does  not  divide  k,  Sk  is  divisible 
by  p;  while  if  p  —  1  divides  k,  Sk=  —  l  (mod  p).  Let  p  =  a''b^ . . . ;  if  no  one 
of  a  — 1,  6  —  1,.  .  .  divides  k,  Sk  is  divisible  by  p;  in  the  contrary  case,  not 
divisible.     If  p  is  a  prime  >2,  and  p  —  1  is  not  a  divisor  of  k-\-l,  then 

^  =  l^(p-l)'+2*(p-2)^+  . .  .+{p-iyv 

is  divisible  by  p;  but,  if  p  —  1  divides  k+l,  S=  —  {  —  iy  (mod  p).    If  k  and  I 
are  of  contrary  parity,  p  divides  S. 

M.  d'Ocagne^^  proved  for  Fergola's^'^"  symbol  the  relation 

(a.  .  .fg.  .  .1.  .  .V.  .  .zr^Xia.  .  .frig.  .  .^)^  ..{v.  .  .z)", 

summed  for  all  combinations  such  that  X+/i+.  .  .-\-p  =  n.     Denoting  by 
a^^^  the  letter  a  taken  p  times,  we  have 

i=0 

"«Nouv.  Ann.  Math.,  (2),  6,  1867,  48. 

»"Nouv.  Correap.  Math.,  4,  1878,  401;  Nouv.  Ann.  Math.,  (3),  2,  1883,  240. 

"8Nouv.  Coiresp.  Math.,  4,  1878,  368. 

"»Amer.  Jour.  Math.,  4,  1881,  25-38. 

"ONouv.  Ann.  Math.,  (3),  2,  1883,  384;  3,  1884,  395-6. 

"»/6id.,  (3),  2,  1883,  220-6.  Cf.  Ces^ro,  (3),  4,  1885,  67-9. 

2«BuU.  Ac.  Sc.  Belgique,  (3),  7,  1884,  448-9. 

"'M6m.  Ac.  R.  Sc.  Belgique,  46,  1886,  No.  1,  16  pp. 

»**Nouv.  Ann.  Math.,  (3),  5,  1886,  257-272. 


Chap.  Ill]  SYMMETRIC   FUNCTIONS   MODULO  p.  99 

It  is  shown  that  (1^^^)"  equals  the  number  of  combinations  of  n-\-p  —  l 
things  p  —  1  at  a  time.  Various  algebraic  relations  between  binomial 
coefficients  are  derived. 

L.  Gegenbauer^^^  considered  the  polynomial 


p-2+k 

f{x)=    S    hix'  {l-p<k^p-l) 

i=0 


and  proved  that 


V/(X)/X^-2=  -h,.2  (mod  p),  k<p-l, 

X=l 

'xf{X)/\^-'=  -6p_2-fc2p-3  (mod  p),  k  =  p-\, 

X=l 

and  deduced  the  theorem  on  the  divisibility  of  s„  by  p. 

E.  Lucas^^^  proved  the  theorem  on  the  divisibility  of  s„  by  p  by  use  of  the 
symbolic  expression  (s+l)"— s"  for  x"  — 1. 

N.  Nielsen^^^"  proved  that  if  p  is  an  odd  prime  and  if  k  is  odd  and 
\<k<p  —  \,  the  sum  of  the  products  of  1, . . .,  p  —  1  taken  A;  at  a  time  is 
divisible  by  p^.     For  k=p—2  this  result  is  due  to  Wolstenholme.^^^ 

N.  M.  Ferrers^^^  proved  that,  if  2n+l  is  a  prime,  the  sum  of  the  products 
of  1,  2, . . . ,  2n  taken  r  at  a  time  is  divisible  by  2n+l  if  r<2n  [Lagrange^^], 
while  the  sum  of  the  products  of  the  squares  of  1, . . . ,  n  taken  r  at  a  time  is 
divisible  by  2n+l  if  r<n.     [Other  proofs  by  Glaisher.^^^] 

J.  Perott^^^  gave  a  new  proof  that  s^  is  divisible  by  p  if  n<p  —  l. 

R.  Rawson^^^  proved  the  second  theorem  of  Ferrers. 

G.  Osborn^^°  proved  for  r<p  —  l  that  s^.  is  divisible  by  p  if  r  is  even,  by 
p^  if  r  is  odd;  while  the  sum  of  the  products  of  1, . . .,  p  —  l  taken  r  at  a 
time  is  divisible  by  p^  if  r  is  odd  and  l<r<p. 

J.  W.  L.  Glaisher^^^  stated  theorems  on  the  sum  Sriai,...,  a,)  of  the 
products  of  fli, . . . ,  rti  taken  r  at  a  time.  If  r  is  odd,  Sr{l, . . . ,  n)  is  divisible 
by  n+1  (special  case  n+1  a  prime  proved  by  Lagrange  and  Ferrers).  If  r 
is  odd  and  >  1,  and  if  n+1  is  a  prime> 3,  Sr{l, . . . ,  n)  is  divisible  by  {n-{-iy 
[Nielsen^^^''].  If  r  is  odd  and  >1,  and  if  w  is  a  prime  >2,  Sril,. . .,  n)  is 
divisible  by  n^.  If  n+1  is  a  prime,  Sr{l^,.  •  ■,  n^)  is  divisible  by  n+1  for 
r  =  l,...,n  —  1,  except  for  r  =  n/2,  when  it  is  congruent  to  ( — 1)1+"/^  j^odulo 
n+1.  If  p  is  a  prime  ^n,  and  k  is  the  quotient  obtained  on  dividing  n+1 
by  p,  then  aSp_i(1,...,  n)=— A;  (mod  p);  the  case  n  =  p  — 1  is  Wilson's 
theorem. 

"^Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  95  II,  1887,  616-7. 

"«Th6orie  des  nombres,  1891,  437. 

286aNyt  Tidsskrift  for  Mat.,  4,  B,  1893,  1-10. 

"'Messenger  Math.,  23,  1893-4,  56-58. 

288BuU.  des  sc.  math.,  18,  I,  1894,  64.     Other  proofs.  Math.  Quest.  Educ.  Times,  58,  1893,  109; 

4,  1903,  42. 
"•Messenger  Math.,  24,  1894-5,  68-69. 
""Ibid.,  25,  1895-6,  68-69. 
"'Ibid.,  28,  1898-9,  184-6.     Proofs"*. 


100 


History  of  the  Theory  of  Numbers. 


[Chap.  Ill 


S.  Monteiro'^-  noted  that  2n+l  divides  {2n)\Z\n/r. 

J.  Westlund-^^  reproduced  the  discussion  by  Serret^^^  and  Tchebychef.'^^ 

Glaisher^^  proved  his^^^  earlier  theorems.    Also,  ii  p  =  27n+l  is  prime, 

{m-t)pS2t{l,.. .,  2m)=S2t+i{l,.  ■ .,  2m)  (mod  f) 

and,  if  i>l,  modulo  p'^.    According  as  n  is  odd  or  even, 

>S2t(l,. . .,  n)=S2t{l,. . .,  n-1)  (mod  n^  or  ^n^). 

For  m  odd  and  >3,  S2m-z0-i- . .,  2m  — 1)  is  divisible  by  m^,  and 

^„_2(1^...,  \m-l\^),  ^2n.-4(l,...,2m-l) 

are  divisible  by  m.  He  gave  the  values  of  Sr{\,.  ■  ■,  n)  and  Ar  =  Sr{l,. . ., 
n  — 1)  in  terms  of  n  for  r  =  l,. . .,  7;  the  numerical  values  of  5^(1,. . .,  n) 
for  n^22,  and  a  list  of  known  theorems  on  the  divisors  of  Ar  and  Sr.  For 
r  odd,  3^r^m  — 2,  Sr{l, .  . .,  2w— 1)  is  divisible  by  m  and,  if  w  is  a  prime 
>3,  by  m.^  He  proved  {ibid.,  p.  321)  that,  if  l^r^  (p-3)/2,  and  5,  is  a 
BernoulU  number, 

2.S2.+i(l,. .  .,  p-l)_-{2r+l)S2r{l,...,  p-1) 


V  V 

^2.(i,...,p-i)_(-ir5: 


V 


2t 


(modp). 


Glaisher^^^  gave  the  residues  of  a^  [Frost^^^]  modulo  p^  and  p^  and  proved 
that  (72,  o'i,  •• . ,  o-p-3  are  divisible  by  p,  and  0-3,  cs, . . . ,  o-p_2  by  p^,  if  p  is 
a  prime. 

Glaisher^^®  proved  that,  if  p  is  an  odd  prime. 


■'•"'   o2n    I    r2n   I     ' 


(p-2) 


2n 


^0  or  —  I  (mod  p), 


according  as  2n  is  not  or  is  a  multiple  of  p  — 1.  He  obtained  (pp.  154-162) 
the  residue  of  the  sum  of  the  inverses  of  like  powers  of  numbers  in  arith- 
metical progression. 

F.  Sibirani^^^"  proved  for  the  Sn,m  of  Sylvester^^^  (designated  Sn,m-\-\)  that 


^n,n  '^n— l,n  •  •  •  *^n— Jfe+l.n 

On+A;— l,n+*— 1       Sn+k—2,n+k—l  ■  ■  ■  ^n.n+k—l 


t 


=  inl)K 


"'Jornal  Sc.  Mat.  Phys.  e  Nat.,  Lisbon,  5,  1898,  224. 
»»Proc.  Indiana  Ac.  Sc,  1900,  103-4, 
"^Quar.  Jour.  Math.,  31,  1900,  1-35. 
»'/Wd.,  329-39;  32,  1901,  271-305. 
"•Messenger  Math.,  30,  1900-1,  26-31. 
'wPeriodico  di  Mat.,  16,  1900-1,  279-284. 


Chap.  Ill]  SyMMETKIC   FUNCTIONS   MODULO  p.  101 

K.  HenseP^^  proved  by  the  method  of  Poinsot^^^  that  any  integral  sym- 
metric function  of  degree  v  of  1,...,  p  —  1  with  integral  coefficients  is 
divisible  by  the  prime  p  if  y  is  not  a  multiple  of  p  —  1. 

W.  F.  Meyer^^^  gave  the  generalization  that,  if  ai, . . . ,  ap_i  are  incongru- 
ent  modulo  p",  and  each  af~^  — 1  is  divisible  by  p",  any  integral  symmetric 
function  of  degree  voiai,...,  ap_i  is  divisible  by  p'*  if  v  is  not  a  multiple  of 
p  —  1.  Of  the  </)(p")  residues  modulo  p",  prime  to  p,  there  are  p'^Cp  — 1)^  for 
which  a^~^  —  l  is  divisible  by  p"~^~'^,  but  by  no  higher  power  of  p,  where 
A;  =  1, . . .,  n— 1;  the  remaining  p  —  1  residues  give  the  above  ai,...,  a^-i. 

J.  W.  Nicholson^^^  noted  that,  if  p  is  a  prime,  the  sum  of  the  nth  powers 
of  p  numbers  in  arithmetical  progression  is  divisible  by  p  if  n<p  — 1,  and 
=  —  1  (mod  p)  if  71  =  p  —  1. 

G.  Wertheim^"''  proved  the  same  result  by  use  of  a  primitive  root. 

A.  Aubry^°^  took  x  =  1,  2, . . . ,  p  —  1  in 

(a:+l)"-rc"  =  nx^-i+Ax"-2+  . . .  -{-Lx+l 
and  added  the  results.    Thus 

p''  =  ns„_i+As„_2+. .  .+Lsi+p. 

Hence  by  induction  Sn_i  is  divisible  by  the  prime  p  if  n<p.  He  attributed 
this  theorem  to  Gauss  and  Libri  without  references. 

U.  Concina^°^  proved  that  s„  is  divisible  by  the  prime  p>2  if  n  is  not 
divisible  by  p  —  1.  Let  5  be  the  g.  c.  d.  of  n,  p  —  1,  and  set  ^i5  =  p  —  1.  The 
)u  distinct  residues  Ti  of  nth  powers  modulo  p  are  the  roots  of  ^"=1  (mod  p), 
whence  Sr,=0  (mod  p)  for  n  not  divisible  by  p  —  1.  For  each  r^-,  x'*=rj  has 
6  incongruent  roots.  Hence  s„=5Sri=0.  He  proved  also  that,  if  p+1  is 
a  prime  >3,  and  n  is  even  and  not  divisible  by  p,  l''+2''+ . . .  +(p/2)'*  is 
divisible  by  p+1. 

W.  H.  L.  Janssen  van  Raay^°^  considered,  for  a  prime  p>3, 

(p-1)!  ^      (P-I)I 

^''~      h      '  ^'~h{v-h) 

and  proved  that  B^-\-B2-\- . . .  +-B(p_i)/2  is  divisible  by  p,  and 

are  divisible  by  p^. 

U.  Concina^o^  proved  that  ^  =  1+2"+ .  .  .+/c"  is  divisible  by  the  odd 
number  A;  if  n  is  not  divisible  by  p  —  1  for  any  prime  divisor  of  p  of  k.  Next, 
let  k  be  even.    For  n  odd  >  1,  ^  is  divisible  by  k  or  only  by  k/2  according 

"'Archiv  Math.  Phys.,  (3),  1,  1901,  319.     Inserted  by  Hensel  in  Kronecker's  Vorlesungen  tiber 

Zahlentheorie  I,  1901,  104-5,  504. 
"sArchiv  Math.  Phys.,  (3),  2,  1902,  141.     Cf.  Meissner'"  of  Ch.  IV. 
"^Amer.  Math.  Monthly,  9,  1902,  212-3.     Stated,  1,  1894,  188. 
^ooAnfangsgninde  der  Zahlentheorie,  1902,  265-6. 
'"iL'enseignement  math.,  9,  1907,  296. 
»°2Periodico  di  Mat.,  27,  1912,  79-83. 
"'Nieuw  Archief  voor  Wiskunde,  (2),  10,  1912,  172-7. 
so^Periodico  di  Mat.,  28,  1913,  164-177,  267-270. 


102  History  of  the  Theory  of  Numbers.  [Chap.  hi 

as  k  is  or  is  not  divisible  by  4.  For  n  even,  S  is  divisible  only  by  k/2  pro- 
vided n  is  not  divisible  by  any  prime  factor,  diminished  by  unity,  of  k. 

N.  Nielsen^"^  wrote  Cp  for  the  sum  of  the  products  r  at  a  time  of  1, ... , 
p  — 1,  and 

s„(p)=2;s^        (7„(p)=S(-l)''-V. 
*-l  «=1 

If  p  is  a  prime  >2m+1, 

o-2n(p-l)=S2n(p-l)=0(modp),  Sgn+iCp "  1)  =  O(niod  p2). 

If  p  =  2n+l  is  a  prime  >3,  and  l^r^n  — 1,  Cf'^^  is  divisible  by  p^. 

Nielsen^''^  proved  that  2Di^'^^  is  divisible  by  2n  for  2p+l^n,  where  D\ 
is  the  sum  of  the  products  of  1,  3,  5, .  .  . ,  2n  —  1  taken  s  at  a  time;  also, 

2^"+'s2fl(n  - 1)  =  2^%,{2n  - 1)         (mod  4n^), 

and  analogous  congruences  between  sums  of  powers  of  successive  even  or 
successive  odd  integers,  also  when  alternate  terms  are  negative.  He  proved 
(pp.  258-260)  relations  between  the  C's,  including  the  final  formulas  by 
Glaisher.29^ 

Nielsen^°^  proved  the  results  last  cited.  Let  p  be  an  odd  prime.  If 
2n  is  not  divisible  by  p  — 1,  S2r.(p  — 1)  =  0  (mod  p),  S2„+i(p  — 1)=0  (modp^). 
But  if  2n  is  divisible  by  p  — 1, 

S2n(p-1)=-1,     S2„+i(p-l)  =  0  (modp),     Sp(p  - 1)  =  0  (mod  p^). 

T.  E.  Mason^"^  proved  that,  if  p  is  an  odd  prime  and  i  an  odd  integer  >  1, 
the  sum  Ai  of  the  products  i  at  a  time  of  1, .  .  . ,  p  —  1  is  divisible  by  p^.  If 
p  is  a  prime  >3,  Sk  is  divisible  by  p^  when  k  is  odd  and  not  of  the  form 
m(p  — 1)  +  1,  by  p  when  k  is  even  and  not  of  the  form  7n{p  —  l),  and  not 
by  p  if  A:  is  of  the  latter  form.  If  A;  =  7n(p  — 1)  +  1,  s^  is  divisible  by  p^  or 
p  according  as  k  is  or  is  not  divisible  by  p.  Let  p  be  composite  and  r  its 
least  prime  factor;  then  r  —  1  is  the  least  integer  t  for  which  At  is  not  divisible 
by  p  and  conversely.  Hence  p  is  a  prime  if  and  only  if  p  —  1  is  the  least  t  for 
which  At  is  not  divisible  by  p.  The  last  two  theorems  hold  also  if  we 
replace  A's  by  s's. 

T.  M.  Putnam^"^  proved  Glaisher's^^^  theorem  that  s_„  is  divisible  by 
p  if  n  is  not  a  multiple  of  p  —  1 ,  and 

(p-l)/2  9  — 9P 

2    jp-2=f_A(inodp). 
y-i  p 

W.  Meissner^^°  arranged  the  residues  modulo  p,  a  prime,  of  the  successive 

•o'K.  Danske  Vidensk.  Selsk.  Skrifter,  (7),  10,  1913,  353. 

"•Annali  di  Mat.,  (3),  22,  1914,  81-94. 

•"Ann.  sc.  I'^cole  norm,  sup.,  (3),  31,  1914,  165,  196-7. 

"*T6hoku  Math.  Jour.,  5,  1914,  136-141. 

»"Amer.  Math.  Monthly,  21,  1914,  220-2. 

"•Mitt.  Math.  Gesell.  Hamburg,  5,  1915,  159-182. 


1 


i 


1 

2 

4 

8 

3 

6 

12 

11 

9 

5 

10 

7 

Chap.  Ill]  SYMMETRIC   FUNCTIONS   MODULO    Jp.  103 

powers  of  a  primitive  root  /i  of  p  in  a  rectangular  table  of  t  rows  and  r  col- 
umns, where  ir  =  p  — 1.  For  p  =  13,  /i  =  2,  i  =  4,  the  table 
is  shown  here.  Let  R  range  over  the  numbers  in  any 
column.  Then  Si2  and  Sl/^R  are  divisible  by  p.  If  Ms 
even,  Sl/i?  is  divisible  by  p^  as  1/1  +  1/8+1/12+1/5  = 
13^/120.  For  t  =  p  —  \,  the  theorem  becomes  the  first  one 
due  to  Wolstenholme.^^^  Generalizations  are  given  at  the  end  of  the 
paper. 

N.  Nielsen^^^  proved  his^^^*"  theorem  and  the  final  results  of  Glaisher.^^^ 
Nielsen^^^  proceeded  as  had  Aubry^°^  and  then  proved 

(p-l)/2  ^_3 

S2„+i^0  (mod  f),       S    r^O  (mod  p),  l^n^  ^  . 

Then  by  Newton's  identities  we  get  Wilson's  theorem  and  Nielsen's^"^  last 
result. 

E.  Cahen^^^  stated  Nielsen's^^^''  theorem. 

F.  Irwin  stated  and  E.  B.  Escott^^*  proved  that  if  Sj  is  the  sum  of  the 
products  J  at  a  time  of  1,  1/2,  1/3,. .,  \/t,  where  ^=  (p  — 1)/2,  then  2S2—Si^, 
etc.,  are  divisible  by  the  odd  prime  p. 

"iQversigt  Danske  Vidensk.  Selsk.  ForhandUnger,  1915,  171-180,  521. 

3i276id.,  1916,  194-5. 

'i^Comptes  Rendus  Stances  Soc.  Math.  France,  1916,  29. 

»"Ainer.  Math.  Monthly,  24,  1917,  471-2. 


I 


CHAPTER  IV. 

RESIDUE  OF  (C/^->-l)/P  MODULO  P. 
N.  H.  Abel^  asked  if  there  are  primes  p  and  integers  a  for  which 

(1)  a^-'=l  (modp'),  l<a<p. 

C.  G.  J.  Jacobi^  noted  that,  for  p^37,  (1)  holds  only  when  p  =  ll, 
a  =  3  or  9;  p  =  29,  a  =  14;  p  =  37,  a  =  18.     Cf.  Thibault^^  of  Ch.  VI. 
G.  Eisenstein^  noted  that,  for  p  a  prime,  the  function 

has  the  properties 

(2)  quv=qu-\-qv,  qu+pv^Qu--  (mod  p), 

2g2=l-Ki-|+  •  •  •  — ^^^\  Md  p), 

P         1  o 

where  s  =  (p+l)/2, . , .,  p  —  1.  All  solutions  of  (1)  are  included  in  a= 
u+puqu,  0<u<p. 

E.  Desmarest^  noted  that  (1)  holds  for  p  =  4S7,  a  =  10,  and  stated  that 
p  =  3  and  p  =  487  are  the  only  primes  <  1000  for  which  10  is  a  solution. 

J.  J.  Sylvester^  stated  that,  if  p,  r  are  distinct  primes,  p>2,  then  g^ 
is  congruent  modulo  p  to  a  sum  of  fractions  with  the  successive  denominators 
p  — 1, .  .  . ,  2, 1  and  (as  corrected)  with  numerators  the  repeated  cycle  of  the 
positive  integers  ^r  congruent  modulo  r  to  1/p,  2/p,. . .,  r/p.  Thus,  for 
r  =  5, 

p  —  1     p  —  2    p  —  6    p  —  4:     p  —  0    p  —  0 

j,^^+.-l-+-i-+^+-^+-l-+...  (p=10fc+7). 

p  —  1    p—2    p  —  S    p—4:    p  —  5    p  —  0 

According  as  p  =  4A;+l  or  4A:  — 1,  ^2  is  congruent  to 
2.22.2.      2 


p—S    p—4:    p-7    p—S    p  —  ll 
2  2  2  2  2 


p-2    p-3    p-Q    p-7     p-10     " 
[the  signs  were  given  +  erroneously].    For  any  p, 

?2= 7-\ ^ 5+...     (modp). 

p—1     p—2    p—o 

iJour.  fiir  Math.,  3,  1828,  212;  Oeuvres,  1,  1881,  619. 

mid.,  301-2;  Werke,  6,  238-9;  Canon  Arithmeticus,  Berlin,  1839,  Introd.,  xxxiv. 
'BerUn  Berichte,  1850,  41. 
*Th6orie  des  nombres,  1852,  295. 

'Comptes  Rendus  Paris,  52,  1861,  161,  212,  307,  817;  Phil.  Mag.,  21,  1861, 136;  Coll.  Math. 
Papers,  II,  229-235,  241,  262-3. 

105 


106  History  of  the  Theory  of  Numbers.  [Chap,  iv 

Jean  Plana^  developed  j  (Af  —  1)  +  1 } "  and  obtained 

M''-M-\{M-iy-{M-l)\=pf{M), 

Take  M  =  m,  7n  —  l,. .  .,  1  m  the  first  equation  and  add.     Thus 

=/(!)  +  .  .  .  4-/(w)  =Si+^^S2-f  •  •  •  +Sp-i, 

where  s,  =  r+2'H-  .  .  .  +  (m  —  1)'.     For  j>  1,  we  may  replace  p  by  j  and  get 


m 


-w=is,_i+  (0s,_2+  Qjsi-aH-  •  •  •  +isi, 


a  result  obtained  by  Plana  by  a  long  discussion  [Euler"].  He  concluded 
erroneously  that  each  Si  is  divisible  by  m  (for  m  =  3,  S2  =  5). 

F.  Proth^  stated  that,  if  p  is  a  prime,  2''— 2  is  not  divisible  by  p^  [error, 
see  Meissner^^]. 

M.  A.  Stern^  proved  that,  if  p  is  an  odd  prime, 

rrf  —  m_        ,      ,  ,  1  _  ,  i  ,  ,1 

— - — =Si-§S2+iS3-.  .  .-— YSp-i=o-p-i+io-p_2+. .  •  +  —^0-1 

for  Si  as  by  Plana  and  0-^  =  l'+2*+  •  • .  +w\  Proof  is  given  of  the  formula 
below  (2)  of  Eisenstein^  and  Sylvester's  formulae  for  q2  (corrected),  as  well 
as  several  related  formulae. 

L.  Gegenbauer^  used  Stern's  congruences  to  prove  that  the  coefficient  of 
the  highest  power  of  a^  in  a  polynomial  f{x)  of  degree  p  —  2  is  congruent  to 
{m^—m)/p  modulo  p  if /(a:)  satisfies  one  of  the  systems  of  equations 

J{\)  =  {-lf^'V-\{m-\),  /(X)=X--VxW  (X  =  l,. . .,  p-1). 

E.  Lucas^°  proved  that  ^2  is  a  square  only  for  p  =  2,  3,  7,  and  stated  the 
result  by  Desmarest.^ 

F.  Panizza^^  enumerated  the  combinations  p  at  a  time  of  ap  distinct 
things  separated  into  p  sets  of  a  each,  by  counting  for  each  r  the  combina- 
tions of  the  things  belonging  to  r  of  the  p  sets : 


(T)=i.(^>a)  CO  ■•(")' 


•Mem.  Acad.  Turin,  (2),  20,  1863,  120. 

^Comptes  Rendus  Paris,  83,  1876,  1288. 

•Jour,  fur  Math.,  100,  1887,  182-8. 

•Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  95,  1887,  II,  616-7. 
">Th6orie  des  nombres,  1891,  423. 
"Periodico  di  Mat.,  10,  1895,  14-16,  54-58. 


Chap.  IV]  RESIDUE   OF    {U^^  —  l)/^   MoDULO   y.  107 

where  ^l^-...^-^r  =  p,  ij>0.  The  term  given  by  r  =  p  is  a^.  For  p  a 
prime,  the  left  member  is  =a  (mod  p)  and  we  have  Fermat's  theorem.  By 
induction  on  r, 

Taking  r  =  p,  we  have 

D.  Mirimanoff^^  wrote  ao  for  the  least  positive  integer  making  aop+1 
divisible  by  the  prime  r<p,  and  denoted  the  quotient  by  r%i,  where  6i  is 
prime  to  r.  Similarly,  let  a^  be  the  least  positive  integer  such  that  a,p+6i  = 
r%ij^i.    We  ultimately  find  an  n  for  which  6n  =  l-   Then  6„4.j  =  6i.    By  (2), 

?b<~r-^i?'-+?6.>"  -  .^  7r=gr2ei(modp). 

Oi  »-0  Oj 

Let  r  belong  to  the  exponent  co  modulo  p  and  set  6w  =  p  — 1.  Then  Sev=co, 
while  1,  61, . . . ,  6„_i  are  the  distinct  residues  of  the  eth  powers  of  the  integers 
<  r  and  prime  to  r.    Thus 

g^=eS  -^  (mod  p). 

The  formula  obtained  by  taking  r  a  primitive  root  of  p  is  included  in  the 
following,  which  holds  also  for  any  prime  r : 

$,=  2  -*  (mod  p), 

ai  being  the  least  positive  integer  for  which  aip+iSi=0  (mod  r).  Set 
/3i  =  p— 5,  p'p=l  (mod  r),  0<p'<r.     Then  ai=p'b  —  \  (mod  r), 

]A;}  being  the  least  positive  residue  modulo  r  of  k.  Whence  Sylvester's^ 
statement. 

J.  S.  Aladow^^  proved  that  (1)  has  at  most  (p=f1)/4  roots  if  p  =  4?n=tl. 

A.  Cunningham^^"  listed  27  cases  in  which  r^~^=l  or  r'=l  (mod  p'), 
r<p^~^,  where  Z  is  a  divisor  of  p  — 1.  For  the  11  cases  of  the  first  kind, 
p  =  5,  7,  17,  19,  29,  37,  43,  71,  487. 

W.  Fr.  Meyer^^  proved  by  induction  that,  if  p  is  a  prime,  x^~^  —  l  is 
divisible  by  p*  {l^k<n),  but  not  by  p^'^^,  for  exactly  p"~^~^  (p  — 1)^  posi- 
tive integers  a:<p"  and  prime  to  p,  and  is  divisible  by  p"  for  the  remaining 
p  —  1  such  integers.     Set 


^=a+MiP+.  ■  ■+MnP"  (l^Q<P,0^iU><p),  Xp=(a^'-a"'  Vp 

"Jour,  fiir  Math.,  115,  1895,  295-300. 

"St.  Petersburg  Math.  Soc.  (Rusaian),  1899,  40-44. 

"<»Messenger  Math.,  29,  1899-1900,  158.     See  Cuniungham"^,  Ch.  VI. 

"Archiv  Math.  Phys.,  (3),  2,  1901,  141-6. 


108  History  of  the  Theory  of  Numbers.  [Chap,  iv 

If  k  is  the  least  index  for  which  fXk^Xk,  M/»=^/.  (mod  p)  for  h<k,  then 
A'~^  —  l  is  divisible  by  p*",  but  not  by  p^"*"^. 

A.  Palmstrom  and  A.  Pollak^^  proved  that,  if  p  is  a  prime  and  n,  m  are. 
the  exponents  to  which  a  belongs  modulo  p,  p~,  respectively,  then  a""  —  ! 
is  divisible  by  p^,  so  that  m  is  a  multiple  of  n  and  a  divisor  of  np,  whence 
m  =  n  OT  pn.  Thus  according  as  a^~^  is  or  is  not  =1  (mod  p^),  m  =  n  or 
m  =  np. 

Worms  de  Romilly^^"  noted  that,  if  co  is  a  primitive  root  of  p^,  the  incon- 
gnient  roots  of  x^^=l  (mod  p^)  are  co^^{j  =  l,. .  .,  p  — 1). 

J.  W.  L.  Glaisher^^  proved  that  if  r  is  a  positive  integer  <p,  p  sl  prime, 

r''-^  =  l+^iP+K^i'-^2)p'+|(^i'-3^i^2+2^3)p'+. . ., 
where  Qn  is  the  sum  of  the  nth  powers  of 

1    _2_  r-1     .     1  2  r-1        .      1 

a   [2(t]'"  ''  [{r-l)a]'  r+cr'  r+[2(T]'"  ''  r+[(r-lV]'  2r+(r'*  '  '' 

a  being  the  least  positive  residue  modulo  r  of  —  p.  If  /i^  is  the  least  positive 
solution  of  ani=i  (mod  r),  viz.,  p/i<H-i=0,  then 

Ml    ,  M2    ,  I    Mr-l     ,       Ml       ,       M2       I  ,      Mr-1      ,        Ml         , 


12  r-1     r  +  1     r+2  '  '  '  '  '  2r-l  ^  2r+l      ' " 

Set  /Ur  =  Oj  Mi+jr  =Mi-    Then 

g^=i\^)\  s'^'^O(modp). 

Sylvester's  corrected  results  are  proved.     From  (1  +  1)^, 

op_2  1  /  1    \ 

=l-^+i-  • .  • 7^2(l+K  .  . .  + -)  (mod  p). 

p  p  — 1       \  p  — 2/ 

For  r'  =  r+A:p,  let  m/  be  the  positive  root  of  p/x/+^=0  (mod  r').     Then 
It  is  shown  that,  for  some  integer  t, 


k  k    k^  2k       k^ 

hi-gi+-  =  tp,  h2-g2=  -2---2+^t-y9i-:^  (mod  p), 

Glaisher,^^  using  the  same  notations,  gave 

^'-'-l+p(^+f +...+^l)  (modp=). 

"L'intermddiaire  des  math.,  8,  1901,  122,  205-6  (7,  1900,  357). 

^Ibid.,  214-5. 

'•Quar.  Jour.  Math.,  32,  1901,  1-27,  240-251. 

"Messenger  Math.,  30,  1900-1,  78. 


Chap.  IV]  RESIDUE   OF    (U^    ^  —  l)/p   MODULO   p.  109 

Glaisher^^  considered  Qu  in  connection  with  BernouUian  numbers  and 
gave 

^-^^-i(}+l+      +1)  (modp  =  3ft+l). 

A.  Pleskot^^  duplicated  the  work  of  Plana. ^ 

P.  Bachmann^°  gave  an  exposition  of  the  work  by  Sylvester,^  Stern,* 
Mirimanoff.-^^ 

M.  Lerch^^  set,  for  any  odd  integer  p  and  for  u  prime  to  p, 

P 

Then,*  as  a  generahzation  of  (2), 

qnv =qu+qv,  Qu+pv =qu-\ (mod  p) , 

"-I^H-  2g.-2i— 2I  (mod  p), 

where  v  ranges  over  the  positive  integers  <p  and  prime  to  p;  X  over  those 
>p/2;  fi  over  those  <p/2.    Henceforth,  let  p  be  an  odd  prime  and  set 

N=\{p-l)\+l\/p.     Then  N^q,+ .  .  .+q,.^, 

[P/4]1  [p/3]i  [p/5]i  [2p/5U 

32^-1  si,  3^3= -2  si,  5^5^-2  2^-2  Si 

»=l^  v=l^  a=l«  5=1 0 

modulo  p.    If  \p(n)  is  the  number  of  sets  of  positive  solutions  <p  oi  ixv  =  n 
and  hence  the  number  of  divisors  between  n/p  and  p  of  n, 

Employing  Legendre's  symbol  and  BernoulHan  numbers,  we  have 
^=  sY^)g  =0  or  (-1)"-^2j5„  (mod  p), 

v=l  \p/ 

according  as  p  =  4n+3  or  4n+l.     In  the  respective  cases, 

p-i 


\{^^Vq.^Cl{-p)oTO{modp), 


where   CZ(— A)    is   the   number   of   classes   of   positive   primitive   forms 
ax^+bxy+cy^  of  negative  discriminant  6^— 4ac=  —A.    Also,  modulo  p, 

y^iv'^LpJ'  ^        ^aaLpJ' 

6  ao  Lp  J  a  0  aabf-pj 

where  a,  a  are  quadratic  residues  of  p,  and  6,  /3  non-residues. 

isProc.  London  Math.  Soc,  33,  1900-1,  49-50. 

"Zeitschrift  fiir  das  Realschulwesen,  Wien,  27,  1902,  471-2. 

"Niedere  Zahlentheorie,  I,  1902,  159-169.  *The  greatest  integer  ^x  is  denoted  by  [x]. 

"Math.  Annalen,  60,  1905,  471-490. 


1 10  History  of  the  Theory  of  Numbers.  [Chap,  iv 

H.  F.  Baker"  extended  Sylvester's  theorem  to  any  modulus  N: 

A'  »-l  N  —  TTli 

where  the  m,  denote  the  integers  <A^  and  prime  to  A'",  N'N=1  (mod  r), 
and  \k\  is  the  least  positive  residue  modulo  r  of  k. 

Lerch^  extended  Mirimanofif's^^  formula  to  the  case  of  a  composite 
modulus  m.     Set 

m 
Let  a  belong  to  the  exponent  <f){m)/e.    Then  q{a,  m)=e'Za/^  (mod  m), 
where  /?  ranges  over  the  residues  of  the  incongruent  powers  of  a,  and 
wa+j3=0  (mod  a),  0^a<a.     As  an  extension  of  Sylvester's  theorem, 

T  T  ' 

q(a,  w)=2-=  —2-^     (mod  m), 

V  V 

where  v  ranges  over  the  integers  <  m  and  prime  to  m,  while 

7nr,-\-v=0,  wr/  — ?'=0  (mod  a),  0^r,<a,  0^r,'<a. 

For  m  =  mi. .  .rtik,  where  the  rrij  are  relatively  prime, 

k 

q{a,  m)  =  2  njn/(l){nj)q{a,  nij)  (mod  m), 

where  m  =  mjnj,  n/n/=l  (mod  mj). 

H.  Hertzer-'*  verified  that,  for  a<p<307,  a^^  — 1  is  di\'isible  by  p^  only 
for  a  =  68,  p  =  113;  a  =  3,  9,  p  =  ll.     He  examined  all  the  primes  between 
307  and  751,  but  only  for  a  and  p  —  a  when  a<y/p,  finding  only  p  =  113, 
a  =  68.     Removing  the  restriction  a<  Vp^  be  found  only  the  solutions 
p  =  ll,a  =  3;         p  =  331,  a  =  18,  71;         p  =  353,a  =  14; 
p  =  487,  a  =  10,  175;  p  =  673,  a  =  22, 
together  with  the  square  of  each  a. 

A.  Friedmann  and  J.  Tamarkine^^  gave  formulas  connecting  q^  with 
Bernoullian  numbers  and  [u/p]. 

A.  Wieferich^®  proved  that  if  x^+y^-\-z^  =  Q  is  satisfied  by  integers 
X,  y,  z  prime  to  p,  where  p  is  an  odd  prime,  then  2""^  =  1  (mod  p^).  Shorter 
proofs  were  given  by  D.  IMirimanoff-^  and  G.  Frobenius.'* 

D.  A.  Grave- ^  gave  the  residue  of  q^  for  each  prime  p<  1000  and  thought 
he  could  prove  that  2^  — 2  is  never  divisible  by  p^  (error,  Meissner^). 

A.  Cunningham^"  verified  that  2^  — 2  is  not  divisible  by  p^  for  any  prime 
p<  1000,  and^^  that  3^ - 3  is  not  divisible  by  p^  for  a  prime  p  =  2''3''+ 1<  100. 

W.  H.  L.  Janssen  van  Raay^^  noted  that  2^  —  2  is  not  divisible  by  p^  in 
general. 

»Proc.  London  Math.  Soc,  (2),  4,  1906,  131-5.        "Comptes  Rendus  Paris,  142, 1906,  35-38. 
"Archiv  Math.  Phys.,  (3),  13,  1908,  107.  «Jour.  fur  Math.,  135,  1909,  146-156. 

»Jour.  fiir  Math.,  136,  1909,  293-302.  "L'enseignement  math.,  11,  1909,  455-9. 

"Sitzungsber.  Ak.  Wiss.  Berlin,  1909,  1222-4;  reprinted  in  Jour,  fiir  Math.,  137,  1910,  314. 
**An  elementary  text  on  the  theory  of  numbers  (in  Russian),  I^ev,  1909,  p.  315;  Kiev  Izv.  Univ., 

1909,  Nos.  2-10. 
"Report  British  Assoc,  for  1910,  530.     L'interm^diaire  des  math.,  18,  1911,  47;  19,  1912,  159. 

Proc.  London  Math.  Soc,  (2),  8,  1910,  xiii. 
"L'interm^diaire  des  math.,  18,  1911,  47.     Cf.,  20,  1913,  206. 
"Nieuw  Archief  voor  Wiskunde,  (2),  10,  1912,  172-7. 


Chap.  IV]  RESIDUE   OF    {U^^  —  l)/p   MODULO   p.  Ill 

'    L.  Bastien^^"  verified  that  (1)  holds  for  p<  50  only  for  p=  43,  a=  19,  and 
for  Jacobi's^  cases.     He  stated  that,  if  p=  4p='=  1  is  a  prime, 

-1^2=1  +  1/3+1/5+... +1/(2/1-1)  (mod  p). 

W.  Meissner^^  gave  a  table  showing  the  least  positive  residue  of  (2'  —  l)/p 
modulo  p  for  each  prime  p<  2000,  where  t  is  the  exponent  to  which  2  belongs 
modulo  p.  In  particular,  2^  —  2  is  divisible  by  the  square  of  the  prime 
p  =  1093,  contrary  to  Proth^  and  Grave,^^  but  for  no  other  p<2000. 

In  the  chapter  on  Fermat's  last  theorem  will  be  given  not  only  the  con- 
dition q2—0  (mod  p)  of  Wieferich^^  but  also  q^^O  (mod  p),  etc.,  with  cita- 
tions to  D.  Mirimanoff,  Comptes  Rendus  Paris,  150,  1910,  204-6,  and  Jour, 
fiir  Math.,  139,  1911,  309-324;  H.  S.  Vandiver,  ibid.,  144,  1914,  314-8; 
G.  Frobenius,  Sitzungsber.  Ak.  Wiss.  Beriin,  1910,  200-8;  1914,  653-81. 
These  papers  give  further  properties  of  q^. 

P.  Bachmann^^  employed  the  identity 

(a-\-h-\-cy-{a+b-cy+ia-h-cy-{a-h+cy 
=  2(^)cl(a+6r-^-(a-6r-n+2(|)c^l(a+6r-^-(a-6r-^f  +  ... 
for  a  =  6  =  1,  c  =  2  or  1  to  get  expressions  for  ^2  or  q^,  whence 

for  an  odd  prime  p.    Comparing  this  with  the  value  of  (3^  — 3)/p  obtained 
by  expanding  (2+1)^,  we  see  that 

?!ll2_2P-i_^i.2P-2_|_i.2P-3+  , . .  +-^-2  (mod  p). 
p  p-l 

Again, 

92^2-  (^)  Vssfirn.(s-0  {-ly+'+'s  (mod  p), 
summed  for  all  sets  of  solutions  of  s^=f^+l  (mod  p).    Finally, 

g2^s'|(r''-r-'')2(r2'"'-l)-i|, 

h=l[  "  J 

where  r  is  a  primitive  pth  root  of  unity. 

*H.  Brocard^^  commented  on  a^^=l  (mod  p"").  *H.  G.  A.  Verkaart^^ 
treated  the  divisibility  of  a^  —  a  by  p.  E.  Fauquembergue^^  checked  that 
2^=2  (mod  p2)  for  p  =  1093. 

N.  G.  W.  H.  Beeger^^  tabulated  all  roots  of  a;^~^=  1  (mod  p^)  for  each 
prime  p<200.     If  w  is  a  primitive  root  of  p^,  the  absolutely  least  residue 

32aSphinx-Oedipe,  7,  1912,  4-6.     It  is  stated  that  G.  Tarry  had  verified  in  1911  that  2P-2  is 

not  divisible  by  a  prime  p  <  1013. 
"Sitzungsber.  Ak.  Wiss.  Berlin,  1913,  663-7. 
"Jour,  fiir  Math.,  142,  1913,  41-50. 

'^Revista  de  la  Sociedad  Mat.  Espanola,  3,  1913-4,  113-4. 
'"Wiskundig  Tijdschrift,  vol.  2,  1906,  238-240. 
"L'interm6diaire  des  math.,  1914,  33. 
38Messenger  Math.,  43,  1913-4,  72-84. 


112  History  of  the  Theory  of  Numbers.  [Chap,  iv 

±xi  modulo  p2  of  CO"  is  a  root,  that  (^xg)  of  xi^  is  a  second  root,  that  ( ^Xg) 
of  X1X2  is  a  third  root,  etc.,  until  the  root  =tx,  is  reached,  where  s  =  (p-l)/2. 
The  remaining  roots  are  p^-x,(i  =  l,. .  .,  s).     He  proved  that 

ixi...xy={-l)-^  (modp2). 
Hence  Xi. .  .x,=  ±l  if  p  =  4n+l. 

W.  Meissner^^  WTote  /i^  for  the  residue  <p'"  of  /i^""'  modulo  p"*.  When 
h  varies  from  1  to  p-1,  we  get  p-1  roots  h^  of  xP-^=1  (mod  p"*).  The 
product  of  the  roots  given  by  h  =  l,..  .,  (p-l)/2,  is  =(-1)'  or  (-l)V 
(mod  p'"),  according  as  p  =  4m-1  or  4n+l,  where  z  is  the  number  of  pairs 
of  integers  <p/2  whose  product  is  =  -1  (mod  p),  and  c  is  the  smaller  of 
the  two  roots  of  x^=-l  (mod  p).  No  number  <p  which  belongs  to  one 
of  the  exponents  2,  3,  4,  6,  modulo  p,  can  be  a  root  of  x^^=l  (mod  p^). 
A  root  of  the  latter  is  given  for  each  prime  p<  300,  and  a  root  modulo  p^  for 
each  p<200;  also  the  exponent  to  which  each  root  belongs. 

N.  Nielsen^^  noted  that,  if  we  select  2r  distinct  integers  a„  &,  (s  =  1, . . .  ,r) 
from  1, .  .  .,  p-1,  such  that  a,+b,  =  p,  then 

"^  =  i-mi-pA),   A^  ^  =  k^lg^-g^^'j     (modp). 

Proof  is  given  of  various  results  by  Lerch,-^  also  of  simple  relations  between 
Qa  and  BernouUian  numbers,  and  of  the  final  formula  by  Plana,^  here  attrib- 
uted to  Euler.^^ 

H.  S.  Vandiver^-  proved  that  there  are  not  fewer  than  [Vp]  and  not 
more  than  p  — (l  +  \/2p— 5)/2  incongruent  least  positive  residues  of 
1,  2^\...,  (p-l)^\  modulo  f. 

N.  Nielsen^^  noted  that,  if  a  is  not  di\'isible  by  the  odd  prime  p, 
a  — I    (p-3)/2  2 
9a=-^+    S^  2j^-ir'i5.(a^-'^-^-l)     (modp),  j 

gi+?2+...+gp-i=(-l)"-'5„+--l     (modp2),    n=ip-l)/2. 

V 

W.  Meissner^  gave  various  expressions  for  ^2  and  ^3. 

A.  G^rardin^^  found  all  primes  p<2000,  including  those  of  the  form 

2"—!,  for  which  52  is  sjTnmetrical  when  written  to  the  base  2. 

H.  S.  Vandiver^^  proved  that  52—0  (mod  p^)  if  and  only  if 

He  gave  various  expressions  for  (n*  — l)/m. 

"Sitzungsber.  Berlin  IMath.  Gesell.,  13,  1914,  96-107. 

"Ann.  sc.  I'^cole  norm,  sup.,  (3),  31,  1914,  171-9. 

"Euler,  Institutiones  Calculi  Diff.,  1755,  406.    Proof,  Math.  Quest.  Educ.  Times,  48, 1888,  48. 

«BuU.  AmQT.  Math.  Soc,  22,  1915,  61-7. 

«K)versigt  Danske  Vidensk.  SeLsk.  ForhandUnger,  1915,  518-9,  177-180;  cf.  Lerch's»»  N. 

♦^Mitt.  Math.  Gesell.  Hamburg,  5,  1915,  172-6,  180. 

«Xouv.  Ann.  Math.,  (4),  17,  1917,  102-8. 

"Annals  of  Math.,  18,  1917,  112. 


CHAPTER  V. 

EULER'S  (^-FUNCTION,  GENERALIZATIONS.  FAREY  SERIES. 
Number  <f){n)  of  Integers  <n  and  Prime  to  n. 

L.  Euler/  in  connection  with  his  generalization  of  Fermat's  theorem, 
investigated  the  number  <j){n)  of  positive  integers  not  exceeding  n  which  are 
relatively  prime  to  n,  without  then  using  a  functional  notation  for  0(n). 
He  began  with  the  theorem  that,  if  the  n  terms  a,  a-\-d,. .  .,  a+(n  — l)d 
in  arithmetical  progression  are  divided  by  n,  the  remainders  are  0,  1,. . ., 
n  — 1  in  some  order,  provided  d  is  prime  to  n;  in  fact,  no  two  of  the  terms 
have  the  same  remainder. 

If  p  is  a  prime,  (^(p"")  =p'"~^(p  — 1),  since  p,  2p,. .  .,  p^~^-p  are  the  only 
ones  of  the  p"*  positive  integers  ^  p^  not  prime  to  p"*.     To  prove  that 

(1)  <t>{AB)  =4>{A)<f>{B)  {A,  B  relatively  prime), 

let  1,  a, . . .,  CO  be  the  integers  <A  and  prime  to  A.    Then  the  integers 
<  AB  and  prime  to  A  are 

1  a     . . .                       CO 

A  +  1  A+a     ...                A+co 

2A  +  1  2A+a     ...              2A+C0 


(B-l)A+co. 


(5-l)A  +  l     {B-l)A-\-a 

The  terms  in  any  column  form  an  arithmetical  progression  whose  difference 
A  is  prime  to  B,  and  hence  include  <^(J5)  integers  prime  to  B.  The  number 
of  columns  is  (f>{A).  Hence  there  are  ({>{A)<f){B)  positive  integers  <AB, 
prime  to  both  A  and  B,  and  hence  prime  to  AB.  If  p, . . . ,  s  are  distinct 
primes,  the  two  theorems  give 

(2)  0(p\  .  .s«)=p^-np-l). .  .s'-\s-l). 

Euler^  later  used  ttN  to  denote  4>{N)  and  gave  a  different  proof  of  (2). 
First,  let  N  =  p'^q,  where  p,  q  are  distinct  primes.  Among  the  N—1  integers 
<A^  there  are  p""—!  multiples  of  q,  and  p'^'^q  —  l  multiples  of  p,  these  sets 
having  in  common  the  p"~^  — 1  multiples  of  pq.    Hence 

<^(iV)=iV-l-(p"-l)-(p'*-^g-l)+p"-i-l=p"-i(p-l)(g-l). 

A  simpler  proof  is  then  given  for  the  modified  form  of  (2) : 

(3)  iV(p-l)to-l)...(.-l), 

pq...s 
where  p,  q,  r, . . . ,  s  are  the  distinct  primes  dividing  N.    There  are  N/p 
multiples  <N  of  p  and  hence  N'  =  N{p  —  \)/p  integers  <A^  and  prime  to  p. 
Of  these,  N' /q  are  divisible  by  q;  excluding  them,  we  have  N"  =  N'{q  —  l)/q 
numbers  <  N  and  prime  to  both  p  and  q.     The  rth  part  of  these  are  said 

^Novi  Comm.  Ac.  Petrop.,  8,  1760-1,  74;  Comm.  Arith.,  1,  274,     Opera  postuma,  I,  492-3. 
«Acta  Ac.  Petrop.,  4  II  (or  8),  1780  (1755),  18;  Comm.  Arith.,  2,  127-133.     He  took  0(1)=O. 

113 


114  History  of  the  Theory  of  Numbers.  [Chap,  v 

[cf.  Poinsot^^]  to  be  divisible  by  r;  after  excluding  them  we  get  N"{r  —  l)/r 
numbers;  etc. 

Euler^  noted  in  a  posthumous  paper  that,  if  p,  q,  r  are  distinct  primes, 
there  are  r  multiples  ^pqr  of  pq,  and  qr  multiples  of  p,  and  a  single  multiple 
of  pqr,  whence 

<t){pqr)=pqr-qr-pr-pq-^r+p-\-q-l  =  {p-l){q-l){r-l). 

In  general,  if  M  is  any  number  not  divisible  by  the  prime  p,  and  if  fx 
denotes  the  number  of  integers  ^M  and  prime  to  M,  there  are  M—fj. 
integers  ^M  and  not  prime  to  M  and  hence  p''{M—ii)  integers  ^Mp"  and 
not  prime  to  M  and  therefore  not  prime  to  Mp".  Of  the  Mp^~^  multiples 
^Mp^  of  p,  exclude  the  p"~^(ilf  — /i)  which  are  not  prime  to  M;  we  obtain 
p^'V  multiples  of  p  which  are  prime  to  M.    Hence 

<^(p"M)=p"M-p''(M-m)-p""V  =  P"~Hp-1)m- 

A.  M.  Legendre^  noted  that,  if  ^, . . . ,  co  are  any  odd  primes  not  dividing 
A,  the  number  of  terms  of  the  progression  A+B,  2A+5, .  . . ,  nA-\-B  which 
are  divisible  by  no  one  of  the  primes  0, . . . ,  co  is  approximately  n(l  — 1/0) . . . 
(1  —  1/co),  and  exactly  that  number  if  n  is  divisible  by  0, . . . ,  co. 

C.  F.  Gauss^  introduced  the  symbol  (i>{N).  He  expressed  Euler's^  proof 
of  (1)  in  a  different  form.  Let  a  be  any  one  of  the  4>{A)  integers  <A  and 
prime  to  A,  while  j8  is  any  one  of  the  4>{B)  integers  <B  and  prime  to  B. 
There  is  one  and  but  one  positive  integer  x<AB  such  that  x=a  (mod  A), 
x=^  (mod  B).     Since  this  x  is  prime  to  A  and  to  B,  it  is  prime  to  AB. 

Making  the  agreement  that  <^(1)  =  1,  Gauss  proved 

(4)  20(d)  =N  {d  ranging  over  the  divisors  of  N). 

For  each  d,  multiply  the  integers  ^d  and  prime  to  d  by  N/d;  we  obtain 
S0(d)  integers  ^N,  proved  to  be  distinct  and  to  include  1,  2, . . .,  iV. 

A.  M.  Legendre^  proved  (3)  as  follows:  First,  let  N  =  pM,  where  p  is  a 
prime  which  may  or  may  not  divide  M;  then  Mp—M  of  the  numbers 
1,.  . .,  N  are  not  divisible  by  p.  Second,  let  N  =  pqM,  where  p  and  q  are 
distinct  primes.  Then  1,. .  .,  N  include  M  numbers  divisible  by  both  p 
and  q;  Mp  —  M  numbers  divisible  by  q  and  not  by  p;  Mq  —  M  numbers 
divisible  by  p  and  not  by  q.  Hence  there  remain  A^(l  —  l/p)(l  —  1/q)  num- 
bers divisible  by  neither  p  nor  q.  Third,  a  like  argument  is  said  to  apply 
to  N  =  pqrM,  etc. 

Legendre  (p.  412)  proved  that  ii  A,C  are  relatively  prune  and  if  0,X,ju, . . . , 
CO  are  odd  primes  not  dividing  A,  the  number  of  terms  kA  —  C{k  =  l,...,n), 
which  are  divisible  by  no  one  of  0, . .  . ,  to,  is 

*Tractatu3  de  numerorum,  Comm.  Arith.,  2,  515-8.     Opera  postuma,  I,  1862,  16-17. 

*Es3ai  siir  la  thdorie  des  nombres,  1798,  p.  14. 

'Disquisitionea  Arithraeticse,  1801,  Arts.  38,  39. 

•Th^orie  des  nombres,  ed.  2,  1808,  7-8;  German  trans,  of  ed.  3  by  Maser,  8-10. 


Chap.  V]  EulER's  (^-FUNCTION.  115 

where  the  summations  extend  over  the  combinations  of  6,. . .,  co  taken 
1,  2, .  .  .,  at  a  time,  while  Aq  is  a  positive  integer  <A  for  which  ^Aq+C  is 
divisible  by  A,  and  [x]  is  the  greatest  integer  ^x.  We  thus  derive  the 
approximation  stated  by  Legendre.^  Taking  A  =  l,  C  =  0  (p.  420),  we  see 
that  the  number  of  integers  ^n,  which  are  divisible  by  no  one  of  the  dis- 
tinct primes  0,  X, . .  . ,  co  is 

A.  von  Ettingshausen^  reproduced  without  reference  Euler's^  proof  of 
(3)  and  gave  an  obscurely  expressed  proof  of  (4) .  Let  A''  =  p'g" .  .  . ,  where 
p,  q,. . . are  distinct  primes.  Consider  first  only  the  divisors  d  =  p^q',  where 
/i>0,  v>0,  so  that  d  involves  the  primes  p  and  q,  but  no  others.    By  (3), 

^(d)=d(i-^)  (i-l),        J^ |^py=(p+p'+.  .  .+r)(g+. .  .+2"), 

S(^(pY)  =  (p"-i)(/-i). 

Similarly,  S0(p'')  =p"— 1.  In  this  way  we  treat  together  the  divisors  of  N 
which  involve  the  same  prime  factors.  Hence  when  d  ranges  over  all  the 
divisors  of  N, 

S<^(d)  =  l+S(p«-l)  +  S(p»-l)(g^-l)+  S  (p»-l)(5''_l)(r-_l)  +  ... 

P  P.Q  P.  3.  r 

=n]i+(p"-i)J=np"=iv, 
p 

where  the  summation  indices  range  over  the  combinations  of  all  the  prime 
factors  of  N  taken  1,  2, .  .  .at  a  time.     [Cf,  Sylvester .^^] 

A.  L.  Crelle^  considered  the  number  Zj  of  integers,  chosen  from  rii, .  . .  ,na, 
which  are  divisible  by  exactly  j  of  the  distinct  primes  Pi, . . .,  Pm',  and  the 
number  Sy  of  the  integers,  chosen  from  rii, .  .  . ,  n^,  which  are  divisible  by  at 
least  j  of  the  primes  Pi.    Then 

Z1  +  Z2+.  .  .+Zm  =  Si-S2  +  Ss-  .  .  .=tS^. 

Let  V  be  the  number  of  the  integers  rii, . . . ,  n^  which  are  divisible  by  no  one 
of  the  primes  Pi.    Then 

a^'Ezi+v,  p  =  a-Si+S2—  . .  .=PSrr,. 

In  particular,  take  nj, ...,  ria  to  be  1,  2,. ..,  iV,  where  N  =  p''q^r^ . . .,  and 

take  Pi,...,Pm  to  be  p,  g,  r, .  . . .     Then 

N    N  ^  N    N  ,  N  , 

P      q  pq     pr  '  pqr 

cf>{N)=N-s,-{-S2-  . . .  =n(i--^  0"-)  •  •  •• 

He  proved  (1)  for  5  =  0",  where  a  is  a  prime  not  dividing  A  (p.  40).  By 
Euler's^  table  there  are  B({)(A)  integers  <AB  and  prime  to  A.     In  Euler's 

^Zeitschrif t  fur  Physik  u.  Math,  (eds.,  Baumgartner  and  Ettmg8hauaen),Wien,  5, 1829,  287-292. 
*Abh.  Akad.  Wiss.  Berlin  (Math.),  1832,  37-50. 


116  History  of  the  Theory  of  Numbers.  (Chap,  v 

notation,  a{kA  +  l),  a{kA-\-a),. . .,  a{kA-{-03)  give  all  the  numbers  between 
kaA  and  {k-\-\)aA  which  are  divisible  by  a  and  are  prime  to  A.  Taking 
A:  =  0,  1,. . .,  a°~^  — 1,  we  see  that  there  are  exactly  a"~V(^)  multiples  of  a 
which  are  <AB  and  prime  to  A.    Hence 

0(a"^)  =aXA)  -o^-VU)  =(t>{a'')(f>{A). 

F.  Minding^  proved  Legendre's  formula  (5).  The  number  of  integers 
^n,  not  divisible  by  the  prime  0,  is  n  —  [n/d].  To  make  the  general  step 
by  induction,  let  Pi, . . . ,  Pk  be  distinct  primes,  and  denote  by  (5;  pi, . . . ,  p^) 
the  number  of  integers  ^  5  which  are  divisible  by  no  one  of  the  primes  pi, .  .  . , 
Pk'    Then,  if  p  is  a  new  prime, 

(B;  pi, . . . ,  Pk,  p)  =  {B;pu...,  Pk)  -  ([B/p] ;  Pi, .  •  • ,  Pk)- 

The  truth  of  (4)  for  the  special  case  N  =  p  —  1,  where  p  is  a  prime,  follows 
(p.  41)  from  the  fact  that  ((){d)  numbers  belong  to  the  exponent  d  modulo  p 
if  d  is  any  divisor  of  p  — 1. 

N.  Druckenmiiller^*^  evaluated  (f>{b),  first  for  the  case  in  which  6  is  a 
product  cd. .  .kl  oi  distinct  primes.  Set  h=^l  and  denote  by  \f/{h)  the  num- 
ber of  integers  <b  having  a  factor  in  common  with  6.  There  are  l\p{^) 
numbers  <  b  which  are  divisible  by  one  of  the  primes  c, . .  . ,  k,  since  there 
are  \p{P)  in  each  of  the  sets 

l,2,...,/3;    ^+1,...,2^;     ...;     (i-l)/3+l,. . .,  Z/3. 

Again,  I,  21,...,  pi  are  the  integers  <b  with  the  factor  I.  Of  these,  0(j3) 
are  prime  to  jS,  while  the  others  have  one  of  the  factors  c, . . . ,  k  and  occur 
among  the  above  lxl/{^).  Hence  xl/{b)=l\l/i^)-\-<i>iP).  But  i/'O3)+0(/3)=/3. 
Hence 

</,(6)  =  a-l)(A(^)  =  (c-l)...(Z-l). 

Next,  let  6  be  a  product  of  powers  oi  c,  d,. .  .,  I,  and  set  b  =  L^,  ^  =  cd. .  .1. 
By  considering  L  sets  as  before,  we  get 

E.  Catalan^^  proved  (4)  by  noting  that 

2(/,(py.  .  .)=n]i+(/)(p)+ . . .  +<f>(p'')\  =np»=Ar, 

where  there  are  as  many  factors  in  each  product  as  there  are  distinct  prime 
factors  of  N. 

A.  Cauchy^^  gave  without  reference  Gauss'^  proof  of  (1). 

E.  Catalan^^  evaluated  <t){N)  by  Euler's^  second  method. 

C.  F.  Arndt^"*  gave  an  obscure  proof  of  (4),  apparently  intended  for 
Catalan's.  ^^  It  was  reproduced  by  Desmarest,  Th^orie  des  nombres,  1852, 
p.  230. 

•Anfangsgriinde  der  Hoheren  Arith.,  1832,  13-15. 
»oTheorie  der  Kettenreihen . .  .Trier,  1837,  21. 
"Jour,  de  Mathdmatiques,  4,  1839,  7-8. 
i»Compte8  Rendus  Paris,  12,  1841,  819-821;  Exercices  d'analyse  et  de  phys.  math.,  Paris,  2, 

1841,  9;  Oeuvres,  (2),  12. 
"Nouv.  Ann.  Math.,  1,  1842,  466-7. 
"Archiv  Math.  Phys.,  2,  1842,  6-7. 


Chap.  V]  EuLER's  (^-FUNCTION.  117 

J.  A.  Grunert^^  examined  in  a  very  elementary  way  the  sets 
jk+1,    jk-{-2,...,    jk+k-l,     {j+l)k     (j  =  0,  l,...,p-l) 
and  proved  that  4){'pk)='p4){k)  if  the  prime  p  divides  k,  while  4){pk)  = 
(p  —  l)<j){k)  if  the  prime  p  does  not  divide  k.    From  these  results,  (2)  is 
easily  deduced  [cf.  Crelle^^  on  (f){Z)]. 

L.  Poinsot^®  gave  Catalan's^^  proof  of  (4)  and  proved  the  statements 
made  by  Euler^  in  his  proof  of  (3) .  Thus  to  show  that,  of  the  N'  =  N{1-  1/p) 
integers  <  N  and  prime  to  p,  exactly  N'/q  are  divisible  by  q,  note  that  the 
set  1,.  .  .,  N  contains  N/q  multiples  of  q  and  the  set  p,  2p, . . .  contains 
{N/p)/q  multiples  of  q,  while  the  difference  is  N'/q. 

If  P,  Q,  R,. . .  are  relatively  prime  in  pairs,  any  number  prime  to 
N  =  PQR . . .  can  be  expressed  in  the  form 

pQR...+qPR...+rPQ...  +  ..., 

where  p  is  prime  to  P,  q  to  Q,  etc.  If  also  p<P,  q<Q,  etc.,  no  two  of  these 
sums  are  equal.  Thus  there  are  0(P)0(Q) . . .  such  sums  [certain  of  which 
may  exceed  N]. 

To  prove  (4),  take  (pp.  70-71)  a  prime  p  of  the  form  kN+l  and  any  one 
of  the  N  roots  p  of  a;^=  1  (mod  p).  Then  there  is  a  least  integer  d,  sl  divisor 
of  N,  such  that  p'^=  1  (mod  p).  The  latter  has  (f)(d)  such  roots.  Also  p  is  a 
primitive  root  of  the  last  congruence  and  of  no  other  such  congruence  whose 
degree  is  a  divisor  of  N. 

A.  L.  Crelle^^  considered  the  product  E  =  eie2. .  .e„  of  integers  relatively 
prime  in  pairs,  and  set  Ej  =  E/ej.  When  x  ranges  over  the  values  1, . . .,  Ci, 
the  least  positive  residue  modulo  E  of  EiXi-\- . . .  +£'„a;„  takes  each  of  the 
values  1, .  .  .,E  once  and  but  once.  In  case  Xi  is  prime  to  ei  for  i  =  1, . .  . ,  n, 
the  residue  of  SE'^Xi  is  prime  to  E  and  conversely.  Let  dn,  di2, ...  be  any 
chosen  divisors  >1  of  e^  which  are  relatively  prime  in  pairs.  Let  \}/{ei) 
denote  the  number  of  integers  ^e^  which  are  divisible  by  no  one  of  the 
^ti,  di2,. .  ..  Let  yl/{E)  be  the  number  of  integers  ^E  which  are  divisible 
by  no  one  of  the  dn,  di2,  c^2ij  ■  ■  •>  including  now  all  the  d's.  Then  \1/{E)  = 
^(ei) .  .  .  i/'(en).  In  case  dn,  di2, .  .  .  include  all  the  prime  divisors  >  1  of  e,-, 
ypie^  becomes  ^(e^).  Of  the  two  proofs  (pp.  69-73),  one  is  based  on  the 
j&rst  result  quoted,  while  the  other  is  like  that  by  Gauss .^ 

As  before,  let  ^{y)  be  the  number  of  integers  '^y  which  are  divisible  by 
no  one  of  certain  chosen  relatively  prime  divisors  di,...,dm  of  y.  By  con- 
sidering the  xy  numbers  ny-\-r  (0^n<x,  I'^r^y),  it  is  proved  (p.  74)  that, 
when  X  and  y  are  relatively  prime, 

ypixy)  =x\p{y),  \p2ixy)  =  {x-l)xl/{y), 

where  \p2{^y)  is  the  number  of  integers  ^xy  which  are  divisible  neither  by 
X  nor  by  any  one  of  the  d's.     These  formulas  lead  (pp.  79-83)  to  the  value 
of0(Z).     Set 
Z  =  p/'...p/M,  z  =  Pi...p^,  n  =  Z/z, 

i^Archiv.  Math.  Phys.,  3,  1843,  196-203. 

"Jour,  de  Math^matiques,  10,  1845,  37-43. 

"Encyklopadie  der  Zahlentheorie,  Jour,  fiir  Math.,  29,  1845,  58-95. 


118  History  of  the  Theory  of  Numbers.  [Chap,  v 

where  Pi,.  . .,  p^  are  distinct  primes.  For  a  prime  p,  not  dividing  y,  we 
have  (f){py)  =  {p- l)<t>iy) .    Take  y  =  Pi,  p  =  P2',  then 

<^(PiP2)  =  (Pi-l)(P2-l)- 
Next,  take  y  =  PiP2,  P  =  P3}  and  use  also  the  last  result;  thus 

<A(PlP2P3)  =  (Pl-l)(P2-l)(P3-l), 

and  similarly  for  <t){z).  When  f  ranges  over  the  integers  <  z  and  prime  to  z, 
the  numbers  vz-\-^  {v  =  0,l,. .  .,n  —  l)  give  without  repetition  all  the  integers 
<Z  and  prime  to  Z.  Hence  (f>{Z)=n<i){z),  which  leads  to  (2).  [Cf.  Guil- 
min,2^  Steggall.'^^] 

The  proofs  of  (4)  by  Gauss^  and  Catalan^  ^  are  reproduced  without  refer- 
ences (pp.  87-90).  A  third  proof  is  given.  Set  N  =  a''h^c' . .  .,  where  a,  b, 
c, . . .  are  distinct  primes.  Consider  any  divisor  e  =  b^"^' .  . .  of  N  such  that  e 
is  not  divisible  by  a.    Then 

<t>i€a'')=a''-\a-l)(t>{e). 

Sum  for  /b  =  0,  1, . . . ,  a;  we  get  a°0(e).  When  k  ranges  over  its  values  and 
/3i  over  the  values  0,  1,. .  .,  j3,  and  71  over  the  values  0,  1,.  . .,  7,  etc.,  ea* 
ranges  over  all  the  divisors  d  of  iV.  Hence  20  (d)  =a''S0(e).  Similarly,  if 
Ci  range  over  the  divisors  not  divisible  by  a  or  b, 

S</)(€)=6^(^(ei),.  .  .,  S<^(d)=a»6^  .  .  =N. 

E.  Prouhet^^  proposed  the  name  indicator  and  symbol  i{N)  for  0(iV). 
He  gave  Gauss'  proof  of  (1)  and  Catalan's  proof  of  (4).  If  5  is  the  product 
of  the  distinct  prime  factors  common  to  a  and  b, 

<j>{ab)  =(i>{a)(}>ib)8/(f){8). 

As  a  generalization,  let  5^  be  the  product  of  the  distinct  primes  common  to 
i  of  the  numbers  Oi, . . . ,  a„;  then 

2         §  2  2  n-l 

</)(ai. .  .a„)  =<j>{ai) . .  .<f){an)      ^        ^ 


Friderico  Arndt^^  proved  (1)  by  showing  that,  if  x  ranges  over  the 
integers  <A  and  prime  to  A,  while  y  ranges  over  the  integers  <B  and  prime 
to  B,  then  Ay-{-Bx  gives  only  incongruent  residues  modulo  AB,  each  prime 
to  AB,  and  they  include  every  integer  <AB  and  prime  to  AB.  [Crelle's^^ 
first  theorem  for  n  =  2.] 

V.  A.  Lebesgue^°  used  Euler's^  argument  to  show  that  there  are 

Nip-l){q-l)...{k-l) 
p-q. .  .k 

integers  < iV  and  prime  top,q,...,k,  the  latter  being  certain  prime  divisors 
of  A''  [Legendre,^  Minding^]. 

"Nouv.  Ann.  Math.,  4,  1845,  75-80. 
"Jour,  fur  Math.,  31,  1846,  246-8. 
"Nouv.  Ann.  Math.,  8,  1849,  347. 


Chap.  V]  EuLER's  0-FuNCTION.  119 

G.  L.  Dirichlet^^  added  equations  (4)  for  iV  =  n, . . .,  2,  1,  noting  that, 
if  s^n,  4>(s)  occurs  in  the  new  left  member  as  often  as  there  are  mul- 
tiples ^n  oi  s.    Hence 


i\-']<f>{s)=Un'+n). 

s=lLsJ 


The  left  member  is  proved  equal  to  XxJ/in/s],  where 

It  is  then  shown  that  \p{n)  —Zn^/ir^  is  of  an  order  6i  magnitude  not  exceed- 
ing that  of  n\  where  2>5>7>1,7  being  such  that 

8=2  S^ 

P.  L.  Tchebychef^^  evaluated  0(n)  by  showing  that,  if  p  is  a  prime  not 
dividing  A,  the  ratio  of  the  number  of  integers  ^  pAN  which  are  prime  to  A 
to  the  number  which  are  prime  to  both  A  and  p  is  p:p  —  l. 

A.  Guilmin^^  gave  Crelle's^^  argument  leading  to  0(Z). 

F.  Landry ^^  proved  (3).  First,  reject  from  1, .  .  .,  iV  the  N/p  multiples 
of  p;  there  remain  A^(l  —  1/p)  numbers  prime  to  p.  Next,  to  find  how  many 
of  the  multiples  q,  2q, . . . ,  N  of  q  are  prime  to  p,  note  that  the  coefficients 
1,  2, . .  .,  N/q  contain  N/q-{l  —  l/p)  integers  prime  to  p  by  the  first  result, 
applied  to  the  multiple  N/q  of  p  in  place  of  N. 

Daniel  Augusto  da  Silva^^  considered  any  set  S  of  numbers  and  denoted 
by  S{a)  the  subset  possessing  the  property  a,  by  S{ab)  the  subset  with  the 
properties  a  and  b  simultaneously,  by  {a)S  the  subset  of  numbers  in  S 
not  having  property  a;  etc.     Then 

{a)S  =  S-S{a)=S\l-{a)\, 

symbolically.     Hence 

(ha)S  =  {b)\(a)S\=S\l-{a)\\l-{h)\, 
{. .  .cba)S^S\l-{a)\\l-{b)\  \l-{c)\  . . .. 

A  proof  of  the  latter  symbohc  formula  was  given  by  F.  Horta.^^" 
With  Silva,  let  *S  be  the  set  1,  2, . . . ,  n,  and  let  A,  ^, . . .  be  the  distinct 
prime  factors  of  n.     Let  properties  a,  6, ...  be  divisibility  by  A,B,. . ..    Then 
there  are  n/A  terms  in  >S(a),  n/{AB)  terms  in  S{ab), . . .,  and  <f){n)  terms  in 
( . .  .cba)S.    Hence  our  symbolic  formula  gives 


*(«)=»(l-i)(l-|). 


"Abhand.  Ak.  Wiss.  Berlin  (Math.),  1849,  78-81;  Werke,  2,  60-64. 

^^Theorie  der  Congruenzen,  1889,  §7;  in  Russian,  1849. 

«Nouv.  Ann.  Math.,  10,  1851,  23. 

^^Troisieme  mlmoire  siir  la  th^orie  des  nombres,  1854,  23-24. 

'^Proprietades  geraes  et  resoluQao  directa  das  Congruencias  binomias,  Lisbon,  1854.  Report 
on  same  by  C.  Alasia,  Rivista  di  Fisica,  Mat.  e  Sc.  Nat.,  Pavia,  4,  1903,  i3-17;  reprinted 
in  Annaes  Scientificos  Acad.  Polyt.  do  Porto,  Coimbra,  4,  1909,  166-192. 

""Annaes  de  Sciencias  e  Lettras,  Lisbon,  1,  1857,  705. 


120  History  of  the  Theory  of  Numbers.  [Chap,  v 

E.  Betti^®  evaluated  0(w),  where  7n  is  a  product  of  powers  of  the  distinct 
primes  ai,  a2>  •  ••  •  Consider  the  set  Ci  of  the  products  of  the  a's  taken  i  at 
a  time  and  their  multiples  ^m.     Thus  Co  is  1, . . . ,  w,  while  C2  is 

0x02,  20x02, .  .  ., 0102;        cii(h,  2ai03,  .  .  ., diCis',-  ■  •• 

aia2  ttitta 

Let  X  be  an  integer  <  w  divisible  by  ai, . . . ,  a„.     Then  x  occurs 

times  in  the  sets  Co,  C2,  C4, . .  . ;  and  2"  ^  times  in  Ci,  C3, . . ..     Summing 

l-(l)  +  (2)-(3)+       =0 

for  each  of  the  m—<t>{m)  integers  ^m  having  factors  in  common  with  m, 
we  get 

m-0(m)-s(j)+s(2)-...=O. 

But  ^i-i)  is  the  niunber  of  integers  having  in  common  with  m  one  of  the 
factors  ai,  02, . . .,  and  hence  equals  S— .     Next,  ^i^j  ^^  *^®  number  of 

integers  having  in  common  with  m  one  of  the  factors  0102,  OiOa, .  . . ,  and  hence 

equals  2  { m/  (0102) } .     Thus 

mm 
4>{m)  =m—L, — 1-2 .... 

R.  Dedekind^^  gave  a  general  theorem  on  the  inversion  of  functions  (to 
be  explained  in  the  chapter  on  that  subject),  which  for  the  special  case  of 
</)(n)  becomes  a  proof  like  Betti's.  Cf.  Chrystal's  Algebra,  II,  1889,  511; 
Mathews'  Theory  of  Numbers,  1892,  5;  Borel  and  Drach,^^  p.  27. 

J.  B.  Sturm^^  evaluated  4>{N)  by  a  method  which  will  be  illustrated  for 
the  case  N  =  \b.  From  1,. . .,  15  delete  the  five  multiples  of  3.  Among 
the  remaining  ten  numbers  there  are  as  many  multiples  of  5  as  there  are 
multiples  of  5  among  the  first  ten  numbers.  Hence  <^(15)  =  10—2  =  8. 
The  theorem  involved  is  the  following.     From  the  three  sets 

1,  2,  3,*  4,  5;  6,*  7,  8,  9,*  10;  11,  12,*  13,  14,  15* 

delete  (by  marking  with  an  asterisk)  the  multiples  of  3.  The  numbers 
11,  13,  14  which  remain  in  the  final  set  are  congruent  modulo  5  to  the  num- 
bers 6,  3,  9  deleted  from  the  earUer  sets. 

J.  Liouville"  proved  by  use  of  (4)  that,  for  |x|<l, 

g  <t>{m)x"'  _      X  (t>{m)x"'_  g  <f>{m)x'^  _x{l-\-x^) 

m^ll-X"^    ~{l-xf  l-X^^'m^ll+X""    ~{1-Xy    ' 

"Bertrand's  Alg^bre,  Ital.  transl.  with  notes  by  Betti,  Firenze,  1856,  note  5.    Proof  reproduced 

by  Fontebasso'*,  pp.  74-77.  ^ 

"Jour,  fvir  Math.,  54,  1857,  21.     Dirichlet-Dedekind,  Zahlentheohe,  §138.  ^ 

"Archiv  Math.  Phys.,  29,  1857,  448-452. 
"Jour,  de  math6matiques,  (2),  2,  1857,  433^40. 


Chap.  V]  EuLER's  (^-FUNCTION.  121 

where  m  in  S'  ranges  only  over  the  positive  odd  integers.  The  final  fraction 
equals  x-\-Sx^-{-5x^+ ....  From  the  coefficient  of  x^  in  the  expansion  of 
the  third  sum,  we  conclude  that,  if  n  is  even, 

where  d  ranges  over  all  the  divisors  of  n.  Let  5i  range  over  the  odd  values 
of  5,  and  82  over  the  even  values  of  5;  then 


0-0- 


the  value  n/2  following  from  (4).  Another,  purely  arithmetical,  proof  is 
given.    Finally,  by  use  of  (4),  it  is  proved  that,  if  s>2, 

n=l     /fr  n=l/t 

A.  Cayley3°  discussed  the  solution  for  N  of  <^(iV)  =  iV^  Set  N  =  a^b^ ..., 
where  a,  6, . . .  are  distinct  primes.     Multiply 

l  +  (a-l)  \a\+a{a-l)  \a^\  +  ...  +a''-\a-l)  {a"}  +  . . . 

by  the  analogous  series  in  5,  etc. ;  the  bracketed  terms  are  to  be  multiplied 
together  by  enclosing  their  product  in  a  bracket.  The  general  term  of  the 
product  is  evidently 

Hence  in  the  product  first  mentioned  each  of  the  bracketed  numbers  which 
are  multiplied  by  the  coefficient  N'  will  be  a  solution  N  of  <f){N)=N'.  We 
need  use  only  the  primes  a  for  which  a  — 1  divides  N',  and  continue  each 
series  only  so  far  as  it  gives  a  divisor  of  N'  for  the  coefficient  of  a"~^(a  — 1). 

V.  A.  Lebesgue^^  proved  4){Z)=n4>{z)  as  had  Crelle^^  and  then  4>{z) 
=n(pi  —  1)  by  the  usual  method  of  excluding  multiples  of  pi, . .  . ,  p„  in  turn. 
By  the  last  method  he  proved  (pp.  125-8)  Legendre's  (5),  and  the  more 
general  formula  preceding  (5). 

J.  J.  Sylvester^^  proved  (4)  by  the  method  of  Ettingshausen,'  using  (2) 
instead  of  (3) .  By  means  of  (4)  he  gave  a  simple  proof  of  the  first  formula 
of  Dirichlet;^^  call  the  left  member  u^',  since  [n/r]  — [(n  — l)/r]  =  l  or  0, 
according  as  n  is  or  is  not  divisible  by  r, 

v^^J^  w(n+l) 

The  constant  c  is  zero  since  Ui  =  l.     He  stated  the  generalization 


2{*(i')(l-+2-+...  +  [?]'")}^ 


r+2'-+...+n'". 


He  remarked  that  the  theorem  in  its  simplest  form  is 

"London  Ed.  and  Dublin  Phil.  Mag.,  (4),  14,  1857,  539-540. 

"Exercicea  d'analyse  niim^rique,  1859,  43-45. 

"Quar.  Jour.  Math.,  3,  1860,  186-190;  CoU.  Math.  Papers,  2,  225-8. 


122  History  of  the  Theory  of  Numbers.  [Chap,  v 

the  example  given  being  r  =  2,  n  =  4,  whence  the  divisors  of  n  are  1-1, 2-1, 4-1, 
1-2,  2-2,  1-4  and  the  above  terms  are 

Ml,         Ml,         M-2,        2-M,        2M,        4-21, 

with  the  sum  4".     [With  this  obscure  result  contrast  that  by  Cantor/^] 

G.  L.  Dirichlet^^  completed  by  induction  Euler's^  method  of  proving  (3), 
obtaining  at  the  same  time  the  generalization  that,  if  p,  g, .  . . ,  s  are  divisors, 
relatively  prime  in  pairs,  of  N,  the  number  of  integers  ^  N  which  are  divi- 
sible by  no  one  of  p, .  .  . ,  s  is 


H-;)04)  0--.) 


A  proof  (§13)  of  (4)  follows  from  the  fact  that,  if  d  is  a  divisor  of  N,  there 
are  exactly  </)(d)  integers  ^N  having  with  N  the  g.  c.  d.  N/d. 

P.  A.  Fontebasso^^  repeated  the  last  remark  and  gave  Gauss'  proof  of  (1). 

E.  Laguerre^^  employed  any  real  number  k  and  integer  m  and  wrote 
(m,  m/k)  for  the  number  of  integers  ^m/k  which  are  prime  to  m.  By 
continuous  variation  of  k  he  proved  that 

i:{d,d/k)  =  [m/k], 

where  d  ranges  over  the  divisors  of  m.     For  k  =  l,  this  reduces  to  (4). 

F.  Mertens^^  obtained  an  asymptotic  value  for  0(1)+ . . .  +(i>iG)  for  G 
large.     He  employed  the  function  ju(n)  [see  Ch.  XIX]  and  proved  that 

I  0(m)  =  |  S  /.(n){r^lVr^l  U^GHA 
TO=i  n=i         iLnJ       Lnjj      tt 

|A|<G(ilog,G+iC+f)  +  l, 

where  C  is  Euler's  constant  0.57721 ....     This  upper  limit  for  A  is  more 
exact  than  that  by  Dirichlet.^^ 

T.  Pepin^''  stated  that,  if  n  =  a"6^. .  .  (a,  6, . .  .distinct  primes), 

n=0(n)+2a»-V(^)  +2a-V-^(/,(^)  +  .  . .  -\-a-'b'-'. . .. 

Moret-Blanc^^  proved  the  latter  by  noting  that  the  first  sum  is  the  num- 
ber of  integers  <  n  which  are  divisible  by  a  single  one  of  the  primes  a,  6, ... , 
the  second  sum  is  the  number  of  integers  <  n  divisible  by  two  of  the  primes, 
. . .,  while  a''~^6^~\  .  .  is  the  number  of  integers  <n  divisible  by  all  those 
primes. 

H.  J.  S.  Smith^^  considered  the  m-rowed  determinant  A„,  having  as  the 
element  in  the  ith.  row  and  jth  column  the  g.  c.  d.  {i,  j)  of  i,  j.     Let  li  =  m, 

"Zahlentheorie,  §11,  1863;  ed.  2,  1871;  ed.  3,  1879;  ed.  4,  1894. 

»*Saggio  di  una  introd.  all'arit.  trascendente,  Treviso,  1867,  23-26. 

«BuU.  Soc.  Math.  France,  1,  1872-3,  77. 

»«Jour.  fiir  Math.,  77,  1874,  289-91. 

»'Nouv.  Ann.  Math.,  (2),  14,  1875,  276. 

"/bid.,  p.  374.     L.  Gegenbauer,  Monatsh.  Math.  Phys.,  4,  1893,  184,  gave  a  generahzation  to 

primary  complex  numbers. 
»»Proc.  London  Math.  Soc,  7,  1875-6,  208-212;  CoU.  Papers,  2,  161. 


Chap.  V]  EulER's  </)-FuNCTION.  123 

I2,  I3,...  be  those  divisors  of  m  =  p  V  •  •  •  ^'^  which  are  given  by  the  expansion 
of  the  product 

0(m)  =  (p--p"-i). .  .(r-r-i)=?i-Z2+?3-. .  .-^v 

It  is  proved  that 

cl>{m,  k)  =  {k,  k)-{k,  k)+  .  .  .  -{I,  k) 

[called  Smith's  function  by  Lucas/^  p.  407]  is  zero  if  k<m,  but  equals 
<^(m)  if  ^  =  m.  Hence  if  to  the  mth  column  of  A^  we  add  the  columns  with 
indices  I3,  I5,...  and  subtract  the  columns  with  indices  I2,  Z4, .  .  . ,  we  obtain 
an  equal  determinant  in  which  the  elements  of  the  mth  column  are  zero 
with  the  exception  of  the  element  <^(m).     Hence  A^=A^_i0(m),  so  that 

(6)  A^=<^(l)0(2)...0(m). 

If  we  replace  the  element  5  =  {i,  j)  by  any  function  /(5)  of  5,  we  obtain  a 
determinant  equal  to  i^(l) . .  .F{m),  where 

nm)=/W-2/g)+S/g)-.,.. 

Particular  cases  are  noted.  For /(§)  =  S''',  F{m)  becomes  Jordan's^""  func- 
tion Jkitn).  Next,  if /(5)  is  the  sum  of  the  kth.  powers  of  the  divisors  of  5, 
then  F{m)='m!'.  Finally,  if  /(6)  =  1^+2^+  . . .  +5^,  it  is  stated  erroneous- 
ly that  F{m)  is  the  sum  (i>k{'m)  of  the  /bth  powers  of  the  integers  ^m  and 
prime  to  m.  [Smith  overlooked  the  factors  o!',  oJ^h^, ...  in  Thacker's^^°  first 
expression  for  <l>k{n),  which  is  otherwise  of  the  desired  form  F{n).  The 
determinant  is  not  equal  to  4>k{\) . .  .^^kim),  as  the  simple  case  k  =  l,  w  =  2, 
shows.] 

In  the  main  theorem  we  may  replace  1,. . .,  m  by  any  set  of  distinct 
numbers  jui, . . . ,  jU;„  such  that  every  divisor  of  each  ju,  is  a  number  of  the 
set;  the  determinant  whose  element  in  the  ith.  row  and  jth  column  is/(5), 
where  5  =  (jUi,  /xy),  equals  F()Ui) . .  .F{fx^.  Examples  of  sets  of  ^t's  are  the 
numbers  in  their  natural  order  with  the  multiples  of  given  primes  rejected; 
the  numbers  composed  of  given  primes;  and  the  numbers  without  square 
factors. 

R.  Dedekind^°  proved  that,  if  n  be  decomposed  in  every  way  into  a 
product  ad,  and  if  e  is  the  g.  o,.  &.  oi  a,  d,  then 

S|.#,(6)  =  nn(l+^), 

where  a  ranges  over  all  divisors  of  n,  and  p  over  the  prime  divisors  of  n. 

P.  Mansion^^  stated  that  Smith's  relation  (6)  yields  a  true  relation  if  we 
replace  the  elements  1,2,. .  .of  the  determinant  A^  by  any  symbols  Xi,X2,. . ., 
and  replace  0(m)  by  Xi^—Xi^-\-Xi  —  ....  [But  the  latter  is  only  another 
form  of  Smith's  F{m)  when  we  write  x^  for  Smith's /(5),  so  that  the  generali- 
zation is  the  same  as  Smith's.] 

"Jour,  ftir  Math.,  83,  1877,  288.     Cf.  H.  Weber,  Elliptische  Functionen,  1891,  244-5;  ed.  2, 

1908  (Algebra  III),  234-5. 
^Messenger  Math.,  7,  1877-8,  81-2. 


124  History  of  the  Theory  of  Numbers.  [Chap,  v 

P.  Mansion^^  proved  (6),  showing  that  </>(m,  k)  equals  <f){m)  or  0,  accord- 
ing as  m  is  or  is  not  a  divisor  of  k.  [Cf.  Bachmann,  Niedere  Zahlentheorie, 
I,  1902,  97-8.]  He  repeated  his*^  ''generalization."  He  stated  that  if  a 
and  b  are  relatively  prime,  the  products  of  the  0(a)  numbers  <a  and  prime 
to  a  by  the  numbers  <b  and  prime  to  b  give  the  numbers  <ab  and  prime 
to  ab  [false  for  a  =4,  6  =  3;  cf.  Mansion^].  His  proof  of  (4)  should  have 
been  credited  to  Catalan." 

E.  Catalan^^  gave  a  condensation  and  shght  modification  of  Mansion's*' 
paper.  C.  Le  Paige  {ibid.,  pp.  176-8)  proved  ]\Iansion's^  theorem  that 
every  product  equals  a  determinant  formed  from  the  factors. 

P.  IMansion""  proved  that  the  determinant  |cy|  of  order  n  equals  rriX2-  •  x^ 
if  Cij=Xxp,  where  p  ranges  over  the  divisors  of  the  g.  c.  d.  of  i,  j.  To  obtain 
a  "generahzation"  of  Smith's  theorem,  set  Zi  =  Xi,  Z2  =  Xi-{-X2,.  . .,  Zi=J>Xi, 
where  d  ranges  over  all  the  divisors  of  i.     Solving,  we  get 

where  the  Vs  are  defined  above.^^    Thus  each  Cy  is  a  z.     For  example,  if  n  =  4,   - 


21 

21 

2l 

2l 

Zi 

22 

2l 

22 

Zl 

2l 

23 

2i 

21 

22 

2l 

24 

Cii    = 


For  Zi  =  i,  Xi  becomes  4>{i)  and  we  get  (6).  [As  explained  in  connection  with 
Mansion's*^  first  paper,  the  generaUzation  is  due  to  Smith.] 

J.  J.  Sylvester^^  called  (/)(7i)  the  totient  T{n)  of  n,  and  defined  the  totitives 
of  n  to  be  the  integers  <  n  and  prime  to  n. 

F.  de  Rocquigny^^  stated  that,  if  ^"(A^)  denotes  <l)\(i>{N)\ ,  etc., 

if  A^  is  a  prime  and  m>2,  p>2.  He  stated  incorrectly  (ibid.,  50,  1879,  604) 
that  the  number  of  integers  ^  P  which  are  prime  to  N  =  a^b^ ...  is  P(l  —  1/a) 
(1-1/6).... 

A.  Minine*^  noted  that  the  last  result  is  correct  for  the  case  in  which 
P  is  divisible  by  each  prime  factor  a,  b,.  . .  oi  N.     He  wrote  symbolically 

nE—  for  [n/x],  the  greatest  integer  ^n/x.    By  deleting  from  1, .  . .,  P  the 

[P/a]  numbers  di\'isible  by  a,  then  the  multiples  of  6,  etc.,  we  obtain  for 
the  number  of  integers  ^  P  which  are  prime  to  N  the  expression 

[equivalent  to  (5)].     If  N,  N',  N", ...  are  relatively  prime  by  twos, 

♦^Annalea  de  la  Soc.  Sc,  Bru.\elles,  2,  II,  1877-8,  211-224.     Reprinted  in  Mansion's  Sur  la 

th^orie  des  nombres,  Gand,  1878,  §3,  pp.  3-16. 
"Nouv.  Corresp.  Math.,  4,  1878,  103-112. 
«Bull.  Acad.  R.  Sc.  de  Belgique,  (2),  46,  1878,  892-9. 

«Amer.  Jour.  Math.,  2,  1879,  361,  378;  Coll.  Papers,  3,  321,  337.     Nature,  37,  1888,  152-3. 
«Le8  Mondes,  Revue  Hebdom.  des  Sciences,  48,  1879,  327. 
"Ibid.,  51,  1880,  333.     Math.  Soc.  of  Moscow,  1880.    Jour,  de  math.  616in.  et  sp^c,  1880,  278. 


Chap.  V]  EulEr's  0-FunCTION.  125 

cf>{N)p-(t>iN')prcf>{N")p.. . .  =<f>(NN'Nr  .  .)p'P'P". . .. 
E.  Lucas^^  stated  and  Radicke  proved  that 

a=l  fc=2  0=1  k=2 

if  ^(a,  n)  is  the  number  of  integers  >  a,  prime  to  a  and  ^  n. 
H.  G.  Cantor^^  proved  by  use  of  ^-functions  that 

Svo^-Vr".  .  .vU24>M4>{vi) .  .  .0(^-i)  ^n", 
summed  for  all  distinct  sets  of  positive  integral  solutions  Vq,...,  v^^i  of 
Vq.  .  .Vp=n,  and  noted  that  this  result  can  be  derived  from  the  special  case  (4). 
0.  H.  MitchelP°  defined  the  a-totient  Taik)  of  k^a'b''. . . (where  a,h,.  .. 
are  distinct  primes)  to  be  the  number  of  integers  <k  which  are  divisible 
by  a,  but  by  no  one  of  the  remaining  prime  factors  6,  c, ...  of  k.  Similarly, 
the  a6-totient  Tabik)  of  k  is  the  number  of  integers  <k  which  are  divisible 
by  a  and  h,  but  not  by  c, . . . ;  etc.     If  /c  =  a'6V, 

tM  =a'-V(&V),         Ta,{k)=a'-'h--'<t>{c^),         TaUk)=a'-'b^-'c'-\ 
<f>ik)  +2t,(A;)  +2ra,(/c)  +Tadk)  =  k. 

3  3 

If  a  contains  the  same  primes  as  s,  but  with  the  same  exponents  as  in  k,  so 
that  o-  =  a'  if  s  =  a,  it  is  stated  (p.  302)  that 


■w=i*a- 


C.  Crone"  evaluated  (^(n)  by  an  argument  valid  only  when  n  is  a  product 
of  distinct  primes  Pi,...,Pq.  The  number  of  integers  <n  having  a  factor 
in  common  with  n  is  then 

A=2(ii-l)-s(^-l)  +  ...+(-l).2(^^ 1). 

The  sum  of  the  second  terms  of  each  sum  is 

-(0+a)-  -(-^)'G^)=-i-(-i)"- 

Hence  the  number  of  integers  <n  and  prime  to  n  is 

n-\-A=n-l^—^^— —  . . .  -(-1)«S +(-!)« 

Pi  V\V2  Pl-Pg-l 

provided  n  =  pi. .  .pg.     [To  modify  the  proof  to  make  it  vahd  for  any  n, 
we  need  only  add  to  A  the  term 

and  hence  replace  (-1)*'  by  (-l)%/(pi. .  .p^)  in  n-l-A.] 

*8Nouv.  Corresp.  Math.,  6,  1880,  267-9.     Also  Lucas, '^  p.  403. 
"Gottingen  Nachrichten,  1880,  161;  Math.  Ann.,  16,  1880,  583-8. 
"Amer.  Jour.  Math.,  3,  1880,  294. 
"Tidsskrift  for  Mathematik,  (4),  4,  1880,  158-9. 


126  History  of  the  Theory  of  Numbers.  [Chap.  V 

Franz  Walla^-  considered  the  product  P  of  the  first  n  primes  >  1.  Let 
a*i, . .  . ,  X,  be  the  integers  <P/2  and  prime  to  P,  so  that  v=4>{P)/2.  Then, 
if  n>2,  half  of  the  x's  are  =1  (mod  4)  and  the  others  are  =3  (mod  4). 
Also,  the  absolute  values  of  \P  —  2Xj  (j  =  1, .  .  . ,  v)  are  the  a:'s  in  some  order. 
Half  of  the  a:'s  are  <P/4. 

J.  Perott^^  proved  that 

the  context  showing  that  the  summations  extend  over  all  the  primes  p<  for 
which  Kpi^N  [Lucas"].     He  proved  that 

lim  ^jN)  _  3 
iV  =  «)    m      7r2 

and  gave  a  table  showing  the  approximation  of  SN^/tt^  to  $(iV)  for  iV^  100. 
The  last  formula,  proved  earlier  by  Dirichlet^^  and  Mertens,^^  was  proved 
by  G.  H.  Halphen^^  by  the  use  of  integrals  and  f -functions. 

Sylvester^*"  defined  the  frequency  5  of  a  divisor  d  of  one  or  more  given 
integers  a,  h, .  .  . ,  I  to  he  the  number  of  the  latter  which  are  divisible  by 
d.     By  use  of  (4)  he  proved  the  generalization 

X8(f>{d)=a+h-\-...-\-l. 

d 

J.  J.  Sylvester^^  stated  that  the  number  of  [irreducible  proper]  fractions 
whose  numerator  and  denominator  are  ^j  is  T{j)  =  <f){l)+  . . .  +<t>{j),  and 

that  3        PoT         •'"      f-'/^]  n^-\-i 

stU^s  S(/,(t)=^, 

k=i    L/CJ      ^=1  i=i  ^ 

whence  T{j)/f  approximates  S/tt^  as  j  increases  indefinitely. 

If  u{x)  denotes  the  sum  of  all  the  integers  <x  and  prime  to  x,  and  if 
U(j)=u(l)+ .  .  .-{-u(j),  then  U{j)  is  the  sum  of  the  numerators  in  the 
above  set  of  fractions,  and* 

When  j  increases  indefinitely,  U{j)/f  approximates  I/tt^.  For  each  integer 
n^  1000  the  values  of  (^(n),  T{n),  Srr/ir'^  are  tabulated, 

Sylvester^^  stated  the  preceding  results  and  noted  that  the  first  formula 
is  equivalent  to 


!I3^^^) 


l(/+i). 


"Archiv  Math.  Phys.,  66,  1881,  353-7. 

"Bull,  des  Sc.  Math,  et  Astr.,  (2),  5,  I,  1881,  37-40. 

"Comptes  Rendua  Paris,  96,  1883,  634-7. 

"«Amer.  Jour.  Math.,  5,  1882,  124;  Coll.  Math.  Papers,  3,  611. 

"Phil.  Mag.,  15,  1883,  251-7;  16,  1883,  230-3;  Coll.  Math.  Papers,  4,  101-9.     Cf.  Sylvester." 

"Comptes  Rendus  Paris,  96,  1883,  409-13,  463-5;  Coll.  Math.  Papers,  4,  84-90.     Proofs  by 

F.  Rogel  and  H.  W.  Curjel,  Math.  Quest.  Educ.  Times,  66,  1897,  62-4;  70,  1899,  56. 
*With  denominator  3,  but  corrected  to  6  by  Sylvester,"  which  accords  with  Ces&,ro."     The 

editor  of  Sylvester's  Papers  stated  in  both  places  that  the  second  member  should  be 

jij  +  l){2j+l)/12,  evidently  wrong  for;  =2. 


Chap.  V]  EulEE's  (^-FUNCTION.  127 

E.  Ces^ro^^  proved  that,  if  /  is  any  function, 

X^i^Xx^Fin),  F(n)^S/(d), 

n  =  ll       X  71=1 

where  d  ranges  over  the  divisors  of  n.  For  / = 0,  we  have  F(x)=x  and  obtain 
Liouville's^^  first  formula.  By  the  same  specialization  (p.  64)  of  another 
formula  (given  in  Chapter  X  on  sums  of  divisors^^),  Cesaro  derived  the 
final  formula  of  Liouville.^^  If  (n,  j)  is  the  g.  c.  d.  of  n  and  j,  then  (p.  77, 
p.  80) 

Mn,  j)  =S#Q),  X-^  =  h:d<t>{d),  X4>{n,  j)  =S(^(d)0(-). 

If  (p.  94)  p  is  one  of  the  integers  a,  /3, . . .  ^  n  and  prime  to  n, 

S^(a)F(a)  =SG(a)/(a),  /^(a:)^S/(d),  G(p)^S^(pa), 

a 

where  d  ranges  over  the  divisors  of  x.    For  g{x)  =  l,  this  gives 

S/(a)0(n,  n/a)=SF(a), 

a 

where  (p.  96)  </)(n,  x)  is  the  number  of  integers  ^x  and  prime  to  n.  Cesaro 
(pp.  144-151,  302-3)  discussed  and  modified  Perott's^^  proof  of  his  first 
formula,  criticizing  his  replacement  of  [n/k]  by  n/k  for  n  large.  He  gave 
(pp.  153-6)  a  simple  proof  that  the  mean^^  of  <^(n)  is  Qn/w^  and  reproduced 
the  proofs  by  Dirichlet^^  and  Mertens,^^  the  last  essentially  the  same  as 
Perott's.     For  Kw)  =  l  +  l/2"^+l/3"+.  .  ., 

s4r(^>l),  2i  2-i-(m>l),  2-^ 


a"*  '  ''  a'  a'"</)(a)  '  ''  0(a) 

equal  asymptotically  (pp.  167-9) 

f(m)/r(m+l),         (6  1og7i)/7r^         r(m+l),         log  n. 

As  a  corollary  (p.  251)  to  Mansion's^^  generalization  of  Smith's  theorem  we 
have  the  result  that  the  determinant  of  order  n^,  each  element  being  1  or  0 
according  as  the  g.  c.  d.  of  its  two  indices  is  or  is  not  a  perfect  square,  equals 
(  — 1)"+^+-  ,  where  pV-  •  •  is  the  value  of  n\  expressed  in  terms  of  its  prime 
factors. 

Ces^ro^*  considered  any  function  F{x,  y)  of  the  g.  c.  d.  of  x,  y,  and  the 
determinant  A„  of  order  n  having  the  element  F{Ui,  u/)  in  the  ith.  row  and 
ith  column,  where  Ui,...,Un  are  integers  in  ascending  order  such  that  each 
divisor  of  every  Ui  is  a  u.  Employing  the  function  ix{n)  [see  Ch.  XIX],  he 
noted  that 


i nO^)F(u„ud=f{u,)  or  0, 


"M6m.  Soc.  R.  Sc.  de  LiSge,  (2),  10,  1883,  No.  6,  74. 
"Atti  Reale  Accad.  Lincei,  (4),  1,  1884-5,  709-711. 


128  History  of  the  Theory  of  Numbers.  [Chap,  v 

according  as  u^  is  or  is  not  divisible  by  w„,  while 

fix)  =Kx)F{l)  +M  (2)  F{2)  +M  (f)  F{S)  +  . . . . 

Hence  if  we  multiply  the  elements  of  the  ith  colmnn  of  A„  by  fx{ujui)  and 
add  the  products  to  the  last  column  for  2  =  1, . .  .,  n  — 1,  the  new  elements  of 
the  last  colunm  are  zero  except  the  final  element,  which  is/(w„).     Thus 

A,=/(ii„)A„_i=/(ui)/(w2) .  .  .fM- 

[These  results  are  due  to  Smith,^^  not  merely  the  case  Ui  =  i  a.s  stated.] 
Cesaro^^  noted  that  |wy|=/(l) . .  ./(n)  if 


«,=  s/wft0ft.Q, 


where  the  function  h  has  the  property  that  the  determinant  with  the  general 
element  h{i/j)  is  unity,  and  similarly  for  hi. 

Cesaro®°  gave  the  last  result  for  the  case  in  which  h{x)=hi{x)  =  l  or  0 
according  as  x  is  or  is  not  an  integer.  P.  Mansion  (p.  250)  stated  that  he** 
had  employed  a  similar  proof. 

Ces^ro®^  duphcated  his  paper^^  and  transformed  its  final  result  into 

/(l)/(2) .  .  .fin) 


F[i,j] 


F\nl) 


where  [i,  j]  =  ij/{i,  j)  is  the  1.  c.  m.  of  i,  j,  and  F{x)  is  a  function  such  that 
F{xy)=F{x)F{y).  In  particular,  if  F{x)  =  \/x,  then  J{x)=4){x)Tr{x)/x^, 
where  7r(n)  is  the  product  of  the  negatives  of  the  distinct  prime  factors  of  n. 
Hence 

|Ki]|n=0(l)...0(n)7r(l)...7r(n). 

Ces^ro^^  investigated  the  r-rowed  minors  of  the  n-rowed  determinant 
whose  general  element  is  F{b)=F{i,  j),  where  5  is  the  g.  c.  d.  of  i,  j.  It  is 
shown  that  the  {n  —  v)-Towed  determinant  whose  general  element  is  F{i-\-v, 
j-\-v)  is  equal  to  the  sum  of  certain  products  of /(I), .  . .,  f{n)  taken  n  —  v&i 
a  time,  the  case  v  =  Q  being  Smith's  theorem.     Here 

/(x)  =^J^F{j),  Fix)  =S/(d)  (d  divisor  of  x). 

Ces^ro^^  stated  that  the  (n  — l)-rowed  determinant,  whose  general  ele- 
ment Uij  equals  the  number  of  divisors  common  to  i+1  and  j  +  1,  equals 
the  number  of  integers  ^  n  deprived  of  square  factors  >  1 . 

"Atti.  Reale  Accad.  Lincei,  (4),  1,  1884-5,  711-5. 

•"Mathesis,  5,  1885,  248-9. 

"Giornale  di  Mat.,  23,  1885,  182-197. 

"Annales  de  I'^cole  normale  sup.,  (3),  2,  1885,  425-435. 

"Nouv.  Ann.  Math.,  (3),  4,  1885,  56. 


Chap.  V]  EulEE's  (^-FUNCTION.  129 

Ces^ro"  employed  F(n)=S/(d),  G(n)='Zg{d),  where  d  ranges  over  the 
divisors  of  n,  and  proved  that 

0         G(l)         G{2)         ...     G{n) 

G{\)     F(l,l)     F(l,2)      ...     F(l,n)    -_j(i)..j(^)p'(^), 


G{n)    F{n,l)     F{n,2)  Fin,  n) 

In  particular,  if  F{n)  is  the  number  of  divisors  of  n  and  if  G{n)  is  the  number 
of  prime  divisors  of  n,  the  determinant,  apart  from  signs,  equals  the  number 
of  primes  ^  n. 

E.  Cesaro^^  wrote  (a,  b)  for  the  g.  c.  d.  of  a,  h.     If  F{n)  =S/(d),  where  d 
ranges  over  the  divisors  of  n,  then 

XF\(n,i)\=i:f(d)N/d. 

In  particular,  if  /,(n)  is  the  number  of  irreducible  fractions  ^e  of  denomi- 
nator n. 


IXn)=i:[j]n{d),  S7,(d)  =  [ne]. 


The  last  formula,  due  to  Laguerre,^^  follows  by  inversion  (Ch.  XIX),  and 
directly  from  the  fact  that  I^d)  is  the  number  of  the  first  [ne]  integers  which 
with  n  have  the  g.  c.  d.  n/d.  The  number  of  irreducible  fractions  ^  e  of 
denominator  ^n  is  $e(^)  =-f «(!)+•••  +-^.(^).     We  have 

00  \n/j]  O, 

^M  =  SmO')  S  [ie],  Imi $,(n)/n2  =  -2     (€>0), 

j  =  l  1=1  n=oo  T 

due  to  Sylvester^^  for  €  =  1.  Let  (^^g^(n)  be  the  sum  of  the  j'th  powers  of 
the  numerators  of  the  irreducible  fractions  <  e  of  denominator  n.     Set 


Then 


$«  in)  =  S  </)(?  ii) ,        sXn)  =  S  ^^ 
1=1  »-i 


i=l  LzJ       i.i 


which  generalizes  the  two  formulas  of  Sylvester .^^    Also, 
$^;^  (w)  =  —  — —  — — ,  asymptotically. 

TT     V-\-l    V-\-2 

Ces^ro^^"  factored  determinants  of  the  tj^e  in  his  paper,^^  the  function  F 
now  being  such  that  Fixy)/ \Fix)Fiy)\  is  a  function  of  the  g.  c.  d.  oi  x,  y. 

L.  Gegenbauer®^''  gave  a  complicated  theorem  involving  several  general 
functions,  special  cases  of  which  give  Sylvester's^^  two  summation  formulas. 

"Nouv.  Ann.  Math.,  (3),  5,  1886,  44-47. 

"Annali  di  Mat.,  (2),  14,  1886-7,  143-6. 

""Giornale  di  Mat.,  25,  1887,  18-19. 

«5fcSitzungsber.  Ak.  Wiss.  Wien  (Math.),  94,  1886,  II,  757-762. 


130  History  of  the  Theory  of  Numbers.  [Chap,  v 

P.  S.  Poretzky^^  gave  a  formula  for  the  function  \l/{m)  whose  values  are 
the  4>{7?i)  integers  <?fi  and  prime  to  m.  For  the  case  w  =  2-3-5. .  .p,  where 
p  is  a  prime, 

lpi-2     Pi  J 

where  K  is  an  integer.  Application  is  made  to  the  finding  of  a  prime 
exceeding  a  given  number,  and  to  a  generalization  of  the  sieve  of  Eras- 
tosthenes. 

E.  Ces^ro^^  gave  a  very  simple  proof  of  the  known  fact  that 

2  2' 

n-00  n^  T 

which  he  expressed  in  words  by  saying  that  0(n)  is  asymptotic  to  6n/7r^ 
(not  meaning  that  the  limit  of  4>{n)/n  is  G/tt").  On  the  distinction  between 
asymptotic  mean  and  median  value,  see  Encyclop^die  des  sc.  math.,  I, 
17  (vol.  3),  p.  347. 

Ces^ro^^  noted  that  if  F{i,  j)  is  a  function  of  the  g.  c.  d.  of  i,  j,  then 
Q=SF(i,  j)  XiXj  {i,  j  =  l,...,  n)  becomes  q='Lf{i)yi^  by  the  substitution 
yk  =  Xk-{-X2k-\-Xsk-\-  ■  ■  -,  provided  F{n)  =2/(d),  d  ranging  over  the  divisors  of 
n.  Since  the  determinant  of  the  substitution  is  unity,  the  discriminants 
of  Q  and  q  are  equal.  Hence  we  have  the  theorem  of  Smith.^^  A  gen- 
eralization is  obtained  by  use  of  2F(e„  e^XiXj,  where  the  numbers  ei,  C2, .  . . 
include  the  divisors  of  each  €. 

E.  Catalan^^  proved  that,  if  d  ranges  over  the  divisors  of  iV  =  a"6^ . . . , 

E.  Busche^°  derived  at  once  from  Dirichlet's^^  formula  the  result 

S0(x))p(^)+p(^)  +  ...(=Snn', 
j=l  \x/  \x  / 

where  p(a)  =a  — [a].     The  case  n  =  n'  =n"  =  .  . .  leads  to 

i:4>{x)  =  {v-\)n^, 

where  x  takes  all  values  for  which  p{n/x)>p{vn/x).  If  we  take  n  =  l  and 
add  </)(!)  =  1,  we  get  (4)  for  N  =  v.  Next,  S0(a;)  =rr'5",  where  x  takes  all 
values  for  which 

yJ±zi^,Q<yyi      (y=i,...,.;,'=i,...,.'). 

r-\-r  \xy     r   r 

6«Math.  phys.  soc.  Kasan,  6,  1888,  52-142  (in  Russian). 

"Comptcs  Rendus  Paris,  106,  1888,  1651;  107,  1888,  81,  426;  Annali  di  Mat.,  (2),  16,  1888-9, 

178  (discussion  with  Jensen  on  terminology). 
•8Atti  Rcale  Accad.  Lincei,  Rendiconti,  2,  1888,  II,  56-61. 

"M6m.  Soc.  Sc.  Li^ge,  (2),  15,  1888,  No.  1,  pp.  21-22;  Melanges  Math.,  Ill,  No.  222,  dated  1882. 
"Math.  Annalen,  31,  1888,  70-74. 


Chap.  V]  EulEr's  0-FuNCTION.  131 

For  d  =  n,r'  =  l,r  =  v  —  l,  this  becomes  the  former  result ;  f or  r  =  r '  =  1 ,  5  =  n, 
it  becomes  20  (x)  =n^,  where  x  takes  the  values  for  which  p(n/rc)^  1/2. 

H.  W.  Lloyd  Tanner^^  studied  the  group  G  of  the  totitives  of  n  (the 
integers  <n  and  prime  to  n),  finding  all  its  subgroups  and  the  simple  groups 
whose  direct  product  is  G. 

E.  Lucas^^  proved  that,  in  an  arithmetical  progression  of  n  terms  whose 
common  difference  is  prime  to  n,  there  are  (ji{d)  terms  having  with  n  the g.  c.  d. 
n/d.  If,  when  d  ranges  over  the  divisors  of  n,  Xxpid)  =n  for  every  integer  n, 
then  (p.  401)  \p{n)=(^{n),  as  proved  by  using  n  =  l,  a,  a^,.  . .,  and  n  =  ah, 
a^b, . .  . ,  where  a,h,.  .  .  are  distinct  primes.  He  gave  (pp.  500-1)  a  proof  of 
Perott's^^  first  formula  by  induction  from  N  —  1  to  N,  communicated  to  him 
by  J.  Hammond.  The  name  "  indicateur  "of  n  is  given  (preface,  xv)  to 
<f){n)   [Prouhet^sj. 

C.  Moreau  (cf .  Lucas,'^^  501-3)  considered  the  C{n)  circular  permutations  of 
n  objects  of  which  a  are  alike,  (3  alike, .  .  . ,  X  alike.  Thus,  if  a  =  2,  /3  =  4,  the 
C(6)  =  3  distinct  circular  permutations  are  aahbhb,  ababbb,  abbabb.    In  general, 

^^^^=n^^^^^(a/d)!...(X/d)r 

where  d  ranges  over  the  divisors  of  the  g.  c.  d.  of  a,  jS, . . . ,  X.  In  the 
example,  d  =  1  or  2,  and  the  terms  of  the  sum  are  15  and  3. 

P.  A.  MacMahon^^  noted  that  C(n)  =  1  if  n  =  a,  so  that  we  have  formula 
(4).  His  expression  for  the  number  of  circular  permutations  of  p  things  n 
at  a  time  is  quoted  in  Chapter  III  on  Fermat's  theorem. 

A.  Berger^^"  evaluated  S^il  k'^%{k).  For  a  =  2  the  result  is  3nV7rH 
\n  log  n,  where  X  is  finite  for  all  values  of  n. 

E.  Jablonski'^^  considered  rectilinear  permutations  of  indices  a, .  .  .,  X, 
with  the  g.  c.  d.  D.  Set  a  =  a'D,-  •  .,\  =  \'D,  a+  • .  .+X  =  m  =  m'Z).  Then 
the  number  of  complete  rectilinear  permutations  of  indices  a'n, . .  . ,  \'n  is 

P{n)=-       ^'^'''^' 


{a'n)\...{\'n)\ 
The  number  of  complete  circular  permutations  is 

where  d  ranges  over  the  divisors  of  D.  If  Q{D/d)  is  the  number  of  rectilinear 
permutations  of  indices  a, .  .  . ,  X  which  can  be  decomposed  into  d  identical 
portions,  ^Q(D/d)=P{D).     Also 

'iProc.  London  Math.  Soc,  20,  1888-9,  63-83. 

"Theorie  des  nombres,  1891,  396-7.     The  first  theorem  was  proved  also  by  U.  Concina,  II 

Boll,  di  Matematica,  1913,  9. 
"Proc.  London  Math.  Soc,  23,  1891-2,  305-313. 
'»«Nova  Acta  Regiae  Soc.  Sc.  UpsaUensis,  (3),  14,  1891,  No.  2,  113. 
'♦Comptes  Rendus  Paris,  114,  1892,  904-7;  Jour,  de  Math.,  (4),  8,  1892,  331-349.     He  proved 

Moreau's'*  formula  for  C{n). 


132  History  of  the  Theory  of  Numbers.  [Chap,  v 


2Q©d'=2pg)/,(d), 


where  Jt{d)  is  Jordan's-""  function. 

S.  Schatunowsky"*  proved  that  30  is  the  largest  number  such  that  all 
smaller  numbers  relatively  prime  to  it  are  primes.  He  employed  Tcheby- 
chef's^"  theorem  of  Ch.  XVIII  that,  if  a>  1,  there  exists  at  least  one  prime 
between  a  and  2a.  Cf.  Wolfskehl,^^  Landau,^^.  113  Maillet,^^  Bonse/"« 
Remak.^^2 

E.  W.  Da\ds''^  used  points  with  integral  coordinates  ^0  to  visualize 
and  prove  (1)  and  (4). 

K.  Zsigmondy^^  wrote  r,  for  the  greatest  integer  ^  r/s  and  proved  that, 
if  a  takes  those  positive  integral  values  ^r  which  are  di\asible  by  no  one 
of  the  given  positive  integers  rii, . .  . ,  n^  which  are  relatively  prime  in  pairs, 

r  rn  rnn' 

2/(a)  =  S f{k) -S  S f{kn)  +22 f{knn') -..., 

k=l  n   k=l  n,  n'  k  =  l 

n,  n',.  .  .  ranging  over  the  combinations  of  rii,.  .  .,  n^  taken  1,  2, .  .  .  at  a 
time.  Taking /(A:)  =  1,  we  obtain  for  the  number  (f>{r;  rii, . . . ,  nj  of  integers 
^r,  which  are  divisible  by  no  one  of  ni, .  .  . ,  n^,  the  expression  (5)  obtained 
by  Legendre  for  the  case  in  which  the  n's  are  all  primes.  By  induction 
from  p  to  p+1,  we  get 

+  2<^(r^;ni,.  ..,n,)-..., 

p 
r=4>{r]  ni, . .  . ,  n,)+  2  </)(r„<;  rii, .  .  . ,  n,_i,  n,+i, .  .  . ,  nj 


t=i 


+  2<^(r„„/;  riiST^n,  n')  + 


r  =  20(r,;ni,..  .,  n,), 


where  c  ranges  over  all  combinations  of  powers  ^r  of  the  n's.  The  last 
becomes  (4)  when  ni,.  .  .,  n^  are  the  different  primes  di\ading  r.  These 
formulas  for  r  were  deduced  by  him  in  1896  as  special  cases  of  his  inversion 
formula  (see  Ch.  XIX). 

J.  E.  Steggair^  evaluated  </)(n)  by  the  second  method  of  Crelle.^^ 

P.  Bachmann^^  gave  an  exposition  of  the  work  of  Dirichlet,^^  Mertens," 

Halphen^  and  Sylvester^^  on  the  mean  of  <p{n),  and  (p.  319)  a  proof  of  (5). 

L.  Goldschmidt^"  gave  an  evaluation  of  <j){n)  by  successive  steps  which 

may  be  combined  as  follows.     Let  p  be  a  prime  not  dividing  k.     Each  of 

"Spaczinakis  Bote  (phys.  math.),  14,  1893,   No.  159,  p.  65;  15,   1893,   No.   180,   pp.  276-8 

(Russian). 
"Amer.  Jour.  Math.,  15,  1893,  84. 
"Jour,  fur  Math.,  Ill,  1893,  344-6. 
"Proc.  Edinburgh  Math.  Soc,  12,  1893-4,  23-24. 
"Die  Anab-tische  Zahlentheorie,  1894,  422^30,  481-4. 
««Zeitschrift  Math.  Phys.,  39,  1894,  203-4. 


Chap.  V]  EulER's  0-FuNCTION.  133 

the  <j)(k)  integers  ^k  and  prime  to  k  occurs  just  once  among  the  residues 
modulo  k  of  the  integers  from  Ik  to  {l-{-l)k;  taking  1  =  0,  1,.  .  .,  p  —  l,  we 
obtain  this  residue  p  times.  Hence  there  are  p({>{k)  numbers  ^pk  and 
prime  to  k.  These  include  <j){k)  multiples  of  p,  whence  4){pk)  =  {p  —  l)(p{k). 
For,  if  r  is  one  of  the  above  residues,  then  r,  r+k,.  .  .,  r-{-{p  —  l)k  form  a 
complete  set  of  residues  modulo  p  and  hence  include  a  single  multiple  of  p. 
Hence 

</)(a6c...)  =  (a-l)(6-l)(c-l)..., 

if  a,  b,  c, . . .  are  distinct  primes.  Next,  for  n  =  a^h^ . . . ,  we  use  the  sets  of 
numbers  from  lab.  .  .to  (l-]-l)ab.  .  .,  for  Z  =  0,  1,.  .  .,  a°-~^b^~^ .  .  .  —  1. 

Borel  and  Drach^^  noted  that  the  period  of  the  least  residues  of  0,  a, 
2a,...  modulo  N,  contains  N/8  terms,  if  d  is  the  g.  c.  d.  of  a,  iV;  conversely, 
if  d  is  any  divisor  of  N,  there  exist  integers  such  that  the  period  has  d  terms. 
Taking  a  =  0,  1,. . .,  iV  — 1,  we  get  (4). 

H.  Weber ^^  defined  0(n)  to  be  the  number  of  primitive  nth  roots  of 
unity.  If  a  is  a  primitive  ath  root  of  unity  and  /3  a  primitive  6th  root,  and 
if  a,  b  are  relatively  prime,  a/3  is  a  primitive  a6th  root  of  unity  and  all  of 
the  latter  are  found  in  this  way.  Hence  0(a6)  =</)(a)0(6).  This  is  also 
proved  for  relatively  prime  divisors  a,  6  of  n  — 1,  where  n  is  a  prime,  by  use 
of  integers  a  and  jS  belonging  to  the  exponents  a  and  b  respectively,  modulo 
n,  whence  a^  belongs  to  the  exponent  ab. 

K.  Th.  Vahlen^^  proved  that,  if  la.^in)  is  the  number  of  irreducible  frac- 
tions between  the  limits  a  and  /3,  a>j8^0,  with  the  denominator  n, 

S/„.,(d)  =  [(a-^)n],  ij~~^h,,{k)=i[{a-m], 

where  d  ranges  over  the  divisors  of  n.  For  /3  =  0,  the  first  was  given  by 
Laguerre.^^  Since  /i,o(^)=<^(^),  these  formulas  include  (4)  of  Gauss  and 
that  by  Dirichlet.2i 

J.  J.  Sylvester^  corrected  his^^  first  formula  to  read 

k^[k\  =  2Ui]H[i]t  ^^U),         r[n]=0(l)+. .  .+c^([n]), 
and  proved  it.    By  the  usual  formula  for  reversion, 

A.  P.  Minin^^  solved  ^(f){m)=R  for  m  when  R  has  certain  values.    The 
equation  determines  the  number  of  regular  star  polygons  of  m  sides. 
Fr.  RogeP®  gave  the  formula  of  Dirichlet.^^ 

*'Introd.  thdorie  des  nombres,  1895,  23. 

«Lehrbuch  der  Algebra,  I,  1895,  412,  429;  ed.  2,  1898,  456,  470. 
"Zeitschrift  Math.  Phys.,  40,  1895,  126-7. 

"Messenger  Math.,  27,  1897-8,  1-5;  Coll.  Math.  Papers,  4,  738-742. 

"Report  of  Phys.  Sec.  Roy.  Soc.  of  Friends  of  Nat.  Sc,  Anthropology,  etc.  (in  Russian),  Mos- 
cow, 9,  1897,  30-33.     Cf.  Hammond."^ 
«»Educat.  Times,  66,  1897,  62. 


134  History  of  the  Theory  of  Numbers.  [Chap,  v 

RogeP^  considered  the  number  of  integers  v<n  such  that  v  and  n  are 
not  both  di\'isible  by  the  rth  power  of  a  prime.  Also  the  number  when 
each  prime  factor  common  to  v  and  n  occurs  in  them  exactly  to  the  rth  power. 

I.  T.  Kaplan  published  at  Odessa  in  1897  a  pamphlet  in  Russian  on  the 
distribution  of  the  numbers  relatively  prime  to  a  given  number. 

M.  Bauer^^  proved  that,  for  x  prime  to  in,  kx-\-l  represents 


\p{m)    4>{d^d2) 


<j) 


integers  relatively  prime  to  m  and  incongruent  modulo  m,  where  di  is  the 
g.  c.  d.  {k,  m)  of  k,  m,  and  c?2=  (/,  m),  {di,  do)  =  1,  w^hile 


^W=0Wn{i-^} 


is  the  number  of  incongruent  integers  prime  to  m  =  pi^ .  .  .  p/*  which  are 
represented  by  kx+l  when  k,  I,  x  are  prime  to  7n.  Of  those  integers, 
\p{m)/\l/{pi. .  .pr)  are  di\'isible  only  by  the  special  prime  factors  Pi, .  .  .,  Pr 
of  m. 

J.  de  Vries^^"  proved  the  first  formula  of  Dirichlet's.^^ 
C.  Moreau^^  evaluated  4){n)  by  the  method  of  Grunert.^^ 
E.  Landau^°  proved  that 

„=i<^(n)  27r^     \    *=  pp2_p+iy 

where  e  is  of  the  order  of  magnitude  of  x~^  log  x,  C  is  Euler's  constant,  and 
f  is  Riemann's  ^-function. 

P.  WolfskehP^  proved  by  Tchebychef's  theorem  that  the  0(n)  integers 
<n  and  prime  to  n  are  all  primes  only  when  n  =  1,  2,  3,  4,  6,  8,  12,  18,  24,  30, 
[Schatunowsky.'°] 

E.  Landau^^  gave  a  proof,  without  the  use  of  Tchebychef's  theorem,  by 
finding  a  lower  limit  to  the  number  of  integers  k  ha\dng  no  square  factor 
>1,  where  t^k>Dt/S. 

E.  Maillet,^^  by  use  of  Tchebychef's  theorem,  proved  the  same  result 
and  the  generaUzation :  Given  any  integer  r,  there  exist  only  a  finite  number 
of  integers  N  such  that  the  <t>{N)  integers  <A^  and  relatively  prime  to  N 
contain  at  most  r  equal  or  distinct  prime  factors. 

Alois  Pichler^'*  noted  that  (}>(x)=n  has  no  solution  if  n  is  odd  and  >1; 
while  (i)(x)  =2"  has  the  solutions  x  =  2''bc.  .  .  (a  =  0,  1, .  .  .,  n  +  1)  if 

«^Sitzungsber.  Bohm.  GeseU.,  Prag,  1897;  1900,  No.  30. 
"Math.  Natur.  Berichte  aua  Ungam,  15,  1897,  41-6. 
""K.  Akad.  Wetenschappen  te  Amsterdam.  Verslagen,  5,  1897,  222. 
"Nouv.  Ann.  Math.,  (3),  17,  1898,  293-5.  ' 
»»G6ttingen  Nachrichten,  1900,  184. 

"L'interm^diaire  des  math.,  7,  1900,  253-4;  Math.  Ann.,  54,  1901,  503-4. 
"Archiv  Math.  Phys.,  (3),  1,  1901,  138-142. 
»»L'interm6diaire  des  math.,  7,  1900,  254. 

"Ueber  die  Auflosimg  der  01.  <p{x)  =n. . .,  Jahres-Bericht  Maximilians-Gymn.  in  Wien,  1900-1, 
3-17. 


Chap.  V]  EulEK's  0-FuNCTION.  135 

6  =  2^^+1,  c  =  22''+l,... 

are  distinct  primes  and  2^+2"^+. .  .  =n  or  n  —  a+1  according  as  a  =  0  or 
a>0.  When  g-  is  a  prime  >3,  <f){x)  =  2q''  is  impossible  if  p  =  2q^-\-l  is  not 
prime;  while  if  p  is  prime  it  has  the  two  solutions  p,  2p.  If  g  =  3  and  p  is 
prime,  it  has  the  additional  solutions  3"+\  2-3''"^^  Next,  4>{x)=2''q  is 
impossible  if  no  one  of  p^  =  2''~''q-\-l{v  =  0,  1,.  .  .,  n  — 1)  is  prime  and  q  is 
not  a  prime  of  the  form  2*+l,  s  =  2^^n;  but  if  q  is  such  a  prime  or  if  at 
least  one  p^  is  prime,  the  equation  has  solutions  of  the  respective  forms  bq^, 
where  (/)(6)  =2""*;  ap„  where  0(o)  =2".  Finally,  (f>{x)=2qr  has  no  solution 
if  p  =  2gr+l  is  not  prime  and  r9^2q-\-l.  If  p  is  a  prime,  but  r9^2q-{-l,  the 
two  solutions  are  p,  2p.  If  p  is  not  prime,  but  r  =  2g+l,  the  two  solutions 
are  r^,  2r^.  If  p  is  prime  and  r  =  2g-|-l,  all  four  solutions  occur.  There  is 
a  table  of  the  values  n<200  for  which  (f){x)=n  has  solutions. 

L.  Kronecker®^  considered  two  fractions  with  the  denominator  m  as 
equivalent  if  their  numerators  are  congruent  modulo  m.  The  number  of 
non-equivalent  reduced  fractions  with  the  denominator  m  is  therefore  4){m). 
If  m  =  m'm",  where  m' ,  m"  are  relatively  prime,  each  reduced  fraction  r/m 
can  be  expressed  in  a  single  way  as  a  sum  of  two  reduced  partial  fractions 
r' /m',  r'  /m".  Conversely,  if  the  latter  are  reduced  fractions,  their  sum 
r/m  is  reduced.  Hence  0(m)  =</)(m')</)(m").  The  latter  is  also  derived 
(pp.  245-6,  added  by  Hensel)  from  (4),  which  is  proved  (pp.  243-4)  by 
considering  the  g.  c.  d.  of  n  with  any  integer  ^n,  and  also  (pp.  266-7)  by 
use  of  infinite  series  and  products.  Proof  is  given  (pp.  300-1)  of  (5).  The 
Gaussian  median  value  (p.  334)  of  (f>{n)/n  is  Q/w^  with  an  error  whose  order 
of  magnitude  is  l/\/n,  provided  we  take  as  the  auxiliary  number  of  values 
of  4>{n)/n  a  value  of  the  order  of  magnitude  ^yn  log^  n. 

E.  B.  Elliott^^  considered  monomials  n  =  p'^q^.  .  .  in  the  independent 
variables  p,q,....  In  the  expansion  of  n(l  —  l/p)"'(l  —  l/g)"* .  .  . ,  the  aggre- 
gate of  those  monomial  terms  whose  exponents  are  all  ^0  is  denoted  by 
Fm{n).  Define  iJi{p'q\  .  .)  to  be  zero  if  any  one  of  r,  s, .  .  .  exceeds  1,  but  to 
be  (  —  1)'  if  no  one  of  them  exceeds  1,  and  t  of  them  equal  1.    Then 

(7)  F^_i(n)  =Si^,,(d),  F^^,{n)  =Sm  Q)  F^((i), 

where  d  ranges  over  the  monomials  pV-  •  •  with  O^a^a,  0^/3^?),.... 
Henceforth,  let  p,  q,...  be  distinct  primes.  Then  Fi{n)=(j){n),  while 
F_i(n)  is  the  sum  o-(n)  of  the  divisors  of  n.  In  (7),  d  now  ranges  over  all 
the  divisors  of  n,  and  ai(/c)  is  Merten's  function  [Inversion].  For  m  =  0,  (72) 
gives  the  usual  expression  for  </)(n),  while  (7i)  defines  o-(n).  For  m  =  l,  (7i) 
becomes  (4). 

If  T''^\n)  ^T(n)  is  the  number  of  divisors  d  of  n,  write 

r(2)(n)=ST(d),.  .  .,  T(^>(n)=ST^^-i>(d). 


'^Vorlesungen  iiber  Zahlentheorie,  I,  1901,  125-6. 
»«Proc.  London  Math.  Soc,  34,  1901,  3-15. 


136  History  of  the  Theory  of  Numbers.  [Chap,  v 

Then 

Generalizing  /x(s),  let  )u-*^(s)  be  zero  if  the  expansion  of  the  product 
n(l— p)*",  extended  over  all  primes  p,  does  not  contain  a  term  equal  to  s, 
but  let  it  equal  the  coefficient  of  s  if  s  occurs  in  the  expansion.     Then 


i?,(n)=SdM'"Q) 


The  7i-rowed  determinant  in  which  the  element  in  the  rth  row  and  sth 
column  is  F„_i(5),  where  5  is  the  g.  c.  d.  of  r,  s,  is  proved  equal  to  F^(l) 
F^(2) . .  .Fm{n),  a  generaUzation  of  Smith's^^  theorem.     Finally, 


isf.,,Q)f_.(d)=siF,(d), 


the  right  member  being  T{n),  20(c?)/c?,  lla{d)/d  for  r  =  0,  1,  —1. 

G.  Landsberg^^"  gave  a  simple  proof  of  Moreau's"^  formula  for  the 
number  of  circular  permutations. 

L.  Carlini^^  proved  Dirichlet's^^  formula  by  noting  that 

(8)  n(x''-l)=0 

has  unity  as  an  n-f old  root,  while  a  root  7^  1  of  x''  —  1  is  a  root  of  [n/h]  factors 
x"*  — 1.  Hence  the  4){h)  primitive  roots  of  x^  =  \  furnish  <l>{h)[n/h]  roots 
of  (8). 

M.  Lerch^^  found  the  number  N  of  positive  integers  '^m  which  have  no 
one  of  the  divisors  a,  6, . . . ,  k,  I,  the  latter  being  relatively  prime  in  pairs 
and  ha\'ing  m  as  their  product.  Let  F{x)  =  1  or  0,  according  as  x  is  frac- 
tional or  integral.     Let  L  =  ab.  .  .k.    Then  [Dirichlet^^] 


^^m(Z-l)^ 


,!/©-©-(-■)    (-i) 


L 
E.  Landau^^  proved  that  the  inferior  limit  for  a:=  00  of 

-<f>{x)  log.  log,  X 

X 

is  e~^,  where  C  is  Euler's  constant.     Hence  <^(j)  is  comprised  between  this 
inferior  limit  and  the  maximum  x  —  1. 

R.  Occhipinti^°°  proved  that,  if  aj  is  an  nth  root  of  unity,  and  if  c?,i,  •  ■  • , 
dat  are  the  divisors  of  i, 

n|s<^(di,)+a,S<^(d2i)+.  .  .+a/-4V(0|  =  i(-l)"-'n(n+l)n"-2. 

j-lU-l  i-l  i-l  J 

••"Archiv  Math.  Phys.,  (3),  3,  1902,  152-4.  "Periodico  di  Mat.,  17,  1902,  329. 

"Prag  Sitzungsber.,  1903,  II.  "Archiv  Math.   Phys.,   (3),  5,  1903,  86-91. 

"«Periodico  di  Mat.,  19,  1904,  93.  Handbuch,"'  I,  217-9. 


Chap.  V] 


Euler's  0-Function. 


137 


G.  A.  Miller^"^  proved  (4)  by  noting  that  in  a  cyclic  group  G  of  order  N 
there  is  a  single  cyclic  subgroup  of  order  d,  a  divisor  of  N,  and  it  contains 
0((i)  operators  of  order  d,  while  the  order  of  any  operator  of  G  is  a  divisor 
of  N.  Thus  (4)  states  merely  that  the  order  of  G  equals  the  sum  of  the 
numbers  of  the  operators  of  the  various  possible  orders.  Next,  (1)  follows 
from  an  enumeration  of  the  operators  of  highest  period  AB  in  a  cyclic  group 
of  order  AB,  which  is  the  direct  product  of  its  cyclic  subgroups  of  orders 
A  and  B.  Finally,  if  p  is  a  prime,  all  the  subgroups  of  a  cyclic  group  of 
order  p"  are  contained  in  its  subgroup  of  order  p"~\  whence  <^(p")  =  p"  —  p"~^ 

G.  A.  Miller^°^  proved  the  last  three  theorems  and  the  fact  that  0(0  is 
even  if  Z>2  by  means  of  the  properties  of  the  abelian  group  whose  elements 
are  the  integers  <m  which  have  with  m  a  g.  c.  d.  equal  to  k. 

K.  P.  Nordlund^"^  proved  4){mn  ...)  =  (m  —  l)(n  —  1)...,  where  m,  n,. . . 
are  distinct  primes,  by  writing  down  the  multiples  <mnp  of  m,  the  multi- 
ples of  mn,  etc.,  whence  the  number  of  integers  Kmnp  and  not  prime  to  it 
is  mnp  —  l  —  {m  —  l){n  —  l){p  —  l), 

E.  Busche^*^  treated  geometrically  systems  (td)  of  four  integers  such 

that  ad  —  hc>0,  evaluated  the  number  $(aS)  of  systems  incongruent  modulo 
S  and  prime  to  S,  and  generalized  (4)  to  2$(*S). 

L.  Orlando^°^  showed  that  0(m)  is  determined  by  (4)  [Lucas''^]. 

H.  Bonse^°^  proved  Maillet's^^  theorem  for  r  =  l,  2,  3  without  using 
•■Tcheby chef's  theorem.     His  lemma  was  generalized  by  T.  Suzuki. ^°^" 

J.  Sommer^^^  gave  without  reference  Crelle's^  final  evaluation  of  (/)(n). 

R.  D.  CarmichaeP*'^  proved  that  if  n  is  such  that  (l){x)=n  is  solvable 
there  are  at  least  two  solutions  x.  He  found  solutions  of  </)(x)  =  2"  [in  accord 
with  Pichler^^]  and  proved  that  there  are  just  n+2  solutions  (a  single  one 
being  odd)  when  n^31  and  just  33  solutions  when  32^ n^  255.  All  the 
solutions  of  <^(x)  =  4n — 2>  2  are  of  the  form  p",  2p",  where  p  is  a  prime  of  the 
form  4s  —  1 ;  for  example,  if  n  =  5,  the  solutions  are  19,  27  and  their  doubles. 

CarmichaeP°®  gave  a  table  showing  every  value  of  m  for  which  0(m) 
has  any  given  value  ^  1000. 

A.  Ranum^^^"  would  solve  4>{x)  =  n  by  resolving  n  in  every  possible  way 
into  factors  no, .,  n^,  capable  of  being  taken  as  the  values  of  0(2*"),  4>{pi'), 
.  .  .,  0(pA))  where  2,  pi, .  . .,  p,  are  distinct  primes.  Then  2'^pi"'. .  .p^^'r  is 
a  value  of  x. 

CarmichaeP^"  gave  a  method  of  solving  (j>(x)=a,  based  on  the  testing 
of  the  equation  for  each  factor  x  of  a  definite  function  of  a. 

M.  Fekete^^^  considered  the  determinant  pkn  obtained  by  deleting  the 
last  row  and  last  column  of  Sylvester's  eliminant  for  a;'''  —  1  =  0  and  a;**  —  1  =  0 


"lAmer.  Math.  Monthly,  12,  1905,  41-43. 
"'Amer.  Jour.  Math.,  27,  1905,  315. 
"'Nyt  Tidsskrift  for  Mat.,  16A,  1905,  15-29. 
iMJour.  fiir  Math.,  131,  1906,  113-135. 
"»Periodico  di  Mat.,  22,  1907,  134-6. 
iwArchiv  Math.  Phya.,  (3),  12,  1907,  292-5. 
i^^Tohoku  Math.  Jour.,  3,  1913,  83-6. 
"'Vorlesungen  liber  Zahlentheorie,  1907,  5. 


"SBull.  Amer.  Math.  Soc,  13,  1907,  241-3. 

"»Amer.  Jour.  Math.,  30,  1908,  394-400. 

lo'^^Trans.  Amer.  Math.  Soc,  9,  1908,  193-4. 

"«BuU.  Amer.  Math.  Soc,  15,  1909,  223. 

"iMath.  6s  Phys.  Lapok  (Math.  Phys.  Soc), 
Budapest,  18,  1909,  349-370.  German 
transl.,  Math.  Naturwiss.  Berichte  aus 
Ungam,  26,  1913  (1908),  196. 


138  History  of  the  Theory  of  Numbers.  [Chap,  v 

{k<n).  Thus  |p;tn|  =  1  or  0  according  as  k  and  n  are  relatively  prime  or  not. 
Hence 

n  n 

(t>{n)  =  2  \pkn\,  (t>i(n)  =  2  fc|pt„|, 

A=l  k=l 

where  <i>i{n)  is  the  sum  of  the  integers  ^n  and  prime  to  n. 

R.  Remak^^-  proved  Maillet's^^  theorem  without  using  Tehebychef's. 

E.  Landau^^^  proved  (5),  Wolfskehl's^^  theorem  and  Maillet's^^  generali- 
zation. 

C.  Orlandi^"  proved  that,  if  x  ranges  over  all  the  positive  integers  for 
wliich  [m/x]  is  odd,  then  20(x)  =  (?w/2)'^  for  7fi  even  (Cesaro,  p.  144  of  this 
History),  while  20 (x)  =  k^  for  m  =  2k  —  l. 

A.  Axer^^°  considered  the  system  (P)  of  all  integers  relatively  prime  to 
the  product  P  of  a  finite  number  of  given  primes  and  obtained  formulas 
and  asymptotic  theorems  concerning  the  number  of  integers  ^x  of  (P) 
which  are  prime  to  x.  Application  is  made  to  the  probability  that  two 
numbers  ^  n  of  (P)  are  relatively  prime  and  to  the  asymptotic  values  of  the 
number  (i)  of  positive  irreducible  fractions  with  numerator  and  denominator 
in  (P)  and  ^n  and  {ii)  of  regular  continued  fractions  representing  positive 
fractions  m  (P)  with  numerator  and  denominator  S  n. 

G.  A.  ]Miller^^^  defined  the  order  of  a  modulo  m  to  be  the  least  positive 
integer  h  such  that  ab=0  (mod  m).  If  p"  is  the  highest  power  of  a  prime 
p  dividing  vi,  the  numbers  ^7n  whose  orders  are  powers  of  p  are  km/p" 
(k  =  l,  2,. . .,  p").  Hence  l^kim/p-'i  {ki  =  \,.  .  .,  p-'i)  form  a  complete  set  of 
residues  modulo  7?i=Ilpi'i.  If  the  orders  of  two  integers  are  relatively 
prime,  the  order  of  their  sum  is  congruent  modulo  77i  to  the  product  of 
their  orders.  But  the  number  of  integers  ^m  whose  orders  equal  m  is 
(t>{7n).  Hence  (/)(np°)  =n0(p°).  Since  all  numbers  ^m  whose  orders 
divide  d,  a  di\'isor  of  7n,  are  multiples  of  7n/d,  there  are  exactly  d  numbers 
^m  whose  orders  di\ide  d,  and  (f){d)  of  them  are  of  order  d.  Hence 
7n  =  'E4>{d). 

S.  Composto^^^  employed  distinct  primes  7n,  n,  r,  and  the  v=<t>{77in) 
integers  P\,...,p^  prime  to  Tnn  and  ^ mn,  and  proved  that 

Pi,  Pi+7nn,  pi+2mn, .  .  . ,  p,+(r- l)wn  {i  =  l,.  .  .,v) 

include  all  and  only  the  numbers  rpi,. . .,  rp,  and  the  numbers  not  exceeding 
and  prime  to  7nnr.  Hence  4>{vmr)=4>{m7i)-{r  —  \).  A  like  theorem  is 
proved  for  two  primes  and  stated  for  any  number  of  primes.  [The  proof  is 
essentially  Euler's^  proof  of  (1)  for  the  case  in  which  J5  is  a  prime  not  divid- 
ing a  product  A  of  distinct  primes.]  Next,  if  d  is  a  prime  factor  of  7i,  the 
integers  not  exceeding  and  prime  to  dn  are  the  numbers  ^  n  and  prime  to  n, 
together  with  the  integers  obtained  by  adding  to  each  of  them  n,  2n, .  .  . , 

"2.\rchiv  Math.  Phys.,  (3),  15,  1909,  186-193. 

i^Handbuch. .  .VerteUung  der  Primzahlen,  I,  1909,  67-9,  229-234. 

"♦Periodico  di  Mat.,  24,  1909,  17&-8. 

"'Monatshefte  Math.  Phys.,  22,  1911,  3-25. 

"«.\mer.  Math.  Monthly,  18,  1911,  204-9. 

"'II  Boll,  di  Matematica  Gior.  Sc.-Didat.,  11,  1912,  12-33. 


Chap.  V]  EulEr's   0-FunCTION.  139 

(d  —  l)n;  whence  4>{dn)  =  d^{n) .  Finally,  let  Pi, ...,  Py  be  the  v  =  (f){n) 
integers  <n  and  prime  to  n.  Then  pi-\-kn  (^  =  l, .  .  .,v;  k  =  0,  1, .  .  .)  give 
all  integers  prime  to  n;  let  Ph{n)  denote  the  hth  one  of  them  arranged  in 
order  of  magnitude.     Then 

P,Xn)=kn-l  (k^l),       P,,+M=kn+pr  {l^r^v-l,  k^O). 

If  h  =  kv-\-r,  r<v,  the  sum  of  the  first  h  numbers  prime  to  n  is 

where  pi, .  .  . ,  p^  are  the  first  r  integers  <n  and  prime  to  n. 

K.  HenseP^^  evaluated  <^(n)  by  the  first  remark  of  Crelle.^'^ 

J.  G.  van  der  Corput  and  J.  C.  Kuyver^^^  proved  that  the  number 
/(a/4)  of  integers  ^  a/4  and  prime  to  a  is  N  =  \aJl{l  —  \/p)  if  a  has  a  prime 
factor  4m+l,  where  p  ranges  over  the  distinct  prime  factors  of  a;  but  is 
N  —  2^~^  if  a  is  a  product  of  powers  of  k  prime  factors  all  of  the  form  4m  —  1. 
Also  /(a/6)  is  evaluated. 

U.  Scarpis^^'^  noted  that  0(p"  — 1)  is  divisible  by  n  if  p  is  a  prime. 

Several  writers^^^  discussed  the  solution  of  4>{x)=4){y),  where  x,  y  are 
powers  of  primes.  SeveraP^^  proved  that  (f){xy)>4>{x)4>{y)  if  x,  y  have  a 
common  factor. 

J.  Hammond^^^  proved  that  there  are  ^^(n)  —  1  regular  star  n-gons. 

H.  Hancock^"^  denoted  by  ^{i,  k)  the  number  of  triples  {i,  k,  1),  {i,  k,  2), 
. . . ,  {i,  k,  i)  whose  g.  c.  d.  is  unity.  Let  i  =  iid,  k  =  kid,  where  ii,  ki  are 
relatively  prime.     Then  ^{i,  k)=ii(i>{d),  $(/c,  i)=ki(j}(d). 

A.  Fleck^^^  considered  the  function,  of  m^Hp", 

<p,{m)  =  n|<^(p«)  -  (J)c/>(p"-^) +...+(-  i)°(^y  (p^-")}. 

Thus  (f)o{'m)  =4){'m),  <^_i(m)  =  m,  <^_2(w)  is  the  sum  of  the  divisors  of  m.    Also 
S  4>k{d)=<i>k-i{m),        (f>kimn)=(l)k{ni)(t)k{n), 

d:m 

if  m,  n  are  relatively  prime.     For  f  (s)  =2m~*, 

^    (f)k-i{m)  ^    4>k{m) 

0.(p)=p-CI^),  0.(p^)=p^-(^l>+Ct')' ■•' 

</).(p'+'+o=p''(p-l)'^'. 

"sZahlentheorie,  1913,  97. 

"^Wiskundige  Opgaven,  11,  1912-14,  483-8. 

i^'oPeriodico  di  Mat.,  29,  1913,  138. 

i2iAmer.  Math.  Monthly,  20,  1913,  227-8  (incomplete);  309-10. 

li'^Math.  Quest.  Educat.  Times,  24,  1913,  72,  106. 

^^'Ibid.,  25,  1914,  69-70. 

i^^Comptes  Rendus  Paris,  158,  1914,  469-470. 

"^Sitzungsber.  Berlin  Math.  Gesell.,  13,  1914,  161-9. 


140  History  of  the  Theory  of  Numbers.  [Chap,  v 

E.  Cahen^^^  gave  F.  Arndt's^^  proof  without  reference. 

A.  Cunningham^"  tabulated  all  solutions  N  of  0(iV)=2'  for  r  =  4,  6,  8, 
9,  10,  11,  12,  16,  each  solution  being  a  product  of  a  power  of  2  by  distinct 
primes  22"+ 1. 

J.  Hanmiond^-^  noted  that,  if  'Zf{k/n)=F{7i)  or  <l>(n),  according  as  the 
summation  extends  over  all  positive  integers  k  from  1  to  n  or  only  over 
such  of  them  as  are  prime  to  n,  then  Z$(d)=F(n).  This  becomes  (4) 
when /is  constant. 

R.  Ratat^29  ^oted  that  0(n)  =  0(n  +  l)  for  n=  1,  3,  15,  104.  For  n<125, 
2n7^2,  4,  16,  104,  he  verified  that  (/)(2n=t l)>0(2n). 

R.  Goormaghtigh^^o  ^^^^^  ^j^j^^  0(^^)  =  <^(^i_|_l)  also  for  n=  164,  194,  255 

and  495.     He  gave  very  special  results  on  the  solution  of  (f){x)  =  2a. 

Formulas  involving  cf)  are  cited  under  Lipschitz,'''°'  ^^  Cesaro,^^  Ham- 
mond,^"  and  Knopp^^^  of  Ch.  X,  Hammond^  of  Ch.  XI,  and  RogeP«  of 
Ch.  XVIII.  Cunningham^^  of  Ch.  VII  gave  the  factors  of  (t>{f).  Dede- 
kind^^  of  Ch.  VIII  generalized  ^  to  a  double  modulus.  Minin^^°  of  Ch. 
X  solved  0(iV)=r(A^). 

Sum  0fc(n)  of  the  A:th  Powers  of  the  Integers  ^n  and  Prime  to  n. 

A.  Cauchy^^^  noted  that  (piin)  is  divisible  by  n  if  n>2,  since  the  integers 
<n  and  prime  to  n  may  be  paired  so  that  the  sum  of  the  two  of  any  pair  is  n. 

A.  L.  Crelle^^  (p.  80,  p.  84)  noted  that  (^i(n)  =  |n<^(n).  The  proof 
follows  from  the  remark  by  Cauchy. 

A.  Thacker^^*^  defined  (f)k{n)  and  noted  that  it  reduces  for  k  =  0  to  Euler's 
<i>{n).  Set  St(2)  =  l'"-|-2^+ .  .  .+2^n  =  a°6V.  . .,  where  a,  6, . .  .  are  distinct 
primes.  By  deleting  the  multiples  of  a,  then  the  remaining  multiples  of 
b,  etc.,  he  proved  that 

Mn)=sM  -2a^s.(^)  +2  aVs,(^)  -  S  ^a*feVs,(^)  +  .  .  . , 

where  the  summation  indices  range  over  the  combinations  of  a,  5,  c, .  .  .  one, 
two, ...  at  a  time.     In  the  second  paper,  he  proved  Bernoulli's^^""  formula 

where  Bi,  Bz,...  are  the  Bernoullian  numbers.     Then,  by  substitution, 
^^(^)=^n(l-i)+§(J)5,n^-^n(l-a)-i(3)53n*-^n(l-a^) 

i^TWorie  des  nombres,  I,  1914,  393. 

i"Math.  Quest.  Educ.  Times,  27,  1915,  103-6. 

128/bid.,  29,  1916,  53. 

i"L'interm(5diaire  des  math.,  24,  1917,  101-2. 

"»/6ui.,  25,  1918,  42-4. 

"'M6m.  Ac.  Sc.  de  I'Institut  de  France,  17,  1840,  565;  Oeuvres,  (1),  3,  272. 

"ojour.  fur  Math.,  40,  1850,  89-92;  Cambridge  and  Dublin  Math.  Jour.,  5,  1850,  243.  Repro- 
duced, with  errors  as  to  signs,  by  Zerr,  Amer.  Math.  Monthly,  5,  1898,  93-5.  Cf.  E. 
Prouhet,  Xouv.  Ann.  Math.,  10,  1851,  324-330. 

"""Jacques  Bernoulli,  Are  conjectandi,  1713,  95-7. 


Chap.  V]  GENERALIZATIONS  OF  EuLER's  0-FuNCTION.  141 


wheren(l-aO  denotes  {l-a^){l-h').  .  .. 

J.  Binet^"  wrote  Vif  ■  ■>  Vn  for  the  integers  <iV  and  prime  to  N^p^q". . .. 
Then,  if  B^,  —B^,  B^,...  are  the  BernouUian  numbers  1/6.  1/30,  1/42, . . ., 

andP,=  (l-p^)(l-g'')..., 

for  X  sufficiently  small  to  insure  convergence.  Expanding  each  member  into 
negative  powers  of  x  and  comparing  coefficients,  we  get 

n =277/  =  P_,N,     2Sr7i  =  P_,N\     SXrjf  =  P_,N^+SB,P,N, 

^n,^  =  P_,N^+QB,P,N^..  . 

the  first  being  equivalent  to  the  usual  formula  for  0(iV).  The  general  law 
can  be  represented  symbolically  by 

givr'=^\{N+Bpy-h{N-Bpy\, 

where,  after  expanding  the  binomials,  we  are  to  replace  N"/{BP)  by  P^iN" 
and  any  other  term  {BPY^~'^  by  B2h-\P2h-\-  It  is  easily  shown  that,  if  k  is 
odd,  Hit]''  is  divisible  by  N. 

Silva^^  used  his  symbolic  formula,  taking  S  to  be  the  sum  of  1, . .  .,  n, 
whence  S{a)  is  the  sum  §n(l+n/A)  of  the  multiples  ^n  of  A.  Thus 
^i(^)  =  2^(^)  •   This  proof  of  Crelle's  result  is  thus  like  that  by  Brennecke.^" 

W.  Brennecke^^^  proved  Crelle's  result  by  means  of 

H-...+n-la(l+2+...+^)+6(l+...+^)  +  ...t 

+  ]4+...+;J  +  . ..!  +  .... 

Set  )Li = 0(n) ,  a  =  ahc ....     He  proved  that 

<i>^{n)=^}xn'-^\aixn^-^n{\-a^){\-h^) .  .  ., 

the  signs  being  +  or  —  according  as  the  number  of  the  distinct  prime 
factors  a,  6, .  .  .  of  n  is  even  or  odd. 

•"Comptes  Rendus  Paris,  32,  1851,  918-921. 
"^Programm  Realschule,  Posen,  1855,  §§5-6. 


142  History  of  the  Theory  of  Numbers.  [Chap,  v 

G.  Oltramare^^  obtained  for  the  sum,  sum  of  squares,  sum  of  cubes,  and 
sum  of  biquadrates,  of  the  integers  <7na  and  relatively  prime  to  a  the 
respective  values 

^m~a<}>{a),  JmV0(a)  +  (-l)»— a0(a,),  ll 

o 

imV</>(«)  +  (-l)"^a2<A(ai), 

Ttl  111 

6  z-O'O 

where  a  is  the  number  and  Oi  the  product  of  the  distinct  prime  factors 
ju,  I', .  .  .  of  a,  while  ^(aO  =  (ju^  — l)(j/^  — 1) .  .  ..  The  number  of  integers  <n 
which  are  prime  to  a  is  4>{a)n/a. 

J.  Liou\'ille^^  stated  that  Gauss'  proof  of  S0(d)  =iV  may  be  extended  to 
the  generalization 


2QWc?)  =  l*+2*+...+iV*, 


where  d  ranges  over  the  di\'isors  of  N.     He  remarked  that  Binet's^"  results 
are  readily  proved  in  various  ways.     Also, 


e);3w={z>wf. 


N.  V.  Bougaief^^^  stated  that,  if  ^(n)  is  the  number  of  distinct  prime 
factors  of  n>\,  and  ^i(n)  is  their  product, 

also  a  result  quoted  below  with  Gegenbauer's^"°  generalization. 

August  BUnd^^^  reproduced  without  reference  the  formulas  and  proofs 
by  Thacker,^^°  and  gave 

0,(m)=w'</)o(7n)-^^^w'-Vi(/7O  +  ('2)^«''-'<A2W-  •  .  . +(-l)'<^.(^0. 

E.  Lucas^^^  indicated  a  proof  that  7?<^„_i(x)  is  given  symbolically  by 
{x+QY-Q\  where,  if  n  =  a°6^  .  .,  0,  =  5,(l-a'-')(l-5'-0  -  •  ••  Thus,  if 
IT  is  the  product  of  the  negatives  of  the  primes  a,  b, .  .  . , 

2</)i(x)  =x4>{x),        3<t>2{x)  =(j>{x)  (x~  +  hA ,        403Ct)  =.T(/)(.T)(x2+7r). 

'"Mdmoires  de  I'lnstitut  Nat.  Gr^nevois,  4,  1856,  1-10. 

""Comptea  Rendus  Paris,  44,  1857,  753-4;  Jour,  de  Math.,  (2),  2,  1857,  393-6. 

i"Nouv.  Ann.  Math.,  (2),  13,  1874,  381-3;  Bull.  Sc.  Math.  Astr.,  10,  I,  1876,  18. 

'**Ueber  die  Potenzsummen  der  iinter  einer  Zahl  ?«  Uegenden  und  zu  ihr  relativ  primen  Zahlen, 

Diss.,  Bonn,  1876,  37  pp. 
i^^Nouv.  Ann.  Math.,  (2),  16,  1877,  159;  Throne  des  nombres,  1891,  394. 


Chap.  V]  GENERALIZATIONS   OF  EuLER's  0-FuNCTION.  143 

Several ^^^^  found  expressions  for  0„=<^„(iV)  and  proved  that 

</>ox'»+n</)ia;"-i+i7i(n- 1)  (/)2a;"-2+  . . .  +<^„=  0     (n  odd) 

has  the  root  —4>\/4>q,  while  the  remaining  roots  can  be  paired  so  that  the 
sum  of  the  two  of  any  pair  is  —  20i/(/)o.  If  n=3  the  roots  are  in  arith- 
metical progression. 

H.  Postula^^^  proved  Crelle's  result  by  the  long  method  of  deleting 
multiples,  used  by  Brennecke.^^^  Catalan  {ibid.,  pp.  208-9)  gave  Crelle's 
short  proof. 

Mennesson^^^  stated  that,  if  q  is  any  odd  number, 

<^»  =  *0(^'+')  (modg), 

and  (Ex.366)  that  the  sum  of  the  products  (/)(n)  —  1  at  a  time  of  the  integers 
^n  and  prime  to  n  is  a  multiple  of  n. 

E.  Cesaro^^°  proved  the  generalization:  The  sum  rprn  of  the  products  m 
at  a  time  of  the  integers  a,  ^,.  .  .^N  and  prime  to  N  is  divisible  by  iV  if  m 
is  odd.     For  by  replacing  a  by  iV— a,  /3  by  A^"— j8, . . .  and  expanding. 


'^^-&^Ht-\y-Mt'-2y 


¥2-... 


where  0=0(iV).     Also  (l>m{N)  is  divisible  by  iV  if  m  is  odd. 

F.  de  Rocquigny^^^  proved  Crelle's  result.  Later,  he"^  employed  con- 
centric circles  of  radii  1,  2,  3, . .  .  and  marked  the  numbers  {m  —  l)N-}-l, 
(m  —  l)N-\-2, .  . .,  mN  at  points  dividing  the  circle  of  radius  m  into  A''  equal 
parts.  The  lines  joining  the  center  to  the  0(iV)  points  on  the  unit  circle, 
marked  by  the  numbers  <N  and  prime  to  N,  meet  the  various  circles  in 
points  marked  by  all  the  numbers  prime  to  N.  He  stated  that  the  sum 
of  the  4>{N)  numbers  prime  to  N  appearing  on  the  circle  of  radius  m  is 
|(2m  — l)0(iV^),  and  [the  equivalent  result]  that  the  sum  of  the  numbers 
prime  to  N  from  0  to  mN  is  ^'m^(i>{N^).  He  later  recurred  to  the  subject 
{HUd.,  54,  1881,  160). 

A.  Minine^®^  noted  that,  if  P>N>  1  and  k  is  the  remainder  obtained  by 
dividing  P  by  N,  the  sum  s{N,  P)  of  the  integers  <P  and  prime  to  N  may 
be  computed  by  use  of 

s{N,  mN+k)=s{N,  k)+^4>{N')+mN4>{N)„ 

where  (Minine^O  <i>{N)k  is  the  number  of  integers  ^k  prime  to  N. 

*A.  Minine^^^  considered  the  number  and  sum  of  all  the  integers  <  P 
which  are  prime  to  N  [Legendre's  (5)  and  Minine^*^^]. 

i"«Matli.  Quest.  Educ.  Times,  28,  1878,  45-7,  103-5. 

i"Nouv.  Corresp.  Math.,  4,  1878,  204-7.     Likewise,  R.  A.  Harris,  Math.  Mag.,  2,  1904,  272. 

^^lUd.,  p.  302. 

i6«76id.,  5,  1879,  56-59. 

"iLes  Mondes,  Revue  Hebdom.  des  Sciences,  51,  1880,  335-6. 

i62/6id.,  52,  1880,  516-9. 

i"/6id.,  53,  1880,  526-9. 

"^Nouveaux  theoremes  de  la  th^orie  des  nombres,  Moscow,  1881. 


144  History  of  the  Theory  of  Numbers.  [Chap,  v 

A.  Minine^^  investigated  the  numbers  N  which  divide  the  sum  of  all 
the  integers  <  N  and  prime  to  N. 

E.  Cesaro^^^  proposed  his  theorems^®^  as  exercises.  Proofs,  by  associa- 
ting a  with  N  —  a,  etc.,  were  given  by  Moret-Blanc  (3,  1884,  483-4). 

Ces^ro"  (p.  82)  proved  the  formula  of  Liouville.^"  Writing  (pp.  158-9) 
<f)„  for  </),„(A0  and  expanding  0„=2(iV— a)"*,  where  a,  /3, .  .  .  are  the  integers 
^  A^  and  prime  to  A'^,  we  get 

whence  <^^  is  di\'isible  by  N  if  m  is  odd,  but  not  if  m  is  even.  This  is  e\'ident 
(p.  257)  since  aJ^ -{- {N  —  a)""  is  di\isible  by  a+A''  — a  if  m  is  odd.  The  above 
formula  gives  A'"  =  (1  —  A)"*,  symboUcally,  where 

"     4>   AT-" 

is  the  arithmetic  mean  of  the  mth  powers  of  a/N,  ^/N, ....  The  mean 
value  of  <j)m{N)  is  6A„A"'"+V'''"^-  He  reproduced  (pp.  161-2)  an  earHer  for- 
mula,^^°  which  shows  that  B"'  =  {l-B)'",  symbolically,  if  B^  is  the  arith- 
metic mean  of  the  products  of  a/N,  ^/N, .  .  .  taken  m  at  a  time.  We  have 
(p.  165)  the  approximation 

X  x"*"^^  6 

2  <f)m(j)  =  7 — rrr} — r^  *  ~2> 
y=i  (7M+l)(m-|-2)  tT 

whence  (p.  261)  the  mean  of  (t>^{N)  is  6A''"+7(m+l)7r2. 
Proof  is  given  (pp.  255-6)  of  Thacker's^^°  formula 

*-«'"'"'*'::;"'*'""-iiiij.cr)'-»-""'"' 

where 

UN)=^d'-'f^(d)=Il{l-u^-'), 

d  ranging  over  the  divisors  of  A^,  and  u  over  the  prime  divisors  of  N.  Here 
nix)  is  Merten's  function  (Ch.  XIX).     It  is  proved  (pp.  258-9)  that 

2d^-Vp(^  =  1,  2^V.(^  =2dV,-.(rf), 

the  first  characterizing  the  function  \pp{N),  and  reducing  to  (4)  for  p  =  0. 
If  a  ranges  over  the  integers  for  which  [2n/a]  is  odd,  then  (p.  293) 

exactly  if  7?7  =  0,  1,  2,  3,  approximately  if  m>3,  where  A^,  is  the  excess  of  the 
sum  of  the  inverses  of  1,.  .  .,  n  over  that  of  n  +  l, .  .  . ,  2n.     In  particular, 

20(a)  =nl 

'"Math.  Soc.  Moscow  (in  Russian),  10,  1882-3,  87-101. 
»«Nouv.  Ann.  Math.,  (3),  2,  1883,  288. 


Chap.  V]  GENERALIZATIONS   OF  EuLER's  </)-FuNCTION.  145 

P.  Nazimov^"  (Nasimof)  noted  that,  when  x  ranges  over  the  integers 
^m  and  prime  to  n,  the  sum  of  the  values  taken  by  any  function /(x)  equals 

\mld\ 

7:ix{d)Xf{dx), 

d  1=1 

where  d  ranges  over  all  divisors  of  n.  The  case  f{x)  =  1  yields  Legendre's 
formula  (5) .  The  case/(a;)  =  xyields  a  result  equivalent  to  that  of  Minine.^^^"'* 
A  generalization  was  given  by  Zsigmondy'^'^  and  Gegenbauer.^'^^ 

E.  Cesaro^^^  noted  that,  if  A^  is  the  arithmetic  mean  of  the  mth  powers 
of  the  integers  ^  N  and  prime  to  N,  and  B^  that  of  their  products  m  at  a 
time,  we  have  the  symbolic  relations 

Cesaro^^^  proved  Thacker's^^'^  formula  expressed  as 

the  last  being  symboHc,  where  f^  is  a  function  such  that  l^^,,{d)='n}~^,  d 
ranging  over  the  divisors  of  n.     By  inversion 

n(n)=2M©<i'-'=;^n  (!-»-'), 

where  u  ranges  over  the  distinct  prime  factors  of  n. 
L.  Gegenbauer^'^°  proved  that,  if  j/=    yln   , 

x-l  n=lL3;  J  ,-1 

For  the  case  /c  =  0,  p  =  2,  this  becomes  Bougaief 's^^^  formula 
ig2{x)=i\^^<i>{x),  v  =  [Vn]. 

C.  Leudesdorf^'^^  considered  for  fx  odd  the  sum  i/'^(iV)  of  the  inverses  of 
the  juth  powers  of  the  integers  <  N  and  prime  to  N.     Then 

^|^,{N)=^kN'-hfiN^P,+,iN), 

where  k  is  an  integer.  Thus,  if  N  =  p^q,  where  q  is  not  divisible  by  the 
prime  p>3,  »/'^(A^)  is  divisible  by  p^'  unless  ju  is  prime  to  p,  and  )U+1  is 
divisible  by  p  — 1;  for  example,  \{/^{p)  is  divisible  by  p^.  If  p  =  3,  ^l/^iN) 
is  divisible  by  p^'  if  fx  is  an  odd  multiple  of  3.  If  p  =  2,  it  is  divisible  by 
2^'~^  except  when  q  =  l. 

Cesaro"^  inverted  his"  symbolic  form  of  Thacker's  formula  for  <l)m{N) 
in  terms  of  xf/'s  and  obtained 

nB,rPp{n)  =  {<f>-nBy. 

i"Matem.  Sbomik  (Math.  Soc.  Moscow),  11,  1883-4,  603-10  (Russian). 

"'Mathesis,  5,  1885,  81. 

"'Giomale  di  Mat.,  23,  1885,  172-4. 

""Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  95,  II,  1887,  219-224. 

i"Proc.  London  Math.  Soc,  20,  1889,  199-212. 

"Teriodico  di  Mat.,  7,  1892,  3-6.     See  p.  144  of  this  history. 


s 


146  History  of  the  Theory  of  Numbers.  [Chap,  v 

Hence  if  a  ranges  over  the  integers  ^  n  and  prime  to  n, 
Z(a  —  nBy  =  0  or  a  multiple  of  mpp 
according  as  p  is  odd  or  even.     By  this  recursion  formula, 

L.  Gegenbauer"^  gave  a  formula  including  those  of  Nazimov^^^  and 
Zsigmondy."    For  any  functions  xid),  Xiid) ,  f  i^i,  ■  ■  ■ ,  x,), 

m  /^\  /^\1 [m/d] 

f{KXi,.  .  .,  /cx,)2x(5)xiM  =^x(d)xA^)  ^  2  ^  fidKX^,.  .  .,  dKX,), 

where  d  ranges  over  all  divisors  of  n  which  have  some  definite  property  P, 
while  5  ranges  over  those  common  divisors  of  n,  Xi,...,  x,  which  have 
property  P.  Various  special  choices  are  made  for  x>  Xi>  /  and  P.  For 
instance,  property  P  may  be  that  d  is  an  exact  pth  power,  whence,  if  p  =  1, 
d  is  any  divisor  of  n.  The  special  results  obtained  relate  mainly  to  new 
number-theoretic  functions  without  great  interest  and  suggested  apparently 
by  the  topic  in  hand. 

T.  del  Beccaro^'^  noted  that  (t>k{n)  is  divisible  by  n  if  A;  is  odd  [Binet^^^]. 
When  n  is  a  power  of  2, 

l^+2*+...4-(n-l)*  =  Oor0(n)  (modn), 

according  as  k  is  odd  or  even.     His  proof  of  (1)  is  due  to  Euler. 

J.  W.  L.  Glaisher^^^  proved  that,  if  a,  h,. .  .  are  any  divisors  of  x  such 
that  their  product  is  also  a  divisor,  the  sum  of  the  nth.  powers  of  the  integers 
<  X  and  not  divisible  by  a  or  6, . .  . ,  is 

where  s  is  the  number  of  the  divisors  a,  6, ... ,  and 

li  a,h,.  .  .  are  all  the  prime  factors  of  x,  this  result  becomes  Thacker's.^^" 
N.  Nielsen^^^  proved  by  induction  on  y  that  the  sum  of  the  nth  powers 
of  the  positive  integers  <mM  and  prime  to  M  =  pi^.  .  .ply  is 

"'""'^'^W+(-i)-'f'^-^'"'  C^+iV  (™m)"--'  n  (P---1). 

n+l  «=in+l         \   zs  y  ,=i 

The  case  m=  1  gives  Thacker's^^°  result.  That  result  shows  {ihid.,  p.  179) 
that  02n(w)  and  <^2n+i(^)  are  divisible  by  m  and  m^  respectively,  for  l^n 
^  (pi— 3)/2,  where  pi  is  the  least  prime  factor  of  m,  and  also  gives  the  resi- 
dues of  the  quotients  modulo  m.  Corresponding  theorems  therefore  hold 
for  the  sum  of  the  products  of  the  integers  <  m  and  prime  to  m,  taken  t  at  a. 
time. 

i"Sitzungsberichte  Ak.  Wiss.  Wien  (Math.),  102,  1893,  Ila,  1265-94. 
"♦Atti  R.  Accad.  Lined,  Mem.  CI.  Fis.  Mat.,  1,  1894,  344-371. 
i«Messenger  Math.,  28,  1898-9,  39-41. 
"»Oversigt  Danske  Vidensk.  Selsk.  Forhandlinger,  1915,  509-12;  cf.  178-9. 


Chap.  V]  GENERALIZATIONS  OF  EuLER's  0-FuNCTION.  147 

Schemmel's  Generalization  of  Euler's  ^-Function. 

V.  SchemmeP^°  considered  the  $„(m)  sets  of  n  consecutive  numbers  each 
<m  and  relatively  prime  to  m.  If  m  =  a'^}f .  .  ,,  where  a,  h,.  .  .  are  distinct 
primes,  and  m,  m'  are  relatively  prime,  he  stated  that 

$„(m)  =a"~^(a  — n)6^~-^(5— n) .  .  .,  <^n{mm')  =<l>„(m)$n(m'), 

Sn"-''V-^'.  .  .<l>„(5)=w,  6  =  a«V.  .  .,  a'^a, /S'^/S, .  . ., 

the  third  formula  being  a  generalization  of  Gauss'  (4) .  If  ^  is  a  fixed  integer 
prime  to  m,  $„(w)  is  the  number  of  sets  of  n  integers  <m  and  prime  to  m 
such  that  each  term  of  a  set  exceeds  by  k  the  preceding  term  modulo  m. 
Consider  the  productPof  the  Xth  terms  of  the  *J>„(m)  sets.  If  n  =  1,  P=  =t  1 
(mod  m)  by  Wilson's  theorem.     If  n>  1, 

P"-i=)(-l)^-ir-i(X-l)!(n-X)!j*>)  (modm). 

For  the  case  A:  =  X  =  1,  n  =  2,  we  see  that  the  product  of  those  integers  <  m 
and  prime  to  m,  which  if  increased  by  unity  give  integers  prime  to  m,  is 
=  1  (mod  m) . 

E.  Lucas^^^  gave  a  generalization  of  Schemmel's  function,  without  men- 
tion of  the  latter.  Let  ei,...,  e^  be  any  integers.  Let  ^(n)  denote  the 
number  of  those  integers  h,  chosen  from  0,  1, .  .  .,  n  —  1,  such  that 

h  —  ei,  h  —  e2,.  .  .,  h  —  Ck 
are  prime  to  n.  For  k<n,  ei  =  0,  62=  —I,.  ..,  ei,—  —  {k  —  1),  we  have  k  con- 
secutive integers  h,  h-j-1,.  .  .,  h+k  —  l  each  prime  to  n,  and  the  number  of 
such  sets  is  */c(n).  Lucas  noted  that  ^(p)'^(g)  =^{pq)  if  p  and  q  are  rela- 
tively prime.  Let  n  =  a°-h^ .  .  .,  where  a,  h,.  .  .  are  distinct  primes.  Let  X 
be  the  number  of  distinct  residues  oi  ei, .  .  . ,  e^  modulo  a;  fx  the  number  of 
their  distinct  residues  modulo  h;  etc.    Then 

^(n)=a»-i(a-X)&^-\6-M). .  .. 
L.  Goldschmidt^®^  proved  the  theorems  stated  by  Schemmel,  and  himself 
stated  the  further  generalization:  Select  any  a  — A  positive  integers  <a, 
any  h—B  positive  integers  <b,  etc.;  there  are  exactly 

a''-\a-A)¥-\h-B)... 
integers  <m  which  are  congruent  modulo  a  to  one  of  the  a  — A  numbers 
selected  and  congruent  modulo  b  to  one  of  the  h  —  B  numbers  selected,  etc. 
P.  Bachmann^^^  proved  the  theorems  due  to  Schemmel  and  Lucas. 

Jordan's  Generalization  of  Euler's  ^-Function. 

C.  Jordan,^*^*^  in  connection  with  his  study  of  linear  congruence  groups, 
proved  that  the  number  of  different  sets  of  k  (equal  or  distinct)  positive 
integers  ^n,  whose  g.  c.  d.  is  prime  to  n,  is* 

^^ ■/>.w=»'(i-^.)-..(i-^j 

"ojour.  fur  Math.,  70,  1869,  191-2.  "'Th^orie  des  nombres,  1891,  p.  402. 

""Zeitschrift  Math.  Phys.,  39,  1894,  205-212.       i^^Niedere  Zahlentheorie,  1, 1902,  91-94, 174-5. 
^iioTraitg  des  substitutions,  Paris,  1870,  95-97. 
*He  used  the  symbol  [n,  k].     Several  of  the  writers  mentioned  later  used  the  symbol  (f>k(n), 
which,  however,  conflicts  with  that  by  Thacker.^^" 


148  History  of  the  Theory  of  Numbers.  [Chap,  v 

if  Pi, .  .  .,  Pg  are  the  distinct  prime  factors  of  n.     In  fact,  there  are  n*"  sets 
of  k  integers  ^n,  while  {n/piY  of  these  sets  have  the  common  divisor  pi, 

etc.,  whence 

k 

+  .... 


«"'"•-©•-©■-  -(^y 


Jordan  noted  the  corollary:  if  n  and  n'  are  relatively  prime, 

(11)  J,{nn')=J,{n)J,{n'). 

A.  Blind^^^  defined  the  function  (10)  also  for  negative  values  of  k,  proved 
(11),  and  the  following  generalization  of  (4): 

(12)  2Ji.(d)  =n'''  (d  ranging  over  the  di\'isors  of  7i). 

W.  E.  Story^°^  employed  the  s>Tnbol  r'^in)  for  Jk{n)  and  called  it  one 
of  the  two  kinds  of  kth  totients.  The  second  kind  is  the  number  </)*(n)  of 
sets  of  k  integers  ^??  and  not  all  di\isible  by  any  factor  of  n,  such  that  we 
do  not  distinguish  between  two  sets  differing  only  by  a  permutation  of 
their  numbers.     He  stated  that 

<t>\n)  =|-,ir*(n)+f,V-nn)+t2V-2(n)+  .  •  •  +<tiT(n)[ , 

where  1,  fi*,  W,.  .  .  are  the  coefficients  of  the  successive  descending  powers 
of  X  in  the  expansion  of  (x+l)(x+2).  .  .{x-\-k  —  \). 

Story-°-  defined  "the  kih.  totient  of  n  to  the  condition  k  to  be  the  num- 
ber of  sets  of  k  numbers  ^  n  which  satisfy  condition  k.  The  number  of  sets 
of  k  numbers  ^n,  all  containing  some  common  di\'isor  of  n  satisfying  the 
condition  k,  but  not  all  containing  any  one  di\'isor  of  n  satisfying  the  con- 
dition X  is  (if  different  permutations  of  k  numbers  count  as  different  sets) 


'^\H''^-y~b,')y~b,'>) 


where  5,  5', .  •  •  are  the  least  divisors  of  n  satisfj-ing  condition  /c,  while 
5i,  5/, .  .  .  are  the  least  di\'isors  of  n  satisfying  condition  x-  Here  a  set  of 
least  divisors  is  a  set  of  divisors  no  one  of  which  is  a  multiple  of  any  other." 
E.  Ces^ro"  (p.  345)  stated  that,  if  $;.(a:)  is  the  number  of  sets  of  k  integers 
^x  whose  g.  c.  d.  is  prime  to  x,  then 

where  J*  is  to  be  replaced  by  J/n),  and  d  ranges  over  the  di\'isors  of  n. 

J.  W.  L.  Glaisher-^^  proved  (12)  by  means  of  a  symbolic  expression  for 
the  infinite  series  2/t(n)/(a:'').     If  ^t(n)  is  Merten's  function, 

JM  -2p,V,(^)  +2pi  V«/.(^^)  -  •    •  =M(n), 
where  the  summations  relate  to  the  distinct  prime  factors  p,  of  n.     Using 

"* Johns  Hopkins  University  Circulars,  1,  1881,  132. 
^^Ihid.,  p.  151.     Cf.  Amer.  Jour.  Math..  3,  1880,  382-7. 
»<»London,  Ed.  Dublin  Phil.  Mag.,  (5),  18,  1884,  531,  537-8. 


Chap.  V] 


Generalizations  of  Euler's  (^-Function. 


149 


these  formulas  for  n  =  l,  2,. 
each  equal  to  {  —  lY~'^Jkin): 

1^     2^     3^     4^ 
1111 
0      10      1 
0      0      10 


n,  we  obtain  two  determinants  of  order  n, 


1 

-1 

-1 

0 

-1 

1       ... 

1 

-2" 

-3^ 

0 

-5^ 

6^      ... 

0 

1' 

0 

-2' 

0 

-3*      . . . 

0 

0 

1* 

0 

0 

-2'      ... 

L.  Gegenbauer^*^  proved  (12).    For  n  =  'pi^ . .  .p/'^,  set 

7r(n)  =  (-irpi...p„  X(n)  =  (-l^+••+^ 

where  w{n)  denotes  the  number  of  distinct  prime  factors  of  n.     By  means 
of  the  series  f  (s)  =Sn~*,  he  proved  that,  when  d  ranges  over  the  divisors  of  r, 

ZF(d)d2'  =  r^  SF(d)d2'  =  r'SdV,(d), 


^F{d)Jk{d)d''  =  0,  S(-l)(^+i)"'(^/V7rM^j  =0, 

the  last  holding  if  r  has  no  square  factor  and  following  from  the  third  in 
view  of  (11), 

Mr)  =2c^m(2)  ,     i:F{d)d'tx{d)  =r''fx{r),    i:Md)Ud)J,(^^  =0  or  J,,{Vr), 

according  as  r  is  or  is  not  a  square, 

S  ( - 1) ^'+^)"'('"V^(m)/fe(m)/2fe(^)w'*  =  r''\{r)Jk{r)  {mn^  =  r) , 

TO,  n 


^JkM .  .  .JkinW-^'W-^'" 


.nti=r"', 


where  rii, .  .  .,  n^  range  over  all  sets  of  solutions  of  nin2.  .  .n,+i  =  n,  the  case 
A;  =  1  being  due  to  H.  G.  Cantor .^^ 

E.  Cesaro^^^  derived  (10)  from  (12),  writing  ^i_k  for  Jk. 

E.  Cesaro^"^  denoted  J  kin)  by  xf^'^in)  and  gave  (12). 

L.  Gegenbauer^^"  gave  the  further  generaUzation 


X{g,{x)f==i: 


i[f]u-), 


[^]. 


J.  Hammond^°^  wrote  \f/(n,  d)  for  2/(5),  where  /  is  an  arbitrary  function 
and  5  ranges  over  all  multiples  ^  n  of  the  fixed  divisor  d  of  n.     Then 

(13) mt)=^^P{n,  l)-2)/^(n,  pi)+2iA(n,  p,P2)-..., 

""Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  89  II,  1884,  37-46.     Cf.  p.  841.     See  Gegenbauer" 

of  Ch.  X. 
"'Annali  di  Mat.,  (2),  14,  1886-7,  142-6; 
"'Messenger  Math.,  20,  1890-1,  182-190. 


150  History  of  the  Theory  of  Numbers.  [Chap,  v 

where  i  ranges  over  the  integers  ^n  which  are  prime  to  n,  while  pi,  P2>  •  •  • 
denote  the  distinct  prime  factors  of  n.  If  f{t)  =  l,  then  \l/{n,  d)=n/d 
and  (13)  becomes 

^in)=n-i:^+X-^-...=n(l-^)(l-^).... 
Pi       P1P2  \      Pi/  V      P2/ 

Next,  take  f(t)  =  ao+ait-\-a2f+  ....     Using  hyperboUc  functions, 

S/(0  =  Jcoth(./2)=l+^-4+..., 

provided  Z  be  replaced  by  nJXn)J_r{n),  where 

/i(n)  =/'(n)  -a„         Mn)  =f{n)  -2a2, .  .  .,         /_i(n)  =  ff{n)dn. 
Hence,  since  Ji(n)  =(j){n), 

2/(0  =^/-i(^)  +^^-i(n)/i(n)  -^V_3(n)/3(n)  +  .  .  . . 

In  particular,  for  f{t)=t'',  we  get  ^^-(n).  In  Prouhet's^^  first  formula,  5 
may  be  replaced  by  the  g.  c.  d.  A,,,  b  of  a  and  h.    The  generalization 

J,{ah)=Ma)J,{h)j^^ 

is  proved.     From  (12)  we  get  by  addition* 


(14)  i\-]j,{j)  =  l'+2' 

y=iLjJ 


+  .  .  .  +n*. 


Taking  n  =  l,  2,...,  n,  we  obtain  equations  whose  solution  gives  Jk(n) 
expressed  as  a  determinant  of  order  n  in  which  the  elements  of  the  last 
coluimi  are  1,  1+2*,  1+2^+3*, .  .  .,  while  for  s<n  the  sth  column  consists 
of  s  — 1  zeros  followed  by  s  units,  then  s  twos,  etc.  For  s>0,  the  element 
in  the  (s  +  l)th  row  and  rth  column  in  Glaisher's^"^  first  determinant  is 
1  or  0  according  as  r/s  is  integral  or  fractional. 

J.  Valyi^°^  used  J2{n)  ■^({>{n)  in  his  enumeration  of  the  n-fold  perspective 
polygons  of  n  sides  inscribed  in  a  cubic  curve. 

H.  Weber^os  proved  (10)  for  k  =  2. 

L.Carlini209  gave  without  references  (10),  (11),  (12),  with</)(^)  for  J„(A;). 

E.  Cesaro^io  noted  that  (12)  implies  (10).     For,  if  2/(d)  =F(n),  we  have 
by  inversion  (Ch.  XIX),  /(n)  =Xii{d)F(n/d).    The  case  f=Ji  gives 

Jijn)  _^IJLid) 

The  latter  is  a  case  of  G{n)  ='2g{d)  and  hence,  with  (12)  and 

W)QQ^2,Wf(^). 

♦This  work,  Mess.  Math.,  20,  1890-1,  p.  161,  for  k  =  l,  is  really  due  to  Dirichlet."    Formula 
(14)  is  the  case  p  =  1  of  Gegenbauer's,  p.  217. 
"^Math.  Nat.  Berichte  aus  Ungarn,  9,  1890,  148;  10,  1891,  171. 
"'EUiptische  Functioncn,  1891,  225;  ed.  2,  1908  (Algebra  III),  215. 
"»Periodico  di  Mat.,  6,  1891,  119-122. 
"o/fcid.,  7,  1892,  1-6. 


Chap.  V]  GENERALIZATIONS   OF  EuLER's  </)-FuNCTION.  151 

Ji+M=i:d'JMJi(^, 

which  is  next  to  the  last  formula  of  Gegenbauer's.^"^    Similarly, 

which  is  the  case  i  =  1  of  Gegenbauer's'^^  fifth  formula  in  Ch.  X,  (Tk{n)  being 
the  sum  of  the  A;th  powers  of  the  divisors  of  n. 

E.  Weyr^^^  interpreted  J2in)  in  connection  with  involutions  on  loci  of 
genus  1.  From  the  same  standpoint,  L.  Gegenbauer^^^  proved  (12)  for 
k  =  2  and  noted  that  the  value  (10)  of  J-zin)  then  follows  by  the  usual  method 
of  number-theoretic  derivatives. 

L.  Gegenbauer^^^"  wrote  cf)kim,  n)  for  the  number  of  sets  of  k  positive 
integers  ^  m  whose  g.  c.  d.  is  prime  to  n  =  pi°' . . .  p/''  and  proved  a  formula 
including 

[mf=Um,  n)+i      S       {\„ . . .  XT  4>k  i  -^ •    ,  -^^ ) 

where  (Xi, .  .  . ,  X,,)  is  the  determinant  derived  from  that  with  unity  through- 
out the  main  diagonal  and  zeros  elsewhere  by  replacing  the  7th  row  by 
the  X^th  row  for  7  =  1,.  .  .,  c.  The  case  m  =  n,  k  —  l,  is  due  to  Pepin.^^ 
There  is  an  analogous  formula  involving  the  sum  of  the  /cth  powers  of  the 
positive  integers  ^m  and  prime  to  n. 

E.  Jablonski^^  used  Jk{n)  in  connection  with  permutations. 

G.  Arnoux^^^  proved  (10)  in  connection  with  modular  space. 

*J.  J.  Tschistiakow^^"*  (or  Cistiakov)  treated  the  function  /^(n). 

R.  D.  von  Sterneck^^^  proved  that 

J,{n)  =SJ,(Xi)J,_,(X2)  =S0(X,) . .  4{\), 

the  X's  ranging  over  all  sets  of  integers  S.  n  whose  1.  c.  m.  is  n.  To  generalize 
this,  let  Jk{n;  mi, ... ,  rrik)  be  the  number  of  sets  of  integers  z'l, .  .  . ,  ik,  whose 
g.  c.  d.  is  prime  to  n,  while  ij^n/mj  for  j  =  1, .  .  . ,  k.     Then 

Jk{n;  Wi, .  .  .,  m^)=SJ,(Xi;  m\,.  . .,  'm'r)Jk-r0^2]  ^'r+i,-  •  •,  rn'k) 
=2:Ji(Xi;  mi). .  . J'i(X^;  m^), 

the  X's  ranging  over  all  sets  of  integers  ^n  whose  1.  c.  m.  is  n,  while  m'l, . . . , 
m'k  form  any  fixed  permutation  of  mi, .  .  . ,  m^t,  and  J"i(n;  m),  designated 
<f)^"'\n)  by  the  author,  is  the  number  of  integers  ^n/m  which  are  prime 
to  n.     Also, 

"iSitzungsberichte  Ak.  Wiss.  Wien  (Math.),  101,  Ila,  1892,  1729-1741. 
2i2Monatshefte  Math.  Phys.,  4,  1893,  330. 
2i2aDenkschr.  Ak.  Wiss.  Wien  (Math.),  60,  1893,  25-47. 
2i'Arithm6tique  graphique;  espaces  arith.  hypermagiques,  1894,  93. 
2"Math.  Soc.  Moscow,  17,  1894,  530-7  (in  Russian). 
"'Monatshefte  Math.  Phys.,  5,  1894,  255-266. 


152  History  of  the  Theory  of  Numbers.  [Chap,  v 

SJ.(d;m„...,m.)  =  [ii]W...[iL], 

where  d  ranges  over  the  divisors  of  n,  the  case  A;  =  1  being  due  to  Laguerre.'* 
In  the  latter  case,  take  n  =  1, .  .  . ,  n  and  add.     Thus 

k=i  LfcJ      -^^LmJ        Ini     m  \     m/ j 

the  last  equality,  in  which  (n,  h)  is  the  g.  c.  d.  of  n,  h,  following  from  expres- 
sions for  (n,  h)  given  by  Hacks^^  of  Ch.  XL  In  the  present  paper  the  above 
double  equation  was  proved  geometrically.  For  m  =  l,  we  get  Dirichlet's^^ 
formula.  The  g.  c.  d.  of  three  numbers  is  expressed  in  terms  of  them  and  [x]. 
The  initial  formulas  were  proved  geometrically,  but  were  recognized  to 
be  special  cases  of  a  more  general  theorem.     Let 

2Md)=FM  (1  =  1, ...,k), 

where  d  ranges  over  all  divisors  of  n.    Then  the  function 

^(n)  =S/i(Xi) . .  .A(X,)  (1.  c.  m.  of  Xj, . . . ,  X*  is  n) 

has  the  property 

S^(d)=Fi(n)...n(n). 

Hence  in  the  terminology  of  Bougaief  (Ch.  XIX)  the  number-theoretic 
derivative  ^{n)  of  Fi{n) . .  .Fk{n)  equals  the  sum  of  the  products  of  the 
derivatives  /»  of  the  factors  Fi,  the  arguments  ranging  over  all  sets  of  k 
numbers  having  n  as  their  g.  c.  d. 

L.  Gegenbauer^^^"  proved  easily  that,  if  [n, .  . ,  t]  is  the  g.  c.  d.  of  n, . . .  ,  f 

2      F{[n,x^,...,x,])  =  'E  F{d)jJ fj, 

where  d  ranges  over  all  divisors  of  n,  and  F  is  any  function. 

K.  Zsigmondy^^^  considered  any  abelian  (commutative)  group  G  with 
the  independent  generators  ^i,. .  .,  Qs  of  periods  ni,  .  .  .,  n^,  respectively. 
Any  element  g'l''' . . .  gj"'  of  G  is  of  period  5  if  and  only  if  5  is  the  least  positive 
value  of  X  for  which  xhi,. . .,  xhs  are  multiples  of  rii,  . .  .,  n^,  respectively. 
The  number  of  elements  of  period  5  of  G  is  thus  the  number  of  sets  of  posi- 
tive integers  hi,. . .,  hg  {hi^rii,.  .  .,  /ij^nj  such  that  5  is  the  least  value  of 
X  for  which  xhi,. . . ,  xhs  are  divisible  by  ni,  . . . ,  n^,  respectively.  The  num- 
ber of  sets  is  shown  to  be 

rPid;ni,...,n,)=lldjIl{l-l/q.'*), 

where  5_,  is  the  g.  c.  d.  of  5  and  Uj]  q\,...,qr  are  the  distinct  prime  factors 
of  5;  while  U  is  the  number  of  those  integers  nx,  .  .  .,  n^  which  contain  q^ 
at  least  as  often  as  5  contains  it.     If  5  and  5'  are  relatively  prime, 

ypib]  ni,.  .  .,  n,)\pi8';  rii,.  .  .,  n,)=i/'(55';  nj.  .  .,  nj. 

"li^Sitzungsber.  Akad.  Wiss.  Wien  (Math.),  103,  Ila,  1894,  115. 

"'Monatshefte  Math.  Phys.,  7,  1896,  227-233.     For  his  0  we  write  \p,  as  did  Carmichael." 


I 


Chap.  V]  GENERALIZATIONS   OF  EuLER's  0-FuNCTION.  153 

If  d  ranges  over  all  divisors  of  the  product  ni . . .  n^, 

Si//(5;ni, .  .  .,  n,)=nin2. .  .n,. 

d 

In  case  5  divides  each  ni{i  =  \, .  .  .,  s),  4/  becomes  Jordan's  Js(5). 

As  a  generalization  (pp.  237-9)  consider  sets  of  positive  integers  ai, . . . ,  a„ 
where  aj  =  l,  2, .  .  . ,  7_,  for  j  =  1,  2, .  .  . ,  s.     Counting  the  sets  not  of  the  form 

n^ai,  nf  a2, .  .  . ,  n^f  a,  (i  =  l,.  ..,r), 

we  get  the  number 


n 7,-2 n \-%\ +s  n f.  J'  ,,,~\ - 


where  (ni,  n2, . . .)  is  the  1.  c.  m.  of  rii,  n2, . . ..     In  particular,  take 

n^P=  . .  .  =n^f  =  ni  (i  =  l,...,r), 

where  ni, . . . ,  n^.  are  relatively  prime  in  pairs,  and  let  iV  be  a  positive  mul- 
tiple of  ni, .  .  . ,  n^  such  that 

Then  the  above  expression  equals 

J/(iV;  mi,. . .,  m,)  =  n  T-l-S  II  [—1  +  2  H  f-^^l  -  .  • ., 
y=iLmyJ     i  j=\unjniA     i,i' j=\unjnini>j 

which  determines  the  number  of  sets 

tti,.  .  .,  a,  (ay  =  l,  2,.  .  .,   —    ;i=l,.  .  .,  s) 

l_A/tyJ 

whose  g.  c.  d.  is  divisible  by  no  one  of  ni,  ^2, . . . ,  n^.     By  inversion, 


S//g;^„...,m.)  =  n[|], 


where  d  ranges  over  the  divisors  of  N  which  are  products  of  powers  of 
rii, .  .  . ,  Ur.  When  ni,...,ns  are  the  distinct  prime  factors  of  N,J/{N;  rrii, .  .  , , 
m,)  becomes  the  function  Js{N;  mi, .  .  .,  Wj)  of  von  Sterneck.^^^  As  in  the 
case  of  the  latter  function,  we  have 

J/{N;  Wi,.  . .,  m,)=SJi'(Xi;  mi). .  .//(X,;  mj, 
the  X's  ranging  over  all  sets  whose  1.  c.  m.  is  N. 

L.  Carhni^^^  proved  that  if  a  ranges  over  the  integers  for  which  [2n/a] 
=  2/c+l,  then 

XJM  =  sg^  -  2s^'J,  s^^  ^1'+...  +m*. 

For  k  =  l,  this  becomes  2(^(a)  =n^  [E.  Cesaro,  p.  144  of  this  History]. 

D.  N.  Lehmer^^^  called  Jmin)  the  m-fold  totient  of  n  or  multiple  totient 
of  n  of  multiplicity  m.     He  proved  that,  if  A:  =  pi"'.  .  .pr°^ 

Jm{k'')=k^'''-''Jm{k),  Jm(ky)=JM  n  \pr^-pr"^-'Xy,  Pi)  \  , 

1=1 
where  \(y,  pj  =0  or  1  according  as  Pi  is  or  is  not  a  divisor  of  y.     In  the 

'"Periodico  di  Mat.,  12,  1897,  137-9. 
"»Amer.  Jour.  Math.,  22,  1900,  293-335. 


154  History  of  the  Theory  of  Numbers.  [Chap,  v 

second  formula  the  product  equals  the  similar  function  of  y'  if  y  and  y'  are 
congruent  modulo  pip^  ■  ■  Pr-     Consider  the  function 

U/k] 

1=1 
where  m,  n,  k  are  positive  integers  and  x  is  a  positive  number.     Then  if 
S{x,  k)  denotes  l*+2''-f  . . .  H-[x]*,  it  is  proved  that 


which  for  m  =  n  =  1  becomes  Sylvester's^^  formula.     By  inversion, 
where  ai(i)  is  Merten's  function.     For  k  as  above  and  k'  =  k/pr''^, 


^, 


.(X,  n,  A;)=p,-(v-i)|(p^-i)ci,^(^^,  n,  A:')+$^(^,  n,  p,k')  | 

=  p,'"(«r-i)(p^_l)Sp^-^-i),^^/'^,  n,  A;'), 

where  I  is  the  least  value  of  j  for  which  [x/p^"'""^-']  =  0.  Hence  $^(x,  n,  k) 
can  be  expressed  in  terms  of  functions  $,„(?/,  n,  1).  True  relations  are 
derived  from  the  last  four  equations  by  replacing  n  by  1  —  n  and  ^m{Xy  1  —  n, 
A:)  by 

\xlk\ 

n^{x,n,k)=i:j^{ik)'(ik)-'''". 
1=1 

Proof  is  given  of  the  asymptotic  formula 

„»n7i+l       p 

^'"(^'  ^'  ^)=:;;;:;rxT  7r^+^'        hl^^x--  log  x, 

wn  +  l  D^+i 

where  A  is  finite  and  independent  of  x,  ??2,  n,  while 

«     1  *■  Pi  — I 

Dm+i  =  2  — q:Y>  P^.  fc  =  n     a^_w    „+i     7TJ  P„,.  1  =  1. 

j  =  U  i=lPi       \Pi        —i-j 

For  m  =  n  =  fc  =  l,  this  result  becomes  that  of  Mertens^*  (and  Dirichlet'^). 

The  asymptotic  expressions  found  for  ^^i^,  n,  k)  are  different  for  the  cases 

n  =  l,  n  =  2,  n>2. 

A  set  of  m  integers  (not  necessarily  positive)  having  no  common  divisor 
>  1  is  said  to  define  a  totient  point.  Let  one  coordinate,  as  x^,  have  a 
fixed  integral  value  5^0,  while  Xi,. . .,  x^-i  take  integral  values  such  that 
[xi/x^],. .  .,  [X;„_i/X;„]  have  prescribed  values;  we  obtain  a  compartment  in 
space  of  m  dimensions  which  contains  /m-i(^m)  totient  points.  For 
example,  if  m  =  3,  X3  =  6,  and  the  two  prescribed  values  are  zero,  there  are 
24  totient  points  (xi,  X2,  6)  for  which  0^Xi<6,  0^X2<6,  while  Xi  and  X2 
have  no  common  divisor  dividing  6.  For  Xi  =  l  or  5,  Xo  has  6  values;  for 
Xi  =  2  or  4,  X2=l,  3  or  5;  for  Xi  =  3,  X2  =  l,  2,  5;  for  Xi  =  0,  X2  =  0,  1,  5. 

Given  a  closed  curve  r=f{d),  decomposable  into  a  finite  number  of  seg- 
ments for  each  of  which  f{d)  is  a  single- valued,  continuous  function.     Let 


I 


Chap.  V]  FarEY  SeRIES.  155 

K  be  the  area  of  the  region  bounded  by  this  curve,  and  N  the  number  of 
points  {x,  y)  within  it  or  on  its  boundary  such  that  a;  is  a  multiple  of  k  and 
is  prime  to  y.    Then  ^     ^ 

lim— =  -2Pi.fc, 

k=aa  A       TT 

where  K  increases  by  uniform  stretching  of  the  figure  from  the  origin. 

In  particular,  consider  the  number  A^  of  irreducible  fractions  x/y^\ 
whose  denominators  are  ^n.  Since  x^y,  the  area  K  of  the  triangular  re- 
gion is  n^/2.  Hence  N  =  {n^/2)  (6/7r^) ,  approximately  (Sylvester^^) .  Again, 
the  number  of  irreducible  fractions  whose  numerators  he  between  I  and 
l-\-m,  and  denominators  between  V  and  I'+m',  is  Qmm'/ir^,  approximately. 

There  is  a  similar  theorem  in  which  the  points  are  such  that  y  is  divisible 
by  k',  while  three  new  constants  obey  conditions  of  relative  primality  to 
each  other  or  to  x,  y,  k,  k'. 

Extensions  are  stated  for  m-dimensional  space. 

E.  Cahen^^^  called  /^(n)  the  indicateur  of  /cth  order  of  n. 

G.  A.  Miller^^°  evaluated  Jk{in)  by  noting  that  it  is  the  number  of 
operators  of  period  m  in  the  abeUan  group  with  k  independent  generators 
of  period  m. 

G.  A.  Miller^^^  proved  (10)  and  (11)  by  using  the  same  abelian  group. 

E.  Busche^^^  indicated  a  proof  of  (10)  and  (12)  by  an  extension  to  space 
oi  k+1  dimensions  of  Kronecker's^^^  plane,  in  which  every  point  whose 
rectangular  coordinates  x,  y  are  integers  is  associated  with  the  g.  c.  d.  of  x,  y. 

A.  P.  Minin^^^  proved  (14)  and  some  results  due  to  Gegenbauer.^"^ 

R.  D.  CarmichaeP^^  gave  a  simple  proof  of  Zsigmondy's^^®  formula  for  ^. 

G.  Metrod^^^  stated  that  the  number  of  incongruent  sets  of  solutions 
of  xy'  —  x'y  =  a  (mod  m)  is  'EdmJ2{m/d),  where  d  ranges  over  the  common 
divisors  of  m  and  a.  When  a  takes  its  m  values,  the  total  number  of  sets 
of  solutions  is  vJ'Ay^    rt'A 

It  is  asked  if  like  relations  hold  for  Jk,  k>2. 

Cordone^^  and  Sanderson^^^  (of  Ch.  VIII)  used  Jordan's  function  in 
giving  a  generalization  of  Fermat's  theorem  to  a  double  modulus. 

Farey  Series. 

Flitcon^^  gave  the  number  of  irreducible  fractions  <1  with  each 
denominator  <100,  stating  in  effect  the  value  of  Euler's  (/)(n)  when 
n  is  a  product  of  four  or  fewer  primes. 

"9Th6orie  des  nombres,  1900,  p.  36;  I,  1914,  396-400. 
«»Amer.  Math.  Monthly,  11,  1904,  129-130. 
2"Amer.  Jour.  Math.,  27,  1905,  321-2. 
222Math.  Annalen,  60,  1905,  292. 
^^'Vorlesungen  iiber  Zahlentheorie,  1901,  I,  p.  242. 
224Matem.  Sbomik  (Moscow  Math.  Soc),  27,  1910,  340-5. 
"^''Quart.  Jour.  Math.,  44,  1913,  94-104. 

«2«L'interm6diaire  des  math.,  20,  1913,  148.     Proof,  Sphinx-Oedipe,  9,  1914,  4. 
^'Ladies'  Diary,  1751.     Reply  to  Question  281,  1747-8.     T.  Leybourn's  Math.  Quest,  pro- 
posed in  Ladies'  Diary,  1,  1817,  397-400. 


156  History  of  the  Theory  of  Numbers.  [Chap,  v 

C.  Haros-"*^  proved  the  results  rediscovered  by  Farey-^°  and  Caiichy.^^- 
J.  Farey-^"  stated  that  if  all  the  proper  vulgar  fractions  in  their  lowest 
terms,  having  both  numerator  and  denominator  not  exceeding  a  given 
number  n,  be  arranged  in  order  of  magnitude,  each  fraction  equals  a  frac- 
tion whose  numerator  and  denominator  equal  respectively  the  sum  of  the 
numerators  and  sum  of  the  denominators  of  the  two  fractions  adjacent  to  it 
in  the  series.     Thus,  for  n  =  5,  the  series  is 

1112      13      2      3      4 
Z'  T'  -JT'  T'  IT'  T'  7'  T'  T' 

and 

1_1  +  1  2_1  +  1. 

4      5+3'  5      3+2* 

Henry  Goodwyn  mentioned  this  property  on  page  5  of  the  introduction 
to  his  "tabular  series  of  decmial  quotients"  of  1818,  published  in  1816  for 
private  circulation  (see  Goodwyn,^^' ^-  Ch.  VI),  and  is  apparently  to  be 
credited  with  the  theorem.  It  was  ascribed  to  Goodwyn  by  C.  W.  Merri- 
field.2" 

A.  L.  Cauchy^^^  proved  that,  if  a/b,  a'/b',  a"/b"  are  any  three  consecu- 
tive fractions  of  a  Farey  series,  b  and  b'  are  relatively  prime  and  a'b—ab'  =  1 
(so  that  a'/b'-a/b  =  l/bb').  Similarly,  a"b'-a'b"  =  l,  so  that  a+a":  b+b" 
=  a':  b',  as  stated  by  Farey. 

StouveneP^^  proved  that,  in  a  Farey  series  of  order  n,  if  two  fractions 
a/b  and  c/b  are  complementary  (i.  e.,  have  the  sum  unity),  the  same  is  true 
of  the  fraction  preceding  a/b  and  that  following  c/b.  The  two  fractions 
adjacent  to  1/2  are  complementary  and  their  common  denominator  is  the 
greatest  odd  integer  ^n.  Hence  1/2  is  the  middle  term  of  the  series  and 
two  fractions  equidistant  from  1/2  are  complementary.  To  find  the  third 
of  three  consecutive  fractions  a/b,  a'/b',  x/y,  we  have  a+x  =  a'z,  b+y  =  b'z 
(Farey),  and  we  easily  see  that  z  is  the  greatest  integer  ^  {n-\-b)/b', 

M.  A.  Stern-^^  studied  the  sets  m,  n,  and  m,  m-\-n,  n,  and  m,  2m-\-n, 
m-\-n,  m-\-2n,  n,  etc.,  obtained  by  interpolating  the  sum  of  consecutive 
terms.     G.  Eisenstein^^"  briefly  considered  such  sets. 

*A.  Brocot^^''  considered  the  sets  obtained  by  mediation  [Farey]  from 
U/1,  1/0:  oil.  01121. 

T'  T'  ITJ  T'  Y'  1'  T'  TJ">-  •  •• 

Herzer^^^  and  Hrabak^^^  gave  tables  with  the  limits  57  and  50. 

G.  H.  Halphen^^^  considered  a  series  of  irreducible  fractions,  arranged 
in  order  of  magnitude,  chosen  according  to  a  law  such  that  if  any  fraction  / 
is  excluded  then  also  every  fraction  is  excluded  if  its  two  terms  are  at  least 

2<9Jour.  de  I'dcole  polyt.,  cah.  11,  t.  4,  1802,  364-8. 

"oPhilos.  Mag.  and  Journal,  London,  47, 1816,  385-6;  [48, 1816,  204];  Bull.  Sc.  Soc.  Philomatique 

de  Paris,  (3),  3,  1816,  112. 
"'Math.  Quest.  Educat.  Times,  9,  1868,  92-5. 
"*Bufl.  Sc.  Soc.  Philomatique  de  Paris,  (3),  3,  1816,  133-5.     Reproduced  in  Exercices  de  Math., 

1,  1826,  114-6;  Oeuvres,  (2),  6,  1887,  146-8. 
"»Jour.  de  mathdmatiques,  5,  1840,  265-275. 

»«Jour.  fur  Math.,  55,  1858,  193-220.  "laBericht  Ak.  Wiss.  Berlin,  1850,  41^2. 

"'Calcul  des  rouages  par  approximation,  Paris,  1862.  Lucas.'" 

2^«Tabellen,  Basle,  1864.  ""Tabellen-Werk,  Leipzig,  1876. 

"'Bull.  Soc.  Math.  France,  5,  1876-7,  170-5. 


Chap.  V]  FaREY  SeRIES.  157 

equal  to  the  corresponding  terms  of  /.  Such  a  series  has  the  properties 
noted  by  Farey  and  Cauchy  for  Farey  series. 

E.  Lucas^^^  considered  series  1,  1  and  1,  2,  1,  etc.,  formed  as  by  Stern. 
For  the  nth  series  it  is  stated  that  the  number  of  terms  is  2"~-^  +  l,  their 
sum  is  3""^  +  l,  the  greatest  two  terms  (of  rank  2""^+l=±=2"~^)  are 

(i+V5r+^-(i-\/5r+^ 

2"+V5 

Changing  n  to  p,  we  obtain  the  value  of  certain  other  terms. 

J.  W.  L.  Glaisher^^°  gave  some  of  the  above  facts  on  the  history  of  Farey 
series.  Glaisher^"  treated  the  history  more  fully  and  proved  (p.  328)  that 
the  properties  noted  by  Farey  and  Cauchy  hold  also  for  the  series  of  irre- 
ducible fractions  of  numerators  ^  m  and  denominators  ^  n. 

Edward  Sang^^"  proved  that  any  fraction  between  A/ a  and  C/y  is'of 
the  form  {'pA-[-qC)/{'pa-{-qy),  where  p  and  q  are  integers,  and  is  irreducible 
if  p,  q  are  relatively  prime. 

A.  Minine^®^  considered  the  number  S{a,  N)  of  irreducible  fractions  a/h 
such  that  h-\-aa^N.  Let  0(6)p  denote  the  number  of  integers  ^p  which 
are  prime  to  h.    Then,  for  a  >  0, 

>S(a,iV)=  S0(6)p,  P=L    a    J' 

since  for  each  denominator  h  there  are  (/)(6)p  integers  prime  to  h  for  which 
h+aa-^N  and  hence  that  number  of  fractions. 

A.  F.  Pullich^^^  proved  Farey's  theorem  by  induction,  using  continued 
fractions. 

G.  Airy^^^  gave  the  3043  irreducible  fractions  with  numerator  and  denom- 
inator ^  100. 

J.  J.  Sylvester^^^  showed  how  to  deduce  the  number  of  fractions  in  a 
Farey  series  by  means  of  a  functional  equation. 

Sylvester,^^'  ^^  Cesaro,^^  Vahlen,^^  Axer,^^^  and  Lehmer^^^  investigated  the 
number  of  fractions  in  a  Farey  series. 

Sylvester^^^"  discussed  the  fractions  x/y  for  which  x<n^  y<n,  x-\-y^n. 

M.  d'Ocagne^^'^  prolonged  Farey's  series  by  adding  1/1  in  the  pth  place, 
where  p=<^(l)+  .  .  .  +(j>{n).  From  the  first  p  terms  we  obtain  the  next  p 
by  adding  unity,  then  the  next  p  by  adding  unity,  etc.  Consider  a  series 
S{a,  N)  of  irreducible  fractions  Ui/hi  in  order  of  magnitude  such  that 
bi+atti^N,  where  a  is  any  fixed  integer  called  the  characteristic.  All 
the  series  S(_a,  N)  with  a  given  base  N  may  be  derived  from  Farey's  series 

««Bull.  Soc.  Math.  France,  6,  1877-8,  118-9.  ^oProc.  Cambr.  Phil.  Soc,  3,  1878,  194. 

MiLondon  Ed.  Dub.  Phil.  Mag.,  (5),  7,  1879,  321-336. 
'"Trans.  Roy.  Soc.  Edinburgh,  28,  1879,  287. 

"'Jour,  de  math.  el6m.  et  spec,  1880,  278.     Math.  Soc.  Moscow,  1880. 

'"Mathesis,  1,  1881,  161-3.  '"Trans.  Inst.  Civil  Engineers;  cf.  Phil.  Mag.,  1881,  175. 

'"Johns  Hopkins  Univ.  Circulars,  2,  1883,  44-5,  143;  Coll.  Math.  Papers,  3,  672-6,  687-8. 
266aAmer.  Jour.  Math.,  5,  1882,  303-7,  327-330;  Coll.  Math.  Papers,  IV,  55-9,  78-81. 
'"Annales  Soc.  Sc.  Bruxelles,  10, 1885-6,  II,  90.     Extract  in  Bull.  Soc.  Math.  France,  14, 1885-6, 
93-7. 


158  History  of  the  Theory  of  Numbers.  [Chap,  v 

5(0,  N)  by  use  of 

a.(a,  N)  =a,(0,  N),  &.(a,  iV)  =6,(0,  N)  -aa,(0,  N). 

Thus  a,6._i  — a,_i6i  =  l,  so  that  the  area  of  OA.A,_i  is  1/2  if  the  point 
Ai  has  the  coordinates  o,,  6,.  All  points  representing  terms  of  the  same 
rank  in  all  the  series  of  the  same  base  he  at  equally  spaced  intervals  on  a 
parallel  to  the  x-axis,  and  the  distance  between  adjacent  points  is  the  num- 
ber of  units  between  this  parallel  and  the  a:-axis. 

A.  Hurwitz-^^  apphed  Farey  series  to  the  approximation  of  numbers 
by  rational  fractions  and  to  the  reduction  of  binary  quadratic  forms. 

J.  Hermes-^^  designated  as  numbers  of  Farey  the  numbers  ri  =  l,  72  =  2, 
X3  =  7^  =  3,  To  =  4,  tq  =  t7  =  5,  t8  =  4,  .  .  .  with  the  recursion  formula 

T„  =  r„_2^+r2^+i-n+i,  2''<n^2'+\ 

and  connected  with  the  representation  of  numbers  to  base  2.  The  ratios 
of  the  r's  give  the  Farey  fractions. 

K.  Th.  Vahlen-^^''  noted  that  the  formation  of  the  convergents  to  a 
fraction  w  by  Farey's  series  coincides  with  the  development  of  w  into  a  con- 
tinued fraction  whose  numerators  are  ±1,  and  made  an  application  to  the 
composition  of  linear  fractional  substitutions. 

H.  Made-'°  apphed  Hurwitz's  method  to  numbers  a+hi. 

E.  Busche"^  apphed  geometrically  the  series  of  irreducible  fractions  of 
denominators  ^a  and  numerators  ^b,  and  noted  that  the  properties  of 
Farey  series  {a  =  h)  hold  [Glaisher-®^]. 

W.  Sierpinski^^^  used  consecutive  fractions  of  Farey  series  of  order  m 
to  show  that,  if  x  is  irrational. 


===«  U=i^     ^  2  2j 


Expositions  of  the  theory  of  Farey  series  were  given  by  E.  Lucas,'" 
E.  Cahen,-'^  Bachmann.^'^ 

An  anonymous  writer,^'^  starting  with  the  irreducible  fractions  <1, 
arranged  in  order  of  magnitude,  with  the  denominators  ^  10,  inserted  the 
fractions  with  denominator  11  by  listing  the  pairs  of  fractions  0/1,  1/10; 
1/6,  1/5;  1/4,  2/7;.  .  .,  the  sum  of  whose  denominators  is  11,  and  noting 
that  between  the  two  of  each  pair  lies  a  fraction  with  denominator  11  and 
numerator  equal  the  sum  of  their  numerators. 

*«8Math.  Annalen,  44,  1894,  417-436;  39,  1891,  279;  45,  1894,  85;  Math.  Papers  of  the  Chicago 
Congress,  1896,  125.  Cf.  F.  Klein,  Ausgewahlte  Kapitel  der  Zahlentheorie,  I,  1896, 
19^210.     Cf.  G.  Humbert,  Jour,  de  Math.,  (7),  2,  1916,  116-7. 

*«9Math.  Annalen,  45,  1894,  371.     Cf.  L.  von  Schrutka,  71,  1912,  574,  583. 

^s'ajour.  fiir  Math.,  115,  1895,  221-233. 

^'^Ueber  Fareysche  Doppelreihen,  Diss.  Giessen,  Darmstadt,  1903. 

"'Math.  Annalen,  60,  1905,  288. 

*"BuU.  Inter.  Acad.  Sc.  Cracovie,  1909,  II,  725-7. 

»"Th6orie  des  nombres,  1891,  467-475,  508-9. 

*'^fil4ments  de  la  theorie  des  nombres,  1900,  331-5. 

"'Niedere  Zahlentheorie,  1,  1902,  121-150;  2,  1910,  55-96. 

i^'OZeitschrift  Math.  Naturw.  Unterricht,  45,  1914,  559-562. 


CHAPTER  VI. 

PERIODIC  DECIMAL  FRACTIONS;  PERIODIC  FRACTIONS:  FACTORS 

OF  I0"±1. 

Ibn-el-Banna^  (Albanna)  in  the  thirteenth  century  factored  10"  — 1  for 
small  values  of  n.  The  Arab  Sibt  el-Maridini^"  in  the  fifteenth  century 
noted  that  in  the  sexagesimal  division  of  47°  50'  by  1°  25'  the  quotient 
has  a  period  of  eight  terms. 

G.  W.  Leibniz^  in  1677  noted  that  \/n  gives  rise  to  a  purely  periodic 
fraction  to  any  base  h,  later  adding  the  correction  that  n  and  h  must  be 
relatively  prime.  The  length  of  the  period  of  the  decimal  fraction  for  1/n, 
where  n  is  prime  to  10,  is  a  divisor  of  n  —  1  [erroneous  for  n  =  21 ;  cf .  Wallis^] . 

John  Wallis^  noted  that,  if  N  has  a  prime  factor  other  than  2  and  5,  the 
reduced  fraction  M/N  equals  an  unending  decimal  fraction  with  a  repetend 
of  at  most  A^  — 1  digits.  If  N  is  not  divisible  by  2  or  5,  the  period  has  two 
digits  if  N  divides  99,  but  not  9;  three  digits  if  A^  divides  999,  but  not  99. 
The  period  of  1/21  has  six  digits  and  6  is  not  a  divisor  of  21  —  1.  The 
length  of  the  period  for  the  reciprocal  of  a  product  equals  the  1.  c.  m.  of 
the  lengths  of  the  periods  of  the  reciprocals  of  the  factors  [cf.  Bernoulli^]. 
Similar  results  hold  for  base  60  in  place  of  10. 

J.  H.  Lambert^  noted  that  all  periodic  decimal  fractions  arise  from 
rational  fractions;  if  the  period  p  has  n  digits  and  is  preceded  by  a  decimal 
with  m  digits,  we  have 


lO'"      '  lO'^lO"     lO'^lO^"  lO'^ClO^-l) 

John  Robertson^  noted  that  a  pure  periodic  decimal  with  a  period  P  of 
k  digits  equals  P/9 ...  9,  where  there  are  k  digits  9. 

J.  H.  Lambert^  concluded  from  Fermat's  theorem  that,  if  a  is  a  prime 
other  than  2  and  5,  the  number  of  terms  in  the  period  of  \/a  is  a  divisor 
of  a  — 1.  If  S'  is  odd  and  \/g  has  a  period  oi  g  —  1  terms,  then  ^  is  a  prime. 
If  \/g  has  a  period  of  m  terms,  but  ^  —  1  is  not  divisible  by  m,  g  is  composite. 
Let  1/a  have  a  period  of  2m  terms;  if  a  is  prime,  A;  =  lO'^+l  is  divisible  by 
a;  if  a  is  composite,  k  and  a  have  a  common  factor;  if  k  is  divisible  by  a 
and  if  m  is  prime,  each  factor  other  than  2^5^  of  a  is  of  period  2m. 

Let  a  be  a  composite  number  not  divisible  by  2,  3  or  5.  If  1/a  has  a 
period  of  m  terms,  where  w  is  a  prime,  each  factor  of  a  produces  a  period 

'Cf.  E.  Lucas,  Arithm^tique  amusante,  1895,  63-9;  Brocard.'o^ 

i«Carra  de  Vaux,  Bibliotheca  Matb.,  (2),  13,  1899,  33-4. 

^Manuscript  in  Bibliothek  Hannover,  vol.  Ill,  24;  XII,  2,  Blatt  4;  also.  III,  25,  Blatt  1,  seq., 

10,  Jan.,  1687.     Cf.  D.  Mahnke,  BibUotheca  Math.,  (3),  13,  1912-3,  45-48. 
^Treatise  of  Algebra  both  historical  &  practical,  London,  1685,  ch.  89,  326-8  (in  manuscript, 

1676). 
*Acta  Helvetica,  3,  1758,  128-132. 
»Phil.  Trans.,  London,  58,  1768,  207-213. 
"Nova  Acta  Eruditorum,  Lipsiae,  1769,  107-128. 

159 


160  History  of  the  Theory  of  Numbers.  [Chap,  vi 

of  m  terms.  If  \/a  has  a  period  of  mn  terms,  where  m  and  n  are  primes, 
while  no  factor  has  such  a  period,  one  factor  of  a  divides  10'"  —  1  and  another 
di\'ides  10"  —  1.  If  \/a  has  a  period  of  mnp  terms,  where  vi,  n,  p  are  primes, 
but  no  factor  has  such  a  period,  any  factor  of  a  divides  10"*- 1,.  .  .,  or 
10""  —  !.     These  theorems  aid  in  factoring  a. 

L.  Euler^  gave  numerical  examples  of  the  conversion  of  ordinary  frac- 
tions into  decimal  fractions  and  the  converse  problem. 

Euler^''  noted  that  if  2p+l  is  a  prime  40?2±1,  ±3,  ±9,  ±13,  it  divides 
lO^-ljif  2p  +  l  isaprime40;?±7,  ±11,  ±17,  ±19,  it  divides  10^+1. 

Jean  BernouUi^  gave  a  r^sum6  of  the  work  by  Wallis,^  Robertson,* 
Lambert^  and  Euler,^  and  gave  a  table  showing  the  full  period  for  1/D  for 
each  odd  prime  D<200,  and  a  like  table  when  Z)  is  a  product  of  two  equal 
or  distinct  primes  <  25.  When  the  two  primes  are  distinct,  the  table  con- 
firms Wallis'  assertion  that  the  length  of  the  period  for  1/D  is  the  1.  c.  m. 
of  the  lengths  of  the  periods  for  the  reciprocals  of  the  factors.  But  for 
l/D^,  where  D  is  a  prime  >  3,  the  length  of  the  period  equals  D  times  that 
for  1/D.  If  the  period  for  1/D,  where  D  is  a  prime,  has  D  —  1  digits,  the 
period  for  ?n/D  has  the  same  digits  permuted  cyclically  to  begin  with  m. 
He  gave  (p.  310)  a  device  communicated  to  him  by  Lambert:  to  find  the 
period  for  1/D,  where  Z)  =  181,  we  find  the  remainder  7  after  obtaining  the 
part  p  composed  of  the  first  15  digits  of  the  period;  multiply  l/D  =  p-\-7/D 
by  7 ;  thus  the  next  15  digits  of  the  period  are  given  by  7p ;  since  7^  =  /)+ 162, 
the  third  set  of  15  digits  is  found  by  adding  unity  to  7~p,  etc.;  since  7 
belongs  to  the  exponent  12  modulo  D,  the  period  for  1/D  contains  15-12 
digits. 

Jean  Bernoulli^  made  use  of  various  theorems  due  to  Euler  which  give 
the  possible  linear  forms  of  the  divisors  of  10*±1,  and  obtained  factors  of 
(10*-l)/9  when  A-^30,  except  for  k  =  ll,  17,  19,  23,  29,  with  doubt  as  to 
the  primality  of  the  largest  factor  when  A' =  13,  15  or  ^19.  He  stated 
(p.  325)  erroneously^^  that  (10^^  +  l)/ll-23  has  no  factor  <3000.     Also, 

10''+1  =  7-1M3-211-9091-520S1. 

He  gave  part  of  the  periods  for  the  reciprocals  of  various  primes  ^601. 

L.  Euler^^  wTote  to  Bernoulli  concerning  the  latter's^  paper  and  stated 
criteria  for  the  divisibility  of  10^±1  by  a  prime  2p  +  l=4n±l.  If  both 
2  and  5  or  neither  occur  among  the  divisors  of  n,  n=F2,  n=F6,  then  10''  — 1 
is  divisible  by  2p-\-l.  But  if  only  one  of  2  and  5  occurs,  then  10^+1  is 
divisible  by  2p+l  [cf.  Genocchi^^]. 

Henry  Clarke^^  discussed  the  conversion  of  ordinary  fractions  into 
decimals  without  dealing  with  theoretical  principles. 

'Algebra,  I,  Ch.  12,  1770;  French  trans!.,  1774. 
'"Opusc.  anal.,  1,  1773,  242;  Comm.  Arith.  Coll.,  2,  p.  10,  p.  25. 
'Nouv.  m6m.  acad.  roy.  Berlin,  ann^e  1771  (1773),  273-317. 
*Ibid.,  318-337. 

"P.  Seelhoff,  Zeitschrift  Math.  Phys.,  31,  1886,  63.  Reprinted,  Sphinx-Oedipc,  5,  1910,  77-8. 
"Nouv.  m(Sm.  acad.  roy.  Berlin,  annde  1772  (1774),  Histoire,  pp.  35-36;  Comm.  Arith.,  1,  584. 
^^he  rationale  of  circulating  numbers,  London,  1777,  1794. 


\ 


Chap.  VI]  PERIODIC   DECIMAL   FRACTIONS.  161 

Anton  FelkeP^  showed  how  to  convert  directly  a  periodic  fraction 
written  to  one  base  into  one  to  another  base.  He  gave  all  primes  <  1000 
which  can  divide  a  period  with  a  prime  number  of  digits  <30,  as  29m +1 
=  59,233,.... 

Oberreit^*  extended  Bernoulli's^  table  of  factors  of  10*=*=!. 

C.  F.  Gauss^^  gave  a  table  showing  the  period  of  the  decimal  fraction  for 
Vp",  p"<467,  V  a  prime,  and  the  period  for  1/p",  467^ p"^  997. 

W.  F.  Wucherer^®  gave  five  places  of  the  decimal  fraction  for  n/d, 
d<1000,  n<dford<50,  n^  10  for  (^^ 50. 

Schroter  published  at  Helmstadt  in  1799  a  table  for  converting  ordinary 
fractions  into  decimal  fractions. 

C.  F.  Gauss^'^  proved  that,  if  a  is  not  divisible  by  the  prime  p  (^7^2,  5), 
the  length  of  the  period  for  a/p"  is  the  exponent  e  to  which  10  belongs 
modulo  p^.  If  we  set  0(p")  =ef  and  choose  a  primitive  root  r  of  p^  such 
that  the  index  of  10  is  /,  we  can  easily  deduce  from  the  periods  for  k/p^, 
where  k  =  \,  r, .  .  .,  r^~\  the  period  for  m/p",  where  m  is  any  integer  not 
divisible  by  p.  For,  if  i  be  the  index  of  m  to  the  base  r,  and  if  i  =  af-\-^, 
where  0^/3</,  we  obtain  the  period  for  m/p"  from  that  for  rVp"  by  carrying 
the  first  a  digits  to  the  end.  He  computed^^  the  necessary  periods  for  each 
p"<1000,  but  published  here  the  table  only  to  100.  By  using  partial 
fractions,  we  may  employ  the  table  to  obtain  the  period  for  a/b,  where  b 
is  a  product  of  powers  of  primes  within  the  limits  of  the  table. 

H.  Goodwyn^^  noted  that,  if  a<17,  the  period  for  a/17  is  derived  from 
the  period  for  1/17  by  a  cyclic  permutation  of  the  digits.  Thus  we  may 
print  in  a  double  line  the  periods  for  1/17, . . . ,  16/17  by  showing  the  period 
for  1/17  and,  above  each  digit  d  of  the  latter,  showing  the  value  of  a  such 
that  the  period  for  a/ 17  begins  with  the  digit  d,  while  the  rest  of  the 
period  is  to  be  read  cyclically  from  that  for  1/17. 

Goodwyn^^  noted  that  when  1/p  is  converted  into  a  decimal  fraction, 
p  being  prime,  the  sum  of  corresponding  quotients  in  the  two  half  periods 
is  9,  and  that  for  remainders  is  p,  if  p^7. 

J.  C.  Burckhardt^"  gave  the  length  of  the  period  for  1/p  for  each  prime 
p^2543  and  for  22  higher  primes.  It  follows  that  10  is  a  primitive  root 
of  148  of  the  365  primes  p,  5<p<2500. 

"Abhand.  Bohmiachen  Gesell.  Wias.,  Prag,  1,  1785,  135-174. 

"J.  H.  Lambert's  Deutscher  Gelehrter  Briefwechsel,  pub.  by  J.  Bernoulli,  Leipzig,  vol.  5, 
1787,  480-1.  The  part  (464-479)  relating  to  periodic  decimals  is  mainly  from  Ber- 
noulli's' paper. 

"Posthumous  manuscript,  dated  Oct.,  1795;  Werke,  2,  1863,  412-434. 

^'Beytrage  zum  allgemeinem  Gebrauch  der  Decimal  Brliche. .  . .,  Carlsruhe,  1796. 

"Disq.  Arith.,  1801,  Arts.  312-8.  A  part  was  reproduced  by  Wertheim,  Elemente  der  Zahlen- 
theorie,  1887,  153-6. 

I'Jour.  Nat.  Phil.  Chem.  Arts  (ed.,  Nicholson),  London,  4,  1801,  402-3. 

"76id.,  new  series,  1,  1802,  314-6.     Cf.  R.  Law,  Ladies'  Diary,  1824,  44-45,  Quest.  1418. 

'"Tables  des  diviseurs  pour  tous  les  nombres  du  premier  milMon,  Paris,  1817,  p.  114.  For  errata 
see  Shanks,"  Kessler,"  Cimningham,^^!  and  G^rardin."^ 


162  History  of  the  Theory  of  Numbers.  [Chap,  vi 

H.  Goodw-jTi-^  gave  for  each  integer  d^  100  a  table  of  the  periods  for 
n/d,  for  the  various  integers  n<d  and  prime  to  d.  Also,  a  table  giving 
the  first  eight  digits  of  the  decimal  equivalent  to  everj^  irreducible  vulgar 
fraction  <  1/2,  whose  numerator  and  denominator  are  both  ^  100,  arranged 
in  order  of  magnitude,  up  to  1/2. 

GoodwATi"'  -^  was  without  doubt  the  author  of  two  tables,  which  refer 
to  the  preceding  ''short  specimen"  by  the  same  author.  The  first  gives 
the  first  eight  digits  of  the  decimal  equivalent  to  every  irreducible  \'ulgar 
fraction,  whose  numerator  and  denominator  are  both  ^  1000,  from  1/1000 
to  99/991  arranged  in  order  of  magnitude.  In  the  second  volume,  the 
"table  of  circles"  occupies  107  pages  and  contains  all  the  periods  (circles) 
of  ever}'  denominator  prime  to  10  up  to  1024;  there  is  added  a  two-page 
table  showing  the  quotient  of  each  number  ^  1024  by  its  largest  factor  2°5''. 

For  example,  the  entry  in  the  "tabular  series"  under  -^^  is  .08689024. 
The  entry  in  the  two-page  table  under  656  is  41.  Of  the  various  entries 
under  41  in  the  "table  of  circles,"  the  one  containing  the  digits  9024  gives 
the  complete  period  90243.     Hence  /^V  =  -086890243. 

Glaisher"^  gave  a  detailed  account  of  Goodwyn's  tables  and  checks  on 
them.  They  are  described  in  the  British  Assoc.  Report,  1873,  pp.  31-34, 
along  -wdth  tables  showing  seven  figures  of  the  reciprocals  of  numbers 
< 100000. 

F.  T.  Poselger'^  considered  the  quotients  0,  a,  h,.  .  .  and  the  remainders 
1,  a,  j8, .  .  .  obtained  by  di\'iding  1,  ^,  A~, ...  by  the  prime  p;  thus 

A  a  A'         .,.B 

—=a-i — ,  —  =  aA+b-\ — ,.... 

P  P  P  P 

Adding,  we  see  that  the  sum  lH-a-}-/3-|-  ...  of  the  remainders  of  the  period 
is  a  multiple  TTzp  of  p;  also,  w(A  — 1)  =a+6-f- . .  ..     Set 

M  =  k+...+hA'-'--\-aA*-\ 
where  A  belongs  to  the  exponent  t  modulo  p.    Then 

—  =  -+MS,  S  =  l+A'+  .  .  .  +A^"-'\ 

P     P 

"The  first  centenarj'  of  a  series  of  concise  and  useful  tables  of  all  the  complete  decimal  quotients 
which  can  arise  from  dividing  a  unit,  or  any  whole  number  less  than  each  divisor,  by  all 
integers  from  1  to  1024.  To  which  is  now  added  a  tabular  series  of  complete  decimal 
quotients  for  all  the  proper  vulgar  fractions  of  which,  when  in  their  lowest  terms,  neither 
the  numerator  nor  the  denominator  is  greater  than  100;  with  the  equivalent  vulgar 
fractions  prefixed.  By  Henry  Goodwyn,  London,  1818,  pp.  xiv  +  lS;  vii+30.  The  first 
part  was  printed  in  1816  for  private  circulation  and  cited  by  J.  Farey  in  Philos.  Mag.  and 
Journal,  London,  47,  1816,  385. 

"A  tabular  series  of  decimal  quotients  for  all  the  proper  vulgar  fractions  of  which,  when  in  their 
lowest  terms,  neither  the  numerator  nor  the  denominator  is  greater  than  1000,  London, 
1823,  pp.  v  +  153. 

"A  table  of  the  circles  arising  from  the  division  of  a  unit,  or  any  other  whole  number,  by  all  the 
integers  from  1  to  1024;  being  all  the  pure  decimal  quotients  that  can  arise  from  this 
source,  London,  1823,  pp.  v  +  118. 

"Abhand.  Ak.  Wiss.  BerUn  (Math.),  1827,  21-36. 


Chap.  VI]  PERIODIC    DECIMAL   FRACTIONS.  163 

If  M  is  divisible  by  p,  we  may  take  n  =  1  and  conclude  that  A^p^  differs 
from  1/p^  by  an  integer.  If  M  is  not  divisible  by  p,  S  must  be,  so  that  n 
is  divisible  by  p  and  the  length  of  the  period  is  pt.  In  general,  for  the  denom- 
inator p^,  we  have  n  =  l  if  M  is  divisible  by  p^~^,  but  in  the  contrary  case 
n  is  a  multiple  of  p'"'^.  If  the  period  for  a  prime  p  has  an  even  number  of 
digits,  the  sum  of  corresponding  quotients  in  the  two  half  periods  is  p. 

An  anonymous  writer^^  noted  that,  if  we  add  the  digits  of  the  period  of 
a  circulating  decimal,  then  add  the  digits  of  the  new  sum,  etc.,  we  finally 
get  9.  From  a  number  subtract  that  obtained  by  reversing  its  digits;  add 
the  digits  of  the  difference;  repeat  for  the  sum,  etc.;  we  get  9. 

Bredow^^  gave  the  periods  for  a/p,  where  p  is  a  prime  or  power  of  a 
prime  between  100  and  200.  He  gave  certain  factors  of  10"  —  1  for  w  =  6-10, 
12-16,  18,  21,  22,  28,  33,  35,  41,  44,  46,  58,  60,  96. 

E.  Midy"  noted  that,  if  a"",  a"', . . .  are  the  least  powers  of  a  which, 
diminished  by  unity,  give  remainders  divisible  by  q^,  qi''', .  . . ,  respectively 
{q,  qi,...  being  distinct  primes),  and  if  the  quotients  are  not  divisible  by 
q,  qi,.  .  .,  respectively,  and  if  t  is  the  1.  c.  m.  of  n,  ni, .  .  .,  then  a  belongs  to 
the  exponent  t  modulo  p  =  q^qi^' .  .  . ,  and  a'  —  1  is  divisible  by  q  only  h  times. 

Let  the  period  of  the  pure  decimal  fraction  for  a/h  have  2n  digits.  If 
h  is  prime  to  10"  — 1,  the  sum  of  corresponding  digits  in  the  half  periods  is 
always  9,  and  the  sum  of  corresponding  remainders  is  h.  Next,  let  6  and 
10"  — 1  have  d>l  as  their  g.  c.  d.  and  set  h'  =  h/d.  Let  a„  be  the  nth  re- 
mainder in  finding  the  decimal  fraction.  Then  a+a„  =  6'A:,  ai+a„+i  =  6'/ci, 
etc.  The  sums  q-\-qn,  5i+g„+i, ...  of  corresponding  digits  in  the  half 
periods  equal 

{\{)k-k^)/d,  il0k,-k2)/d,.. .,  {10k,_r-k)/d. 

Similar  results  hold  when  the  period  of  mn  digits  is  divided  into  n  parts  of 
m  digits  each.     For  example,  in  the  period 

002481389578163771712158808933 
for  1/403,  the  two  halves  are  not  complementary  (10^^  —  1  being  divisible 
by  31);  for  i  =  l,  2,  3,  the  sum  of  the  digits  of  rank  i,  i-\-3,  i+6, .  .  .,  i+27 
is  always  45,  while  the  corresponding  sums  of  the  remainders  are  2015. 

N.  Druckenmiiller^^"  noted  that  any  fraction  can  be  expressed  as  a/x-\- 
ai/x^-l- .... 

J.  Westerberg^^  gave  in  1838  factors  of  10"±  1  for  nS  15. 

G.  R.  Perkins^^  considered  the  remainder  r^  when  N'^  is  divided  by  P, 
and  the  quotient  q  in  Nrj._i  =  Pqx-\-rj..  If  Tk'^P—l,  there  are  2k  terms  in 
the  period  of  remainders,  and 

rk+x+r^  =  P,  qk+x+qx  =  N-l. 

[These  results  relate  to  1/P  written  to  the  base  N.] 

^^Polytechnisches  Journal  (ed.,  J.  G.  Dingier),  Stuttgart,  34, 1829, 68;  extract  from  Mechanics' 

Magazine,  N.  313,  p.  411. 
^*Von  den  Perioden  der  Decimalbriiche,  Progr.,  Oels,  1834. 

'^'De  quelques  propriet^s  des  nombres  et  des  fractions  d^cimales  p^riodiques,  Nantes,  1836,21  pp. 
""T.heorie  der  Kettenreihen .  .  .,  Trier,  1837. 
28See  Chapter  on  Perfect  Numbers."* 
2»Amer.  Jour.  Sc.  Arts,  40,  1841,  112-7. 


164  History  of  the  Theory  of  Numbers.  [Chap,  vi 

E.  Catalan^"  converted  periodic  decimals  into  ordinary  fractions  without 
using  infinite  progressions.  When  1/13  is  converted  into  a  decimal,  the 
period  of  remainders  is  1,  10,  9,  12,  3,  4;  repeat  the  period;  starting  in 
the  series  of  12  terms  with  any  term  (as  10),  take  the  fourth  term  (4)  after 
it,  the  fourth  term  (12)  after  that,  etc.;  then  the  sum  26  of  the  three  is  a 
multiple  of  13.  In  general,  if  D  is  a  prime  and  D  —  l=mn,  the  sum  of  n 
terms  taken  w  by  m  in  the  period  for  N/D  is  a  multiple  of  D  [cf.  Thibault'^]. 

If  the  sum  of  two  terms  of  the  period  of  remainders  for  N/D  is  D,  the 
same  is  true  of  the  terms  following  them.  Hence  the  sum  of  corresponding 
terms  of  the  two  half  periods  is  D.  This  happens  if  the  number  of  terms 
of  the  period  is  <f){D). 

Thibault^^  denoted  the  numbers  of  digits  in  the  periods  for  l/d  and 
1/d'  by  m  and  m'.  If  d'  is  divisible  by  d,  m'  is  divisible  by  m.  If  d  and  d' 
have  no  common  prime  factor  other  than  2  or  5,  the  number  of  digits  in 
the  period  for  \/dd'  is  the  1.  c.  m.  of  m,  m'.  Hence  it  suffices  to  know  the 
length  of  the  period  for  1/p",  where  p  is  a  prime.  If  1/p  has  a  period  of  m 
digits  and  if  1/p"  is  the  last  one  of  the  series  1/p,  1/p^, .  . .  which  has  a 
period  of  m  digits,  then  the  period  for  1/p"  for  a  >n  has  mp"'^  digits.  For 
p  =  3,  we  have  w  =  2;  hence  1/3'^  for  r^2  has  a  period  of  y~^  digits.  For 
any  prime  p  for  which  7^p^  101,  we  have  n  =  1,  so  that  1/p"  has  a  period 
of  mp°-~^  digits.  Note  that  \/p  and  1/p^  have  periods  of  the  same  length 
to  base  h  if  and  only  if  h^~^  =  1  (mod  p^).  Proof  is  given  of  Catalan's^"  first 
theorem,  which  holds  only  when  10"' ^1  (mod  D),  i.  e.,  when  m  is  not  a 
multiple  of  the  number  of  digits  in  the  period.  For  example,  the  sum  of 
the  /cth  and  (6+A;)th  remainders  for  1/13  is  not  a  multiple  of  13. 

E.  Prouhet^^  proved  Thibault's"  theorem  on  the  period  for  l/p".  He^^" 
noted  that  multiples  of  142857  have  the  same  digits  permuted. 

P.  Lafitte^^  proved  Midy's^^  theorem  that,  if  p  is  a  prime  not  dividing 
m  and  if  the  period  for  m/p  has  an  even  number  of  digits,  the  sum  of  the 
two  halves  of  the  period  is  9 ...  9. 

J.  Sornin^^  investigated  the  number  m  of  digits  in  the  period  for  1/Z), 
where  D  is  prime  to  10.  The  period  is  a;  =  (10"*  -  l)/D.  First,  let  D  =  lOA: + 1 . 
Then  x  =  \Qy  —  \,  where 

10*"-^+ A;     ,^    ,  ,                   lO'-^-A:^ 
y  = ^ =  lOz+k,  z  = 

Finally,  we  reach  v=  \l  —  {  —  k)'^\/D,  and  x  is  an  integer  if  and  only  if  v 
is.  Hence  if  we  form  the  powers  of  the  number  k  of  tens  in  D,  add  1  to 
the  odd  powers,  but  subtract  1  from  the  even  powers  of  k,  the  first  exponent 
giving  a  result  divisible  by  D  is  the  number  m  of  digits  in  the  period. 

»»Nouv.  Ann.  Math.,  1,  1842,  464-5,  467-9. 

*nhid.,  2,  1843,  80-89. 

"/bid.,  5,  1846,  661. 

^IhU.,  3,  1844,  376;  1851,  147-152. 

'Vbid.,  397-9.     Cf.  Araer.  Math.  Monthly,  19,  1912,  130-2. 

w/Wd.,  8,  1849,  50-57. 


Chap.  VI]  PeEIODIC   DECIMAL  FRACTIONS.  165 

Next,  if  D  =  10k  —  1,  we  have  a  like  rule  to  be  applied  only  to  the  ^"^  —  1.  If 
D  =  10k=^3,  1/(3Z))  has  a  denominator  lOZ^l,  and  the  length  of  its  period, 
found  as  above,  is  shown  to  be  not  less  than  that  for  1/D. 

Th.  Bertram^^  gave  certain  numbers  p  for  which  l/p  has  a  given 
length  k  of  period  for  k^  100.     Cf.  Shanks.^^ 

J.  R.  Young^^  took  a  part  of  a  periodic  decimal,  as  .1428571  428  for  1/7, 
and  marked  off  from  the  end  a  certain  number  (three)  of  digits.  We  can 
find  a  multipHer  (as  6)  such  that  the  product,  with  the  proper  carrying 
(here  2)  from  the  part  marked  off,  has  all  the  digits  of  the  abridged  number 
in  the  same  cyclic  order,  except  certain  of  the  leading  digits.  In  the  special 
case  the  product  is  .8571428. 

W.  Loof"  gave  the  primes  p  for  which  the  period  for  l/p  has  a  given 
number  n  of  digits,  n^  60,  with  no  entry  for  n  =  17,  19,  37-40,  47,  49,  57,  59, 
and  with  doubt  as  to  the  primality  of  large  numbers  entered  for  various 
other  n's. 

E.  Desmarest^^  gave  the  primes  P<  10000  for  which  10  belongs  to  the 
exponent  {P  —  l)/t  for  successive  values  of  t.  The  table  thus  gives  the 
length  of  the  period  for  1/P.  He  stated  (pp.  294-5)  that  if  P  is  a  prime 
<  1000,  and  if  p  is  the  length  of  the  period  for  A/P,  then  except  for  P  =  3 
and  P  =  487  the  length  of  the  period  for  A/P^  is  pP. 

A.  Genocchi^^  proved  Euler's^^  rule  by  use  of  the  quadratic  reciprocity 
law.  Thus  5  is  a  quadratic  residue  or  non-residue  of  N  according  as 
N  =  5m=^l  or  5m±3;  for  4n+l  =  5m='=l,  n  or  n  —  2  is  divisible  by  5;  for 
4n  — l  =  5m='=l,  n  or  n-\-2  is  divisible  by  5.  Also,  2  is  a  residue  of  4n±l 
for  n  even,  a  non-residue  for  n  odd.  Hence  10  is  a  residue  of  A^  =  4n='=  1  for 
n  even  if  n  orn  =f2  is  divisible  by  5,  and  for  n  odd  if  neither  is.  Thus  Euler's 
inclusion  of  n=F6  is  superfluous.  By  a  similar  proof,  10  is  quadratic  non- 
residue  of  A/'  =  4n±l  if  both  2  and  5  occur  among  the  divisors  of  n±2, 
n±6,  or  if  neither  occurs;  a  residue  if  a  single  one  of  them  occurs. 

A.  P.  Reyer^^"  noted  that  the  period  for  a/3^  has  3^~^  digits  and  gave  the 
length  of  the  period  for  a/p  for  each  prime  p<  150. 

*F.  van  Henekeler^^^  treated  decimal  fractions. 

C.  G.  Reuschle^"  gave  for  each  prime  p<  15000  the  exponent  e  to  which 
10  belongs  modulo  p.  Thus  e  is  the  length  of  the  period  for  l/p.  He  gave 
all  the  prime  factors  of  lO'^-l  for  n^l6,  n  =  lS,  20,  21,  22,  24,  26,  28,  30, 
32,  36,  42;  those  of  10"+1  for  n^l8,  n  =  21;  also  cases  up  to  n  =  243  of 
the  factors  of  the  quotient  obtained  by  excluding  analytic  factors. 

"Einige  Satze  aus  der  Zahlenlehxe,  Progr.  Coin,  Berlin,  1849,  14-15. 

»«London,  Ed.  Dublin  Phil.  Mag.,  36,  1850,  15-20. 

»^Archiv  Math.  Phys.,  16,  1851,  54-57.     French  transl.  in  Nouv.  Ann.  Math.,  14,  1855,  115-7. 

Quoted  by  Brocard,  Mathesis,  4,  1884,  38. 
'^Th^orie  des  nombres,  Paris,  1852,  308.     For  errata,  see  Shanks*^  and  G^rardin.^'^ 
"Bull.  Acad.  Roy.  Sc.  Belgique,  20,  II,  1853,  397-400. 
"<^Archiv  Math.  Phys.,  25,  1855,  190-6. 
'^''Ueber  die  primitiven  Wurzeln  der  Zahlen  und  ihre  Anwendung  auf  Dezimalbriicbe,  Leyden, 

1855  (Dutch). 
"Math.  Abhandlung..  .Tabellen,  Progr.  Stuttgart,  1856.     Full  title  in  Ch.  I."*      Errata, 

Bork,i''5  Hertzer,ii3  Cunningham. i" 


I 


166  History  of  the  Theory  of  Numbers.  [Chap,  vi 

W.  Stammer^^  noted  that  n/p  =  0.di . .  .  a^  implies 

-(10'-l)=ai...a,. 
V 

J.  B.  Sturm^^  used  this  result  to  explain  the  conversion  of  decimal  into 
ordinary  fractions  without  the  use  of  series. 

M.  Collins'^^  stated  that,  if  we  multiply  any  decimal  fraction  having  m 
digits  in  its  period  by  one  with  n  digits,  we  obtain  a  product  with  Own 
digits  in  its  period  if  vi  is  prime  to  n,  but  with  71(10™  — 1)  digits  if  n  is 
divisible  by  m. 

J.  E.  Oliver^^  proved  the  last  theorem.  If  x'/x  gives  a  periodic  fraction 
to  the  base  a  with  a  period  of  ^  figures,  then  a^  =  1  (mod  x)  and  conversely. 
The  product  of  the  periodic  fractions  for  x'/x, . . . ,  z'/z  with  period  lengths 
^, . .  . ,  f  has  the  period  length 

•M(^,...,f), 


M{x,...,z) 


where  M{x, . . .,  z)  is  the  1.  c.  m.  of  x, . . . ,  z.  He  examined  the  cases  in 
which  the  first  factor  in  the  formula  is  expressible  in  terms  of  ^, .  .  . ,  f . 

Fr.  Heime'*^  and  M.  Pokorny^^  gave  expositions  without  novelty. 

Suffield*^  gave  the  more  important  rules  for  periodic  decimals  and  indi- 
cated the  close  connection  with  the  method  of  synthetic  division. 

W.  H.  H.  Hudson*^  called  d  a  proper  prime  if  the  period  for  n/d  has  d  —  1 
digits.  If  the  period  for  r/p  has  n  =  ip  —  l)/\  digits,  there  are  X  periods 
for  p.  The  sum  of  the  digits  in  the  period  for  a  proper  prime  p  is  9{p  — 1)/2. 
If  1/p  has  a  period  of  2n  digits,  the  sum  of  corresponding  digits  in  the  two 
half  periods  is  9,  and  this  holds  also  if  p  is  composite  but  has  no  factor 
dividing  10"  —  1  [Midy"].  If  lOp+l  is  a  proper  prime,  each  digit  0,  1, .  .  . ,  9 
occurs  p  times  in  its  period.  If  a,  h  are  distinct  primes  with  periods  of 
a,  /3  digits,  the  number  of  digits  in  the  period  for  ab  is  the  1.  c.  m.  of  a,  /8 
[Bernoulli^].  Let  p  have  a  period  of  n  digits  and  l/p  =  A-/(10"  — 1).  Let  m 
be  the  least  integer  for  which 

\ljp^-'^\2)p^-''^-    ^\x-l)   p 
is  an  integer;  then  1/p^  has  a  period  of  mn  digits. 

"Archiv  Math.  Phys.,  27,  1856,  124. 

«/6id.,  33,  1859,  94-95. 

«Math.  Monthly  (ed.,  Runkle),  Cambridge,  Mass.,  1,  1859,  295. 

**Ibid.,  345-9. 

♦'Ueber  relative  Prim-  und  correspondirende  Zahlen,  primitive  und  sekundare  Wurzeln  und 

periodische  Decimalbriiche,  Progr.,  BerUn,  1860,  18  pp. 
"Ueber  einige  Eigenschaften  periodischer  Dezimalbriiche,  Prag,  1864. 
♦^Synthetic  division  in  arithmetic,  with  some  introductory  remarks  on  the  period  of  circulating 

decimals,  1863,  pp.  iv-|-19. 
*80xford,  Cambridge  and  Dublin  Messenger  of  Math.,  2,  1864,  1-6.     Glaisher"  atrributed  this 

useful  anonymous  paper  to  Hudson. 


Chap.  VI]  PERIODIC   DECIMAL  FRACTIONS.  167 

V.  A.  Lebesgue^^  gave  for  iV^347  the  periods  for  1/iV,  r/N,. .  .[cf. 
Gauss^l. 

Sanio^°  stated  that,  if  m,  n,.  . .  are  distinct  primes  and  1/m,  1/n, . . . 
have  periods  of  length  q,  q',.  .  .,  then  l/(wV. . .)  has  the  period  length 
^a-i^b-i  qq'  ^  He  gave  the  length  of  the  period  for  l/p  for  each 
prime  p^700,  and  the  factors  of  10"  — 1,  n^  18. 

F.  J.  E.  Lionnet^^  stated  that,  if  the  period  for  a/h  has  n  digits,  that  for 
any  irreducible  fraction  whose  denominator  is  a  multiple  of  h  has  a  multiple 
of  n  digits.  If  the  periods  for  the  irreducible  fractions  a/6,  a'/h', .  .  .  have 
n,  n', .  •  •  digits,  every  irreducible  fraction  whose  denominator  is  the  1.  c.  m. 
of  b,  h',.  .  .  has  a  period  whose  length  is  the  1.  c.  m.  of  n,n',.  .  ..  If  the  period 
for  1/p  has  n  digits  and  if  p"  is  the  highest  power  of  the  prime  p  which  divides 
10"  — 1,  any  irreducible  fraction  with  the  denominator  p"'^^  has  a  period 
of  np^  digits. 

C.  A.  Laisant  and  E.  Beaujeux^^  proved  that  if  g  is  a  prime  and  the 
period  for  1/q  to  the  base  B  is  P  =  ab.  .  .h,  with  q  —  1  digits,  then 


P-{a+h+...+h)  =  {B-l)a,  ^{^+^-y)  = 


B'-'-l 


and  stated  that  a  like  result  holds  for  a  composite  number  q  if  we  replace 
q—1  by/=</)(g).  Their  proof  of  the  generaUzed  Fermat  theorem  5^=1 
(mod  q)  is  quoted  under  that  topic. 

C.  Sardi^^  noted  that  if  10  is  a  primitive  root  of  a  prime  p  =  lOn+1,  the 
period  for  1/p  contains  each  digit  0,...,  9  exactly  n  times  [Hudson^^]. 
For  p  =  10n-f-3,  this  is  true  of  the  digits  other  than  3  and  6,  which  occur 
n+1  times.     Analogous  results  are  given  for  lOn+7,  lOn+9. 

Ferdinand  Meyer^^  proved  an  immediate  generalization  from  10  to 
any  base  k  prime  to  6,  6', ...  of  the  statements  by  Lionnet.^^ 

Lehmann^"  gave  a  clear  exposition  of  the  theory. 

C.  A.  Laisant  and  E.  Beaujeux^^  considered  the  residues  Vq,  ri, . . .  when 
A,  AB,  AB^,. . .  are  divided  by  A-  Let  ri_iB  =  QiZ)i+r,.  When  written 
to  the  base  B,  let  Di  =  ap. .  .02^1,  and  set  Di  =  ap. .  Mi.    Then 

airi+  . . .  +0prp  =  Z)i(ri-Q2A-  •  •  •  -QpDp). 

The  further  results  are  either  evident  or  not  novel. 

For  G.  Barillari^°"  on  the  length  of  the  period,  see  Ch.  VII. 

*'M6m.  soc.  sc.  phys.  et  nat.  de  Bordeaux,  3,  1864,  245. 

^"Ueber  die  periodischen  Decimalbrtiche,  Progr.,  Memel,  1866. 

"Algebre  61em.,  ed.  3, 1868.     Nouv.  Ann.  Math.,  (2),  7,  1868,  239.     Proofs  by  Morel  and  Pellet, 

(2),  10,  1871,  39-42,  92-95. 
MNouv.  Ann.  Math.,  (2),  7,  1868,  289-304. 
"Giornale  di  Mat.,  7,  1869,  24-27. 
"Archiv  Math.  Phys.,  49,  1869,  168-178. 

""Ueber  Dezimalbriiche,  welche  aus  gewohnUchen  Briichen  abgeleitet  sind,  Progr.,  Leipzig,  1869. 
66N0UV.  Ann.  Math.,  (2),  9,  1870,  221-9,  271-281,  302-7,  354-360. 


168  History  of  the  Theory  of  Numbers.  [Chap,  vi 

*Th.  Schroder^^  and  J.  Hartmann^^  treated  periodic  decimals. 

W.  Shanks^^  gave  Lambert's  method  (Bernoulli,^  end)  for  shortening 
the  work  of  finding  the  length  of  the  period  for  1/A^. 

G.  Salmon^^  remarked  that  the  number  71  of  digits  in  the  period  is  known 
if  we  find  two  remainders  which  are  powers  of  2,  since  10"  =  2''  and  10'' =  2' 
imply  10"^"''^=  1;  also  if  we  find  three  remainders  which  are  products  of 
powers  of  2  and  3.  Muir''^  noted  that  it  is  here  impUed  that  aq  —  bp  equals 
n,  whereas  it  is  merely  a  multiple  of  n. 

J.  W.  L.  Glaisher^"  proved  that,  for  any  base  r, 

1 


ir-iy 


.012...r-3r-l, 


a  generalization  of  1/81  =  .012345679. 

W.  Shanks^^  gave  the  length  of  the  period  for  1/p,  when  p  is  a  prime 
<  30000,  and  a  list  of  69  errors  or  misprints  in  the  table  by  Desmarest,^^ 
and  11  in  that  by  Burckhardt.^° 

Shanks^-  gave  primes  p  for  which  the  length  n  of  the  period  for  1/p  is  a 
given  number  ^  100,  naturally  incomplete.  Shanks^^  gave  additional 
entries  p  for  n  =  26,  ?7  =99;  noted  corrections  to  his  former  table  and  stated 
that  he  had  extended  the  table  to  40000.  Shanks^^  mentioned  an  extension 
in  manuscript  from  40000  to  60000.  An  extension  to  120000  in  manu- 
script was  made  by  Shanks,  1875-1880.  The  manuscript,  described  by 
Cunningham, ^-^  who  gave  a  list  of  errata,  is  in  the  Archives  of  the  Royal 
Society  of  London. 

Shanks®^  stated  that  if  a  is  the  length  of  the  period  for  1/p,  where  p  is  a 
prime  >5,  that  for  1/p'*  is  ap""'^  [^vithout  the  restriction  by  Thibault,^^ 
Muir^^]. 

G.  de  Coninck^®  stated  that,  if  the  last  digit  (at  the  right)  of  A  is  1  or  9, 
the  last  digit  of  the  period  for  1/A  is  9  or  1 ;  while,  if  A  is  a  prime  not  ending 
in  1  or  9,  its  last  digit  is  the  same  as  the  last  in  the  period. 

Moret-Blanc^^  noted  that  the  last  property  holds  for  any  A  not  divisible 
by  2  or  5.  For,  if  a  is  the  integer  defined  by  the  period  for  1/A,  that  for 
{A  —  l)/A  is  {A  —  l)a,  whence  a+ (A  — 1)0  =  10"  — 1,  if  n  is  the  length  of  the 
periods.  He  noted  corrections  to  the  remaining  nine  laws  stated  by  Coninck 
and  implied  that  when  corrected  they  become  trivial  or  else  known  facts. 

"Progr.  Ansbach,  1872, 

"Progr.  Rinteln,  1872. 

"Messenger  Math.,  2,  1873,  41-43. 

"/bid.,  pp.  49-51,  80. 

^^Ibid.,  p.  188. 

«iProc.  Roy.  Soc.  London,  22,  1873-4,  200-10,  384-8.    Corrections  by  Workman.*" 

"/bwi.,  pp.  381^.     Cf.  Bertram",  Loof." 

*Hbid.,  23,  1874-5,  260-1. 

^Ibid.,  24,  1875-6,  392. 

"Messenger  Math.,  3,  1874,  52-55. 

"Nouv.  Ann.  Math.,  (2),  13,  1874,  569-71;  errata,  14,  1875,  191-2. 

*Ubid.,  (2),  14,  1875,  229-231. 


Chap.  VI]  PERIODIC   DECIMAL  FRACTIONS.  169 

Karl  Broda^^  considered  a  periodic  decimal  fraction  F  having  an  even 
number  r  of  digits  in  the  period  and  a  number  m  of  p  digits  preceding  the 
period.     Let  x  be  the  first  half  of  the  period,  y  the  second  half.     Then 


10"*     10"*+'"     10"'+^'"     io"'+3^^-    ■"10"'  '  10'"(102'"-1) 

_9(jp40'"+a;+p)+q 
9-10"'(10^+l) 

ifx+2/  =  o(10''  —  l)/9  =  a...a(tor  terms).  The  first  paper  treated  the  case 
p  =  m  =  0,  and  gave  the  generalization  to  base  a  in  place  of  10: 

£  ,^  ,  _£  ,  a+(a-l)x       -r      ,     _    a'-l 

a^^a"-^a'''^--  '  "  (a-l)(a'-+l)     "  ^+2/-^  ^_j- 

The  case  a  =  a  —  l  shows  that  a  purely  periodic  fraction  to  the  base  a  equals 
(a:+l)/(a'"+l)  if  the  sum  of  the  half  periods  has  all  its  digits  (to  base  a) 
equal  to  a— 1.  Returning  to  the  base  10,  and  taking  A'=  9(10'"H-1), 
Z  =  9x+a,  where  each  digit  of  x  is  ^a,  we  see  that  Z/N  equals  a  decimal 
fraction  in  which  x  is  the  first  half  of  the  period  of  r  digits,  while  the  second 
half  is  such  that  the  sum  of  corresponding  digits  in  it  and  x  is  a.  If  R  is  the 
remainder  after  r  digits  of  the  period  have  been  obtained,  R-\-Z  =  a  (10''+ 1). 

C.  G.  Reuschle^^  gave  tables  which  serve  to  find  numbers  belonging  to 
a  given  exponent  <  100  with  respect  to  a  given  prime  modulus  <  1000. 

P.  Mansion''"  gave  a  detailed  proof  that,  if  n  is  prime  to  2,  3,  5,  and  if 
the  period  for  1/n  has  n  —  1  digits,  the  sum  of  corresponding  digits  in  the 
half  periods  is  9. 

T.  Muir^^  proved  that,  if  p  is  a  prime,  either  of 

N'  =  1  (mod  f) ,  iV^P" = 1  (mod  p''+") 

follows  from  the  other.  If  Xi  is  the  least  positive  integer  x  for  which  the 
first  holds  and  if  p'  is  the  highest  power  of  p  dividing  N""'  —  !,  then  Xip"  is 
the  least  positive  integer  y  for  which  N^  =  l  (mod  p'+").  Hence  the  known 
theorem:  If  N=JIpi'^,  where  Pi,P2,--  are  distinct  primes,  and  if  the  period 
for  \/pi  has  m^  digits,  and  if  Pi'  is  the  highest  power  of  Pi  dividing  10"^  — 1, 
the  number  of  digits  in  the  period  for  \/N  is  the  1.  c.  m.  of  the  niip^^i'^*.  He 
asked  if  6  =  1  when  p>3,  as  affirmed  by  Shanks.^^ 

Mansion's  proof  {ibid.,  5,  1876,  33)  by  use  of  periodic  decimals  of  the 
generalized  Fermat  theorem  is  quoted  under  that  topic. 

D.  M.  Sensenig'^  noted  that  a  prime  p?^2,  5,  divides  iV  if  it  divides  the 
sum  of  the  digits  of  N  taken  in  sets  of  as  many  figures  each  as  there  are 
digits  in  the  period  for  l/p. 

«»Archiv  Math.  Phys.,  56,  1874,  85-98;  57,  1875,  297-301. 

"Tafeln  complexer  Primzahlen,  Berlin,  1875.     Errata  by  Cunningham,  Mess.  Math.,  46, 

1916,  60-1. 
'"Nouv.  Corresp.  Math.,  1,  1874-5,  8-12. 
"Messenger  Math.,  4,  1875,  1-5. 
"The  Analyst,  Des  Moines,  Iowa,  3,  1876,  25. 


i 


170  History  of  the  Theory  of  Numbers.  [Chap,  vi 

*A.  J.  M.  Brogtrop^'  treated  periodic  decimals. 

G.  Bellavitis^*  noted  that  the  use  of  base  2  renders  much  more  com- 
pact and  convenient  Gauss' ^^  table  and  hence  constructed  such  a  table. 

W.  Shanks'^  found  that  the  period  for  1/p,  where  p  =  487,  is  divisible 
by  p,  so  that  the  period  for  1/p^  has  p  —  l  digits. 

J.  W.  L.  Glaisher^^  formed  the  period  05263.  . .  for  1/19  as  follows: 
List  5;  divide  it  by  2  and  list  the  quotient  2;  since  the  remainder  is  1, 
divide  12  by  2  and  list  the  quotient  6;  divide  it  by  2  and  list  the  quotient, 
etc.  To  get  the  period  for  1/199,  start  with  50.  To  get  the  period,  apart 
from  the  prefixed  zero,  for  1/49,  start  with  20  and  divide  always  by  5;  for 
1/499,  start  wath  200. 

Glaisher^^  noted  that,  if  we  regard  as  the  same  periods  those  in  which 
the  digits  and  their  cyclic  order  are  the  same,  even  if  commencing  at  differ- 
ent places,  a  number  q  prime  to  10  will  have/  periods  each  of  a  digits,  where 
af=4){q).  This  was  used  to  check  Goodwyn's  table. ^^  If  g  =  39,  there  are 
four  periods  each  of  six  digits,  li  q  —  1  belongs  to  the  period  for  1/q,  the 
two  halves  of  every  period  are  complementary;  if  not,  the  periods  form 
pairs  and  the  periods  in  each  pair  are  complementary.  For  each  prime 
N<  1000,  except  3  and  487,  the  period  for  l/N"  has  nA^*"^  digits  if  that 
for  1/iV  has  n  digits. 

Glaisher'^^  collected  various  known  results  on  periodic  decimals  and 
gave  an  account  of  the  tables  relating  thereto.  If  q  is  prime  to  10  and  if 
the  period  for  1/q  has  (/)(g)  digits,  the  products  of  the  period  by  the  4>{q) 
integers  <q  and  prime  to  q  have  the  same  digits  in  the  same  cyclic  order; 
for  example,  if  g  =  49.  He  gave  (pp.  204-6)  for  each  g<1024  and  prime 
to  10  the  number  a  of  digits  in  the  period  for  1/q,  the  number  n  of  periods 
of  irreducible  fractions  p/q,  not  regarding  as  distinct  two  periods  having 
the  same  digits  in  the  same  cychc  order,  and,  finally  Euler's  (f>(,q).  The 
values  of  a  and  n  were  obtained  by  mere  counting  from  the  entries  in  Good- 
wyn's^^  "table  of  circles";  in  every  case,  an  =  <j){q).  For  the  prime  p  =  487, 
he  gave  the  full  periods  for  1/p  and  1/p",  each  of  486  digits,  thus  verifying 
Desmarest's^^  statement  of  the  exceptional  character  of  this  p  [cf.  Shanks'^]. 

Glaisher^^  again  stated  the  chief  rules  for  the  lengths  of  periods. 

The  problem  was  proposed^"  to  find  a  number  whose  products  by  2, . . . ,  6 
have  the  same  digits,  but  in  a  new  order. 

Birger  Hausted^^  solved  this  problem.  Start  with  any  number  a  of 
one  digit,  multiply  it  by  any  number  p  and  let  b  be  the  digit  in  the  units 

"Nieuw  Archief  voor  VViskunde,  Amsterdam,  3,  1877,  58-9. 

7«Atti  Accad.  Lincei,  Mem.  Sc.  Fis.  Mat.,  (3),  1,  1877,  778-800.     Transunti,  206.     See  62a 

of  Ch.  VII. 
"Proc.  Roy.  Soc.  London,  25,  1877,  551-3. 
^^Messenger  Math.,  7,  1878,  190-1.     Cf.  Desmarest." 
"Report  British  Assoc,  1878,  471-3. 
"Proc.  Cambridge  Phil.  Soc,  3,  1878,  185-206. 

"Solutions  of  the  Cambridge  Senate-House  Problems  and  Riders  for  1878,  pp.  8-9. 
""Tidsskrift  for  Math.,  Kjobenhavn,  2,  1878,  28. 
"/bid.,  pp.  180-3.     Jornal  de  Sc.  Math,  e  Ast.,  2,  1878,  154-6. 


Chap.  VI]  PeEIODIC   DECIMAL  FRACTIONS.  171 

place  of  the  product  ap,  /S  the  digit  in  the  tens  place.  Write  the  digit  b 
to  the  left  of  digit  a  to  form  the  last  two  digits  of  the  required  number  P. 
The  number  c  in  the  units  place  in  6p+j8  is  written  to  the  left  of  digit  h  in 
P.  To  cp  add  the  digit  in  the  tens  place  of  bp  and  place  the  unit  digit  of 
the  sum  to  the  left  of  c  in  P.  The  process  stops  with  the  kth.  digit  t  if  the 
next  digit  would  give  a.  Then  P  =  t. .  .  cba  and  its  products  by  k  integers  or 
fractions  has  the  same  k  digits  in  the  same  cyclic  order.  For  a  =  2,  p  =  3, 
we  get  A;  =  28  and  see  that  P  is  the  period  of  2/27,  and  the  k  multipliers 
are  m/2,  m  =  l,.  .  .,  28.  [To  have  an  example  simpler  than  the  author's, 
take  a  =  7,  p  =  5;  then  P  =  142857,  the  period  of  1/7;  the  multipliers  are 
1, . . . ,  6.]     For  proof,  we  have 

P  =  10''-H-\- . .  .-\-10h+10b+a,         pP  =  10''-'a+10''-H+  . .  .+10c-\-h, 

pP  =  10^-a+^,  10^  =  10^' 


so  that  P  is  the  period  with  k  digits  for  a/{lOp  —  l). 

E.  LucasS2  gave  the  prime  factors  of  lO'^^l,  10'^±1,  lO^^^l,  10^^+1, 
10^^+1,  communicated  to  him  by  W.  Loof,  with  the  remark  that  (10^^  — 1)/9 
has  no  prime  factor  <  3035479.     Lucas  gave  the  factors  of  10^^+1. 

J.  W.  L.  Glaisher^^  proved  his^^  earlier  statements,  repeated  his"  earher 
remarks,  and  noted  that,  if  g  is  a  prime  such  that  the  period  for  1/q  has  q  —  1 
digits,  the  products  of  the  period  for  1/q  by  1,  2, .  .  .,  g  — 1  have  the  same 
digits  in  the  same  cyclic  order.  This  property,  well  known  for  q  =  7,  holds 
also  for  g  =  17,  19,  23,  29,  47,  59,  61,  97  and  for  q  =  7\ 

0.  Schlomilch^^  stated  that,  to  find  every  N  for  which  the  period  for 
1/N  has  2k  digits  such  that  the  sum  of  the  sth  and  (fc+s)th  digits  is  9  for 
s  =  1, . . . ,  A;,  we  must  take  an  integer  iV=  (10*^+l)/r;  then  the  first  k  digits 
of  the  period  are  the  k  digits  of  T  —  1. 

C.  A.  Laisant^^  extended  his  investigations  with  Beaujeux^^'^^  and  gave  a 
summary  of  known  properties  of  periodic  fractions;  also  his^^  process  to 
find  the  period  of  simple  periodic  fractions  without  making  divisions. 

V.  Bouniakowsky^^  noted  that  the  property  of  the  period  of  1/N, 
observed  by  Schlomilch^^  for  iV  =  7,  11,  13,  77,  91,  143,  holds  also  for  the  pe- 
riods of /c/iV,  for  A;  =  iV  —  1  and  (iV  — 1)/2,  with  the  same  values  of  AT.  Consider 
the  decimal  fraction  Q.yiy2-  ■  ■  with  ym  —  ym-i+ym-2  (mod  9),  replacing  any 
residue  zero  by  9,  and  taking  yi  >  0, 1/2  >  0-  The  fraction  is  purely  periodic 
and  is  either  0.9  or  0.33696639  or  has  the  same  digits  permuted  cyclically, 
or  else  has  a  period  of  24  digits  and  begins  with  1,  1  or  2,  2  or  4,  4,  or  has  the 
same  24  digits  permuted  cyclically  or  by  the  interchange  of  the  two  halves 

s^Nouv.  Corresp.  Math.,  5,  1879,  138-9. 

8'Nature,  19,  1879,  208-9. 

s^Zeitschrift  Math.  Phys.,  25,  1880,  416. 

8*M6m.  Soc.  Sc.  Phys.  et  Nat.  de  Bordeaux,  (2),  3,  1880,  213-34. 

86Le8  Mondes,  19,  1869,  331. 

"BuU.  Acad.  Sc.  St.  P^tersboiirg,  27,  1881,  362-9. 


172  History  of  the  Theory  of  Numbers.  [Chap,  vi 

of  the  period.  The  property  of  Schlomilch  holds  for  these  and  the  generali- 
zation to  any  base,  as  well  as  for  those  with  the  law  xjm  =  '^ym-\-\-ym-2'    But  if 

ym  =  ^ym-i-2y^.2  (mod  9),  ?/.  =  (2"'-'-l)(!/2-2/i)+2/i  (mod  9), 

the  fact  that  2^  =  1  (mod  9)  shows  that  the  period  has  at  most  six  digits. 
Those  with  six  reduce  by  cychc  permutation  to  nine  periods : 
167943,    235986,    278154,    197346, 
265389,    218457,    764913,    329568,    751248. 

In  the  A-th  of  these  the  sum  of  corresponding  digits  in  the  two  half  periods 
is  always  =A-  (mod  9). 

Karl  Broda^^  examined  for  small  values  of  r  and  certain  primes  p  the 
solutions  a:  of  x''=  1  (mod  p)  to  obtain  a  base  x  for  which  the  periodic  frac- 
tion for  1/p  has  a  period  of  r  digits,  and  similariy  the  condition  x^=—\ 
(mod  p)  for  an  even  number  of  digits  in  the  period  (Broda^^). 

F.  Kessler^^  factored  10"-1  forn  =  ll,  20,  22,  30. 

W.  W.  Johnson^°  formed  the  period  for  1/19  by  placing  1  at  the  extreme 

right,  next  its  double,  etc.,  marking wdth  a  star  a  digit  when  there  is  1  to  carry: 

«      *         *  *  *  *      *      «« 

05263157894736842  1.  ;^ 

To  deduce  the  value  of  1/19  written  to  the  base  2,  use  1  for  each  digit 
starred  and  0  for  the  others,  reversing  the  order: 

.6 0001101011110010  i. 

If  we  apply  the  first  process  with  the  multipUer  m,  we  get  the  period  for  the 
reciprocal  of  10?7i  — 1. 

E.  Lucas^^  gave  the  prime  factors  of  10"  — 1  for  n  odd,  n^l7,  7i  =  21, 
and  certain  factors  forn  =  19, .  . . ,  41 ;  those  of  10"  + 1  for  n^  18  and  n  =  21. 
He  stated  that  the  majority  of  the  results  were  given  by  Loof  and  pubUshed 
by  Reuschle.     In  1886,  Le  Lasseur  gave 

10^7-1  =3--2071723-5363222[3]57, 

said  by  Loof  to  have  no  divisor  <  400,000  other  than  3,9.  On  the  omission 
of  the  digit  3,  see  Cunningham. ^-^ 

F.  Kessler^-  listed  nine  errors  in  Burckhardt's-"  table  and  described  his 
own  manuscript  of  a  table  to  p  =  12553,  i.  e.,  for  the  first  1500  primes. 

Van  den  Broeck^^  stated  that  10^" -1  is  divisible  by  3"+^ 
A.  Lugli^^  proved  that,  if  p  is  a  prime  5^2,  5,  the  length  of  the  period 
of  1/p  is  a  divisor  of  p  — 1.  If  the  number  of  digits  in  the  period  of  a/p  is 
an  even  number  2t,  the  ^th  remainder  on  dividing  a  by  p  is  p  — 1,  and  con- 
versely. Hence,  if  r^  is  the  hth  remainder,  rh+rh+i  =  p  {h  =  l,.  . .,  t),  and 
the  sum  of  all  the  r's  is  tp.     If  the  period  of  1/p  has  s  digits,  s<p  — 1,  then 

".\rchiv  Math.  Phys.,  68,  1882,  85-99. 

«»Zeitschrift  Math.  Naturw.  Unterricht,  15,  1884,  29. 

•"Messenger  of  Math.,  14,  1884-5,  14-18. 

"Jour,  de  math.  6Um.,  (2),  10,  1886,  160.    Cf .  rinterm^diaire  des  math.,  10,  1903, 183.    Quoted 

by  Brocard,  Mathesis,  6,  1886,  153;  7,  1887,  73  (correction,  1889,  110). 
"Archiv  Math.  Phys.,  (2),  3,  1886,  99-102. 

"Mathesis,  6,  1886,  70.     Proofs,  23.5-6,  and  Math.  Quest.  Educ.  Times,  54,  1891,  117. 
"Periodico  di  Mat.,  2,  1887,  161-174. 


Chap.  VI]  PERIODIC   DECIMAL  FRACTIONS.  173 

p  —  l=sh  and  we  have  h  sets  of  s  fractions  whose  periods  differ  only  by  the 
cycHc  permutation  of  the  digits.  If  p  is  a  product  of  distinct  primes  pi,  P2  •  •  • 
and  if  the  lengths  of  the  periods  of  1/p,  1/pi,  l/p2,  ■  ■  ■  are  s,  Si,  S2, .  . . ,  then 
s  is  the  1.  c.  m.  of  Si,  S2,....  If  p  =  Pi"P2^-  •  •>  and  s,  Si,  s'  are  the  lengths  of 
the  periods  of  1/p,  1/pi,  l/pi",  then  s'  is  one  of  the  numbers  Si,  Sip,.  .  ., 
SiPi°~^  and  hence  divides  (pi  — l)pi"~^;  and  s  is  a  divisor  of  <f){p).  Thus  p 
divides  lO^^^^-l. 

C.  A.  Laisant^^  used  a  lattice  of  points,  whose  abscissas  are  a+r,a-\-2r,..., 
a-\-pf  and  ordinates  are  their  residues  <p  modulo  p,  to  represent  graphically 
periodic  decimal  fractions  and  to  expand  fractions  into  a  difference  of  two 
series  of  ascending  powers  of  fixed  fractions. 

*A.  Rieke^®  noted  that  a  periodic  decimal  with  a  period  of  2m  digits  equals 
(i4.  +  l)/(10"*+l),  where  A  is  the  first  half  of  the  period.  He  discussed  the 
period  length  for  any  base. 

W.  E.  HeaP^  noted  that,  if  B  contains  all  the  prime  factors  of  N,  the 
number  of  digits  in  the  fraction  to  the  base  B  for  M/N  is  the  greatest  integer 
in  (n+n'  —  l)/n',  where  n—n'  is  the  greatest  difference  found  by  subtracting 
the  exponent  of  each  prime  factor  of  N  from  the  exponent  of  the  same  prime 
factor  of  B.  If  B  contains  no  prime  factor  of  N,  the  fraction  for  M/N  is 
purely  periodic,  with  a  period  of  ^(A'')  digits.  If  B  contains  some,  but  not 
all,  of  the  prime  factors  of  N,  the  number  of  digits  preceding  the  period  is 
the  same  as  in  the  first  theorem.  The  proofs  are  obscure.  There  is  given 
the  period  for  1/p  when  p<100  and  has  10  as  a  primitive  root  [the  same 
p's  as  by  Glaisher^^].     Likewise  for  base  12,  with  p<50. 

R.  W.  Genese^^  noted  that,  if  we  multiply  the  period  for  1/81  [Glaisher^"] 
by  m,  where  m<81  and  prime  to  it,  we  get  a  period  containing  the  digits 
0,  1, . . .,  9  except  9n—m,  where  9n  is  the  multiple  of  9  just  exceeding  m. 

Jos.  Mayer^^  investigated  the  moduli  with  respect  to  which  10  belongs 
to  a  given  exponent,  and  gave  the  factors  of  10"— 1,  n<  12.  He  discussed 
the  determination  of  the  exponent  to  which  10  belongs  for  a  given  modulus 
by  use  of  the  theory  of  indices  and  by  the  methods  of  quadratic,  cubic, 
biquadratic,...  residues.  He  used  also  the  fact  that  there  are  (a  — a') 
08-/3') . . .  divisors  of  Pi''p2^Vz  ■  ■  •  which  divide  no  one  of  the  fixed  factors 
ViVVi  ■  •  ■,  Pi>2W> •  •  •  J  where  a<a,b<^,...,  and  pi,  P2,--  are  distinct 
primes.  He  gave  the  length  of  the  period  for  1/p,  for  each  prime  p^2543 
and  22  higher  primes  [Burckhardt^^]. 

L.  Contejean^°°  proved  that,  in  the  conversion  of  an  irreducible  fraction 
a/h  into  a  decimal  fraction,  if  the  remainders  o^  and  a^  are  congruent 
modulo  b,  so  that  lO'a^lO^'a,  then  10"'~''-1  is  divisible  by  the  quotient 
h'  of  b  by  the  highest  factor  2*5'  of  b.     Thus  the  length  of  the  period  is 

"Assoc,  fran?.  avanc.  sc,  16,  1887,  II,  228-235. 

•'Versuch  iiber  die  periodischen  Bniche,  Progr.,  Riga,  1887. 

•^Annals  of  Math.,  3,  1887,  97-103. 

•^Report  Britiah  Assoc,  1888,  580-1. 

"Ueber  die  Grosse  der   Periode  eines  unendlichen  Dezimalbruches,  oder  die   Congruence 

lO^Sl  (mod  P).     Progr.  K.  Studienanstalt  Burghausen,  Munchen,  1888,  52  pp. 
""Bull.  Boc.  philomathique  de  Paris,  (8),  4,  1891-2,  64-70. 


174  History  of  the  Theory  of  Numbers.  [Chap,  vi 

m—r,  while  r  digits  precede  the  period.  The  condition  that  the  length  of 
the  period  be  the  maximum  0(6')  is  that  10  be  a  primitive  root  of  h',  whence 
5'  =  p",  since  6'?^  4  or  2^",  p  being  an  odd  prime. 

P.  Bachmann^°^  used  a  primitive  root  g  of  the  prime  p  and  set 

to  the  base  g.  We  get  the  multiples  Q,  2Q, . .  . ,  (p  —  1)Q  by  cyclic  permuta- 
tion of  the  digits  of  Q.     For  p  =  7,  ^  =  10,  Q  =  142857. 

J.  Kraus^"^  generaUzed  the  last  result.  When  ri/n  is  converted  into  a 
periodic  fraction  to  base  g,  prime  to  n,  let  ai, .  .  . ,  Ck  be  the  quotients 
and  ri, . . . ,  r^  the  remainders.     Then 

<7*-l 

rx  =  ax^^"^+ax+i/"^+-  •    -f«x-i  (X  =  l,.  •  ■,  k), 

n 

whence 

^x(aiS'*"^+  •  •  •  +(ik)  =n(«x9'*"^+  •  ■  •  4-ax-i). 

In  particular,  let  n  be  such  that  it  has  a  primitive  root  g,  and  take  ri  =  1. 

Then 

ft 

and  if  rx  is  prime  to  n,  the  product  ry,Q  has  the  same  digits  as  Q  permuted 
cyclically  and  beginning  with  a^. 

H.  Brocard^"^  gave  a  tentative  method  of  factoring  10"  — 1. 

J.  Mayer^°^  gave  conditions  under  which  the  period  of  z/P  to  base  a, 
where  z  and  a  are  relatively  prime  to  P,  shall  be  complete,  i.  e.,  corresponding 
digits  of  the  two  halves  of  the  period  have  the  sum  a  — 1. 

Heinrich  Bork^°^  gave  an  exposition,  without  use  of  the  theory  of  num- 
bers, of  kno^n  results  on  decimal  fractions.  There  is  here  first  published 
(pp.  36-41)  a  table,  computed  by  Friedrich  Kessler,  showing  for  each  prime 
p<  100000  the  value  of  q={p  —  l)/e,  where  e  is  the  length  of  the  period 
for  1/p.  The  cases  in  which  ^  =  1  or  2  were  omitted  for  brevity.  He 
stated  that  there  are  many  errors  in  the  table  to  15000  by  Reuschle.'*" 
Cunningham^^^  listed  errata  in  Kessler's  table. 

L.  E.  Dickson^"^  proved,  without  the  use  of  the  concept  of  periodic 
fractions,  that  every  integer  of  D  digits  written  to  the  base  N,  which  is 
such  that  its  products  by  D  distinct  integers  have  the  same  D  digits  in 
the  same  cyclic  order,  is  of  the  form  A{N^  —  1)/P,  where  A  and  P  are 
relatively  prime.     A  number  of  this  form  is  an  integer  only  when  P  is  prime 

"iZeitschrift  Math.  Phys.,  36,  1891,  381-3;  Die  Elemente  der  Zahlentheorie,  1892,  95-97. 

Alike  discussion  occurs  in  l'interm(5diaire  des  math.,  5,  1898,  57-8;  10,  1903,  91-3. 
"'Zeitschrift  Math.  Phys.,  37,  1892,  190-1. 
"'El  Progreso  Matematico,  1892,  25-27,  89-93,  114-9.    Cf.  rinterm^diaire  des  math.,  2,  1895, 

323-4. 
'"Zeitschrift  Math.  Phys.,  39, 1894,  376-382. 

losperiodische  Dezimalbriiche,  Progr.  67,  Prinz  Heinrichs-Gymn.,  Berlin,  1895,  41  pp. 
looQuart.  Jour.  Math.,  27,  1895,  366-77. 


Chap.  VI]  PERIODIC  DECIMAL  FRACTIONS.  175 

to  N,  and  D  is  a  multiple  of  the  exponent  d  to  which  N  belongs  modulo  P. 
The  further  discussion  is  limited  to  the  case  D  =  d,  to  exclude  repetitions 
of  the  period  of  digits.  Then  the  multipUers  which  cause  a  cyclic  permuta- 
tion of  the  digits  are  the  least  residues  of  N,  N^, . . . ,  A^^  modulo  P.  For 
A  =  1,  we  have  a  solution  for  any  N  and  any  P  prime  to  N.  There  are  listed 
the  19  possible  solutions  with  A>1,  N^QS,  and  having  the  first  digit  >0. 
The  only  one  with  A^=  10  is  142857.     General  properties  are  noted. 

A  like  form  is  obtained  (pp.  375-7)  for  an  integer  of  D  digits  written  to 
the  base  A^,  such  that  its  quotients  by  D  distinct  integers  have  the  same 
D  digits  in  the  same  cyclic  order.  The  divisors  are  the  least  residues  of 
N^,  N^-\.  . .,  N  modulo  P.  For  example,  if  N  =  n,  P  =  7,  A=4:,  we  get 
4(11^  — 1)/7,  or  631  to  base  11,  whose  quotients  by  2  and  4  are  316  and  163, 
to  base  11.    Another  example  is  512  to  base  9. 

E.  Lucas^  gave  all  the  prime  factors  of  10"— 1  forn^  18. 

F.  W.  Lawrence^"^  proved  that  the  large  factors  of  10^^  —  1  and  10^^  —  1 
are  primes. 

C.  E.  Bickmore^'^^  gave  the  factors  of  10" -1,  n^  100.  Here  (1023-l)/9 
is  marked  prime  on  the  authority  of  Loof ,  whereas  the  latter  regarded  its 
composition  as  unknown  [Cunningham^^^].  There  is  a  misprint  for  43037 
in  10^^-1. 

B.  Bettini^"^  considered  the  number  n  of  digits  in  the  period  of  the  deci- 
mal fraction  for  a/b,  i.  e.,  the  exponent  to  which  10  belongs  modulo  h.  If 
10  is  a  quadratic  non-residue  of  a  prime  b,  n  is  even,  but  not  conversely 
(p.  48).    There  is  a  table  of  values  of  n  for  each  prime  6^277. 

V.  Murer^^"  considered  the  n  =  mq  remainders  obtained  when  a/b  is 
converted  into  a  decimal  fraction  with  a  period  of  length  n,  separated  them 
into  sets  of  m,  starting  with  a  given  remainder,  and  proved  that  the  sum 
of  the  sets  is  a  multiple  of  9 ...  9  (to  m  digits) .  Further  theorems  are  found 
when  q  =  l,  2  or  3. 

J.  Sachs ^^^^  tabulated  all  proper  fractions  with  denominators  <250  and 
their  decimal  equivalents. 

B.  Reynolds^ ^^  repeated  the  rules  given  by  Glaisher'^^' '^^  for  the  length 
of  periods.  He  extended  the  rules  by  Sardi^^  and  gave  the  number  of  times 
a  given  digit  occurs  in  the  various  periods  belonging  to  a  denominator  N, 
both  for  base  10  and  other  bases. 

Reynolds^^^  gave  numerical  results  on  periodic  fractions  for  various 
bases  the  lengths  of  whose  period  is  3  or  6,  and  on  the  length  of  the  period  for 
1/A^  for  every  base  <N—1,  when  A^  is  a  prime. 

A.  Cunningham^ ^^  applied  to  the  question  of  the  length  of  the  period 
of  a  periodic  fraction  to  any  base  the  theory  of  binomial  congruences  [see 

i"Proc.  London  Math.  Soc,  28,  1896-7,  465.     Ci.  Bickmore"  of  Ch.  XVI. 

"'Nouv.  Ann.  Math.,  (3),  15,  1896,  222-7. 

"Teriodico  di  Mat.,  12,  1897,  43-50.  "o/bid.,  142-150. 

uoaprogr.  632,  Baden-Baden,  Leipzig,  1898. 

""Messenger  Math.,  27,  1897-8,  177-87. 

»"/feid.,  28,  1898-9,  33-36,  88-91. 

"'/bid.,  29,  1899-1900,  145-179.     Errata.i" 


176  History  of  the  Theory  of  Numbers.  [Chap.vi 

201  of  Ch.  VII].  He  gave  extensive  tables,  and  references  to  papers  on 
higher  residues  and  to  tables  relating  to  period  lengths. 

O.  Fujimaki^^*  noted  that  if  10'"  — 1  is  exactly  divisible  by  n,  and  the 
quotient  is  Qi.  .  .a^  of  jn  digits,  the  numbers  obtained  from  the  latter  by 
cycHc  permutations  of  the  digits  are  all  multiples  of  Ci .  .  .  a^. 

J.  Cullen,  D.  Biddle,  and  A.  Cunningham^ ^^  proved  that  the  large  factor 
of  14  digits  of  (10-^+l)/(10Hl)  is  a  prime. 

L.  Kronecker^^^  treated  periodic  fractions  to  any  base. 

W.  P.  Workman^^^  corrected  three  errors  in  Shanks'^^  table. 

D.  Biddle^^^  concluded  erroneously  that  (10^^  — 1)/9  is  a  prime. 

H.  Hertzer"'  extended  Kessler's^"'^  table  from  100000  to  112400,  noted 
Reuschle's'*'^  error  on  the  conditions  that  10  be  a  biquadratic  residue  of  a 
prime  p  and  gave  the  conditions  that  10  be  a  residue  of  an  8th  power 
modulo  p.     For  errata  in  the  table,  see  Cunningham. ^^ 

P.  Bachmann^'°  proved  the  chief  results  on  periodic  fractions  and  cyclic 
numbers  to  any  base  g. 

A.  Tagiuri^^^  proved  theorems  [F.  Meyer ,^  Perkins^^]  on  purely  periodic 
fractions  to  any  base  and  on  mixed  fractions. 

E.  B.  Escott^^^  noted  a  misprint  in  Bickmore's^^^  table  and  two  omissions 
in  Lucas'^^  table,  but  described  inaccurately  the  latter  table,  as  noted  by 
A.  Cunningham. ^^^ 

A.  Cunningham^^^  described  various  tables  (cited  above)  which  give 
the  exponent  to  which  10  belongs,  and  listed  many  errata. 

J.  R.  Akerlund^^°  gave  the  prime  factors  of  11 ...  1  (to  n  digits)  for  n^  16, 
n  =  18. 

K.  P.  Nordlund^^^  applied  to  periodic  fractions  the  theorem  that,  if 
Til, .  .  .,  rir  are  distinct  odd  primes,  no  one  dividing  a,  then  N  =  ni"''.  .  .  Ur""^ 
di\'ides  a^  —  l,  where  A:  =  0(iV)/2'""\  He  gave  the  period  of  \/p  for  p  a 
prime  <  100  and  of  certain  a/p. 

T.  H.  Miller, ^-^  generalizing  the  fact  that  the  successive  pairs  of  digits 
in  the  period  for  1/7  are  14,  28, ... ,  investigated  numbers  n  to  the  base  r 
for  which 

1  _2n    4n    8n 

~  —     2"~1       4    I       6  ~r  •  •  •  ) 

n     r       r       r" 

»"Jour.  of  the  Physics  School  in  Tokio,  7,  1897,  16-21;  Abh.  Gesch.  Math.  Wiss.,  28,  1910,  22. 

i"Math.  Quest.  Educat.  Times,  72,  1900,  99-101. 

"'Vorlesungen  iiber  Zahlentheorie,  I,  1901,  428-437. 

"'Messenger  Math.,  31,  1901-2,  115. 

"«7&id.,  p.  34;  corrected,  ibid.,  33,  1903^,  126  (p.  95). 

"•Archiv  Math.  Phys.,  (3),  2,  1902,  249-252. 

""Niedere  Zahlentheorie,  I,  1902,  351-363. 

"iPeriodico  di  Mat.,  18,  1903,  43-58. 

»«Xouv.  Ann.  Math.,  (4),  3,  1903,  136;  Messenger  Math.,  33,  1903-4,  49. 

'"Messenger  Math.,  33,  1903-4,  95-96. 

^^Ibid.,  14.5-155. 

'"Nj-t  Tidsskrift  for  Mat.,  Kjobenhavn,  16  A,  1905,  97-103. 

'"Goteborgs  Kungl.  Vetenskaps-Handlingar,  (4),  VII-VIII,  1905. 

"'Proc.  Edinburgh  Math.  Soc,  26,  1907-8,  95-6. 


Chap.  VI]  PERIODIC   DECIMAL  FRACTIONS.  177 

whence  r^  —  2n^  =  2.     Besides  the  case  r  =  10,  n  =  7,  he  found  r  =  58,  n  =  41 , 
etc. 

A.  Cunningham^^*  noted  two  errors  in  his  paper"^  and  added 
252^2  =  ^j^o^  9972>)^  390112^  =  1  (mod  17«) 

and  cases  modulo  p^,  where  p  =  103,  487,  attributed  to  Th.  Gosset. 

A.  Cunningham^^^  gave  tables  of  the  periods  of  \/N  to  the  bases  2,  3,  5 
for  N^  100. 

H.  Hertzer^^"  noted  three  errors  in  Bickmore's^°*  table. 

A.  Gerardin^^^  gave  factors  of  10"  — 1,  n<100,  and  a  table  of  the  expo- 
nents to  which  10  belongs  modulo  p,  a  prime  <  10000,  with  a  list  of  errors 
in  the  tables  by  Burckhardt  and  Desmarest. 

A.  Filippov^^^  gave  two  methods  of  determining  the  generating  factor 
for  the  periodic  fraction  for  1/6  (cf.  Lucas,  Th^orie  des  nombres,  p.  178). 

G.  C.  Cicioni^^^  treated  the  subject. 

E.  R.  Bennett^^^  proved  the  standard  theorems  by  means  of  group 
theory. 

W.  H.  Jackson^^^  noted  that,  if  a  is  prime  to  10  and  if  h  is  chosen  so  that 
h<  10,  a&  =  10m  — 1,  the  period  for  \/a  may  be  written  as 

6]l  +  10m+(10m)2+. .  .+(10m)'-it  -A;-10', 

where  s  is  the  exponent  to  which  10  belongs  modulo  a,  and  /c  is  a  positive 
integer.    Thus  for  a  =  39,  6  =  1,  we  have  m  =  4,  s  =  6,  and  the  period  is 

1+40+. .  .  +  (40)^-A:-10^  ^  =  .025641. 

G.  Mignosi^^^  discussed  the  logic  underlying  the  identification  of  an 
unending  decimal  with  its  generator  y/q. 

A.  Cunningham^^^  treated  periodic  decimals  with  multiples  having  the 
same  digits  permuted  cyclically. 

F.  Schuh^^^  considered  the  length  g^  of  the  period  for  1/p"  for  the  base  g, 
where  p  is  a  prime.  He  proved  that  qa  is  of  the  form  qip%  where  0^  c^  a  —  2 
when  p  =  2,  a>2,  while  O^c^a  — 1  in  all  other  cases.     For  a>2, 

?a-l  =  giP""\  •  •  • ,  qa-c+1  =  qiVj  Qa-c  =  •  •  •  =  ?2  =  ?, 

where  q  =  qi  txcept  when  p  =  2,  gr  =  4m  — 1,  and  then  g  =  2.  Equality  of 
periods  for  moduli  p"  and  p''  can  occur  for  an  odd  prime  p  only  when  this 
period  is  gi,  and  for  p  =  2  only  when  it  is  1  or  2.  It  is  shown  how  to  find 
the  numbers  g  which  give  equal  periods  for  p"  and  p,  and  the  odd  numbers 
g  which  give  the  period  2  for  2". 

"8Math.  Gazette,  4,  1907-8,  209-210.     Sphinx-Oedipe,  8,  1913,  131. 

"9Math.  Gazette,  4,  1907-8,  259-267;  6,  1911-12,  63-7,  108-116. 

"OArchiv  Math.  Phys.,  (3),  13,  1908,  107. 

"^Sphinx-Oedipe,  Nancy,  1908-9,  101-112. 

"'Spaczinskis  Bote,  1908,  pp.  252-263,  321-2  (Russian). 

"'La  divisibiht^  dei  numeri  e  la  teoria  delle  decimaU  periodiche,  Perugia,  1908,  150  pp. 

"«Amer.  Math.  Monthly,  16,  1909,  79-82. 

"'Annals  of  Math.,  (2),  11,  1909-10,  166-8. 

"'II  Boll.  Matematica  Gior.  Sc.-Didat.,  9,  1910,  128-138. 

"^Math.  Quest.  Educat.  Times,  (2),  18,  1910,  25-26. 

"'Nieuw  Archief  Wiskunde,  (2),  9,  1911,  408-439.    Cf.  Schuh,"'"*,  Ch.  VII. 


178  History  of  the  Theory  of  Numbers.  [Chap,  vi 

T.  Ghezzi^^^  considered  a  proper  irreducible  fraction  m/p  with  p  prime 
to  the  base  b  of  numeration.     Let  h  belong  to  the  exponent  n  modulo  p.     In 

7nb  =  pqi-\-ri,  rih  =  pq2+r2,.  .  .,  0<ri<p,  0<r2<p,.  .  ., 

fi,. . .,  r„  are  distinct  and  r„  =  w.  Multiply  the  respective  equations  by 
6""^,  6""^,. . .  and  add;  we  see  that 

p  hr-i 

A  similar  proof  shows  that  m/p  equals  a  fraction  ^xith.  the  denominator 
b'{b'*  —  l)  when  6  =  aia2a3,  p^piCL^a^a^^  the  a's  being  primes  and  pi  rela- 
tively prime  to  b,  while  6'  is  the  least  power  of  b  having  the  di\isor  ai^a^a^, 
and  n  is  the  exponent  to  which  b  belongs  modulo  pi. 

F.  Stasi^*°  gave  a  long  proof  showing  that  the  length  of  the  period  for 
b/a  does  not  exceed  that  for  1/a.  If  the  period  A  for  1/p  has  m  digits  and 
n  =  p5  is  prime  to  10,  the  length  of  the  period  for  \/n  is  m  if  A  is  divisible 
by  q;  is  mi  if  A  is  prune  to  q  and  if  the  least  A(10'"^*"^^+  .  .  .  +1)  divisible 
by  q  has  m  =  i;  and  is  mj  if  A=A'a,  q  =  aq',  with  A',  q'  relatively  prime, 
while  the  least  A'  (10"'^*'-^^+  ...  +1)  divisible  by  q'  has  k=j.  For  a  prime 
p5^2,  5,  let 


1         A 


h 


i 


p''     10"* -1' 

and  let  A^  be  the  first  of  the  periods  of  successive  powers  of  1/p  not  divisible 
by  p;  then  the  period  for  l/p''+^'  has  wp^'  digits.  If  p,  is  a  prime  9^2,  5, 
and  Ti  is  the  length  of  the  period  for  l/p„  and  if  l/pj^<  is  the  highest  power 
of  1/pi  with  a  period  of  Ti  digits,  the  length  of  the  period  for  l/p,"*  is 
T-' =  r{pi''i~^i  and  that  for  l/II Pi"<  is  a  multiple  of  the  1.  c.  m.  of  the  r/. 

If  n  is  prime  to  10  and  if  ri, .  . . ,  r;„  =  1  are  the  successive  remainders  on 
reducing  \/n  to  a  decimal,  then  r^=r2i  (mod  n).  Hence  if  1/n  has  a  period 
of  2i  digits,  r^  =  \  (mod  n)  and  conversely.  But  if  it  has  a  period  of 
2i+l  digits,  r-+i  =  10  and  conversely. 

*K.  W.  Lichtenecker^^^  gave  the  length  of  the  period  for  1/p,  when  p  is 
a  prime  ^307,  and  the  factors  of  10^  —  1,  r?^  10. 

L.  Pasternak^^^  noted  that,  after  multiplying  the  terms  of  a  fraction  by 
9,  3  or  7,  we  may  assume  the  denominator  iV'  =  10m  — 1.  To  convert  Rq/N 
into  a  decimal,  we  have  10Rk-i  =  Nyk+Rk  (^  =  1,  2, . . .).  Set  7?^.  =  lOz^+e^t, 
ek^  9.  Since  Vk'^  9,  e^  =  Vk  and  Rk-i  =  mCk+Zk.  Hence  the  successive  digits 
of  the  period  are  the  unit  digits  of  the  successive  remainders. 

E.  Maillet^^^  defined  a  unique  development  Oo+ai/n+ 02/11^+ ...  of  an 
arbitrary  number,  where  the  Oi  are  integers  satisfjdng  certain  conditions. 
He  studied  the  conditions  that  the  development  be  limited  or  periodic. 

"»I1  Boll.  Matematica  Gior.  Sc.-Didat.,  9,  1910,  263-9. 
""/bid.,  11,  1912,  226-246. 

i"Zeit3chr.  fur  das  Realschulwesen,  37,  1912,  338-349. 
i^L'enseignement  math.,  14, 1912,  285-9. 
»"L'interm6diaire  des  math.,  20,  1913,  202-6. 


Chap.  VI]  PERIODIC    DECIMAL   FRACTIONS.  179 

Welsch^**  discussed  briefly  the  length  of  the  period  of  a  decimal  fraction. 

B.  Howarth^^^  noted  that  D^  is  not  a  factor  of  (10^"-1)/(10'*-1)  if  D  is 
a  prime  and  n  is  not  a  multiple  of  the  length  of  the  period  for  1/D.  Again/^^ 
(^IQmnp^-  _i)/9  is  not  divisible  by  (lO'"^-!)  (10"^-1)/81. 

A.  Cunningham^^^  factored  10^^  ±  1.     Known  factors  of  lO"^  1  are  given. 

Cunningham^^^  gave  factors  of  10"*^'*  —  !. 

A.  Leman^^^  gave  an  elementary  exposition  and  inserted  proofs  of  Fer- 
mat's  theorem  and  related  facts,  with  the  aim  to  afford  a  concrete  introduc- 
tion to  the  more  elementary  facts  of  the  theory  of  numbers. 

S.  Weixer^^''  would  compute  the  period  P  for  1/p  by  multiplication, 
beginning  at  the  right.  Let  c  be  the  final  digit  of  P,  whence  pc  =  10z  —  l. 
Then  c  is  the  first  digit  of  the  period  P^  for  z/p.  The  units  digit  Ci  of 
cz  =  10zi-\-Ci  is  the  tens  digit  of  P  and  the  units  digit  of  P^.  In  CiZ-\-Zi  = 
1022+^2,  C2  is  the  hundreds  digit  of  P  and  the  tens  digit  of  P\  etc. 

A.  Leman^^^  discussed  the  preceding  paper. 

Problems^^^  on  decimal  fractions  may  be  cited  here. 

O.  Hoppe^^^  proved  that  (10^^  — 1)/9  is  a  prime. 

M.  Jenkins^^^  noted  that  if  iV=  a^6^.  .  . ,  where  a,h,.  .  .are  distinct  primes 
9^2,  5,  the  period  for  1/N  is  complementary  (sum  of  corresponding  digits  of 
the  half  periods  is  9)  if  and  only  if  the  lengths  of  the  periods  for  1/a,  1/b,. . . 
contain  the  same  power  of  2. 

Kraitchik^^^  of  Ch.  VII  and  Levanen^^  of  Ch.  XII  gave  tables  of  ex- 
ponents to  which  10  belongs.  Bickmore  and  Cullen^^^  of  Ch.  XIV  factored 
10^^+1. 

Further  Papers  Involving  No  Theory  of  Numbers. 

J.  L.  Lagrange,  Legons  61em.  k  I'^cole  normale  en  1795,  Oeuvres  7,  200. 
James  Adams,  Annals  Phil.,  Mag.  Chem.  (Thompson),  (2),  2,  1821,  16-18. 

C.  R.  Telosius  and  S.  Morck,  Disquisitio.  .  .  .  Acad.  Carolina,  Lundae, 
1838  (in  Meditationum  Math.  .  .  .  PubHce  Defendant  C.  J.  D.  Hill,  1831, 
Pt.  II). 

J.  A.  Arndt,  Archiv  Math.  Phys.,  1,  1841,  101-4. 

J.  Dienger,  ibid.,  11,  1848,  232;  Jour,  fur  Math.,  39,  1850,  67. 

Wm.  Wiley,  Math.  Magazine,  1,  1882,  7-8. 

A.  V.  Filippov,  Kagans  Bote,  1910,  214-221  (pedagogic). 

i*^L'intermediaire  des  math.,  21,  1914,  10. 

"sMath.  Quest.  Educat.  Times,  28,  1915,  101-4. 

"»76id.,  27,  1915,  33-4. 

^"Ibid.,  29,  1916,  76,  88-9. 

i"Math.  Quest,  and  Solutions,  3,  1917,  59. 

"'Vom  Periodischen  Dezimalbruch  zur  Zahlentheorie,  Leipzig,  1916,  59  pp. 

""Zeitschrift  Math.  Naturw.  Unterricht,  47,  1916,  228-230. 

i"/6id.,  230-1. 

i"Zeitschrift  Math.  Naturw.  Unterricht,  12,  1881,  431;  20,  188;  23,  584. 

i^Proc.  London  Math.  Soc,  Records  of  Meeting,  Dec.  6,  1917,  and  Feb.  14,  1918,  for  a  revised 

proof. 
i"Math.  Quest.  Educ.  Times,  7,  1867,  31-2.     Minor  results,  32,  1880,  69;  34,  1881,  97-8:  37, 

1882,  44;  41,  1884,  113-4;  58,  1893,  108-9;  60,  1894,  128;  63,  1895,  34;  72,  1900,  75-6; 

74,  1901,  35;  (2),  2,  1902,  65-6,  84-5;  4,  1903,  29,  65-7,  95;  7,  1905,  97,  106,  109-10;  8, 

1905,  57;  9,  1906,  73.     Math.  Quest,  and  Solutions,  3,  1917,  72  (table);  4,  1917,  22. 


I 


4 


I 


CHAPTER  VII. 

PRIMITIVE  ROOTS.  BINOMIAL  CONGRUENCES. 
Primitive  Roots,  Exponents,  Indices. 

J.  H.  Lambert^  stated  without  proof  that  there  exists  a  primitive  root 
g  of  any  given  prime  p,  so  that  g^  —  \  is  divisible  by  p  for  e  =  p  — 1,  but  not 
for  0<e<p  — 1. 

L.  Euler^  gave  a  proof  which  is  defective.  He  introduced  the  term 
primitive  root  and  proved  (art.  28)  that  at  most  n  integers  a;<p  make 
x"  — 1  divisible  by  p,  the  proof  applying  equally  well  to  any  polynomial 
of  degree  n  with  integral  coefficients.  He  stated  (art.  29)  that,  for  n<p, 
x"  — 1  has  all  n  solutions  "real"  if  and  only  if  n  is  a  divisor  of  p  — 1;  in  par- 
ticular, x^~^  —  l  has  p  — 1  solutions  (referring  to  arts.  22,  23,  where  he 
repeated  his  earlier  proof  of  Fermat's  theorem).  Very  likely  Euler  had 
in  mind  the  algebraic  identity  a;^"^  — l  =  (x"  — 1)Q,  from  which  he  was  in  a 
position  to  conclude  that  Q  has  at  most  n— p+1  solutions,  and  hence  x"  — 1 
exactly  n.  By  an  incomplete  induction  (arts.  32-34),  he  inferred  that  there 
are  exactly  </)(n)  integers  x<p  for  which  x"  — 1  is  divisible  by  p,  but  x^  —  \ 
not  divisible  by  p  for  0<Z<n,  n  being  a  divisor  of  p  — 1  (as  the  context 
indicates).  In  particular,  there  exist  <f>{p  —  l)  primitive  roots  of  p  (art.  46). 
He  listed  all  the  primitive  roots  of  each  prime  ^  37. 

J.  L.  Lagrange^  proved  that,  if  p  is  an  odd  prime  and 

a:P-i-l=Z^+pF, 
where  X,  ^,  F  are  polynomials  in  x  with  integral  coefficients,  and  if  x""  and  x" 
are  the  highest  powers  of  a;  in  X  and  ^  with  coefficients  not  divisible  by  p, 
there  are  m  integral  values,  numerically  <p/2,  of  x  which  make  X  a  mul- 
tiple of  p,  and  fi  values  making  ^  a  multiple  of  p.  For,  by  Fermat's  theorem, 
the  left  member  is  a  multiple  of  p  for  a;  =  ±  1,  ±  2, . . . ,  =*=  (p  — 1)/2,  while  at 
most  m  of  these  values  make  X  a  multiple  of  p  and  at  most  fx  make  ^  a 
multiple  of  p. 

L.  Euler^  stated  that  he  knew  no  rule  for  finding  a  primitive  root  and 
gave  a  table  of  all  the  primitive  roots  of  each  prime  ^41. 

Euler^  investigated  the  least  exponent  x  (when  it  exists)  for  which 
fa'+g  is  divisible  by  N.  Find  X  such  that  —g^\N  is  a  multiple,  say  aV, 
of  a.  Then  fa'-^-r  is  divisible  by  N.  Set  r=FX'iV  =  a^s,  ^^1.  Then 
j'gx-a-ff_^  is  divisible  by  N;  etc.  If  the  problem  is  possible,  we  finally  get 
/  as  the  residue  of  /a"^"""  •  •  •  "^,  whence  x  =  a+. . .  +f .  For  example,  to  find 
the  least  x  for  which  2""  — 1  is  divisible  by  iV  =  23,  we  have 

1+23  =  2^3,      3-23= -2^5,       -5-23= -2^7,       -7-^23  =  2^, 
whence  a^  =  3+2+2+4  =  ll. 

^Nova  Acta  Eruditomm,  Leipzig,  1769,  p.  127. 
''Novi  Comm.  Acad.  Petrop.,  18,  1773,  85;  Comm.  Arith.,  1,  516-537. 
^Nouv.  M6m.  Ac.  Roy.  Berlin,  ann^e  1775  (1777),  p.  339;  Oeuvres  3,  777. 
<0pu8c.  Anal.,  1,  1783  (1772),. 121;  Comm.  Arith.,  1,  506. 

^Opusc.  Anal.,  1,  1783  (1773),  242;  Comm.  .\i-ith.,  2,  p.  1;  Opera  postuma,  I,  172-4. 

181 


r 


182  History  of  the  Theory  of  Numbers.  [Chap,  vii 

A.  M.  Legendre^  started  with  Lagrange's'  result  that,  if  p  is  a  prime 
and  n  is  a  divisor  of  p  — 1, 
(1)  a:"  =  l(modp) 

has  n  incongruent  integral  roots.  Let  n  =  v^v'^ . .  .,  where  v,  v',.  . .  are  dis- 
tinct primes.  A  root  a  of  (1)  belongs*  to  the  exponent  n  if  no  one  of 
a"/',  a"^', .  . .  is  congruent  to  unity  modulo  p.  For,  if  a*  =  l,  0<d<n,  let 
a  be  the  g.  c.  d.  of  6,  n,  so  that  a  =  ny—dz  for  integers  y,  z;  then 

contrary  to  hypothesis.  Next,  of  the  n  roots  of  (1),  n/v  satisfy  x''^''  =  l 
(mod  p),  and  n{l  —  l/v)  do  not.  Likewise,  n{l  —  l/v')  do  not  satisfy 
2.n/«''  =  1 .  qIq     i^  jg  gaj(j  ^Q  follow  that  there  are 


*(„)=„  (i-i)(i-i) 


numbers  belonging  to  the  exponent  n  modulo  p.    If 
^•^^1,  ^-^-^Kmodp), 

/3  belongs  to  the  exponent  v"".  If  j8'  belongs  to  the  exponent  v'^,  etc.,  the 
product  /3]S' ...  is  stated  to  belong  to  the  exponent  n. 

C.  F.  Gauss^  gave  two  proofs  of  the  existence  of  primitive  roots  of  a 
prime  p.  If  d  is  a  divisor  of  p  —  1,  and  a''  is  the  lowest  power  of  a  congruent 
to  unity  modulo  p,  a  is  said  to  belong  to  the  exponent  d  modulo  p.  Let 
ypid)  of  the  integers  1,  2, . .  . ,  p  — 1  belong  to  the  exponent  d,  a  given  divisor 
of  p  —  1 .  Gauss  showed  that  i/'  (c?)  =  0  or  0  (d) ,  2 1/'  (c^)  =  p  —  1  =  2  <^  (d) ,  whence 
i^(d)  —(t>{d).  In  his  second  proof,  Gauss  set  p  — 1  =a"6'^.  .  .,  where  a,  5, . . . 
are  distinct  primes,  proved  the  existence  of  numbers  A,  B,. .  .  belonging 
to  the  respective  exponents  a",  h^,.  .  .,  and  showed  that  AB . .  .  belongs  to 
the  exponent  p  — 1  and  hence  is  a  primitive  root  of  p. 

Let  a  be  a  primitive  root  of  p,  h  any  integer  not  divisible  by  p,  and  e 
the  integer,  uniquely  determined  modulo  p  — 1,  for  w^hich  o*  =  6  (mod  p). 
Gauss  (arts.  57-59)  called  e  the  index  of  h  for  the  modulus  p  relative  to  the 
base  a,  and  wrote  e  =  ind  h.    Thus 

a'^^^'^b  (mod  p),  ind  66'=ind  6+ind  h'  (mod  p-1). 

Gauss  (arts.  69-72)  discussed  the  relations  between  indices  for  different 
bases  and  the  choice  of  the  most  convenient  base. 

In  articles  73-74,  he  gave  a  convenient  tentative  method  for  finding  a 
primitive  root  of  p.  Form  the  period  of  2  (the  distinct  least  positive  resi- 
dues of  the  successive  powers  of  2);  if  2  belongs  to  an  exponent  ^<p  — 1, 
select  a  number  6<p  not  in  the  period  of  2,  and  form  the  period  of  6;  etc. 

If  a  belongs  to  the  exponent  t  modulo  p,  the  product  of  the  terms  in  the 
period  of  a  is  =  (  — 1)'+^  (mod  p),  while  the  sum  of  the  terms  is  =0  unless 
a=l  (arts.  75,  79). 

•M6m.  Ac.  R.  Sc,  Paris,   1785,  471-3.     Thdorie  des  nombres,  1798,  413-4;   ed.  3,  1830, 

Nos.  341-2;  German  transl.  by  Maser,  2,  pp.  17-18. 
*This  term  was  introduced  later  by  Gauss.'' 
^Disquisitiones  Arith.,  1801,  arts.  52-55. 


Chap.  VII]  PRIMITIVE   RoOTS,   EXPONENTS,    INDICES.  183 

The  product  of  all  the  primitive  roots  of  a  prime  p^3  is  =1  (mod  p) ; 
the  sum  of  the  primitive  roots  of  p  is  =0  if  p  — 1  is  divisible  by  a  square, 
but  is  =(  — 1)"  if  p  — 1  is  the  product  of  n  distinct  primes  (arts.  80,  81). 

If  p  is  an  odd  prime  and  e  is  the  g.  c.  d.  of  ^(p")=p''~^(p  — 1)  and  t, 
then  x'  =  l  (mod  p")  has  exactly  e  incongruent  roots.  It  follows  that  there 
exist  primitive  roots  of  p",  i.  e.,  numbers  belonging  to  the  exponent  0(p") 
(arts.  85-89). 

For  n>2,  every  odd  number  belongs  modulo  2"  to  an  exponent  which 
divides  2""^,  so  that  primitive  roots  of  2"  are  lacking;  however,  a  modified 
method  of  employing  indices  to  the  base  5  may  be  used  (arts.  90,  91). 

If  w  =  A"5^..,  where  A,  B,...  are  distinct  primes,  and  a=0(A"), 
^=4>{B^), . .  .,  and  if  ii  is  the  1.  c.  m.  of  a,  jS, .  . .,  then  ^''  =  1  (mod  m)  for  z 
prime  to  m.  Now  fiKa-^.  .  .  =4>{m)  except  when  m  =  2",  p"  or  2p",  where 
p  is  an  odd  prime.  Thus  there  exist  primitive  roots  of  m  only  when  m  =  2, 
4,  p"or2p"  (art.  92). 

Table  I,  at  the  end  of  Disq.  Arith.,  gives  on  one  page  the  indices  of  each 
prime  <p  for  each  prime  and  power  of  prime  modulus  <  100.  Gauss  gave 
no  direct  table  to  determine  the  number  corresponding  to  a  given  index, 
but  indicated  (end  of  art.  316)  how  his  Table  III  for  the  conversion  of  ordi- 
nary into  decimal  fractions  leads  to  the  number  having  a  given  index  (cf. 
Gauss,i^'i^Ch.  VI). 

S.  F.  Lacroix^  reproduced  Gauss'  second  proof  of  the  existence  of  primi- 
tive roots  of  a  prune,  without  a  reference. 

L.  Poinsot^  argued  that  the  primitive  roots  of  a  prime  p  may  be  obtained 
from  the  algebraic  expressions  for  the  imaginary  (p  — l)th  roots  of  unity 
by  increasing  the  numbers  under  the  radical  signs  by  such  multiples  of  p 
that  the  radicals  become  integral.  The  (/)(p  — 1)  primitive  roots  of  p  may 
be  obtained  by  excluding  from  1, .  .  . ,  p  —  1  the  residues  of  the  powers  whose 
exponents  are  the  distinct  prime  factors  of  p  — 1;  while  symmetrical,  this 
method  is  unpractical  for  large  p. 

Fregier^°  proved  that  the  2"th  power  of  any  odd  number  has  the  remainder 
unity  when  divided  by  2""*"^,  if  n>0. 

Poinsot^^  developed  the  first  point  of  his  preceding  paper.  The  equa- 
tion for  the  primitive  18th  roots  of  unity  is  x^—x^-\-l=0.    The  roots  are 


:  =  a^^Kl  +  ^^'=^  (a'  =  l). 


But  \/^=  ±4,  -¥^=4:,  ^-11  =  2  (mod  19).  Thus  the  six  primitive 
roots  of  19  are  x=  —4,  2,  —9,  —5,  —6,  3.  In  general,  the  algebraic  expres- 
sions for  the  nth  roots  of  unity  represent  the  different  integral  roots  of 
a;"  =  l  (mod  p),  where  p  is  a  prime  kn-\-\,  after  suitable  integers  are  added 
to  the  numbers  under  the  radical  signs.     Since  unity  is  the  only  (integral) 

sCompldment  des  ^l^mens  d'alg^bre,  Paris,  ed.  3,  1804,  303-7;  ed.  4,  1817,  317-321. 

«M6m.  Sc.  Math,  et  Phys.  de  I'Institut  de  France,  14,  1813-5,  381-392. 
"Annales  de  Math,  (ed.,  Gergonne),  9,  1818-9,  285-8. 
"M6m.  Ac.  Sc.  de  I'Institut  de  France,  4,  1819-20,  99-183. 


184  History  of  the  Theory  op  Numbers.  [Chap,  vii 

root  of  x^=l  (mod  p),  if  p  is  a  prime  >2,  he  concluded  (p.  165)  that  p  is 
a  factor  of  the  numbers  under  the  radical  signs  in  the  formula  for  a  primitive 
pth  root  of  unity.     Cf.  Smith^^^  of  Ch.  VIII. 

Poinsot^^"  again  treated  the  same  subject. 

J.  Ivory^^  stated  that  a  primitive  root  of  a  prime  p  satisfies  x^^~^^^^^  —  1, 
but  no  one  of  the  congruences  x'=—l  (mod  p),  <=(p  — l)/(2a),  where  a 
ranges  over  the  odd  prime  factors  of  p  —  1 ;  while  a  number  not  a  primitive 
root  satisfies  at  least  one  of  the  a;'  =  —  1 .  Hence  if  each  a'  ^  —  1  and 
^(p-i)/2^  —  1,  then  a  is  a  primitive  root. 

V.  A.  Lebesgue^^  stated  that  prior  to  1829  he  had  given  in  the  Bulletin 
du  Nord,  Moscow,  the  congruence  X  =  0  of  Cauchy^^  for  the  integers 
belonging  to  the  exponent  n  modulo  p. 

A.  Cauchy^^  proved  the  existence  of  primitive  roots  of  a  prime  p,  essen- 
tially as  in  Gauss'  second  proof.  If  p  —  1  is  divisible  by  n  =  a"6V .  . . ,  where 
a,  b,  c,. .  .  are  distinct  primes,  he  proved  that  the  integers  belonging  to  the 
exponent  n  modulo  p  coincide  with  the  roots  of 

Y        (a^"-l)(a:"^°^-l)(a;"/°--l)...     _^  ,       ,    , 
^=(x"/"-l)(a:''/^-l)...(a:"/-^-l)...=^  ^^^^  P^' 

The  roots  of  the  equation  X  =  0  are  the  primitive  nth  roots  of  unity.  For 
the  above  divisor  n  of  p  — 1,  the  sum  of  the  Zth  powers  of  the  primitive 
roots  of  a;"  =  1  (mod  p)  is  divisible  by  p  if  Z  is  divisible  by  no  one  of  the 
numbers 

n,  n/a,  n/h, .  .  . ,  n/ab, . .  . ,  n/ahc, .... 

But  if  several  of  these  are  divisors  of  I,  and  if  we  replace  n,  a,  b,. . .  by 
<t>{n),  1—a,  1  — 6, .  .  .  in  the  largest  of  these  divisors  in  fractional  form,  we 
get  a  fraction  congruent  to  the  sum  of  the  Ith.  powers.  In  case  x'"  =  l 
(mod  p)  has  m  distinct  integral  roots,  the  sum  of  the  lib.  powers  of  all  the 
roots  is  congruent  modulo  p  to  m  or  0,  according  as  I  is  or  is  not  a  multiple 
of  m. 

M.  A.  Stern^^  proved  that  the  product  of  all  the  numbers  belonging  to 
an  exponent  d  is  =  1  (mod  p) ,  while  their  sum  is  divisible  by  p  if  d  is  divisible 
by  a  square,  but  is  =  ( —  1)"  if  d  is  a  product  of  n  distinct  primes  (generaliza- 
tions of  Gauss,  D.  A.,  arts.  80,  81).  If  p  =  2n+l  and  a  belongs  to  the  expo- 
nent n,  the  product  of  two  numbers,  which  do  not  occur  in  the  period  of  a, 
occurs  in  the  period  of  a.  To  find  a  primitive  root  of  p  when  p  —  1  =  2ab . . . , 
where  a,b,.  .  .  are  distinct  odd  primes,  raise  any  number  as  2  to  the  powers 
(p  —  l)/a,  (p  — 1)/6, . . . ;  if  no  one  of  the  residues  modulo  p  is  1,  the  negative 
of  the  product  of  these  residues  is  a  primitive  root  of  p;  in  case  one  of  the 
residues  is  1,  use  3  or  5  in  place  of  2.  If  p  =  2g+l  and  q  are  odd  primes,  2 
or  —  2  is  a  primitive  root  of  p  according  as  p  =  8n + 3  or  8n + 7 .     If  p  =  4^^  -f-1 

"''Jour,  de  I'dcole  polytechnique,  cah.  18,  t.  11,  1820,  345-410. 
"Supplement  to  Encyclopaedia  Britannica,  4,  1824,  698. 
"Jour,  de  Math.,  2,  1837,  258. 

"Exercices  de  Math.,  1829,  231;  Oeuvrea,  (2),  9,  266,  278-90. 
"Jour,  fiir  Math.,  6,  1830,  147-153. 


Chap.  VII]  PRIMITIVE   RoOTS,   EXPONENTS,    INDICES.  185 

and  q  are  primes,  2  and  —2  are  primitive  roots  of  p.  If  p  =  4g+l  and 
g  =  3n+l  are  primes,  3  and  —3  are  primitive  roots  of  p. 

F.  Minding^^  gave  without  reference  Gauss'  second  proof  of  the  exist- 
ence of  primitive  roots  of  a  prime. 

F.  J.  Richelot^^  proved  that,  if  p  =  2"'+l  is  a  prime,  every  quadratic 
non-residue  (in  particular,  3)  is  a  primitive  root  of  p. 

A.  L.  Crelle^^  gave  a  table  showing  all  prime  numbers  ^  101  having  a 
given  primitive  root;  also  a  table  of  the  residues  of  the  powers  of  the 
natural  numbers  when  divided  by  the  primes  3, . .  .,  101.  His  device^^ 
for  finding  the  residues  modulo  p  of  the  powers  of  a  will  be  clear  from  the 
example  p  =  7,  a  =  3.  Write  under  the  natural  numbers  <7  the  residues 
of  the  successive  multiples  of  3  formed  by  successive  additions  of  3 ;  we  get 

12    3    4    5    6 
3    6    2    5     14. 

Then  the  residues  3,  2,  6, .  .  .  of  3,  3^,  3^, . . .  modulo  7  are  found  as  follows: 
after  3  comes  the  number  2  below  3  in  the  above  table;  after  2  comes  the 
number  6  below  2  in  the  table;  etc. 

Crelle^°  proved  that,  if  p  is  a  prime  and  X  is  prime  to  p  — 1  and  <p  — 1, 
the  residues  modulo  p  of  z^  range  with  z  over  the  integers  1,  2,.  .  .,  p  — 1. 
His  proof  that  there  exist  (^(n)  numbers  belonging  to  the  exponent  n 
modulo  p,  if  n  divides  p  — 1,  is  like  that  by  Legendre.^ 

G.  L.  Dirichlet^^  employed  0(A:)  systems  of  indices  for  a  modulus 
/j  =  2^p'p"'.  . .,  where  p,  p', .  . .  are  distinct  primes,  and  X^3.  Given  any 
integer  n  prime  to  k,  and  primitive  roots  c,  c', .  •  •  of  P'>  v" \  •  •  •  >  we  can 
determine  indices  a,  /3,  7,  7', .  .  .  such  that 

n=(-l)''5^  (mod  2^^),  n  =  c^  (mod  p'),  n  =  c"''  (mod  p"'),-  •  •• 

Michel  Ostrogradsky^^  gave  for  each  prime  p<200  all  the  primitive 
roots  of  p  and  companion  tables  of  the  indices  and  corresponding  numbers. 
(See  Jacobins  and  Tchebychef  .3^) 

C.  G.  J.  Jacobi^^  gave  for  each  prime  and  power  of  a  prime  <  1000  two 
companion  tables  showing  the  numbers  with  given  indices  and  the  index 
of  each  given  number.  In  the  introduction,  he  reproduced  the  table  by 
Burckhardt,  1817,  of  the  length  of  the  period  of  the  decimal  fraction  for 
1/p,  for  each  prime  p^2543,  and  22  higher  primes.  Of  the  365  primes 
<2500,  we  therefore  have  148  having  10  as  a  primitive  root,  and  73  of  the 
form  4w+3  having  —10  as  a  primitive  root.  Use  is  made  also  of  the 
primes  for  which  10  or  — 10  is  the  square  or  cube  of  a  primitive  root. 

"Anfangsgrlinde  der  hoheren  Arith.,  1832,  36-37. 

"Jour,  ftir  Math.,  9,  1832,  p.  5. 

^Hhid.,  27-53. 

"Also,  ihid.,  28,  1844,  166. 

"Abh.  Ak.  Wiss.  BerUn,  1832,  Math.,  p.  57,  p.  65. 

^HhU.,  1837,  Math.,  45;  Werke,  1,  1889,  333. 

"Lectures  on  alg.  and  transc.  analysis,  I-II,  St.  P^tersbourg,  1837;  M6m.  Ac.  Sc.  St.  P6tera- 

bourg,  s6r.  6,  sc.  math,  et  phys.,  1,  1838,  359-85. 
2'Canon  Arithmeticus,  Berlin,  1839,  xl+248  pp.     Errata,  Cunningham.""-"" 


186 


History  of  the  Theory  of  Numbers. 


[Chap.  VII 


To  find  a  primitive  root  g  of  p,  select  any  convenient  integer  a  and  form 
the  residues  of  a,  or,  a^,. . .  [as  by  Crelle^®].  Let  n  be  the  exponent  to 
which  a  belongs.  Set  nn'  =  p  —  \.  If  n<p  — 1,  select  an  integer  h  not  in 
the  period  a, .  .  . ,  a".  The  residue  of  6"  is  in  this  period  of  a.  If  y  is  the 
least  power  of  6  whose  residue  is  in  the  period  of  a,  then  /  divides  n',  say 
w'=i/'  (P-  xxiii).     Since  a=g'^',  h^=a\  we  have 

y^gf^''^g"''^''"'\  h  =  g^'^'+'^^  (mod  p), 

for  some  value  0,  1, . .  . ,  /—I  of  k.  But  A:  must  be  chosen  so  that  i+nk  is 
prime  to  /.  For,  if  i-\-nk  =  du,  where  d  is  a  divisor  of  /,  we  would  have 
5^''  =  a".  The  nf  residues  of  a'b'  (r  =  0,.  .  .,  n-1;  s  =  0,.  .  .,  /-I)  are  dis- 
tinct ;  their  indices  to  base  g  are/',  2f', .  .  . ,  nff  in  some  order  and  are  known. 
If  nf'<p  —  l,  we  employ  an  integer  not  in  the  set  a^b'  and  proceed  similarly. 
Ultimately  we  obtain  a  primitive  root  and  at  the  same  time  the  index  of 
everj"  number.     This  method  was  used  for  the  primes  between  200  and  1000. 

For  primes  <  200,  the  tables  by  Ostrogradsky^^  w^ere  reprinted  with  the 
same  errors  (noted  at  the  end  of  the  Canon). 

Jacobi  proved  that,  if  n  is  an  odd  prime,  any  primitive  root  of  n^  is  a 
primitive  root  of  any  higher  power  of  n  (p.  xxxv). 

For  the  modulus  2",  4^iu^9,  the  final  tables  give  the  index  /  of  any 
positive  odd  number  to  base  3,  where 

(_l)(Ar-l)W-3)/8^  =  37  (jjjQ^  2"). 

Robert  ]\Iurphy-^  stated  the  empirical  theorem  that  every  prime 
anr+p  has  a  as  a  primitive  root  if  p>a/2,  p  is  a  prime  <a,  and  if  a  is  a 
primitive  root  of  p.  For  example,  a  prime  10nr-{-7  has  10  as  a  primitive 
root. 

H.  G.  Erlerus'^  considered  two  odd  primes  p  and  p'  and  a  number  m 
such  that  m=a  (mod  p),  m  =  a'  (mod  p').  Let  a  belong  to  the  exponent 
e  modulo  p,  and  a'  to  the  exponent  e'  modulo  p\  If  8  is  the  g.  c.  d.  of 
e  and  e',  then  m  belongs  to  the  exponent  ee'/8  modulo  pp'.  He  discussed 
at  length  the  number  of  integers  belonging  to  the  exponent  n  for  a  com- 
posite modulus. 

A.  Cauchy^^  called  the  least  positive  integer  i  for  which  m'  =  1  (mod  n) 
the  indicator  relative  (or  corresponding)  to  the  base  m  and  modulus  n, 
which  are  assumed  relatively  prime.  If  the  base  m  is  constant,  and  ii,  12 
are  the  indicators  corresponding  to  moduli  nj,  112,  and  if  n  =  nin2  is  prime 
to  772,  then  the  1.  c.  m.  of  I'l  and  {2  is  the  indicator  corresponding  to  modulus 
n.  If  the  modulus  n  is  constant,  and  ii,  io  are  the  indicators  corresponding 
to  bases  Wi,  ^2,  and  if  I'l,  1*2  are  relatively  prime,  then  1*112  is  the  indicator 
corresponding  to  the  base  7^17^2. 

Let  I'l,  io  be  the  indicators  corresponding  to  the  bases  mi,  7722  and  same 
modulus  n.  The  g.  c.  d.  0;  of  I'l,  2*2  can  be  expressed  (often  in  several  ways) 
as  a  product  uv  such  that  ii/u,  io/v  are  relatively  prime.     For,  if  co  =  a/3. . . , 

"Phil.  Mag.,  (3),  19,  1841,  369. 

"Elementa  Doctrinse  Numerorum,  Diss.,  Halis,  1841,  18-43. 

»«Comptes  Rendus  Paris,  12,  1841,  824-845;  Oeuvres,  (1),  6,  124-146. 


Chap.  VII]  PRIMITIVE   RoOTS,   EXPONENTS,   INDICES.  187 

where  a,  j8, . . .  are  powers  of  distinct  primes,  use  a  as  a  factor  in  forming  u 
in  case  a  is  prime  to  ii/a,  but  as  a  factor  of  y  in  case  a  is  prime  to  zVa,  and 
as  a  factor  of  either  u  or  v  indifferently  in  case  a  is  prime  to  both  ii/a  and 
12/0.  Since  ii/u  and  i2/v  are  relatively  prime  indicators  corresponding  to 
bases  mi"  and  m2^  it  follows  from  the  preceding  theorem  that  the  indicator 
corresponding  to  base  mi"-W2''  and  modulus  n  is 

ii   ^2      iii2      1  r  •      • 
=  —  =  1.  c.  m.  of  ii,  i2. 

U     V         £0 

Hence,  given  several  bases  mi,  m2, . . .  and  a  single  modulus  n,  we  can 
find  a  new  base  relative  to  which  the  indicator  is  the  1.  c.  m.  of  the  indicators 
corresponding  to  mi,  m2, ....  If  the  latter  bases  include  all  the  integers  <n 
and  prime  to  n,  the  corresponding  indicators  give  all  indicators  which  can 
correspond  to  modulus  n,  so  that  all  of  them  divide  a  certain  maximum 
indicator  I.  Then  for  every  integer  m  relatively  prime  to  n,m^  =  l  (mod  n) . 
If  n  =  v°',  where  v  is  an  odd  prime,  or  if  n  =  2  or  4,  l=^{n).  If  n  =  2^  k>2, 
I=(f}{n)/2.  If  Ij  is  the  maximum  indicator  corresponding  to  a  power  Uj 
of  a  prime,  and  if  n  =  llnj,  then  I  is  the  1.  c.  m.  of  /i,  /2,  •  •  ••  The  equation 
mx  —  ny  =  l  has  the  solution  x  =  7n^~^  (mod  n). 

Cauchy^^  republished  the  preceding  paper,  but  with  an  extension  from  the 
limit  n  =  100  to  the  limit  n  =  1000  for  his  table  of  the  maximum  indicator  I. 

C.  F.  Arndt^^  gave  (without  reference)  Gauss'  second  proof  of  the  exist- 
ence of  a  primitive  root  of  an  odd  prime  p,  and  proved  the  existence  of  the 
<^(p")  primitive  roots  of  p'*  or  2p'',  and  that  there  are  no  primitive  roots  for 
moduli  other  than  these  and  4.  If  i  is  a  divisor  of  2""^,  n>2,  exactly  t 
numbers  belong  to  the  exponent  t  modulo  2''  (p.  18).  If,  for  a  modulus 
p",  2p",  a  belongs  to  the  exponent  t,  then  a-a^ .  .  .a'  is  congruent  to  (  —  1)'+^ 
(pp.  26-27),  while  the  product  of  the  numbers  belonging  to  the  exponent  t 
is  congruent  to  +1  if  ^?^  2  (pp.  37-38).  He  proved  also  Stern's^^  theorem 
on  the  sum  of  these  numbers.  He  gave  the  same  two  theorems  also  in  a 
later  paper.  ^^ 

L.  Poinsot^"  used  the  method  of  Legendre^  to  prove  the  existence  of 
4>{n)  integers  belonging  to  the  exponent  n,  sl  divisor  of  p  — 1,  where  p  is  a 
prime.  He  gave  (pp.  71-75)  essentially  Gauss'  first  proof,  and  gave  his 
own^  method  of  finding  primitive  roots  of  a  prime.  The  existence  of 
primitive  roots  of  p",  2p",  4,  but  of  no  further  moduli,  is  established  by  use 
of  the  number  of  roots  of  binomial  congruences  (pp.  87-101). 

C.  F.  Arndt^^  noted  that  if  a  belongs  to  an  even  exponent  t  modulo  2", 
then  ±a,  ±a^, ...,  ±a'~^  give  the  t  incongruent  numbers  belonging  to 
the  exponent  t,  and  are  congruent  to  A;  •  2""  =f  1  (A;  =  1 ,  3, 5, .  . . ) .  The  product 
of  the  numbers  belonging  to  the  exponent  t  modulo  2",  n>2,  is  =  +1. 

•      "Exercices  d' Analyse  et  de  Phys.  Math.,  2,  1841,  1-40;  Oeuvres,  (2),  12. 
"Archiv  Math.  Phys.,  2,  1842,  9,  15-16. 
"Jour,  flir  Math.,  31,  1846,  326-8. 
3«Jour.  de  Math^matiques,  (1),  10,  1845,  65-70,  72. 
"Archiv  Math.  Phys.,  6,  1845,  395,  399. 


188 


History  of  the  Theory  of  Numbers. 


[Chap.  VII 


E.  Prouhet^"  gave,  without  reference,  Crelle's^'  method  of  forming  the 
residues  of  the  powers  of  a  number.  The  object  of  the  paper  is  to  give  a 
uniform  method  of  proof  of  theorems,  given  in  various  places  in  Legendre's 
text,  relating  to  the  residues  of  the  first  n  powers  of  an  integer  belonging 
to  the  exponent  n  modulo  P,  especially  when  P  is  a  prime  or  a  power  of  a 
prime,  and  the  existence  of  primitive  roots.  He  gave  (p.  658)  the  usual 
proof  that  =•=  2  is  a  primitive  root  of  a  prime  2^+ 1  if  5  is  a  prime  4/c±  1  (with 
a  misprint). 

C.  F.  Arndt^^  proved  that  if  ^  is  a  primitive  root  of  the  odd  prime  p  and 
if  p^  (SKn)  is  the  highest  power  of  p  dividing  G  =  g^~^  —  1,  then  g  belongs 
to  the  exponent  p'*~''(p  — 1)  modulo  p".  Conversely,  if  the  last  is  true  of  a 
primitive  root  g  of  p,  then  G  is  divisible  by  p^  and  not  by  p^"'"^  The  first 
result  with  X  =  1  shows  that  any  primitive  root  of  p^  is  a  primitive  root  of 
p",  n>2.  Let  g  he  a,  primitive  root  of  p;  if  G  is  not  divisible  by  p^,  g  is  a. 
primitive  root  of  p^;  but  if  G  is  divisible  by  p^,  and  h  is  not  divisible  by  p, 
then  g+hp  is  a  primitive  root  of  p^.  Any  odd  primitive  root  of  p**  is  a 
primitive  root  of  2p".  If  gr  is  a  primitive  root  of  p'*  or  2p'*,  and  t  is  a  divisor 
of  p"~^(p  — 1),  then  if  a  ranges  over  the  integers  <t  and  prime  to  t,  the 
<f>{t)  integers  belonging  to  the  exponent  t  modulo  p"  or  2p"  are  g%  where 
e  =  p"~^(p  — l)a/<.  The  numbers  belonging  to  the  exponent  2"""*  modulo 
2"  are  found  more  simply  than  by  Gauss'^  and  Jacobi^^  (p.  37). 

P.  L.  Tchebychef^^  proved  that  if  a,  /3, .  . .  are  the  distinct  prime  factors 
of  p  — 1,  where  p  is  a  prime,  then  a  is  a  primitive  root  of  p  if  and  only  if  no 
one  of  the  congruences  x''  =  a,  xP  =  a,. . .  (mod  p)  has  an  integral  root. 
This  furnishes  a  method  (usually  impracticable)  of  finding  all  primitive 
roots  of  p.  A  second  method  uses  a  number  a  belonging  to  the  exponent  n, 
and  a  number  h  not  congruent  to  a  power  of  a,  and  deduces  a  number 
belonging  to  an  exponent  >n.  In  the  second  supplement,  he  proved  that 
3  is  a  primitive  root  of  any  prime  2^"+ 1 ;  that  =*=  2  is  a  primitive  root  of  any 
prime  2a +1  such  that  a  is  a  prime  4A:±  1 ;  3  is  a  primitive  root  of  4iV2"'4-l 
if  w>0  and  iV  is  a  prime  >9^  /(4-2'");  2  is  a  primitive  root  of  any  prime 
4iVH-l  such  that  A^  is  an  odd  prune.  The  last  result  was  later  proposed'^ 
as  a  question  for  solution  (with  reference  to  this  text) .  There  is  given  the 
table  of  primitive  roots  and  indices  for  primes  <  200,  due  to  Ostrogradsky^^. 
Schapira  (p.  314)  noted  that  in  the  list  of  errata  in  Jacobi's^^  Canon  (p.  222) 
there  is  omitted  the  error  8  for  6  in  ind  14  for  p  =  25. 

V.  A.  Lebesgue^*^  remarked  that  Cauchy's^^  congruence  X=0  shows 
the  existence  of  0(n)  integers  belonging  to  the  exponent  n  modulo  p,  a 
prime. 

»«Nouv.  Ann.  Math.,  5,  1846,  175-87,  659-62,  675-83. 

»3Jour.  fiir  Math.,  31,  1846,  259-68. 

"Theory  of  Congruences  (m  Russian),  1849.     German  translation  by  Schapira,  Berlin,  1889, 

p.  192.     Italian  translation  by  Mile.  Massarini,  Rome,  1895,  with  an  extension  of  the 

tables  of  indices  to  353. 
»Nouv.  Ann.  Math.,  15,  1856,  353.     Solved  by  use  of  Euler's  criterion  by  P.  H.  Rochette, 

and.,  16,  1857,  159.     Also  proved  by  Desmarest,*^  p.  278. 
»*Nouv.  Ann.  Math.,  8,  1849,  352;  11,  1852,  420. 


Chap.  VII]  PRIMITIVE  RoOTS,   EXPONENTS,   INDICES.  189 

E.  Desmarest"  devoted  the  last  86  pages  of  his  book  to  primitive  roots; 
the  70  pages  claimed  to  be  new  might  well  have  been  reduced  to  five  by 
the  omission  of  trivial  matters  and  the  use  of  standard  notations.  To  find 
(pp.  267-8)  a  primitive  root  of  the  prime  P  =  6g+l,  where  q  is  an  odd  prime, 
seek  an  odd  solution  of  ^^^+3  =  0  (mod  P)  and  set  w  =  2/2  — 1;  then  R^=—l 
and  R  belongs  to  the  exponent  6;  thus  we  know  the  solutions  of  x^  =  l\ 
let  a  be  any  integer  prime  to  P  and  not  such  a  solution;  if  a^=±l,  then 
=ta  belongs  to  the  exponent  q,  and  ±ai2  is  a  primitive  root  of  P;  but,  if 
a^«  7^  1 ,  then  a^^=  =f  1  (mod  P) ,  and  =*=  a  is  a  primitive  root  of  P.  If  P  =  8Q + 1 
and  Q  are  primes,  then  P=5  (mod  12)  and  3  is  a  quadratic  non-residue  and 
hence  a  primitive  root  of  P. 

Let  P  be  a  prime  of  the  form  5q^2.  Then  u^=  5  (mod  P)  is  not  solvable. 
Thus,  if  a  is  a  primitive  root  of  P,  5  =  a%  where  e  is  odd.  Thus  if  e  is  prime 
to  P  — 1,  5  is  a  primitive  root  of  P.  It  is  recommended  that  5  be  the  first 
number  used  in  seeking  by  trial  a  primitive  root.  And  yet  he  announced 
the  theorem  (p.  283)  that  5  is  in  general  a  primitive  root.  If  P  is  a  prime 
5g±2  also  of  the  form  2"Q+1,  where  Q  is  an  odd  prime  including  1,  then 
(pp.  284-6)  5  is  a  primitive  root  of  P  provided  P  is  not  a  factor  of  5^  —1. 
He  gave  the  factors  of  the  latter  and  of  10^"  —  1  for  n  =  1, . . . ,  5.       ' 

Results,  corresponding  to  those  just  quoted  for  5,  are  stated  for  p  =  7,  —  7, 
10,  17.  What  is  really  given  is  a  Hst  of  the  linear  forms  of  the  primes  P 
for  which  p  is  a  quadratic  non-residue.  If,  in  addition,  P  =  2''Q  +  1,  where 
Q  is  an  odd  prime,  then  p  is  a  primitive  root,  provided  p^^^^l  (mbd  P). 
The  last  condition  is  ignored  in  his  statement  of  his  results  and  again  in  his 
collection  (pp.  297-8)  of  "principles  which  give  primitive  roots"  entered  in 
his  table  (pp.  298-300)  giving  a  primitive  root  of  each  prime  <  10000. 

V.  A.  Lebesgue^^  proved  that,  if  a  and  p  =  2'a+l  are  primes,  any  quad- 
ratic non-residue  x  of  p  is  a  primitive  root  of  p  if 

a;2*-'+1^0(modp). 

J.  P.  Kulik^^  gave  for  each  prime  p  between  103  and  353  the  indices  and 
all  the  primitive  roots  of  p.  His  manuscript  extended  to  1000.  There  is 
an  initial  table  giving  the  least  primitive  root  of  the  primes  from  103  to  1009. 

G.  01tramare^°  called  x  a  root  of  order  or  index  m  of  a  prime  piix  belongs 
to  the  exponent  {p  —  l)/m  modulo  p.  Let  Xm{x)  =  0  (mod  p)  be  the  con- 
gruence whose  roots  are  exclusively  the  roots  of  order  moi  p.  By  changing 
X  to  x^^"",  we  obtain  Xmn=<l>{^) ^0.     li  rii,  n2, . . . ,  n  are  the  divisors  >  1  of  w. 

Am  —  ■ 


Y  Y 


'^Th^orie  des  nombres.  Traits  de  I'analyse  ind6terminee  du  second  degr6  k  deux  inconnues 
suivi  de  I'application  de  cette  analyse  k  la  recherche  des  racines  primitives  avec  une  table 
de  ces  racines  pour  tous  les  nombres  premiers  compris  entre  1  et  10000,  Paris,  1852, 
308  pp.     For  errata,  see  Cunningham,  Mess.  Math.,  33,  1903,  145. 

58Nouv.  Ann.  Math.,  11,  1852,  422-4. 

'9 Jour,  fur  Math.,  45,  1853,  55-81. 

"7Wd.,  303-9. 


190  History  of  the  Theory  of  Numbers.  [Chap,  vii 

V.  A.  Lebesgue^^  noted  that,  given  a  primitive  root  g  (g<p)  of  the 
prime  p,  we  can  find  at  once  the  primitive  roots  of  p".  Let  g'  be  the  positive 
residue  <p~  when  g^  is  divided  by  p^  and  set  h  =  {g'  —  g)/p.     Then 

g+px+p'^y  {y  =  0,.  .  .,  p""^-!;  x  =  0,.  . .,  p-1;  x^h) 

give  p"~^(p  — 1)  primitive  roots.  Replacing  g  by  g\  where  i  is  less  than 
and  prime  to  p  —  1,  we  obtain  0  ]0(p") }  primitive  roots  of  p".  In  particular, 
a  primitive  root  of  p~  is  a  primitive  root  of  p"  (Jacobi^).  But,  if  h  =  0,  g 
is  not  a  primitive  root  of  p".     Since 

ginda+e^p_^  (mod  p") ,  e  =  ip"-np-l), 

we  can  reduce  by  half  the  size  of  Jacobi's  Canon. 

D.  A.  da  Silva^^  gave  two  proofs  that  x'^  =  l  (mod  p)  has  (f)(d)  primitive 
roots,  if  d  divides  p  — 1,  and  perfected  the  method  of  Poinsot^'^"  for  finding 
the  primitive  roots  of  a  prime. 

F.  Landry^^"  was  led  to  the  same  conclusion  as  Ivory.^^  In  particular, 
if  p  =  2*  +  l,  or  if  p  =  2n+l  (n  an  odd  prime)  and  a7^p  —  l,  any  quadratic 
non-residue  a  of  p  is  a  primitive  root.  For  each  prime  p<  10000,  at  least 
one  prime  ^  19  is  a  quadratic  non-residue  of  p.  Cauchy's^*  congruence  for 
the  primitive  roots  is  derived  and  proved. 

G.  Oltramare*^  proved  that  —  3°2^''  is  a  primitive  root  of  the  prime 
p  =  2a/3  +  l,  if  a^3,  /3f^3,  S'^^l,  22^^1  (mod  p);  that,  if 

p  =  3-2"'-M=g2+3r2,  qx-]-ry  =  l, 

{  —  l+qy  —  3rx)5i^/2  is  a  primitive  root  of  p;  and  analogous  theorems.  If 
a  and  2a-^l  are  primes,  2  or  a  is  a  primitive  root  of  2a -fl,  according  as  a 
is  of  the  form  4n-[-l  or  4n+3.  If  a  is  a  prime  9^3  and  if  p  =  2a4-l  is  a 
prime  and  m>  1,  then  3  is  a  primitive  root  of  p  unless  3^'"~^-|-l=0  (mod  p). 
[Cf.  Smith.''^] 

P.  Buttel^  attributed  to  Scheffler  (Die  unbestimmte  Analytik,  1854, 
§142)  the  method  of  Crelle^^  for  finding  the  residues  of  powers. 

C.  G.  Reuschle's^^  table  C  gives  the  Haupt-exponent  {i.  e.,  exponent  to 
which  the  number  belongs)  (a)  of  10,  2,  3,  5,  6,  7  with  respect  to  all  primes 
p<  1000,  and  the  least  primitive  root  of  p;  (b)  of  10  and  2  for  1000<  p<  5000 
and  a  convenient  primitive  root;  (c)  of  10  for  5000<p<  15000  (no  primitive 
root  given).     Numerous  errata  have  been  listed  by  Cunningham."" 

Allegret^^  stated  that  if  n  is  odd,  n  is  not  a  primitive  root  of  a  prime 
2^^n-f-l,  X>0;  proof  can  be  made  as  in  Lebesgue.^^ 

"Comptes  Rendus  Paris,  39,  1854,  1069-71;  same  in  Jour,  de  Math.,  19,  1854,  334-6. 
**Proprietades  geraes  et  resolu^ao  directa  das  Congruencias  binomias,  Lisbon,  1854.     Report 

by  C.  Alasia,  Rivista  di  Fisica,  Mat.  e  Sc.  Nat.,  Pavia,  4,  1903,  25,  27-28;  and  Annaes 

Scientificos  Acad.  Polyt.  do  Porto,  Coimbra,  4,  1909,  166-192. 
**'Troi8i&me  m6moire  sur  la  thdorie  des  nombres,  Paris,  1854,  24  pp. 
"Jour,  fiir  Math.,  49,  1855,  161-86. 
"Archiv  Math.  Phys.,  26,  1856,  247. 
"Math.  Abhandlung. .  .Tabellen,  Prog.  Stuttgart,  1856;  full  title  in  the  chapter  on  perfect 

numbers. i''^ 
"Nouv.  Ann.  Math.,  16,  1857,  309-310. 


Chap.  VII]  PRIMITIVE   RoOTS,   EXPONENTS,   INDICES.  191 

H.  J.  S.  Smith^^  stated  that  some  of  Oltramare's^^  general  results  are 
erroneous  at  least  in  expression,  and  gave  a  simple  proof  that  0^"^= 1  (mod  p**) 
has  exactly  d  roots  if  d  divides  0(p"). 

V.  A.  Lebesgue^^  proved  that,  if  p  is  an  odd  prime  and  a,  b  belong  to 
exponents  a,  (3,  there  exist  numbers  belonging  to  the  1.  c.  m.  m  of  a,  (3,  as 
exponent.  Hence  if  neither  a  nor  /3  is  a  multiple  of  the  other,  w  exceeds 
a  and  /3.  If  d<p  —  l  is  the  greatest  of  the  exponents  to  which  1, .  . .,  p  — 1 
belong,  the  latter  do  not  all  belong  to  exponents  dividing  d,  since  otherwise 
they  would  give  more  than  d  roots  of  x'^=l  (mod  p).  Hence  there  exist 
primitive  roots  of  p.  If  a  is  odd,  ±l+2°a  belongs  to  the  exponent  2™~" 
modulo  2""  (p.  87).  If  h  belongs  to  the  exponent  k  modulo  p,  a  prime,  then 
h+Pz  belongs  modulo  p"  to  an  exponent  which  divides  A;p"~^  (p.  101).  If 
/  is  a  primitive  root  of  p,  and  f^~^  —  l=pz,  then  /  is  a  primitive  root  of  p™ 
if  and  only  if  z  is  not  divisible  by  p  (p.  102). 

G.  L.  Dirichlet^^  proved  the  last  theorem  and  explained  his^^  system  of 
indices  for  a  composite  modulus. 

V.  A.  Lebesgue^°  published  tables,  constructed  by  J.  Hoiiel,^^  of  indices 
and  corresponding  numbers  for  each  prime  and  power  of  prime  modulus 
<  200,  which  differ  from  Jacobi's^^  only  in  the  choice  of  the  least  primitive 
root.  There  is  an  auxiliary  table  of  the  indices  of  x\  for  prime  moduli 
<200. 

V.  A.  Lebesgue^^  stated  that,  if  g'<p  is  a  primitive  root  of  the  prime  p 
and  if  g'=g^~^  (mod  p),  then  g'  is  a  primitive  root  of  p;  at  least  one  of  g  and 
g'  is  a  primitive  root  of  p"  for  n  arbitrary. 

V.  Bouniakowsky^^  proved  in  a  new  way  the  theorems  of  Tchebychef^* 
that  2  is  a  primitive  root  of  p  =  8n+3  if  p  and  4n+l  are  primes,  and  of 
p  =  4nH-l  if  p  and  n  are  primes.  He  gave  a  method  to  find  the  exponent 
to  which  2  or  10  belongs  modulo  p. 

A.  Cayley^^  gave  a  specimen  table  showing  the  indices  a,  j3,. . .  for  every 
number  M  =  a"6^.  .  .(modiV),  where  ilf<iV and  prime  to  iV,  for  iV  =  l,. . .,  50. 
There  is  no  apparent  way  of  forming  another  single  table  for  all  A^'s  analo- 
gous to  Jacobi's  tables  (one  for  each  N)  of  numbers  corresponding  to  given 
indices. 

F.  W.  A.  Heime^^  gave  the  least  primitive  root  of  each  prime  <  1000. 
His  other  results  are  not  new.  A  secondary  root  of  a  prime  p  is  one  belong- 
ing to  an  exponent  <  p  —  1  modulo  p. 

"British  Assoc.  Report,  1859,  228;  1860,  120,  §73;  Coll.  Math.  Papers,  1,  50,  158  (Report  on 
theory  of  numbers). 

**Introd.  th^orie  des  nombres,  1862,  94-96. 

"Zahlentheorie,  §§128-131, 1863;  ed.  2, 1871;  ed.  3, 1879;  ed.  4, 1894. 

soM^m.  soc.  sc.  phys.  et  nat.  de  Bordeaux,  3,  cah.  2, 1864-5,  231-274. 

"Formiiles  et  tables  numer.,  Paris,  1866.     For  moduli  ^  347. 

^Comptes  Rendus  Paris,  64,  1867,  1268-9. 

"Bull.  Ac.  Sc.  St.  Petersbourg,  11,  1867,  97-123. 

"Quart.  Jour.  Math.,  9,  1868,  95-96. 

"Untersuchungen,  besonders  in  Bezug  auf  relative  Primzahlen,  primitive  u.  secundare  Wurzeln, 
quadratische  Reste  u.  Nichtreste;  nebst  Berechnung  der  kleinsten  primitiven  Wurzeln 
vorf alien  Primzahlen  zwischen  1  und  1000.     BerUn,  1868;  ed.  2,  1869. 


192  History  of  the  Theory  of  Numbers.  [Chap,  vii 

C.  J.  D.  Hill^^  noted  that  his  tables  of  indices  for  the  moduU  2"  and  5" 
(n^5)  give  the  residues  of  numbers  modulo  10",  i.  e.,  the  last  n  digits. 
Using  also  tables  for  the  moduli  9091  and  9901,  as  well  as  a  table  of  loga- 
rithms, we  are  able  to  determine  the  last  22  digits. 

B.  M.  Goldberg^^  gave  the  least  primitive  root  of  each  prime  <  10160. 

V.  Bouniakowsky^^  proved  that  3  is  a  primitive  root  of  p  if  p  =  24n+5 
and  (p  — 1)/4  are  primes;  —3  is  a  primitive  root  of  p  if  p  =  12n+ll  and 
(p  — 1)/2  are  primes;  if  p  is  a  primitive  root  of  the  prime  p  =  4n+l,  one 
(or  both)  of  p,  p—p  is  a  primitive  root  of  p"'  and  of  2p"';  5  is  a  primitive 
root  of  p  =  20?i+3  or  20n4-7  if  p  and  {p  — 1)/2  are  primes,  and  of  p  =  40n  + 13 
or  4071-1-37  if  p  and  (p  — 1)/4  are  primes;  6  is  a  primitive  root  of  a  prime 
24n-|-ll  and  —6  of  24n-|-23  if  (p  — 1)/2  is  a  prime;  10  is  a  primitive  root 
of  p  =  40n+7,  19,  23,  and  -10  of  p  =  40n+3,  27,  39,  if  (p-l)/2  is  a  prime; 
10  is  a  primitive  root  of  a  prime  80n-(-73,  n>0,  or  80n+57,  n>l,  if 
(p  — 1)/8  is  a  prime.  If  p  =  8an-h2a  — 1  or  8an+a— 2  and  (p  — 1)/4  are 
primes,  and  if  a^-f-1  is  not  divisible  by  p,  a  is  a  primitive  root  of  p. 

V.  A.  Lebesgue^^  proved  certain  theorems  due  to  Jacobi^^  and  the 
following  theorem  which  gives  a  method  different  from  Jacobi's  for  forming 
a  table  of  indices  for  a  prime  modulus  p:  If  a  belongs  to  the  exponent  n, 
and  if  6  is  not  in  the  period  of  a,  and  if  /  is  the  least  positive  exponent  for 
which  h^=a\  then  x^=a  has  the  root  a'6",  where  ft-\-iu  —  l=nv;  the  root 
belongs  to  the  exponent  nf  if  and  only  if  u  is  prime  to  /. 

Consider  the  congruence  x*"  =  a  (mod  p) ,  where  a  belongs  to  the  exponent 
n  =  (p  — l)/n',  and  m  is  a  divisor  of  n'.  Every  root  r  has  a  period  of  mn 
terms  if  no  one  of  the  residues  of  r,  r^,. .  .,  r*""^  is  in  the  period  of  a.  If  all 
the  prime  divisors  of  m  divide  n,  the  m  roots  have  a  period  of  mn  terms; 
but  if  m  has  prime  divisors  g,  r, . .  . ,  not  dividing  n,  there  are  only 


-(^X^)- 


roots  having  a  period  of  mn  terms.  The  existence  of  primitive  roots  follows; 
this  is  already  the  case  if  m  =  n'. 

Mention  is  made  of  companion  tables  in  manuscript  giving  indices  of 
numbers,  and  numbers  corresponding  to  indices,  constructed  by  J.  Ch. 
Dupain  in  full  for  p<200,  but  from  200  to  1500  with  reduction  to  one-half 
in  view  of  ind  p  — a=ind  a=t(p  — 1)/2  modulo  p  — 1. 

L.  Kronecker^^  proved  the  existence  of  two  series  of  positive  integers 
Qj,  m,  {j=l,. .  .,  p)  such  that  the  least  positive  residues  modulo  A:>2  of 
^1 V2*' •  •  ■  ^p*"  give  all  the  (f){k)  positive  integers  <A:  and  prime  to  k,  if 
ii=0,  1,.  . .,  mi  — 1;  i2  =  0,  1,.  .  .,  m2  — 1;  etc.  [cf.  Mertens^^]. 

G.  Barillari^""  proved  that,  if  a  is  prime  to  h  and  belongs  to  the  exponent 

"Jour,  fur  Math.,  70,  1869,  282-8;  Acta  Univ.  Lundensis,  Lund,  1,  1864  (Math.),  No.  6,  18  pp. 

"Rest-  und  Quotient-Rechnung,  Hamburg,  1869,  97-138. 

"BuU.  Ac.  Sc.  St.  P6tersbourg,  14,  1869,  375-81. 

»«Compte8  Rendua  Paris,  70,  1870,  1243-1251. 

"Monatsber.  Ak.  BerUn,  1870,  881.     Cf.  Traub,  Archiv  Math.  Phys.,  37,  1861,  278-94. 

•KJiomale  di  Mat.,  9,  1871,  125-135. 


Chap.  VII]  PrBIITIVE   RoOTS,   EXPONENTS,   INDICES.  193 

m  modulo  h,  and  if  h^  is  the  highest  power  of  h  which  divides  a"*  — 1,  and  if 
n^/i,  then  6"  divides  a*  — 1  where  e  =  m6"~\  Further,  if  6  is  a  prime,  a 
belongs  to  the  exponent  e  modulo  6".  For  a  new  prime  6',  let  m',  n',  h' 
have  the  corresponding  properties.  Then  the  exponent  to  which  a  belongs 
modulo  B  = })%'"' ...  is  the  1.  c.  m.  L  of  m6''-\  m'b'"'-''', ....  For  a  =  10,  we 
see  that  L  is  the  length  of  the  period  for  the  irreducible  fraction  N/B. 

L.  Sancery^^  proved  that  if  p  is  a  prime  and  a<p  belongs  to  the  exponent 
6  modulo  p,  there  exists  an  infinitude  of  numbers  a-\-px  =  A  such  that  A^—1 
is  divisible  by  p^,  but  not  by  p'''^^,  where  k  is  any  assigned  positive  integer. 
If  A  belongs  to  the  exponent  6  modulo  p>2,  A  will  belong  to  the  exponent 
6  modulo  p"  if  the  highest  power  of  p  which  divides  .A^  — 1  is  ^p";  but  if  it 
be  p"'^,  A  belongs  to  the  exponent  dp^  modulo  p"  [Barillari^°"].  Hence  A 
is  a  primitive  root  of  p"  if  a  primitive  root  of  p  and  if  A^~^  —  1  is  not  divisible 
by  p^,  and  there  are  ^j^CpOj  primitive  roots  of  p"  or  2p\  [Generalization 
of  Arndt.^^j 

C.  A.  Laisant®^  noted  that  if  a  belongs  to  the  exponent  3  modulo  p,  a 
prime,  then  a  + 1  belongs  to  the  exponent  6,  and  conversely.  If  a  belongs  to 
the  exponent  6,  a+1  will  not  belong  to  the  exponent  3  unless  p  =  7,  a  =  3. 
Hence  if  p  is  a  prime  6m +1,  there  are  two  numbers  a,  h  belonging  to  the 
exponent  3,  and  two  numbers  a  +  1,  6+1  belonging  to  the  exponent  6;  also, 
a+6  =  p  — 1.  If  (p.  399)  p+5  is  an  odd  prime  and  p  is  even,  then  pV— 9> 
p^qP  =  p  (mod  p+g). 

G.  Bella vitis^^''  gave,  for  each  power  p'^383  of  a  prime  p,  the  periodic 
fraction  for  1/p'  to  the  base  2  and  showed  how  to  deduce  the  indices  of 
numbers  for  the  modulus  p\  Let  ?  =  p'~^(p  — 1)  and  let  2  belong  to  the 
exponent  q/r  modulo  p\  A  root  b  of  6'"=  2  (mod  p')  is  the  base  of  the 
system  of  indices. 

G.  Frattini^^  proved  by  the  theory  of  roots  of  unity  that,  if  p  is  a  prime, 
the  number  of  interchanges  necessary  to  pass  from  1,  2, .  .  .,  p  — 2  to  ind  2, 
ind  3, . . . ,  ind  (p  —  1)  and  to 

ind  1— ind  2,       ind  2  — ind  3, .  . .,       ind  (p  — 2)— ind  (p  — 1) 

are  both  even  or  both  odd. 

Fritz  Hofmann^^  used  rotations  of  regular  polygons  to  prove  theorems 
on  the  sum  of  the  primitive  roots  of  a  prime  (Gauss^). 

A.  R.  Forsyth^^  found  the  sum  of  the  cth  powers  of  the  primitive  roots 
of  a  prime  p.  The  sum  is  divisible  by  p  if  p  — 1  contains  the  square  of  a 
prime  not  dividing  c  or  if  it  contains  a  prime  dividing  c  but  with  an  exponent 
exceeding  by  at  least  2  its  exponent  in  c.  If  neither  of  these  conditions  is 
satisfied,  the  result  is  not  so  simple. 

"BuU.  Soc.  Math,  de  France,  4,  1875-6,  23-29. 

fi^M^m.  Soc.  So.  Phys.  et  Nat.  de  Bordeaux,  (2),  1,  1876,  400-2. 

"^lAtti  Accad.  Lincei,  Mem.  Sc.  Fis.  Mat.,  (3),  1,  1876-7,  778-800. 

"Giornale  di  Mat.,  18,  1880,  369-76. 

"Math.  Annalen,  20,  1882,  471-86. 

^'Messenger  of  Math.,  13,  1883-4,  180-5. 


194  History  of  the  Theory  of  Numbers.  [Chap,  vii 

J.  Perott^^  gave  a  simple  proof  that  x^  =  l  (mod  p")  has  p''  roots.  Thus 
there  exists  an  integer  b  belonging  to  the  exponent  p""^  modulo  p"*.  Assum- 
ing the  existence  of  a  primitive  root  of  p,  we  employ  a  power  of  it  and  obtain 
a  number  a  belonging  to  the  exponent  p  — 1  modulo  p".  Hence  ab  is  a 
primitive  root  of  p". 

Schwartz^"  stated,  and  Hacken  proved,  the  final  theorem  of  Cauchy.^* 

L.  Gegenbauer^^  stated  19  theorems  of  which  a  specimen  is  the  follow- 
mg:  If  p  =  8a(8/3+l) +  24/3+5  and  (p-l)/4  are  primes  and  if  64a2+48a 
+  10  is  relatively  prime  to  p,  then  8a+3  is  a  primitive  root  of  p. 

G.  Wertheim^^  gave  the  least  primitive  root  of  each  prime  <  1000  and 
companion  tables  of  indices  and  numbers  for  primes  <  100.  He  reproduced 
(pp.  125-130)  arts.  80-81  of  Gauss^  and  stated  the  generalization  by 
Stern.i^ 

H.  Keferstein'''  would  obtain  all  primitive  roots  of  a  prime  p  by  excluding 
all  residues  of  powers  with  exponents  dividing  p  — 1  [Poinsot^]. 

IM.  F.  Daniels"^  gave  a  proof  like  Legendre's^  that  there  are  <f>{n)  num- 
bers belonging  to  the  exponent  n  modulo  p,  a  prime,  if  n  divides  p  — 1. 

*K.  Szily^-  discussed  the  "comparative  number"  of  primitive  roots. 

E.  Lucas"^  gave  the  name  reduced  indicator  of  n  to  Cauchy's^^  maximum 
indicator  of  n,  and  noted  that  it  is  a  divisor  <4){n)  of  0(n)  except  when 
n  =  2,  4,  p*  or  2p^',  where  p  is  an  odd  prime,  and  then  equals  (f>{n).  The 
exponent  to  which  a  belongs  modulo  m  is  called  the  "gaussien"  of  a  modulo 
m  (preface,  xv,  and  p.  440). 

H.  Scheffler"'*  gave,  without  reference,  the  theorem  due  to' Richelot^'^  and 
the  final  one  by  Prouhet.^-  To  test  if  a  proposed  number  a  is  a  primitive 
root  of  a  prime  p,  note  whether  p  is  of  one  of  the  linear  forms  of  primes  for 
which  a  is  a  quadratic  non-residue,  and,  if  so,  raise  a  to  the  powders  whose 
exponents  divide  (p  — 1)/2. 

L.  Contejean^^  noted  that  the  argument  in  Serret's  Algebre,  2,  No.  318, 
leads  to  the  following  result  [for  the  case  a  =  10]:  If  p  is  an  odd  prime  and 
a  belongs  to  the  exponent  e  =  {p  —  l)/q  modulo  p,  it  belongs  to  the  exponent 
p-'^e  modulo  p"  when  (a*— l)/p  is  not  di\dsible  by  p,  but  to  a  smaller 
exponent  if  it  is  divisible  by  p  [Sancery®^]. 

P.  Bachmann^^  proved  the  existence  of  a  primitive  root  of  a  prime  p 
by  use  of  the  group  of  the  residues  1, .  .  . ,  p  —  1  under  multiplication. 

**Bull.  des  Sc.  Math.,  9,  I,  1885,  21-24.     For  k  =  n  —  l  the  theorem  is  contained  imphcitly  in  a 

posthumous  fragment  by  Gauss,  Werke,  2,  266. 
"Mathesis,  6,  1886,  280;  7,  1887,  124-5. 
«8Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  95,  II,  1887,  843-5. 
"Elemente  der  Zahlentheorie,  1887,  116,  375-381. 
'"Mitt.  Math.  Gesell.  Hamburg,  1,  1889,  256. 
^'Lineaire  Congruenties,  Diss.,  Amsterdam,  1890,  92-99. 
"Math,  ^s  termes  ^rtesito  (Memou^  Hungarian  Ac.  Sc),  9,  1891,  264;  10,  1892,  19.     Magyar 

Tudom.  Ak.  Ertesitoje  (Report  of  Hungarian  Ac.  Sc),  2,  1891,  478. 
"Th^orie  des  nombres,  1891,  429. 
'*Beitrage  zur  Zahlentheorie,  1891,  135-143. 
"Bull.  Soc.  Philomathique  de  Paris,  (8),  4,  1891-2,  66-70. 
"Die  Elemente  der  Zahlentheorie,  1892,  89. 


Chap.  VII]  PeIMITIVE   RoOTS,    EXPONENTS,    INDICES.  195 

G.  B.  Mathews'^^  reproduced  art.  81  of  Gauss'''  and  gave  a  second  proof 
by  use  of  Cauchy's^^  congruence  X=0  for  n  =  p  — 1. 

K.  Zsigmondy^^  treated  the  problem  to  find  all  integers  K,  relatively 
prime  to  given  integers  a  and  6,  such  that  a''=b''  (mod  K)  holds  for  the 
given  integral  value  <T  =  y,  but  for  no  smaller  value.  For  6  =  1,  it  is  a 
question  of  the  moduli  K  with  respect  to  which  a  belongs  to  the  exponent  y. 
Set  y=Ilqi*,  where  the  q's  are  distinct  primes  and  qi  the  greatest.  Then 
all  the  primes  K  for  which  a''=b''  (mod  K)  holds  for  (T  =  7,  but  for  no  smaller 
a,  coincide  with  the  prime  factors  of 

y  y 

(a^-6^)n(a««'-6««').  .  . 
A  = 


n(a^/«-6^/«)... 

in  which  the  products  extend  over  the  combinations  of  qi,q2,---  one,  two, . . . 
at  a  time,  provided  that,  if  a''=h''  (mod  qi)  for  (J=^ylqi\  but  for  no  smaller 
(T,  we  do  not  include  among  the  K's  the  prime  q^,  which  then  occurs  in  A 
to  the  first  power  only.  If  the  prime  p  is  a  K  and  if  p^  is  the  highest  power 
of  p  dividing  A,  then  p*  is  the  highest  power  of  p  giving  a  K.  The  com- 
posite i^'s  are  now  easily  found.  If  a  and  6  are  not  both  numerically  equal 
to  unity,  it  is  shown  that  there  is  at  least  one  prime  K  except  in  the  following 
cases:  7  =  1,  a-6  =  l;  7  =  2,  0+6  =  ^2"  (/x^l);  7  =  3,  a  =  ±2,  6==f1; 
7  =  6,  a  =  ='=2,  6  =  ±l.  The  case  h  =  \  shows  that,  apart  from  the  corre- 
sponding exceptions,  there  exists  a  prime  with  respect  to  which  the  given 
integer  aj^^^X  belongs  to  the  given  exponent  7.  As  a  corollary,  every 
arithmetical  progression  of  the  type  mT+I  ()"  =  1?  2, .  .  .)  contains  an  infini- 
tude of  primes. 

Zsigmondy^^  considered  the  function  A^(a)  obtained  from  the  above  A 
by  setting  6  =  1.  If  a  is  a  primitive  root  of  the  prime  p  =  l+7,  the  main 
theorem  of  the  last  paper  shows  that  p  divides  A^(a).  Conversely,  I+7  is 
a  prime  if  it  divides  A.  Thus,  if  all  the  primes  of  a  set  of  integers  possess 
the  same  primitive  root  a,  any  integer  p  of  the  set  is  a  prime  if  and  only  if 
Ap_i(a)  is  divisible  by  p.  Hence  theorems  due  to  Tchebychef^^  imply 
criteria  for  primes.  For  example,  a  prime  2^"+l  has  the  primitive  root  3 
implies  that  2^"+l  is  a  prime  if  and  only  if  it  divides  3^  +  1,  where  k  =  2^'' . 
Since  ±2  is  a  primitive  root  of  any  prime  2q-\rX  such  that  g-  is  a  prime 
4/c±  1,  we  infer  that,  if  g  is  a  prime  4/c±  1,  then  2g+l  is  a  prime  if  and  only 
if  it  divides  (2^±1)/(2±1).  Since  2  is  a  primitive  root  of  a  prime  4A''+1 
such  that  N  is  an  odd  prime,  we  infer  that,  if  N  is  an  odd  prime,  4A^+1  is  a 
prime  if  and  only  if  it  divides  (2^^-|-l)/5. 

G.  F.  Bennett^°  proved  (pp.  196-7)  the  first  theorem  of  Cauchy,^^  and 
(pp.  199-201)  the  results  of  Sancery.^^  If  a  and  a'  belong  to  exponents 
t  and  t'  which  contain  no  prime  factor  raised  to  the  same  power  in  each, 
then  the  exponent  to  which  aa'  belongs  is  the  1.  c.  m.  of  t  and  t'  (p.  194). 

"Theory  of  Numbers,  1892,  23-25. 

"Monatshefte  Math.  Phys.,  3,  1892,  265-284. 

'Hhid.,  4,  1893,  79-80. 

8»Phil.  Trans.  R.  Sec.  London,  184  A,  1893,  189-245. 


196  History  of  the  Theory  of  Numbers.  [Chap,  vii 

If  2*'^^  is  the  highest  power  of  2  dividing  a^  —  1,  where  a  is  odd,  the  exponent 
to  which  a  belongs  modulo  2^  is  2^~'  if  X>s,  but,  if  X^s,  is  1  if  a=l,  2  if 
o^=l,  a^l  (mod  2^^);  the  result  of  Lebesgue'*^  (p.  87)  now  follows  (pp. 
202-6) .  In  case  a  is  not  prime  to  the  modulus,  there  is  an  evident  theorem 
on  the  earliest  power  of  a  congruent  to  a  higher  power  (p.  209).  If  e  is  a 
given  divisor  of  0(w),  there  is  determined  the  number  of  integers  belonging 
to  exponent  e  modulo  m  [cf.  Erlerus^^].  If  a,  a',. . .  belong  to  the  exponents 
t,  t',...  and  if  no  two  of  the  «' .  .  .  numbers  a'a"' .  .  .  {0^r<t,0^r'<t',.  .  .) 
are  congruent  modulo  m,  then  a,  a', .  .  .  are  called  independent  generators 
of  the  4>{m)  integers  <m  and  prime  to  m  (p.  195);  a  particular  set  of 
generators  is  given  and  the  most  general  set  is  investigated  (pp.  220-241) 
[a  special  problem  on  abelian  groups]. 

J.  Perott^^  found  a  number  belonging  to  an  exponent  which  is  the  1.  c.  m. 
of  the  exponents  to  which  given  numbers  belong.  If,  for  a  prime  modulus  p, 
a  belongs  to  an  exponent  t>l,  and  b  to  an  exponent  which  divides  t,  then  b 
is  congruent  to  a  power  of  a  (proof  by  use  of  Newton's  relations  between 
the  sums  of  like  powers  of  a, .  .  . ,  a'  and  their  elementary  symmetric  func- 
tions).    Hence  there  exists  a  primitive  root  of  p. 

M.  Frolo .  ^-  noted  that  all  the  quadratic  non-residues  of  a  prime  modulus 
m  are  primitive  roots  of  m  if  m  =  2^''4-l,  m  =  2n+l  or  4n  +  l  with  n  an  odd 
prime  [Tchebychef^^].  To  find  primitive  roots  of  m  "without  any  trial," 
separate  the  m  —  1  integers  <m  into  sets  of  fours  a,  b,  —a,  —b,  where 
a6=l  (mod  m):  Begin  with  one  such  set,  say  1,  1,  —1,  —1.  Either  a  or 
m  — a  is  even;  divide  the  even  one  by  2  and  multiply  the  corresponding 
=t  6  by  2 ;  we  get  another  set  of  four.  Repeat  the  process.  If  the  resulting 
series  of  sets  contains  all  m  —  1  integers  <m,  2  and  —2  are  primitive  roots 
if  w  =  4/i+l,  and  one  of  them  is  a  primitive  root  if  m  =  4/i  — 1.  If  the  sets 
just  obtained  do  not  include  all  m  —  1  integers  <m,  further  theorems  are 
proved. 

G.  Wertheim^^  gave  the  least  primitive  root  of  each  prime  p  <3000. 

L.  Gegenbauer^^"  gave  two  expressions  for  the  sum  Sk  of  those  terms  of  a 
complete  set  of  residues  modulo  p  which  belong  to  the  exponent  k,  and 
evaluated  l>Sk/t  fit)  with  t  ranging  over  the  divisors  of  k. 

G.  Wertheim^^  proved  that  any  prime  2'*"  +  l  has  the  primitive  root  7. 
If  p  =  2"g-|-l  is  a  prime  and  ^  is  a  prime  >2,  any  quadratic  non-residue  m 
of  p  is  a  primitive  root  of  p  if  m""  — 1  is  not  divisible  by  p.  As  corollaries, 
we  get  primes  q  of  certain  linear  forms  for  which  2,  5,  7  are  primitive  roots 
of  a  prime  2^-1-1  or  4g-f-l;  also,  3  is  a  primitive  root  of  all  primes  8g-|-l 
or  16g+l  except  41;  and  cases  when  5  or  7  is  a  primitive  root  of  primes 
8^+1,  lQq+1.  There  is  given  a  table  showing  the  least  primitive  root  of 
each  prime  between  3000  and  3500. 

"BuU.  des  Sc.  Math.,  (2),  17,  I,  1893,  66-83. 

""BuU.  Soc.  Math,  de  France,  21,  1893,  113-128;  22,  1894,  241-5. 

"Acta  Mathematica,  17,  1893,  315-20;  correction,  22,  1899,  200  (10  for  p  =  1021). 

8"Denkschr.  Ak.  Wiss.  Wien  (Math.),  60,  1893,  48-60. 

"Zeitschrift  Math.  Naturw.  Unterricht,  25,  1894,  81-97. 


Chap.  VII]  PRIMITIVE  RoOTS,   EXPONENTS,   INDICES.  197 

J.  Perott^^  employed  the  sum  Sk  of  the  ^th  powers  of  1,  2, . . .,  p  — 1,  and 
gave  a  new  proof  that  Si=0, . . .,  Sp_2=0,  Sp_i=  —1  (mod  p).  If  m  is  the 
1.  c.  m.  of  the  exponents  to  which  1,2,. .  .,  p  —  1  belong,  evidently  Sm=p  —  1, 
whence  m>p  —  2.  If  A  belongs  to  the  exponent  m,  then  A,  A^, . . .,  A""  are 
incongruent,  whence  mSp  —  1-     Thus  A  is  a  primitive  root. 

N.  Amici^^  proved  that,  if  j'>2,  a  number  belongs  to  the  exponent  2""^ 
modulo  2"  if  and  only  if  it  is  of  the  form  8/i±3,  and  called  such  numbers 
quasi  primitive  roots  of  2\  For  a  base  Sh=^S,  numbers  of  the  two  forms 
8A:+1  or  8^=*=3,  and  no  others,  have  indices.  The  product  of  two  numbers 
having  indices  has  an  index  which  is  congruent  modulo  2""^  to  the  sum  of 
the  indices  of  the  factors.  The  product  of  two  numbers  6i  and  62?  neither 
with  an  index,  has  an  index  congruent  modulo  2""^  to  the  sum  of  the  indices 
of  —61  and  —62-  The  product  of  a  number  with  an  index  by  one  without 
an  index  has  no  index. 

K.  Zsigmondy^^  proved  by  use  of  abelian  groups  that,  if  8  =  qi^\  .  .Qr'"', 
m  =  pi'^K  .  .ps^s,  where  Qi,.  .  .,  Qr  are  distinct  primes,  and  Pi,...,  Ps  are  dis- 
tinct primes,  the  number  of  incongruent  integers  belonging  to  the  exponent 
5  modulo  m  is 

5i...5,n(l-l/g/0, 
1=1 

where  d,  is  the  g.  c.  d.  of  5  and  tj=(f>{pp),  while  li  is  the  number  of  the 
integers  ti,...,ts  which  contain  the  factor  ql'K 

E.  de  Jonquieres^^  proved  that  the  product  of  an  even  number  of  primi- 
tive roots  of  a  prime  p  is  never  a  primitive  root,  while  the  product  of  an 
odd  number  of  them  is  either  a  primitive  root  or  belongs  to  an  exponent  not 
dividing  {p  —  l)/2.  Similar  results  hold  for  products  of  numbers  belonging 
to  like  exponents.  Certain  of  the  n  integers  r,  for  which  f  is  a  given  num- 
ber belonging  to  the  exponent  e  =  {p  —  \)/n,  belong  to  the  exponent  ne, 
while  the  others  (if  any  are  left)  belong  to  an  exponent  ke,  where  k  divides  n. 
He  conjectured  that  2  is  not  a  primitive  root  of  a  prime  p=l,  7,  17  or  23 
(mod  24);  3  not  of  p=l,  11,  13  or  23  (mod  24);  5  not  of  p=\,  11,  19,  or 
29  (mod  30).  These  results  and  analogous  ones  for  7  and  11  were  shown 
by  him  and  T.  Pepin^^  to  follow  from  the  quadratic  reciprocity  law  and 
Gauss'  theorems  on  the  divisors  oi  x^^A. 

G.  Wertheim^°.  added  to  his^*  corollaries  cases  when  6,  10,  11,  13  are 
primitive  roots  of  primes  2^+1,  4^+1;  also,  10  is  a  primitive  root  of  all 
primes  8g+l?^137  for  which  g-  is  a  prime  10A;+7  or  lOyc+9,  and  of  primes 
IGg+l  for  which  g  is  a  prime  10/c+l  or  lO/c+7. 

Wertheim^^  gave  the  least  primitive  root  of  each  prime  between  3000  and 
5000  and  of  certain  higher  primes.     He  noted  errata  in  his^^  table  to  3000. 

85BuU.  des  Sc.  Math6matiques,  18,  I,  1894,  64-66. 

8«Rendiconti  Circolo  Mat.  di  Palermo,  8,  1894,  187-201. 

"Monatshefte  Math.  Phys.,  7,  1896,  271-2. 

88Compte8  Rendus  Paris,  122,  1896,  p.  1451,  p.  1513;  124,  1897,  p.  334,  p.  428. 

8»Comptes  Rendus  Paris,  123,  1896,  pp.  374,  405,  683,  737. 

'"Acta  Math.,  20,  1896,  143-152. 

"/bid.,  153-7;  corrections,  22,  1899,  200. 


198  History  of  the  Theory  of  Numbers.  [Chap,  vn 

F.  Mertens^-  called  I'l,.  .  .,  ip  the  system  of  indices  of  n  modulo  k  if 
n=gi\  .  .qj"  (mod  A:)  for  the  g's  of  Kronecker.^°  Such  systems  of  indices 
differ  from  Dirichlet's. 

C.  Moreau^^  set  A^  =  pV . . . ,  v  =  p''~^q^~^ . . . ,  where  p,  q,...  are  distinct 
primes.  Take  €  =  1  if  iV"  is  not  divisible  by  4  or  if  N  =  4,  but  e  =  2  if  iV  is 
divisible  by  4  and  A'' >  4.  Let  \p{N)  denote  the  1.  c.  m.  of  v/e,  p  —  l,q  —  l,.  .  . 
[equivalent  to  Cauehy's-^  maximum  indicator  for  modulus  N].  For  A 
prime  to  N,  A*^^^=  1  (mod  N) .  If  A^  =  p'',  2p^  or  4  (so  that  N  has  primitive 
roots),  yp{N)  =4>{N)  [Lucas^^j^  ^j^^^.^  -^  ^  ^^^^^  ^^  values  of  A^<  1000  and 
certain  higher  values  for  which  \p{N)  has  a  given  value  <  100. 

A.  Cunningham^"*  noted  that  we  may  often  abbre\iate  Gauss'  method 
of  finding  a  primitive  root  of  a  prime  p  by  testing  whether  or  not  the  trial 
root  a  is  a  primitive  root  before  computing  the  residues  of  all  powers  of  a. 
The  tests  are  the  simple  rules  to  decide  whether  or  not  a  is  a  quadratic  or 
cubic  residue  of  p.  If  a  is  both  a  quadratic  non-residue  and  a  cubic  non- 
residue  of  p  =  3co+l,  and  if  a^^l  for  every/  dividing  p  —  1  except /=p  —  l, 
then  a  is  a  primitive  root. 

A.  Cunningham^^  gave  tables  showing  the  residues  of  the  successive 
powers  of  2  when  divided  by  each  prime  or  power  of  prime  <  1000,  also 
companion  tables  showing  the  indices  x  of  2""  whose  residues  modulo  p''  are 
1,  2,  3, .  .  ..  The  tables  are  more  convenient  than  Jacobi's  Canon-^  (errata 
given  here)  for  the  problem  to  find  the  residue  of  a  given  number  with 
respect  to  a  given  power  of  a  prime,  but  less  convenient  for  finding  all  roots 
of  a  given  order  of  a  given  prime.  There  are  given  (p.  172)  for  each  power 
p^<  1000  of  a  prime  p  the  factors  of  0(p^"),  the  exponent  ^  to  which  2  belongs 
modulo  p'',  and  the  quotient  0/^. 

E.  Cahen^^  proved  that  if  p  is  a  prime  >(32"'^'-l)/2'"+^  and  if  5  = 
2^+'^p-\-l    (7«>0)  is  a  prime,  then  3  is  a  primitive  root  of  q,  whereas 
Tchebychef^^  had  the  less  advantageous  condition  p>3^^V2'"+^.    Other 
related  theorems  by  Tchebychef  are  proved.     There  are  companion  tables 
of  indices  for  primes  <  200. 

G.  A.  Miller^^  appUed  the  theory  of  groups  to  prove  the  existence  of 
primitive  roots  of  p",  to  show  that  the  primitive  roots  of  p^  are  primitive 
roots  of  p",  and  to  determine  primitive  roots  of  the  prime  p. 

L.  Kronecker^^  discussed  the  existence  of  primitive  roots,  defined  sys- 
tems of  indices  and  appHed  them  to  the  decomposition  of  fractions  into 
partial  fractions.  He  developed  (pp.  375-388)  the  theor>^  of  exponents  to 
which  numbers  belong  modulo  p,  a  prime,  by  use  of  the  primitive  factor 


"Sitzungsber.  Ak.  Wien  (Math.),  106,  II  a,  1897,  259. 

«Nouv.  Ann.  Math.,  (3),  17,  1898,  303. 

"Math.  Quest.  Educat.  Times,  73,  1900,  45,  47. 

•'A  Binary  Canon,  showing  residues  of  powers  of  2  for  divisors  under  1000,  and  indices  to 

residues,  London,  1900,  172  pp.     Manuscript  was  described  by  author.  Report  British 

Assoc,  i895,  613.     Errata,  Cunningham.'" 
"filaments  de  la  th^orie  des  nombres,  1900,  335-9,  375-390. 
•'BuU.  Amer.  Math.  Soc,  7,  1901,  350. 
•'Vorlesungen  liber  Zahlentheorie,  I,  1901,  416-428. 


Chap.  VII]  PRIMITIVE   RoOTS,   EXPONENTS,    INDICES.  199 

Fd{x)  of  a;'*— 1  (dividing  the  last  but  not  x'— 1  iorKd).  To  every  divisor 
d  of  p  — 1  belong  exactly  4>{d)  numbers  which  are  the  roots  of  i^d(^)=0 
(modp). 

P.  G.  Foglini^^  gave  an  exposition  of  known  results  on  primitive  roots, 
indices,  linear  congruences,  etc.  In  applying  (p.  322)  Poinsot's^  method  of 
finding  the  primitive  roots  of  a  prime  p  to  the  case  p  =  13,  it  suffices  to  exclude 
the  residues  of  the  cubes  of  the  numbers  which  remain  after  excluding  the 
residues  of  squares;  for,  if  a;  is  a  residue  of  a  square,  (x^)®=l  and  x^  is  the 
residue  of  a  square. 

R.  W.  D.  Christie^""  noted  that,  if  7  is  a  primitive  root  of  a  prime 
p  =  4A;  —  l,the  remaining  primitive  roots  are  congruent  to  p  —  7    (n  =  1, 2, . . . ) 

A.  Cunningham^°^  noted  that  3,  5,  6,  7,  10  and  12  are  primitive  roots  of 
any  prime  /^,  =  22"+l>5.     Also  7^/^^+1  =  0  (mod  F^+,  >5). 

E.  I.  Grigoriev^"^  noted  that  a  primitive  root  of  a  prime  p  can  not  equal 
a  product  of  an  even  number  of  primitive  roots  [evident]. 

G.  Wertheim^°^  treated  the  problem  to  find  the  numbers  belonging  to 
the  exponent  equal  to  the  1.  c.  m.  of  m,  n,  given  the  numbers  belonging  to 
the  exponents  m  and  n,  and  proved  the  first  theorem  of  Stern. ^^  He  dis- 
cussed (pp.  251-3)  the  relation  between  indices  to  two  bases  and  proved 
(pp.  258,  402-3)  that  the  sum  of  the  indices  of  a  number  for  the  various 
primitive  roots  of  w  =  p"  or  2p"  equals  ^4){m)4>  ]0(w)  \  ■  If  «  belongs  to  the 
exponent  45  modulo  p,  the  same  is  true  of  p  — a  (p.  266).  He  gave  a  table 
showing  the  least  primitive  root  of  each  prime  <  6200  and  for  certain  larger 
primes;  also  tables  of  indices  for  primes  <  100. 

P.  Bachmann^°^  gave  a  generalization  (corrected  on  p.  402)  of  Stern's^^ 
first  theorem. 

G.  Arnoux^°^  constructed  tables  of  residues  of  powers  and  tables  of 
indices  for  low  composite  moduli. 

A.  Bindoni^°^  noted  that  a  table  showing  the  exponent  to  which  a  belongs 
modulo  p,  a  prime,  can  be  extended  to  a  table  modulo  N  by  means  of  the 
following  theorems.  Let  a,  61,...,  &„  be  relatively  prime  by  twos.  A 
number  belonging  to  the  exponent  ti  modulo  bi  belongs  modulo  6162 ■  ■  -K 
to  the  1.  c.  m.  of  ^1, .  .  . ,  ^^  as  exponent.  If  ti  is  the  least  exponent  for  which 
a'''+l=0  (mod  bi)  and  if  the  ti  are  all  odd,  the  least  t  for  which  a'+l  is 
divisible  by  6], ... ,  6„  is  the  1.  c.  m.  of  ^i, .  .  . ,  i„.  If  p  is  an  odd  prime  not 
dividing  a  and  if  a  belongs  to  the  exponent  t  modulo  p,  and  a'  =  pg+l,  and 
if  p"  is  the  highest  power  of  p  dividing  q,  then  a  belongs  to  the  exponent 
lpn~i-u  jjiojuio  p".     Hence  if  a  is  a  primitive  root  of  p,  it  is  one  of  p"  if 

s'Memorie  Pont.  Ac.  Nuovi  Lincei,  18,  1901,  261-348. 
^""Math.  Quest.  Educat.  Times,  1,  1902,  90. 
"i/6id.,  pp.  108,  116. 

"^Kazani  Izv.  fiz.  mat.  obsc,  BuU.  Phys.  Math.  Soc.  Kasan,  (2),  12,  1902,  No.  1,  7-10. 
"'Anfangsgrunde  der  Zahlenlehre,  1902,  236-7,  259-262. 
ii^Niedere  Zahlentheorie,  1,  1902,  333-6. 
"'Assoc.  fran§.  av.  sc,  32,  1903,  II,  65-114. 
"«I1  Boll,  di  Matematica  Giorn.  Sc.  Didat.,  Bologna,  4,  1905,  88-92. 


200  History  of  the  Theory  of  Numbers.  [Chap,  vii 

and  only  if  a""^  —  1  is  not  divisible  by  p^.     If  t  is  even,  the  least  x  for  which 
a"+l  =  0  (mod  p")  is  l^p""'"". 

]\I.  Cipolla^°^gave  a  historical  report  on  congruences  (especially  binomial), 
primitive  roots,  exponents,  indices  (in  Peano's  symboUsm). 

K.  P.  Nordlund^°^  proved  by  use  of  Fermat's  theorem  that,  if  rij, . . .,  n,. 
are  distinct  odd  primes,  no  one  dividing  a,  then  A^"  =  ni"*' . . .  n^*"'  divides 
a'-l,  where  A;=0(iV)/2^-^ 

R.  D.  Carmichael^°^  proved  that  the  maximum  indicator  of  any  odd 
number  is  even;  that  of  a  number,  whose  least  prime  factor  is  of  the  form 
4ZH-1,  is  a  multiple  of  4;  that  of  p(2p  — 1)  is  a  multiple  of  4  if  p  and  2p  — 1 
are  odd  primes. 

A.  Cunningham^^°  gave  a  table  of  the  values  of  v,  where  {p  —  l)/v  is  the 
exponent  to  which  2  belongs  modulo  p"<  10000,  the  omitted  values  of  p 
being  those  for  which  i'  =  1  or  2  and  hence  are  immediately  distinguished 
by  the  quadratic  character  of  2  (extension  of  his  Binary  Canon^^).  A  list 
is  given  of  errata  in  the  table  by  Reuschle.^^  An  announcement  is  made  of 
the  manuscript  of  tables  of  the  exponents  to  which  3,  5,  6,  7,  10,  11,  12 
belong  modulo  p"<  10000,  and  the  least  positive  and  negative  primitive 
roots  of  each  prime  <  10000  [now  in  type  and  extended  in  manuscript  to 
p"<  22000]. 

A.  Cunningham^  ^^  defined  the  sub-Haupt-exponent  ^i  of  a  base  q  to 
modulus  m  =  q°-°y]Q  (where  770  is  prime  to  q,  and  ao^O)  to  be  the  exponent  to 
which  q  belongs  modulo  r^o-  Similarly,  let  ^2  be  the  exponent  to  which  q 
belongs  modulo  771,  where  ^i  =  5'''i?i;  etc.  Then  the  ^'s  are  the  successive 
sub-Haupt-exponents,  and  the  train  ends  with  ^,.+1  =  1,  corresponding  to 
77;.  =  1 .  His  table  I  gives  these  ^k  for  bases  g  =  2,  3,  5  and  for  various  moduli 
including  the  primes  <  100. 

Paul  Epstein^ ^^  desired  a  function  ^{m),  called  the  Haupt-exponent  for 
modulus  m,  such  that  a'''^'"^  =  1  (mod  m)  for  every  integer  a  prime  to  m  and 
such  that  this  will  not  hold  for  an  exponent  <\p{m).  Thus  \f/{m)  is  merely 
Cauchy's^^  maximum  indicator.  Although  reference  is  made  to  Lucas, ^^ 
who  gave  the  correct  value  of  4^(ni),  Epstein's  formula  requires  modification 
when  m  =  4  or  8  since  it  then  gives  \p  =  l,  whereas  \p  =  2.  The  number 
x(w,  m)  of  roots  of  x''=  1  (mod  m)  is  2dodi .  .  .d„  if  m  is  divisible  by  4  and  if 
H  is  odd,  but  is  di . .  . ci„  in  the  remaining  cases,  where,  for  m  =  2'*°pi*i . .  .pn'"'*, 
di  is  the  g.  c.  d.  of  jj,  and  4>{pi°-^),  and  do  the  g.  c.  d.  of  fx  and  2°""^,  when 
ao>l.  The  number  of  integers  belonging  to  the  exponent  /x  =  pV-- 
modulo  m  is 
\x{m,  p°)-x(m,  p°-^)[  \x{m,  q^)-x{m,  (f-^)\.  .  .. 

"^Revue  de  Math.  (Peano),  Turin,  8,  1905,  89-117. 

"8G6teborgs  Kungl.  Vetenskaps-Handlingar,  (4),  7-8,  1905,  12-14. 

"»Amer.  Math.  Monthly,  13,  1906,  110. 

"OQuar.  Jour.  Math.,  37,  1906,  122-145.  Manuscript  announced  in  Mess.  Math.,  33,  1903-4, 
145-155  (with  list  of  errata  in  earUer  tables);  British  Assoc.  Report,  1904,  443;  I'inter- 
m^diaire  des  math.,  16,  1909,  240;  17,  1910,  71.     CI.  Cunningham."^ 

"iProc.  London  Math.  Soc,  5,  1907,  237-274. 

"^Archiv  Math.  Phys.,  (3),  12,  1907,  134-150. 


Chap.  VII]  PRIMITIVE  RoOTS,  EXPONENTS,  INDICES.  201 

This  formula  is  simplified  in  the  case  tx  =  \l/{m)  and  the  numbers  belonging 
to  this  Haupt-exponent  are  called  primitive  roots  of  m.  The  primitive 
roots  of  m  divide  into  families  of  0(i/'(m))  each,  such  that  any  two  of  one 
family  are  powers  of  each  other  modulo  m,  while  no  two  of  different  families 
are  powers  of  each  other.  Each  family  is  subdivided.  In  general,  not 
every  integer  prime  to  m  occurs  among  the  residues  modulo  m  of  the  powers 
of  the  various  primitive  roots  of  m. 

A.  Cunningham"^  considered  the  exponent  ^  to  which  an  odd  number  q 
belongs  modulo  2"*;  and  gave  the  values  of  ^  when  m^  3,  and  when  q  =  2^12='=  1 
(fi  odd),  m>3.  When  g'  =  2''=Fl  and  m>x-\-l,  the  residue  of  q^^^^  can 
usually  be  expressed  in  one  of  the  forms  1=f2",  1=f2"=f2^. 

G.  Fontene"^  determined  the  numbers  N  which  belong  to  a  given 
exponent  p"'~''8  modulo  p"",  where  5  is  a  given  divisor  of  p  —  l,  and  h^l, 
without  employing  a  primitive  root  of  p"".  li  p>2,  the  conditions  are  that 
N  shall  belong  to  the  exponent  8  modulo  p  and  that  the  highest  power  of 
p  dividing  N^  —  1  shall  he  p^,  l^h^m. 

*M.  Demeczky"^  discussed  primitive  roots. 

E.  Landau"^  proved  the  existence  of  primitive  roots  of  powers  of  odd 
primes,  discussed  systems  of  indices  for  any  modulus  n,  and  treated  the 
characters  of  n. 

G.  A.  Miller"^  noted  that  the  determination  of  primitive  roots  of  g 
corresponds  to  the  problem  of  finding  operators  of  highest  order  in  the 
cyclic  group  G  of  order  g.  By  use  of  the  group  of  isomorphisms  of  G  it  is 
shown  that  the  primitive  roots  of  g  which  belong  to  an  exponent  2q,  where 
q  is  an  odd  prime,  are  given  by  —a",  when  a  ranges  over  those  integers 
between  1  and  g/2  which  are  prime  to  g.  As  a  corollary,  the  primitive 
roots  of  a  prime  2g+l,  where  q  is  an  odd  prime,  are  —  a^,  l<a<g+l. 

A.  N.  Korkine"^  gave  a  table  showing  for  each  prime  p<4000  a  primitive 
root  g  and  certain  characters  which  serve  to  solve  any  solvable  congruence 
x^=a  (mod  p),  where  g  is  a  prime  dividing  p  —  l.  Let  q"  be  the  highest 
power  of  q  dividing  p  —  1.     The  characters  of  degree  q  are  the  solutions  of 

M«  =  l,  u''  =  u,  u"'  =  u',...,  (w^"-")'  =  w(''-2)  (mod  p) 

and  hence  are  the  residues  of  the  powers  of  g^p~'^^^^  for  k  =  l,. . .,  a.  There 
are  noted  some  errors  in  the  Canon  of  Jacobi^^  and  the  table  of  Burckhardt. 
Korkine  stated  that  if  p  is  a  prime  and  a  belongs  to  the  exponent  e  =  {p  — 1)/5, 
exactly  (f){p  —  l)/<l>{e)  of  the  roots  of  a;*  =  a  (mod  p)  are  primitive  roots  of  p. 
K.  A.  Posse"^  remarked  that  Korkine  constructed  his  table  without 
knowing  of  the  table  by  Wertheim,^^and  extended  Korkine's  tables  to  10000. 

"^Messenger  of  Math.,  37,  1907-8,  162-4. 

"*Nouv.  Ann.  Math.,  (4),  8,  1908,  193-216. 

"^Math  6s  Phys.  Lapok,  Budapest,  17,  1908,  79-86. 

ii«Handbuch  .  .  .Verteilung  der  Primzahlen,  I,  1909,  391-414,  478-486. 

ii'Amer.  Jour.  Math.,  31,  1909,  42-4. 

iisMatem.  Shorn.  Moskva  (Math.  Soc.  Moscow),  27,  1909,  28-115,  120-137  (in  Russian).     Cf. 

D.  A.  Grave,  29,  1913,  7-11.     The  table  was  reprinted  by  Posse."* 
iio/bid.,  116-120,  175-9,  238-257.    Reprinted  by  Posse.»" 


202  History  of  the  Theory  of  Numbers.  [Chap,  vii 

R.  D.  CarmichaeP^®  called  a  number  a  primitive  X-root  modulo  n  if  it 
belongs  to  the  exponent  X(?i),  defined  in  Ch.  Ill,  Lucas. ""^  The  existence 
of  primitive  X-roots  g  is  proved.  The  product  of  those  powers  of  g  which 
are  prhnitive  X-roots  is  =  1  (mod  n)  if  X(n)  >2.  A  method  is  given  to  solve 
X(x)  =a,  and  the  solutions  tabulated  for  a ^24. 

C.  Posse^-^  noted  that  in  Wertheim's^'^^  table,  the  primitive  root  14 
of  2161  should  be  replaced  by  23,  while  10  is  not  a  primitive  root  of  3851. 

E.  Maillet^^^  described  the  manuscript  table  by  Chabanel,  deposited  in 
the  library  of  the  University  of  Paris,  giving  the  indices  for  primes  under 
10000  and  data  to  determine  the  number  having  a  given  index. 

F.  Schuh^^^  showed  how  to  form  the  congruence  for  the  primitive  roots 
of  a  prime  and  gave  two  further  proofs  of  the  existence  of  primitive  roots. 
He  treated  binomial  congruences,  quadratic  residues  and  made  applica- 
tions to  periodic  fractions  to  any  base.  For  any  modulus  n,  he  found  the 
least  m  for  which  x"'  =  1  (mod  n)  holds  for  every  x  prime  to  n,  and  derived 
the  solutions  ?i  of  4>{n)  =m,  i.  e.,  n's  having  primitive  roots. 

F.  Schuh^^^  discussed  the  solution  of  a;'  =  1  (mod  p")  with  the  least  com- 
putation. If  X  belongs  to  the  exponent  q  modulo  n,  the  powers  of  x  give 
a  cycle  of  0(g)  numbers  each  with  the  "period"  q.  The  numbers  prime  to 
n  and  having  the  period  q  may  form  several  such  cycles — more  than  one  if 
n  has  no  primitive  root  and  q  is  the  maximum  period.  If  n  =  2"  (a>2),  then 
g  =  2*  (s^a  — 2)  and  the  number  of  cycles  is  1,  3  or  2  according  as  s  =  0,  s  =  1 
or  s>l.     In  the  last  case,  the  cj^cles  are  formed  by  2''~^(2fc+l)  =f1. 

When  q  is  even,  x  is  said  to  be  of  the  first  or  second  kind  according  as 
x'''^=  —  1  (mod  n)  or  not.  If  the  numbers  of  a  cycle  are  of  the  second  kind, 
we  get  a  new  cycle  of  the  second  kind  by  changing  the  signs  of  the  numbers 
of  the  first  cycle.  While  for  moduli  n  having  primitive  roots  there  exist  no 
numbers  of  the  second  kind,  when  n  has  no  primitive  roots  and  g  is  a  possible 
even  period,  there  exist  at  least  two  cycles  of  the  second  kind  and  of  period 
q.  Finally,  there  is  given  a  table  showing  the  number  of  cycles  of  each 
kind  for  moduli  ^  150. 

M.  Kraitchik^^^  gave  a  table  showing  for  each  prime  p<  10000  a  primi- 
tive root  of  p  and  the  least  solutions  of  2""=!,  10"=  1  (mod  p). 

*J.  Schumacher^^*^  discussed  indices. 

L.  von  Schrutka^^^  noted  that,  if  g,  r, . .  .  are  the  distinct  primes  dividing 
p  —  l,  where  p  is  a  prime,  all  non-primitive  roots  of  p  satisfy 


(a;V-l)(xV_i)  .  .  .=0  (mod  p). 


""Bull.  Amer.  Math.  Soc,  16,  1909-10,  232-7.    Also,  Theory  of  Numbers,  pp.  71-4. 

i^iActa  Math.,  33,  1910,  405-6. 

i=»L'interm6diaire  des  math.,  17,  1910,  19-20. 

i23Supplement  de  Vriend  derWiskunde,  Culemborg,  22, 1910,  34-114,  166-199,  252-9;  25,  1913, 

33-59,  143-159,  228-259. 
"*Ibid.,  23,  1911,  39-70,  130-159,  230-247. 
'"Sphinx-Oedipe,  May,  1911,  Num^ro  Special,  pp.  1-10;  errata  listed  p.  122  by  Cunningham  and 

Woodall.     Extension  to  25000,  1912,  25-9,  39-42,  52-5;  errata,  93-4,  by  Cunningham. 
"'Blatter  Gymnasiaj-Schulwesen,  Miinchen,  47,  1911,  217-9, 
"^Monatshefte  Math.  Phys.,  22,  1911,  177-186. 


Chap.  VII]  PRIMITIVE    RoOTS,    EXPONENTS,    INDICES.  203 

To  this  congruence  he  appHed  Hurwitz's^^  method  (Ch.  VIII)  of  finding  the 
number  of  roots  and  concluded  that  there  are  p  —  l—(f>(p  —  l)  roots. 
Hence  there  exist  0(p  — 1)  primitive  roots  of  p. 

A.  Cunningham  and  H.  J.  WoodalP^^  continued  to  p<  100000  the  table 
of  Cunningham""  of  the  maximum  residue  indices  j^  of  2  modulo  p. 

C.  Posse^^^  reproduced  Korkine's"^  and  his  own"^  tables  and  explained 
their  use  in  the  solution  of  binomial  congruences. 

C.  Krediet^^o  treated  x*'=l  (mod  n)  of  Lucas/^"  Ch.  Ill,  and  called  x 
a  primitive  root  if  it  belongs  to  the  exponent  cp.  The  powers  of  such  a 
root  are  placed  at  equal  intervals  on  a  circle  for  various  n's. 

G.  A.  Miller^^^  proved  by  use  of  group  theory  that,  if  m  is  arbitrary, 
the  sum  of  those  integers  <  m  and  prime  to  m  which  belong  to  an  exponent 
divisible  by  4  is  =  0  (mod  m) ,  and  the  sum  of  those  belonging  to  the  expo- 
nent 2  is  =  —  1  (mod  m),  and  proved  the  corresponding  theorem  by  Stern^^ 
for  a  prime  modulus. 

A.  Cunningham^^^  tabulated  the  number  of  primes  p<10^  for  which 
y  belongs  to  the  same  exponent  modulo  p,  for  y  =  2,  3,  5,  6,  7,  10,  11,  12; 
and  the  number  of  primes  p  in  each  10000  to  10^  for  which  y  (2/  =  2  or  10) 
belongs  to  the  same  exponent  modulo  p.  Also,  for  the  same  ranges  on 
p  and  y,  the  number  of  primes  p  for  which  y''^  1  (mod  p)  is  solvable,  where 
A;  is  a  given  divisor  of  p  —  1 . 

A.  Cunningham^^^  stated  that  he  had  finished  the  manuscript  of  a  table 
of  Haupt-exponents  to  bases  3,  5,  6,  7,  11,  12  for  all  prime  powers  <  15000; 
also  canons  giving  at  sight  the  residues  of  z"  modulo  p'''<  10000  for  z  =  2, 
r^l00;2  =  3,  5,  7,  10,  11,  r^30. 

J.  Barinaga^^^  considered  a  number  a  belonging  to  the  exponent  g 
modulo  p,  a  prime.  If  a  is  not  divisible  by  g,  the  sum  of  the  ath  powers 
of  the  numbers  forming  the  period  of  a  modulo  p  is  divisible  by  p.  The 
sum  of  their  products  n  at  a  time  is  congruent  to  zero  modulo  p  ii  n<g, 
but  to  =^"1  ii  n  =  g,  according  as  g  is  even  or  odd. 

A.  Cunningham^^^  listed  errata  in  his  Binary  Canon^^  and  Jacobi's  Canon. ^^ 

G.  A.  Miller^^^  employed  the  group  formed  by  the  integers  <m  and 
prime  to  m,  combined  by  multiplication  modulo  m,  to  show  that,  if  a 
number  is  =  ±  1  (mod  2"^),  but  not  modulo  2^+\  where  l<7</3,  it  belongs 
to  the  exponent  2^~^  modulo  2^.  Also,  if  p  is  an  odd  prime,  and  A^=  1 
(mod  p),  N  belongs  to  the  exponent  p^~^  modulo  p^  if  and  only  if  A^"  — 1  is 
divisible  by  p^,  but  not  by  p^+\  where  /3>7^  1. 

»8Quar.  Jour.  Math.,  42,  1911,  241-250;  44,  1913,  41-48,  237-242;  45,  1914,  114-125. 

i^Acta  Math.,  35,  1912,  193-231,  233-252. 

""Wiskundig  Tijdskrift,  Haarlem,  8,  1912,  177-188;  9,  1912,  14-38;  10,  1913,  40-46,  87-97. 

(Dutch.) 
"lAmer.  Math.  Monthly,  19,  1912,  41-6. 
"2Proc.  London  Math.  Soc,  (2),  13,  1914,  258-272. 
"'Messenger  Math.,  45,  1915,  69.     Cf.  Cunningham."" 
"<Annaes  Sc.  Acad.  Polyt.  do  Porto,  10,  1915,  74-6. 
"^Messenger  Math.,  46,  1916,  57-9,  67-8. 
"s/Wd.,  101-3. 


204  History  of  the  Theory  of  Numbers.  [Chap,  vil 

A.  Cunningham^^^  gave  five  primes  p  for  which  there  is  a  maximum 
number  of  exponents  to  which  the  various  numbers  belong  modulo  p. 

On  exponents  and  indices,  see  Lebesgue^"'*^  and  Bouniakowsky^^^;  also 
Reuschle^^  of  Ch.  YI,  Bouniakowsky"^  of  Ch.  XIV,  and  Calvitti^^  of  Ch.  XX. 

Binomial  Congruences. 

Bhdscara  Achd,rya^*^  (1150  A.  D.)  found  y  such  that  y^  —  SO  is  di\'isible 
by  7  by  solving  ?/"  =  7c+30.  Changing  30  by  multiples  of  7,  we  reach  a 
perfect  square  16  with  the  root  4.     Hence  set 

7c+30  =  (7n+4)2,  c  =  7n'+8n-2,  y  =  7n-\-4. 

Taking  n  =  1,  we  get  y  =  ll.  Such  a  problem  is  impossible  if,  after  abrading 
the  absolute  term  (30  above)  by  the  divisor  (7  above)  and  the  addition  of 
multiples  of  the  divisor,  we  do  not  reach  a  square. 

Similarly  for  the  case  of  a  cube,  with  corresponding  conditions  for  impos- 
sibihty  (§206,  p.  265).  For  y^  =  5e+Q,  abrade  6  by  the  divisor  5  to  get 
the  cube  1;  adding  43-5,  we  get  216  =  6^.     Hence  set  y  =  5n-\-Q. 

An  anonymous  Japanese  manuscript^^°  of  the  first  part  of  the  eighteenth 
century  gave  a  solution  of  x^  —  ky  =  a  by  trial.  The  residues  Oi, . .  .,  ak-\  of 
1", . . .,  (^  — 1)"  modulo  k  are  formed;  if  a^^a,  then  x  =  r.  It  was  noted 
that  ak-r  =  0'r  or  k—Qr  according  as  k  is  even  or  odd,  and  that  the  residue 
of  r"  is  r  times  that  of  r"~^ 

Matsunaga,^^*^"  in  the  first  half  of  the  eighteenth  century,  solved 
a}-\-hx=  y^  by  expressing  6  as  a  product  mn  and  finding  p,  q  and  A  so  that 
mp  —  nq=l,  2pa=A  (mod  n).  Then  x={Am  —  2a)A/n  [and  y=a  —  7nb]. 
But  if  Am=  2a,  write  A-\-n  in  place  of  A  and  proceed  as  before.  Or  write 
2a+h  in  the  form  bQ+R,  whence  x=2a+b-{Q+l)R.  To  solve  69+ 
llx=y'^,  consider  the  successive  squares  until  we  reach  5^=3  (mod  11). 
Write  2-5+11  in  the  form  1-11  +  10.  Then  for  a=5,  6=  11,  Q=  1,  i2=  10, 
the  preceding  expression  for  x  becomes  1,  whence  5^+11-1  =  6^.  Then 
write  2-6+11  in  the  form  2-11  +  1.  Then  23-(2+l)-l  =  20  gives  6"+ 
20-11=  16^,  and  a;=  (256-69)/ll=  17. 

L.  Euler^^^  proved  that,  if  n  divides  p  —  l,  where  p  is  a  prime,  and  if 
a  =  c''-\-kp,  then  (by  powering  and  using  Fermat's  theorem),  a^^~^^^"  —  l  is 
divisible  by  p.  Conversely,  if  a'"  — 1  is  divisible  by  the  prime  p  =  w7i+l, 
we  can  find  an  integer  y  such  that  a — ?/"  is  divisible  by  p.     For, 

o'"-2/'"''  =  (a-2/")Q(^), 

and  the  differences  of  order  mn—n  of  Q(l),  Q(2),. . .,  Q{mn)  are  the  same 

>"Math.  Quest,  and  Solutions  (Ed.  Times),  3,  1917,  61-2;  corrections,  p.  65. 

'"Vlja-ganita,  §§  204-5;  Algebra,  with  arith.  and  mensuration,  from  the  Sanscrit  of  Brahmegupta 

and  Bhdscara,  transl.  by  H.  T.  Colebrooke,  London,  1817,  pp.  263-4. 
""Abhand.  Geschichte  Math.  Wiss.,  30,  1912,  237. 
^'^Ibid.,  234-5. 
i"Novi  Comm.  Acad.  Petrop.,  7,  1758-9  (1755),  p.  49,  eeq.,  §64,  §72,  §77;  Comm.  Arith.,  1, 

270-1,  273.    In  Novi  Comm.,  1,  1747-8,  p.  20;  Comm.  Arith.,  1,  p.  60,  he  proved  the  first 

statement  and  stated  the  converse 


Chap.  VII]  BiNOMIAL  CONGRUENCES.  205 

as  those  of  the  term  t/'""-"  for  ?/  =  !,...,  mn,  and  hence  equal  {mn  —  n)\, 
so  that  Q(y)  is  not  divisible  by  p  for  some  values  1, .  . . ,  mn  of  y. 

Euler^^^  recurred  to  the  subject.  The  main  conclusion  here  and  from 
his  former  paper  is  the  criterion  that,  if  p  =  mn-]-l  is  a  prime,  x''=a  (mod  p) 
has  exactly  n  roots  or  no  root,  according  as  a"^=l  (mod  p)  or  not.  In 
particular,  there  are  just  m  roots  of  a""^!,  and  each  root  a  is  a  residue  of 
an  nth  power. 

Euler^^^"  stated  that,  if  aq-\-h=p'^,  all  the  values  of  x  making  ax-\-b  b, 
square  are  given  by  a;=  ay'^^2py-\-q. 

J.  L.  Lagrange^^^  gave  the  criterion  of  Euler,  and  noted  that  if  p  is  a 
prime  4n+3,  B'-^'^^^^  —  l  is  divisible  by  p,  so  that  x=B'''^^  is  a  root  of 
x^=B  (mod  p).  Given  a  root  ^  of  the  latter,  where  now  p  is  any  odd  prime 
not  dividing  B,  we  can  find  a  root  of  x^=B  (mod  p^)  by  setting  x  =  ^-\-\p, 
i^-B  =  poi.  Then  x^-B  =  {\^-{-n)p'^  if  2|X+co=MP.  The  latter  can  be 
satisfied  by  integers  X,  jjl  since  2^  and  p  are  relatively  prime.  We  can  pro- 
ceed similarly  and  solve  x^=B  (mod  p"). 

Next,  consider  ^^=B  (mod  2"),  for  n>2  and  B  odd  (since  the  case  B 
even  reduces  to  the  former).  Then  ^  =  2z-\-l,  ^^  —  B  =  Z-\-\—B,  where 
Z  =  4:z{z-\-l)  is  a  multiple  of  8.  Thus  1—B  must  be  a  multiple  of  8.  Let 
w>3  and  1-B  =  2'^,r>3.  If  r^n,  it  suflaces  to  take  2  =  2""^,  where  f  is 
arbitrary.  If  r<n,  Z  must  be  divisible  by  2'',  whence  2  =  2'""^^  or  2*""^^  —  1. 
Hence  w=^{2'-^i:=i=l)-\-p  must  be  divisible  by  2"-''.  If  n-r^r-2,  it 
suffices  to  take  f  =^iS  divisible  by  2""''.  The  latter  is  a  necessary  condition 
if  n-r>r-2.  Thus  ^  =  2'-^p=F^,  w  =  2'-\f=t=p).  Hence  f  ±p  must  be 
divisible  by  2""^'^+^.  We  have  two  sub-cases  according  as  the  exponent  of 
2  is  ^  or  >r  — 1;  etc. 

Finally,  the  solution  of  x^=B{mod  m)  reduces  to  the  case  of  the  powers  of 
primes  dividing  m.  For,  if  /  and  g  are  relatively  prime  and  ^^  —  Bis  divisible 
by  /,  and  \p^—B  by  g,  then  x^—B\b  divisible  by  fg  ii  x= jif^  ^  =  vg^\l/.  But 
the  final  equality  can  be  satisfied  by  integers  /z,  v  since  /  is  prime  to  g. 

A.  M.  Legendre^^^  proved  that  if  p  is  a  prime  and  co  is  the  g.  c.  d.  of  n 
and  p  —  \  =  oip',  there  is  no  integral  root  of 

(1)  a:"=j5(modp) 

unless  B^'=  1  (mod  p) ;  if  the  last  condition  is  satisfied,  there  are  co  roots 
of  (1)  and  they  satisfy 

(2)  x'^^B^  (mod  p), 
where  I  is  the  least  positive  integer  for  which 

(3)  ln  —  q{p  —  \)=o}. 

For,  from  (1)  and  x''-^=l,  we  get  x^''=B\  x^^^-^^^l,  and  hence  (2),  by  use 
of  (3).     Set  n  =  oin'.     Then,  by  (2)  and  (1), 

^n'l^^n^2,  5P''=a;P'"=<rP-l=l    (mod  p) . 

"2Novi  Comm..Petrop.,  8,  1760-1,  74;  Opusc.  Anal.  1,  1772,  121;  Comm.  Arith.,  1,  274,  487. 

"^aOpera  postuma,  I,  1862,  213-4  (about  1771). 

"^Mem.  Acad.  R.  Sc.  Berlin,  23,  ann6e  1767,  1769;  Oeuvres,  2,  497-504. 

"^M6m.  Ac.  R.  Sc.  Paris,  1785,  468,  476-481.     (Cf.  Legendre.i^^) 


206  History  of  the  Theory  of  Numbers.  [Chap,  vii 

Since  In'—qp'  =  1,  the  first  gives  5^"'=  1.     Hence 

Conversely,  if  B^'=l, 

a^p-i_l=a^p'-_j5p'/  (mod  p) 

has  the  factor  x"  —  B\  so  that  (Lagrange^)  congruence  (2)  has  co  roots. 

If  4n  divides  p  — 1,  the  roots  of  x^"=  —1  (mod  p)  are  the  odd  powers  of 
an  integer  belonging  to  the  exponent  4n  modulo  p. 

Let  n  divide  p  —  l,  and  7n  divide  {p  —  l)/n.  Let  co  be  the  g.  c.  d.  of 
m ,  n  and  set  n  =  cov.  Determine  positive  integers  I  and  q  such  that  lv  —  qm  =  l. 
If  5'"=  =•=  1  (mod  p),  (1)  is  satisfied  by  the  roots  of  x'^^B'y  (mod  p),  where 
y  ranges  over  the  roots  of  ?/"=  (=t  1)^  (mod  p).  For,  the  last  two  congruences 
give 

x''  =  x'"'=B''^y''=B'"^+\=i=iy=B  (mod  p). 

Hence  by  means  of  the  roots  of  ?/''=±l,  we  reduce  the  solution  of  (1)  to 
binomial  congruences  of  lower  degrees.  In  particular,  let  n  =  2,  m  =  (p  — 1)/2, 
and  let  2  be  prune  to  ?«,  so  that  p  =  4:a  —  l,l  =  a,q  =  l.  Then  x^  =  B  (mod  p) 
requires  that  5"*  =  1 ,  so  that  we  have  the  solutions  .t  =  =*=  5"  without  trial 
(Lagrange^^^).  Next,  if  n  =  2  and  5^'^+^=  —1,  the  theorem  gives  x  =  B'''^^y, 
where  ?/-  =  —  L  But  we  may  generalize  the  last  result.  Consider  x" + c^  =  0 
(mod  p).  Since  p  must  have  the  form  4a+l,  we  have  p=f^-\-g^.  Deter- 
mine u  and  z  so  that  c  =  gu—pz.     Then  x  =  fu  (mod  p). 

Let  a  belong  to  the  exponent  nw  modulo  p,  where  w  divides  (p  — l)/n. 
Then  the  roots  of  B"'  =  l  (mod  p)  are  B  =  a"*'  (m  =  1,-  •  •,  w)  — 1),  and,  for  a 
fixed  B,  the  roots  of  (1)  are  x  =  0'"""+"  (m  =  0,  1, .  .  . ,  n  - 1).  For,  a"  belongs 
to  the  exponent  w,  whence  B  =  a'"'. 

Legendre^^^  gave  the  same  theorems  in  his  text.  He  added  that,  know- 
ing a  root  6  of  (1),  it  is  easy  to  find  a  root  of  x"  =  B  (mod  p"),  with  the 
possible  exception  of  the  case  in  which  n  is  divisible  by  p.  Let6"—B  =  Mp 
and  set  a:=^+i4p.     Then  a;"  — 5  is  divisible  by  p^  if 

M+'nB''-^A=pM', 

which  can  be  satisfied  by  integers  A,  Af'  if  n  is  not  divisible  by  p.  To  solve 
(1)  when  p  is  composite,  p  =  a°6^ .  .  . ,  where  a,  6, .  .  .  are  distinct  primes,  deter- 
mine all  the  roots  X  of  X"  =  B  (mod  a°),  all  the  roots  ^  of  ijl"  =  B  (mod  b^), 

Then  if  x=\  (mod  a°),  x=iJi  (mod  6^),.  .  .,  x  will  range  over  all  the  roots 
of  (1). 

Legendre^^^  noted  that  if  p  is  a  prime  8n+5  we  can  give  explicitly  the 
solutions  of  a:^+a  =  0  (mod  p)  when  it  is  solvable,  viz.,  when  a'*"''"^  =  1.  For, 
either  «-"+'  + 1=0  and  x  =  a"+'  is  a  solution,  or  a-"+^-l=0  and  (9  =  a"+^ 
satisfies  d^  —  a^O  (mod  p),  so  that  it  remains  only  to  solve  x^-\-d'  =  0,  which 
was  done  at  the  end  of  his^^*  memoir.  For  p  =  8?i+l,  let  n  =  a^,  where  a 
is  a  power  of  2  and  /3  is  odd;  if  0"=  ±  1,  x^-\-a  =  0  can  be  solved  as  in  the 

i"Th(5orie  des  nombres,  1798,  411-8;  ed.  2,  1808,  349-357;  ed.  3,  1830,  Nos.  339-351;  German 

transl.  by  Maser,  1893,  2,  pp.  15-22. 
^"Ibid.,  231-8;  ed.  2,  1808,  pp.  211-219;  Maser,  I,  pp.  246-7. 


Chap.  VII]  BiNOMIAL  CONGEUENCES.  207 

case  p  =  8n+5;  but  in  general  no  such  direct  solution  is  known,  and  it  is 
best  to  represent  some  multiple  of  p  by  the  form  y'^+az^. 

If  we  have  found  6  such  that  ^^+a  is  divisible  by  the  prime  p,  not  dividing 
a,  we  readily  solve  x^+a  =  0  (mod  p").    For,  from 

r^-\-as^  is  divisible  by  p^.  Now  s  is  not  divisible  by  p.  Thus  we  may  take 
r  =  sx+p''y,  whence  x^+a  is  divisible  by  p".  [Cf.  Tchebychef,  Theorie  der 
Congruenzen,  §30.] 

The  case  of  any  composite  modulus  N  is  easily  reduced  to  the  preceding 
(end  of  Lagrange's^^^  paper).  Legendre  proved  that,  if  N  is  odd  and  prime 
to  a,  the  number  of  solutions  of  a:^+a  =  0  (mod  N)  is  2'"^  where  i  is  the 
number  of  distinct  prime  factors  of  N;  the  same  is  true  for  modulus  2N. 
Henceforth  let  N  be  odd  or  the  double  of  an  odd  number  and  let  d  be  the 
g.  c.  d.  of  N  and  a.  If  d  has  no  square  factor,  the  congruence  has  2'"^  roots, 
where  i  is  the  number  of  distinct  odd  prime  factors  of  N  not  dividing  a. 
But  if  d=o)\(/^,  where  co  has  no  square  factor,  the  congruence  has  2'~V 
roots  where  i  is  the  number  of  distinct  odd  prime  factors  of  N/d. 

C.  F.  Gauss^"  treated  congruence  (1)  by  the  use  of  indices.  However, 
we  can  give  a  direct  solution  (arts.  66-68)  when  a  root  is  known  to  be  con- 
gruent to  a  power  of  B.  For,  by  (1)  and  x  =  B^,  B^B^"".  If  therefore  a 
relation  of  the  last  iy^e  is  known,  a  root  of  (1)  is  B''.  The  condition  for 
the  relation  is  l  =  A:n  (mod  t),  where  t  is  the  exponent  to  which  B  belongs 
modulo  p.  It  is  shown  that  t  must  divide  m  =  (p  —  l)/n.  We  may  discard 
from  m  any  factor  of  n;  if  the  resulting  number  is  m/q,  the  unique  solution 
k  of  1  =  1:11  (mod  m/q)  is  the  desired  k.     [Cf.  Poinsot^^^] 

Gauss  (arts.  101-5)  gave  the  usual  method  of  reducing  the  solution  of 
x^=  A  (mod  m)  for  any  composite  modulus  to  the  case  of  a  prime  modulus 
and  gave  the  number  of  roots  modulo  p'*  in  the  various  possible  subcases. 
His  well-known  and  practical  ''method  of  exclusion"  (arts.  319-322)  employs 
successive  small  powers  of  primes  as  moduU.  Another  method  (arts. 
327-8)  is  based  on  the  theory  of  binary  quadratic  forms  [cf.  Smith^^°]. 

The  congruence  ax^-\-'bx-\-c=0  (mod  m)  is  reduced  (art.  152)  to  y^=}?  — 
Aac  (mod  4am).     For  each  root  y,  it  remains  to  solve  2ax-{-b=y  (mod  4aw). 

Gauss^^^  showed  in  a  somewhat  incomplete  posthumous  paper  that,  if 
t  is  a  prime  and  f~'^{t  —  l)=a''¥.  .  .,  where  a,h,.  .  .  are  distinct  primes,  the 
solution  of  a:"=  1  (mod  t")  may  be  made  to  depend  upon  the  solution  of  a 
congruences  of  degree  a,  jS  congruences  of  degree  h,  etc.  Use  is  made  of  the 
periods  formed  of  the  primitive  roots  of  the  congruence,  as  in  Gauss'  theory 
of  roots  of  unity. 

Legendre^^^  solved  x^+a=0  (mod  2'")  when  a  is  of  the  form  —  1  =F8a  by 


i"Disquis.  Arith.,  1801,  Arts.  60-65. 

"sWerke,  2,  1863,  199-211.     Maser's  German  transl.  of  Gauss'  Disq.  Arith.,  etc.,  1889,  589-601 

(comments,  p.  683). 
"'Theorie  des  nombres,  ed.  2,  1808,  pp.  358-60  (Nos.  350-2).     Maser,  2,  1893,  25-7. 


208 


History  of  the  Theory  of  Numbers. 


[Chap.  VII 


use  of  the  expansion  of  (1+2)^'^: 


M 


M-3 


Vl±8a  =  l±|2^a-— 7  2V=t— —  2V-  .  .  .  =tiV23"a"+ . 


N  = 


M-3-5 


2-4 

(2n 


2-4-6 


■3) 


2-4-6-8.  ..2n 


The  coefficient  of  a"  is  an  integer  divisible  by  2"^^  Retain  only  the  terms 
whose  coefficients  are  not  divisible  by  2'""^  and  call  their  sum  6.  Hence 
every  term  of  6~-^a  is  divisible  by  2'".  Thus  the  general  solution  of  the 
proposed  congruence  is  x=2'^~^x'^d. 

P.  S.  Laplace^*^"  attempted  to  prove  that,  if  p  is  a  prime  and  p  —  l=ae, 
there  exists  an  integer  x<e  such  that  x'  —  l  is  not  divisible  by  p.  For,  if 
x  =  e  and  all  earlier  values  of  x  make  a:*  —  1  divisible  by  p, 


/=(e^-l)-e^ 
would  be  divisible  by  p. 


,{e-lY-l\  +  {^^y,{e-2Y-l\-... 

The  sum  of  the  second  terms  of  the  binomials  is 


+  ...  =  -(1-1)^  =  0, 


while  the  sum  of  the  first  terms  of  the  binomials  is  e !  by  the  theory  of  differ- 
ences, and  is  not  divisible  by  p  since  e<p.  [But  the  former  equality  implies 
that  the  last  term  of  /  is  (  — 1)''(0— 1),  whereas  the  theorem  is  trivial  if  x 
is  allowed  to  take  the  value  0.  Again,  nothing  in  the  proof  given  prevents 
a  from  being  unity;  then  the  statement  that  there  is  a  positive  integer 
x<p  —  l  such  that  x^~^  —  1  is  not  divisible  hyp  contradicts  Fermat's  theorem.] 

L.  Poinsot^^  deduced  roots  of  a;"=  1  (mod  p)  from  roots  of  unity. 

M.  A.  Stern^^  (p.  152)  proved  that  if  n  is  odd  and  p  is  a  prime,  rc"=  —1 
(mod  p)  is  solvable  and  the  number  of  roots  is  the  g.  c.  d.  of  n  and  p  —  l; 
while,  if  n  is  even,  it  is  solvable  if  and  only  if  the  factor  2  occurs  in  p  —  1  to  a 
higher  power  than  in  n. 

G.  Libri^^^  gave  a  long  formula,  involving  sums  of  trigonometric  func- 
tions, for  the  number  of  roots  of  x^+c=0  (mod  p). 

V.  A.  Lebesgue^^  applied  a  theorem  on/(a:i, . .  .,  Xk)  =  0  to  derive  Legen- 
dre's^^^  condition  B^'=l  for  the  existence  of  roots  of  (1),  and  the  number 
of  roots.     Cf.  Lebesgue^^  of  Ch.  VIII. 

Erlerus^^  (pp.  9-13)  proved  that,  if  pi, .  . . ,  p^  are  distinct  odd  primes, 

x~=l  (mod2''p/'...p/) 

has  2",  2",  2"+^  or  2"+^  roots  according  as  j/  =  0,  1,  2  or  >2. 

For  the  last  result  and  the  like  number  of  roots  of  x^=a,  see  the  reports, 
in  Ch.  Ill  on  Fermat's  theorem,  of  the  papers  by  Brennecke^^  and  Crelle^* 
of  1839,  Crelle,^^  Poinsot"  (erroneous)  and  Prouhet^^  of  1845,  and  Schering^''* 
of  1882. 

C.  F.  Arndt^^^  proved  that  the  number  of  roots  of  x'=  1  (mod  p")  for 

"»Communication  to  Lacroix,  Traitd  Calcul  Diff.  Int.,  ed.  2,  vol.  in,  1818,  723. 
'"Jour,  fur  Math.,  9,  1832,  175-7.     See  Libri,"  Ch.  VIII. 
"*Archiv  Math.  Phys.,  2,  1842,  10-14,  21-22. 


Chap.  VII]  BiNOMIAL  CONGRUENCES.  209 

p  an  odd  prime  is  the  g.  c.  d.  of  t  and  (/)(p") ;  the  same  holds  for  modulus  2p". 
He  found  the  number  of  roots  of  x'^=r  (mod  m),  m  arbitrary.  By  using 
S</>(0  =5,  if  i  ranges  over  the  divisors  of  5,  he  proved  (pp.  25-26)  the  known 
result  that  the  number  of  roots  of  x"=  1  (mod  p)  is  the  g.  c.  d.  5  of  n  andp  —  1. 
The  product  of  the  roots  of  the  latter  is  congruent  to  (  —  1)*"^^;  the  sum  of 
the  roots  is  divisible  by  p;  the  sum  of  the  squares  of  the  roots  is  divisible 
bypif  6>2. 

P.  F.  Arndt^^^  used  indices  to  find  the  number  of  roots  of  x^  =  a. 

A.  L.  Crelle^^^gave  an  exposition  of  known  results  on  binomial  congruences. 

L.  Poinsot^^^  considered  the  direct  solution  of  x"=A  (mod  p),  where  p 
is  a  prime  and  n  is  a  divisor  of  p  — l=nm  (to  which  the  contrary  case 
reduces).  Let  the  necessary  condition  ^""=1  be  satisfied.  Hence  we  may 
replace  A  by  A^+"''^  and  obtain  the  root  rc=A^  if  l-\-mk  =  ne  is  solvable  for 
integers  k,  e,  which  is  the  case  if  m  and  n  are  relatively  prime  [cf.  Gauss^^^]. 
The  fact  that  we  obtain  a  single  root  x=A^  is  explained  by  the  remark  that 
it  is  a  root  common  to  a:"=A  and  x"'=l,  which  have  a  single  common  root 
when  n  is  prime  to  m.  Next,  let  n  and  m  be  not  relatively  prime.  Then 
there  is  no  root  A'  if  A  belongs  to  the  exponent  m  modulo  p.  But  if  A 
belongs  to  a  smaller  exponent  m'  and  if  m'  is  prime  to  n,  there  exists  as 
before  a  root  A",  where  l-\-m'k  =  ne'.  The  number  of  roots  of  a;"=l 
(mod  N)  is  found  (pp.  87-101). 

C.  F.  Arndt^^^  proved  that  x'=l  (mod  2"),  n>2,  has  the  single  root  1  if 
t  is  odd;  while  for  t  even  the  number  of  roots  is  double  the  g.  c.  d.  of  i  and 
2n-2^  The  sum  of  the  A:th  powers  of  the  roots  of  x'=  1  (mod  p)  is  divisible 
by  the  prime  p  if  A:  is  not  a  multiple  of  t.  By  means  of  Newton's  identities 
it  is  shown  that  the  sum,  sum  of  products  by  twos,  threes,  etc.,  of  the  roots 
of  x*=  1  (mod  p)  is  divisible  by  the  prime  p,  while  their  product  is  =  + 1  or 
—  1  according  as  the  number  of  roots  is  odd  or  even.  If  the  sum,  sum  of 
products  by  twos,  threes,  etc.,  of  m  integers  is  divisible  by  the  prime  p, 
while  their  product  is  =—(  —  1)'",  the  m  integers  are  the  roots  of  x'"=l 
(mod  p). 

A.  Cauchy^"  stated  that  if  I  =  p\'' .  .  .,  where  p,  q,...  are  m  distinct 
primes,  and  if  n  is  an  odd  prime,  x"=  1  (mod  7)  has  rf  distinct  roots,  includ- 
ing primitive  roots,  i.  e.,  numbers  belonging  to  the  exponent  n.  [But 
x^=  1  (mod  5)  has  a  single  root.] 

Cauchy^^^  later  restricted  p,  q,.  . .  to  be  primes  =1  (mod  n).  Then 
a:"=l  (mod  p^)  has  a  primitive  root  ri,  and  rc^^l  (mod  q")  has  a  primitive 
root  7-2,  so  that  x''^  1  (mod  /)  has  a  primitive  root,  viz.,  an  integer  =ri  (mod 
p*")  and  =r2  (mod  q"),  etc.;  but  no  primitive  root  ii  p,  q,.  .  .  are  not  all  =1 
(mod  n). 

i«3Von  den  Kubischen  Resten,  Torgau,  1842,  12  pp. 

»*Jour.  fiir  Math.,  28,  1844,  111-154. 

»«Jour.  de  Math^matiques,  (1),  10,  1845,  77-87. 

""Archiv  Math.  Phys.,  6,  1845,  380,  396-9. 

"'Comptes  Rendus  Paris,  24,  1847,  996;  Oeuvres,  (1),  10,  299. 

"sComptes  Rendus  Paris,  25,  1847,  37;  Oeuvres,  (1),  10,  331. 


210  History  of  the  Theory  of  Numbers.  [Chap,  vii 

Hoen4  Wronski^^^  stated  without  proof  that,  if  a;'"=a  (mod  M), 

a  =  {-iy+'\hK+{-lY+''rA[M/K,o)Y-^+Mi, 
x  =  h  +  (-lY+'A[M/K,  iry-'+Mj, 

and  that  M  must  be  a  factor  of  aK"" -  \hK-{-iy+^\'".  Here  the  "alephs" 
A[M/K,  o)Y,  for  r  =  0,  1,.  .  .,  are  the  numerators  of  the  reduced  fractions 
obtained  in  the  development  of  M/K  as  a  continued  fraction.  In  place  of 
K,  Wronski  wrote  the  square  of  l''^^  =  k\.  Concerning  these  formulas,  see 
Hanegraeff,"^  Bukaty,!^^  Dickstein.^^^    Cf.  Wronski^^^  of  Ch.  VIII. 

E.  Desmarest"  noted  that,  if  x'^+D=0  (mod  p)  is  solvable,  x^+Dy^  =  mp 
can  be  satisfied  by  a  value  of  m<3+y>/16  and  a  value  of  2/^3.  His  proof 
is  not  satisfactory. 

D.  A.  da  Silva''^  (Alasia,  p.  31)  noted  that  x^^l  (mod  m),  where 
m  =  Pi*'P2^* ••  ■ .  has  the  roots  'Zxiqi7n/p{^  where  Xi  is  a  root  of  x^'=  1 
(mod  Pi"^),  Di  being  the  g.  c.  d.  of  D  and  <t){pr),  while  the  g's  are  integers 
such  that  Xqim/p['=\  (mod  m). 

Da  Silva^^^"  proved  that  a  solvable  congruence  a:"=r  (mod  m)  can  be 
reduced  to  the  case  r  prime  to  m  and  then  to  the  case  m  =  p'',p  a  prime  >  2. 
Then,  if  5  is  the  g.  c.  d.  of  n  and  (f){p'')=8di,  there  is  a  root  if  and  only  if 
r*'=l  (mod  p")  and  hence  if  and  only  if  r'^=l  (mod  p"'"^^),  where  p"'  is  the 
g.  c.  d.  of  n  and  p"~"\  while  d  is  the  quotient  of  p  — 1  by  its  g.  c.  d.  with  n. 

H.  J.  S.  Smith^'^"  indicated  a  simplification  in  Gauss'^"  second  method 
of  solving  x^^A,  If  r^-\-D=0  (mod  P)  is  solvable,  mP  =  x^-\-Dy'^  is  solvable 
for  some  value  of  ?72<  2V-D/3.  Employing  all  values  of  m  under  that  limit 
for  which  also 


(i)=S> 


and  finding  with  Gauss  all  prime  representations  of  the  resulting  products 
by  the  form  x^-\-Dy^,  we  get  ±r=x'/y',  x"/y",.  .  .(mod  P),  where  x',  y'; 
x",  y" \. .  .  denote  the  sets  of  solutions  of  mP  =  x^-\-Dy^. 

Eg.  Hanegraeff^^^  reduced  x"'=r  to  d"'r=l  (mod  p)  by  use  of  6x=l. 
When  p/d  is  developed  into  a  continued  fraction,  let  /x  and  P^_i  be  the 
number  of  quotients  and  number  of  convergents  preceding  the  last.  Let  v, 
P^_i  be  the  corresponding  numbers  for  p/O"".     Then 

x^i-iy-'P,_„  r=(-ir^P,_i  (mod  p). 

For  p  a  prime,  we  get  all  roots  by  taking  6  =  1,. .  . ,  (p  — 1)/2.  By  starting 
with  d{x  —  h)^l  in  place  of  6x=  1,  we  get 

"'R6forme  des  Math^matiques,  being  Vol.  i  of  R6forme  du  savoir  humain,  1847.  Wronski's 
mathematical  discoveries  have  been  discussed  by  S.  Dickstein,  Bibliotheca  Math.,  (2), 
6,  1892,  48-52,  85-90;  7,  1893,  9-14  [on  analysis,  (2),  8,  1894,  49,  85;  (2),  10,  1896,  5]. 
Bull.  Int.  Ac.  Sc.  Cracovie,  1896;  Rozprawy,  Krakow,  4,  1913,  73,  396.  Cf.  I'intermd- 
diaire  des  math.,  22,  1915,  68;  23,  1916,  113,  164-7,  181-3,  199,  231-4;  25,  1918,  55-7. 

"'<KU.  Alasia,  Annaes  Sc.  Acad.  Polyt.  do  Porto,  9,  1914,  65-95.  There  are  many  confusing 
misprints;  for  example,  five  at  the  top  of  p.  76. 

""British  Assoc.  Report,  1860,  120-,  §68;  CoU.  M.  Papers,  1,  148-9. 

"*Note  BUT  r^quation  de  congruence  x^=r  (mod  p),  Paris,  1860. 


Chap.  VII]  BiNOMIAL  CONGRUENCES.  211 

x-h={-iy-'P,-u  r^{-iy-\dh+irP^.^  (mod  p). 

By  taking  ^=(1^/')^  and  replacing  1  by  (-1)^+^  in  0(a;-/i)  =  l,  the  last 
results  become  the  fundamental  formula  given  without  proof  by  Wronski^^^ 
in  his  Reforme  des  Mathematiques. 

G.  L.  Dirichlet^^^  discussed  the  solution  of  x^^D  for  any  modulus. 

G.  F.  Meyer"^  gave  an  elementary  discussion  of  the  solution  of  x^=b 
(mod  k),  for  k  a  prime,  power  of  prime,  or  any  integer. 

V.  A.  Lebesgue^^^  employed  a  prime  p,  a  divisor  n  of  p  —  l=nn',  and  a 
number  a  belonging  to  the  exponent  n'  modulo  p.  Then  the  roots  of 
a;"  =  a  (mod  p)  are  a"6^,  where  h  is  not  in  the  period  of  a,  and  6  is  a  quadratic 
non-residue  of  p  if  a  is  a  quadratic  residue,  and  6"  is  the  least  power  of  h 
congruent  to  a  term  of  the  period  of  a.  If  we  set  6"  =  a"  (mod  p),  then 
must  na-\-v^  =  l  (mod  n').  The  roots  x  are  primitive  roots  of  p.  In  the 
construction  of  a  table  of  indices,  his  method  is  to  seek  a  primitive  root 
giving  to  ±2  the  minimum  index  (rather  than  to  ±10,  used  by  Jacobi); 
thus  we  use  the  theorem  for  a=  ±2. 

Lebesgue^'^^  gave  reasons  why  the  conditions  imposed  on  h  in  his  pre- 
ceding paper  are  necessary.  He  added  that  when  we  have  found  that 
x"  =  a  (mod  p)  leads  to  a  primitive  root  x  =  g  oi  p,\i  is  easy  to  solve  x'"=r 
(mod  p)  when  m  divides  p  —  1,  by  expressing  r  as  a  power  of  g  by  the  equiva- 
lent of  an  abridged  table  of  indices. 

Lebesgue^^*^  noted  that  the  usual  method  of  solution  by  indices  leads 
to  the  theorem:  If  a  belongs  to  the  exponent  e  modulo  p,  and  if  n  divides 
p  — 1,  and  we  set  n  =  e'm,  where  e'  has  only  prime  factors  which  divide  e, 
while  m  is  prime  to  e,  then,  for  every  divisor  M  of  m,  x'^^a  (mod  p)  has 
e'(j){M)  roots  belonging  to  the  exponent  M. 

If  a  belongs  to  the  exponent  e  modulo  p,  there  are  e0(n)  numbers  h,  not 
in  the  period  of  a,  for  which  6"=  a'  (mod  p),  with  n  sl  minimum.  A  common 
divisor  of  n  and  i  does  not  divide  e.  Then  the  n  roots  of  x"=  a  (mod  p)  are 
a'6",  where  nt  —  iu  —  l  =  ev,  t<e,  u<n.  This  generahzation  of  his^^''  earlier 
theorem  is  used  to  find  the  period  of  a  primitive  root  of  p  from  the  period  of  2. 

R.  Gorgas"^  stated  that,  if  p  is  the  residue  modulo  M  of  the  pth  term  of 
](M-l)/2[^. .  .,2^  1^,  then  p(p-l)=p±m+ikf  a,  according  as  ilf  =  4m=t:l. 
Take  the  lower  signs  and  solve  for  p ;  we  get 

2p  =  l±6,  62  =  M(4a-l)+4p. 

Set  4p  =  Mc+p'.  Hence  the  initial  equation  x^  =  My+p  has  been  replaced 
by  6^  =  M(4a-fc  — l)+p'  of  like  form.  Let  p'  be  the  p'th  place  from  the 
end.  The  process  may  be  repeated  until  we  reach  an  equation  P(P  — 1) 
=  MA-\-p^—m  solvable  by  inspection. 

"^Zahlentheorie,  1863,  §§32-7;  ed.  2,  1871;  ed.  3,  1879;  ed.  4,  1894. 

»"Archiv  Math.  Phys.,  43,  1865,  413-36. 

"^Comptes  Rendus  Paris,  61,  1865,  1041-4. 

"'Ihid.,  62,  1866,  20-23. 

"«/6id.,  63,  1866,  1100-3. 

"^Ueber  Losung  dioph.  Gl.  2.  Gr.,  Progr.,  Magdeburg,  1867. 


212  History  of  the  Theory  of  Numbers.  [Chap,  vii 

Ladrasch^^^  obtained  known  results  on  x^=a  for  any  modulus. 

V.  Bouniakowsky^"^  gave  a  method  of  sohdng  q-S'^  ^r  (mod  P),  where 
P  is  odd.  His  first  illustration  is  3'==±=1  (mod  25).  Write  the  integers 
^(25  — 1)/2  in  a  line.  Under  the  first  four  wTite  in  order  the  integers 
=  0  (mod  3) ;  under  the  next  four  write  in  reverse  order  those  =  1 ;  under 
the  last  four  write  in  order  those  =  2. 


1*  2*  3*     4* 
3    6    9     12 


5    6*  7*  8* 
10    7    4     1 


9*  10     11*  12* 
2      5      8     11 


Mark  with  an  asterisk  1  in  the  first  line;  below  it  lies  3;  mark  with  an 
asterisk  3  in  the  first  line;  etc.  The  number  10  of  the  integers  marked 
with  an  asterisk  is  the  least  solution  x  of  3""=  —1  (mod  25).  The  sign  is 
determined  by  the  number  of  integers  in  the  second  set  marked  by  an 
asterisk.  The  method  applies  to  any  P  =  6n+1.  But  for  P  =  6n+5,  we 
use  for  the  second  set  of  numbers  in  the  second  line  those  =2  (mod  3)  in 
reverse  order,  and  for  the  third  set  those  =1  in  order.  If  P  =  23,  we  see 
that  each  of  the  11  numbers  in  the  first  line  are  marked  with  an  asterisk, 
whence  3^^=-l  (mod  23).  A  like  marking  occurs  for  P  =  5,  11,  17,  29. 
For  P  =  35,  12  numbers  are  marked,  whence  12  is  the  least  x  for  which 
3''=1  (mod  35).  Starting  with  the  unmarked  number  5,  we  get  the  cycle 
5,  15,  10,  whence  3^=  —1  (mod  7);  similarly,  the  cycle  7,  14  gives  3"=  —1 
(mod  5). 

For  g'-3'^=='=4  (mod  25),  we  begin  with  4  in  the  second  row.  Since  it 
hes  below  7,  we  mark  7  with  an  asterisk  in  the  second  row;  etc.  We  use  an 
affix  n  on  the  number  which  is  the  nth  marked  by  an  asterisk. 


12    3      4 

5 

6     7    8 

9     10 

11     12 

3*6  g*3  9*5  12*10 

10 

7*2^*1  ]^*7 

2*^   5 

8*8  11*9 

For  5  =  11,  we  have  the  entry  8*^  below  11;  hence  11-3^=— 4,  the  sign 
following  from  the  number  of  entries  ^  8  in  the  second  set  which  are  marked 
with  an  asterisk.     Similarly  for  any  5^  12,  except  g  =  5,  10. 

Bukaty^^"  discussed  the  formula  of  Wronski.^^^ 

T.  N.  Thiele^^^  used  a  mosaic  (empty  and  filled  squares  on  cross-section 
paper)  to  test  y^=d{T[\od  c),  where  c  is  an  integer  or  Gauss  complex  integer 
a  +  5v— 1,  employing  the  graph  oi  y'^  —  cx  =  d. 

Dittmar^^^  discussed  a;^=r  (mod  p).  Using  Cauchy's^*  explicit  con- 
gruence for  the  numbers  belonging  to  a  given  exponent,  he  gave  the  expanded 
form  of  the  congruence  with  the  roots  belonging  to  the  successive  exponents 
1,.  ..,21. 

"*Von  den  Kubischen  Resten  u.  Nichtresten,  Progr.,  Dortmund,  1870. 

i-'Bull.  Ac.  Sc.  St.  Pdterebourg,  14,  1870,  356-375. 

i*'>D6duction  et  demonstration  de  trois  lois  primordiales  de  la  congruence  des  nombres,  Paris, 

1873. 
*""0m  Talmonstre,"  Forhandl.  Skandinaviske  Naturforskeres,  Kjobenhavn,  11,  1873,  192-5. 
"^Die  Theorie  der  Reste,  insbesondere  derer  vom  3.  Grade,  nebst  einer  Tafel  der  Kubischen 

Reste  aller  Primzahlen  der  Form  6m +  1  zwischen  den  Grenzen  1  und  100.     Progr.  Koln 

Gym.,  Berlin,  1873. 


^li      Chap.  VII]  BiNOMIAL  CONGRUENCES.  213 

L.  Sancery"  (pp.  17-23)  employed  the  modulus  M  =  p''  or  2^',  where  p 
is  an  odd  prime.  Let  a  belong  to  the  exponent  n  modulo  M.  Let  A  be  the 
g.  c.  d.  of  m  and  4>{M)/n.  Set  A=AiA2  where  Ai  =  pi*'p2*' •  •  •  >  and  p,- 
is  a  prime  dividing  both  A  and  n,  and  p/<  is  the  power  of  Pi  dividing  A. 
Let  b  be  any  divisor  of  Ag.  Then  a:"'=a  (mod  M)  has  0(nAi5)/0(n)  roots 
belonging  to  the  exponent  nAjS ;  the  power  aAi5  of  such  a  root  is  congruent 
to  a,  where  a  can  be  found  by  means  of  a  linear  congruence.  Given  a 
number  belonging  to  the  exponent  nAi5,  we  can  find  Ai5  roots  of  the  con- 
gruence. 

C.  G.  Reuschle^^^"  tabulated  the  roots  of /=0  (mod  p),  where  p  =  wX+l 
and  X  are  primes  and  /  is  the  maximum  irreducible  algebraic  prime  factor 
of  a^  — 1;  also  the  roots  of 

T^Hc^O,     r/Hc^^O,     TyHc^sO,     rf^'n-\-d=Q, 

for  c<13,  d= —1  to  —26,  d=+2  to  +21,  and  for  various  cubic  and 
quartic  congruences. 

A.  Kunerth's  method  for  ^^=c  (mod  h)  will  be  given  in  Vol.  2,  Ch.  XII. 
E.  Lucas^^^^  treated  a;^+l=0  (mod  p"),  where  p  is  a  prime  >2,  for  use 

in  the  question  of  the  number  of  satins.     Given  a^+l=0  (mod  p),  set 

{a+i)"'  =  A+Bi,  ^B=l  (mod  p'"). 

Then  A^  is  a  root  x  of  the  proposed  congruence. 

B.  Stankewitsch^^^  proved  that  if  x^^q  (mod  p)  is  solvable,  p  being  an 
odd  prime,  the  positive  root  <p/2  is  =B/A  (mod  p),  where 

1-2  i 

A=Si_,+qSi.3+q%_s+  ...+q^  S^,  B  =  Si+qSi_2+  ■  ■  •  +q^ 

where  i  =  {p  —  l)/2  and  Sk  denotes  the  sum  of  the  products  of  1,  2, . . .,  i 
taken  k  at  a  time.  Let  n  be  a  divisor  of  p  — 1.  Let  F{x)  be  the  g.  c.  d. 
modulo  p  of  x"  —  !  and  Il(x''^"-  —  1),  where  a  ranges  over  the  distinct 
prime  factors  of  n.  Call  f{x)  the  quotient  of  x''  —  l  by  i^(a;).  Then  the 
roots  of  f{x)  =  0  (mod  p)  are  the  primitive  roots  of  x"=  1  (mod  p) .  [Cf . 
Cauchy.14] 

N.  V.  Bougaief^^^  noted  that  if  p  =  8n+5  is  a  prime  and  if  x^=q 
(mod  p)  is  solvable,  it  has  the  root  g(p+3)/8  ^j.  (pzl)!  g(p+3)/8  according  as 
q2n+i^-^  or  -1.  If  p  =  2^Z+l,  I  odd,  and  q'=l,  it  has  the  root  x=q^'+'^/\ 
[Legendre.^^®] 

T.  Pepin^^^  treated  x^=  2  by  tables  of  indices. 

P.  Gazzaniga^^^  gave  a  generalization  of  Gauss'  lemma  (the  case  n  =  8  =  2, 

1820  Tafeln  Complexer  Primzahlen . . . ,  Berlin,  1875.     Errata,  Cunningham."* 

^^^^  G6om6trie  des  tigsus,  Assoc,  fran^.,  40,  1911,  83-6;  French  transl.  of  his  Italian  paper  in 

I'Ingegnere  Civile,  1880,  Turm. 
is'Moscow  Math.  Soc,  10,  1882-3,  I,  112  (in  Russian). 
^<^Ibid.,  p.  103. 

"»Atti  Accad.  Pont.  Nuovi  Lincei,  38,  1884-5,  201. 
"»Atti  Reale  Istituto  Veneto,  (6),  4,  1885-6,  1271-9. 


214  History  of  the  Theory  of  Numbers.  [Chap,  vii 

^  =  0).     Separate  the  residues  modulo  p  of  kq,  for  k  =  l,  2, . .  .,  {p  —  l)/d, 

into  three  sets:  .': 

P  5-1  i 

0<ri,.  .  .,  r,<-<Si,.  .  .,  s,<-^p<tu.  •  .,  t^<p  j 

and  form  the  differences  mi  =  p  —  t,.  From  the  set  1,. .  .,  (p  — 1)/5,  delete 
the  r,  and  ?n,;  there  remain  v  numbers  i\.  If  ?/,  is  a  root  of  s,?/,=  yi  (mod  p), 
then  x"=5  (mod  p)  is  solvable  if  and  only  if  (  — l)"?/!.  .  .2/„=l  (mod  p), 
where  5  is  the  g.  c.  d.  of  n  and  p  —  1 . 

P.  Seelhoff^^^  gave  the  known  cases  in  which  x^=r  (mod  p)  can  be 
solved  explicitly  [Lagrange,  ^^^  Legendre^^^].  In  the  remaining  cases,  one 
uses  Gauss'  method  of  exclusion,  the  process  of  Desmarest,^^  or,  with 
Seelhoff,  use  various  quadratic  residues  of  p  {ibid.,  p.  306).  Here  x^=41 
(mod  120097)  is  treated. 

A.  Berger^^^  considered  a  quadratic  congruence  reducible  to  a:^=D  (mod 
4n),  where  D=0  or  1  (mod  4).     If  D  is  prime  to  n,  the  number  of  roots  is 

^(D,  in)  =  2n{l  +  (f )  }  =  2S  (f )  f .  =  22  (f )  f,, 

where  p  ranges  over  the  distinct  prime  factors  of  n,  while  d  and  di  range 
over  the  pairs  of  complementary  divisors  of  n,  and  f ^  =  0  or  1  according  as 
d  has  a  square  factor  or  not.  If  g{nm)=g{n)g{m)  for  all  integers  n,  m, 
ands'(l)  =  l, 

zgy  (Z),  4n)^(n)  =  22  Q  ^(n)  -S  Q  ^(n)  -^2  Q  ^(n)^ 
where  n  ranges  over  all  positive  integers.     Mean  values  are  found : 

J,(?)*»«-;b5ti7S.?,©I«-" 

it=i  TT   h=i\n/n 

where  A  is  a  fundamental  discriminant  according  to  Kronecker,  X,  Xi  are 
finite  for  all  n's,  and  p  ranges  over  all  primes. 

G.  Wertheim^^^  presented  the  theory  of  a;^=a  (mod  m). 

R.  Marcolongo^®°  treated  x^-\-P=0  (mod  p)  in  the  usual  manner  when 
explicit  solutions  are  known.  Next,  from  a  particular  set  of  solutions 
X,  y  of  x^+p'"?/+P  =  0,  where  p  is  a  prime  >2,  we  get  the  solution 

=i=x,=x-p'"y[ai..  .o„_i]  (mod  p"'+') 

of  Xi^-\-p"''^^yi-{-P  =  0,  where  [ai.  .  .a„_i]  is  the  numerator  of  next  to  the 
last  convergent  to  the  continued  fraction  for  p"'/{2x).  The  method  is 
Serret's,  Alg.  Sup^r.,  II.  For  p  =  2  the  results  obtained  are  the  same  as  in 
Dirichlet's  Zahlentheorie,  §36. 

i"Zeitschrift  Math.  Phys.,  31,  1886,  378-80. 

"SQfversigt  K.  Vetenskaps-Ak.  Forhandlingar,  Stockholm,  44,   1887,  127-153.     Nova  Acta 

regise  soc.  sc.  Upsalensis,  (3),  12,  1884. 
"»Elemente  der  Zahlentheorie,  1887,  182-3,  207-217.      ""Giomale  di  Mat.,  25,  1887,  161-173. 


Chap.  VII]  BiNOMIAL  CONGRUENCES.  215 

F.  J.  Studnicka^^^  treated  at  length  the  solution  in  integers  x,  y  (y<h) 
of  hx-{-l  =  y^,  discussed  by  Leibniz  in  1716. 

L.  Gegenbauer^^^  gave  a  new  derivation  of  the  equations  of  Berger^^^ 
leading  to  asymptotic  expressions  for  the  number  of  solutions  of  x^=Z). 

A.  Tonelli^^^  gave  a  method  of  solving  x^=c  (mod  p),  when  p  is  a  prime 
4/i+l  and  some  quadratic  non-residue  g'  of  p  is  known.  Set  p  =  2'y-{-l, 
where  y  is  odd.  By  Euler's  criterion,  the  power  72^"^  of  c  and  g  are  con- 
gruent to  +1,  —1.  Set  €0  =  0  or  1,  according  as  the  power  72^"^  of  c  is 
congruent  to  +1  or  —1.     Then 

For  s^3,  set  ei  =  0  or  1  according  as  the  square  root  of  the  left  member  is 
=  -f-lor-l.     Then 

Proceeding  similarly,  we  ultimately  get 

g2eycy=-^l  (mod  p),  e  =  €o+2€i+  . .  .  +2'-\_2- 

Thus  a;=  ±^'^c^^+^^/2  (mod  p).    Then  Z^=c  (mod  p^)  has  the  root 

X=x^'-'c(^^-'^^"'+i^/2  (mod  p^). 

G.  B.  Mathews'^^  (p.  53)  treated  the  cases  in  which  x^^a  (mod  p)  is 
solvable  by  formulas.     Cf.  Legendre.-^^^ 

S.  Dickstein^^^  noted  that  H.  Wronski^^^  gave  the  solution 

rM       -l(7r-l) 

y==hK+{-iy+'+Mi,  0  =  /i+(-1)'+'A|  ^,  ttJ         +Mj 

of  2"— a?/"=0  (mod  M)  with  (iV^)^  in  place  of  K,  and  gave,  as  the  condition 
for  solvability, 

a(lV^)2"-l=0(modM). 

But  there  may  be  solutions  when  the  last  condition  is  satisfied  by  no 
integer  A;.  This  is  due  to  the  fact  that  the  value  assigned  to  y  imposes  a 
limitation,  which  may  be  avoided  by  using  the  same  expressions  for  y,  z 
in  a  parameter  K,  subject  to  the  condition  aK"  — 1  =  0  (mod  M). 

M.  F.  J.  Mann"^'^  proved  that,  if  n=2^XV. .  .,  where  X,  m,  •  •  •  are  dis- 
tinct odd  primes,  the  number  of  solutions  of  x^=  1  (mod  n)  is  GGiGi . . . 
QiQi . . . ,  where  G=  1  if  n  or  p  is  odd,  otherwise  G  is  the  g.  c.  d.  of  2p  and  2^~'^, 
and  where  Gi,  Gi,..,  gi,  g2,. .  are  the  g.  c.  d.'s  of  p  with  X"~\  m''~\-  •  •> 
X  — l,jLi— 1,. . .,  respectively. 

A.  Tonelli^^^  gave  an  explicit  formula  for  the  roots  of  x^=c  (mod  p^), 

"iCasopis,  Prag,  18,  1889,  97;  cf.  Fortschritte  Math.,  1889,  30. 

"^Denkschriften  Ak.  Wiss.  Wien  (Math.),  57,  1890,  520. 

"'Gottingen  Nachrichten,  1891,  344-6. 

I'^BuU.  Internat.  de  I'Acad.  Sc.  de  Cracovie,  1892,  372  (64-65);  Berichte  Krakauer  Ak.  Wiss., 

26,  1893,  155-9. 
"^aMath.  Quest.  Educ.  Times,  56,  1892,  24-7. 
"^AttiR.  Accad.  Lincei,  Rendiconti,  (5),  1,  1892,  116-120. 


216  History  of  the  Theory  of  Numbers.  [Chap,  vii 

when  p  is  an  odd  prime,  and  a  quadratic  non-residue  ^  of  p  is  known. 
Set  p  =  2'a+l,  where  s^  1  and  a  is  odd.  Then  y  =  ap^~^  is  odd,  and 
{}>(p^)  =  2'y.  Tonelli's  earlier  work  for  modulus  p  now  holds  for  modulus 
p^  and  we  get  x=  ±  g'^c^'^'^^^^.  If  s  =  1,  then  e  =  0  and  the  root  is  that  given 
by  Lagrange  if  X  =  l.  If  s  =  2,  whence  p  =  4a+l  =  8Z+5,  the  expression 
for  X  is  given  a  form  free  of  e  =  cq: 

x=  ±  (c»+3)V^+^^/^  y  =  ap^-\ 

A.  Tonelli^^^  expressed  the  root  x  in  a  form  free  of  e  for  every  s: 

7+1 

where  the  v's  are  given  by  the  recursion  formula 

v.-H  =  c''-S?.t^'\  .  .t^--,%+k  (/i  =  2,  3,.  .  .). 

Here  k  is  an  existing  integer  such  that  A;-|-l  is  a  quadratic  residue  of  p, 
and  A:  —  1  a  non-residue.     Thus,  if  s  =  3, 

7+1 

x=^{c^''+ky {{c-'+ky^C+kl^'c  2  , 

where  we  may  take  A:  =  —2  if  a  is  not  divisible  by  3,  but  A;  =  —4  if  a  is  divi- 
sible by  3,  while  neither  a  nor  4a +1  are  di\'isible  by  5. 

N.  Amici^^  proved  that  a;^*=6  (mod  2"),  h  odd,  k^v  —  2,  is  solvable  only 
when  h  is  of  the  form  2^'"^^/i+l  and  then  has  2^+^  roots,  as  shown  by  use  of 
indices.  For  (x'")^  =  5,  the  same  condition  on  b  is  necessary ;  thus  it  remains 
to  solve  x'"=j8  (mod  2')  when  m  is  odd.  If  i3  =  8A;+l  or  8A:+3,  it  has  an 
index  to  the  base  8/z  +  3  and  we  get  an  unique  root.  If  /3  =  8/j  — 3  or  8A:  — 1, 
then  x'"=  — j3  has  a  root  a  by  the  preceding  case,  and  —a  is  a  root  of  the 
proposed  congruence. 

Jos.  Mayer^^^  found  the  number  of  roots  of  x^=a  (mod  p''),  for  the 
primes  2,  3,  p  =  6m='=  1.  If  fli,  Go,.  .  .  are  residues  of  nth  powers  modulo  p, 
and  if  g  is  the  g.  c.  d.  of  n  and  p  — 1,  then  0102-  .  .  =  -f-l  or  —1  (modp), 
according  as  p'  =  (p  — l)/g  is  odd  or  even.  If  p'  is  even,  we  can  pair  the 
numbers  belonging  to  the  exponent  p'  so  that  the  sum  of  a  pair  is  0  or  p; 
hence  there  exists  a  residue  of  an  nth  power  =  —  1  (mod  p) ;  but  none  if 
p'  is  odd. 

K.  Zsigmondy^^  obtained  by  the  use  of  abelian  groups  known  theorems 
on  the  number,  product  and  sum  of  the  roots  of  x*=  1  (mod  m). 

G.  Speckmann^^^  considered  x^=a  (mod  p),  where  p  is  an  odd  prime. 
Set  P=(p  — 1)/2.  When  they  exist,  the  roots  may  be  designated  P  —  k, 
P-\-l-\-k,  whose  sum  is  p.  The  successive  differences  of  P^,  (P-|-l)^ 
(P+2)2,.  .  .  arep,  p+2,  p+4, .  .  ..  Thesumof  2  =  s+l  termsof  2,4,  6, .  . .  is 
s'^+Ss+2  =  z^+z.  Adding  to  the  latter  the  remainder  r  obtained  by  di\'iding 
P^  by  p,  we  must  get  pn-{-a.    Hence  in  pn-\-a—r  we  give  to  n  the  values 

i»«Atti  R.  Accad.  Lincei,  Rendiconti,  (5),  2,  1893,  259-265. 

"'Ueber  nte  Potenzreate  und  binomische  Congruenzen  dritten  Grades,  Progr.,  Freising,  1895. 

>»^\rcliiv  Math.  Phys.,  (2),  14,  1896.  445-8;  15, 1897,  335-6. 


Chap.  VII]  BiNOMIAL  CONGRUENCES.  217 

0,  1,2,...  until  we  reach  a  number  of  the  form  ^-\-z  (found  by  extracting 
the  square  root).  Then  fc  =  s,  so  that  the  roots  P-A^,  P+l+Zc  are  found. 
N.  Amici^^^  proved  that  if  neither  m  nor  6  is  divisible  by  the  prime  p, 
and  if  a  is  a  given  root  of  x'^=h  (mod  p),  and  if  /3,  g  are  (existing)  integers 
such  that 

i3(/)(p')-p'-i  +  l=mg, 

then  of^'^"^  is  a  root  of  x™=6  (mod  y^).  Hence  we  limit  attention  to  the 
case  X  =  l.  Consider  henceforth  x'^=h  (mod  p),  where  p  =  2''/i+l  is  an 
odd  prime,  h  being  odd,  and  6  not  divisible  by  p.  First,  let  ¥^8.  Then 
6''=1  (mod  p)  is  a  necessary  and  sufficient  condition  for  solvabihty  and 
x=  ±  y^  are  roots,  where  q  is  such  that  2^q  —  1  is  divisible  by  /i.  If  gr  is  a 
quadratic  non-residue  of  p,  all  2"  roots  are  given  by  ±  6''gr''',  where  e  =  6i+2e2 
4- . . .  +2*~^€^,_i,  the  ei  taking  the  values  0  and  1  independently.  Finally, 
let  A:<s.  Then  two  roots  ±(3  are  determined  by  the  method  of  TonelU, 
while  all  the  roots  are  given  by 

x==^^g'',  t  =  e,+2e2+  .  .  .  +2'-\_„  €^  =  0  or  1. 

R.  Alagna^°°  considered  a  prime  p  =  4/c+l  for  which  /b  is  a  prime.  Since 
2  is  known  to  be  a  primitive  root  of  p,  it  is  easy  to  write  down  those  powers 
of  2  which  give  all  the  roots  of  x'^=l  (mod  p),  where  d  is  one  of  the  six 
divisors  2'  or  2'k  of  p  — 1,  likewise  of  x'^^N,  since  N  must  be  congruent  to 
an  even  power  of  2.  For  the  modulus  p^,  we  may  apply  the  first  theorem 
of  Amici  or  proceed  directly.  The  same  questions  are  treated  for  a  prime 
4A;+3  for  which  2A;  +  1  is  a  prime. 

A.  Cunningham^"^  treated  at  length  the  solution  of  x^=\  (mod  iV'), 
where  iV  is  a  prime,  and  gave  tables  showing  all  incongruent  roots  when 
<  =  1,  2,  N-^  101,  I  any  admissible  divisor  of  iV  —  1 ;  also  for  a  few  additional 
f's  when  N  is  small. 

Cunningham^oi"  treated  a^=  1  (mod  q^)  and  3.2^=  ±  1  (mod  p).  He^oi*- 
treated  the  problem  to  find  5''=+l  or  ±a,  given  a^=\,  a''=^h  (mod  p), 
where  ^  is  odd  and  ^,  x,  17  are  the  least  values  of  their  kind;  also  given 
a*=l,  a'^^^h,  a'=^c,  to  find  the  least  /?  and  7  such  that  h^=c,  c^=6 
(mod  p). 

W.  H.  Besant^°^  would  solve  y^  =  ax+h  by  finding  the  roots  s  of  s^=6 
(mod  a).     Then  y  =  ar-\-s,  x  =  ar^-\-2rs+{s'^  —  h)/a. 

G.  Speckmann^°^  replaced  x"=A;  (mod  p)  by  the  pair  of  congruences 
x"~^=r,  xr^A;  (mod  p).  In  np+/c  give  to  n  the  values  0,  1,  2, .  . .  until  we 
find  one  for  which  np+k  =  rx  such  that,  by  trial,  x"~^=r.  The  method  is,  of 
course,  impractical. 

"'Rendiconti  Circolo  Mat.  di  Palermo,  11,  1897,  43-57. 

""Rendiconti  Circolo  Mat.  di  Palermo,  13,  1899,  99-129. 

"^Messenger  of  Math.,  29,  1899-1900,  145-179.     Errata,  Cunmngham226,  p.  155.    See  13a  of 

Ch.  IV. 
"i^Math.  Quest.  Educ.  Times,  71,  1899,  43-4;  75,  1901,  52-4. 
2"&/6td.,  (2),  1,  1902,  70-2. 
^o^Math.  Gazette,  1,  1900,  130. 
'"'Archiv  Math.  Phys.,  (2),  17,  1900,  110-2,  120-1. 


218  History  of  the  Theory  of  Numbers.  [Chap,  vii 

G.  Picou^*^  applied  to  the  case  n  =  2  Wronski's^^^  formula  for  the  resi- 
dues of  71  th  powers  modulo  M,  M  arbitrary.     For  example,  if  M  =  \Qa^\, 

(h=^Sa)-^  =pa{^h-iy  (mod  M). 

[If  8a  were  replaced  by  4a,  we  would  have  an  identity  in  h.] 

P.  Bachmann^^  (pp.  344-351)  discussed  x"'=a  (mod  p"),  p>2,  p  =  2. 

G.  Arnoux-°^  solved  x^^=79  (mod  3-5-7)  by  getting  the  residue  2  of  79 
modulo  7  and  that  of  14  modulo  0(7)  =6  and  solving  x'^=2  (mod  7)  by  use 
of  a  table  of  residues  of  powers  modulo  7.  Similarly  for  moduli  3,  5.  Take 
the  product  of  the  roots  as  usual. 

M.  Cipolla-"'^  generahzed  the  results  of  Alagna-*^"  to  the  case  of  a  prime 
p  =  2"'q-\-l,  7n>0,  q  an  odd  prime,  including  unity.  For  any  divisor  d  of 
p  — 1,  the  roots  of  x'^=N  (mod  p)  are  expressed  as  given  powers  of  a  primi- 
tive root  a  of  p.  If  2  belongs  to  the  exponent  2''co  modulo  p,  where  w  is 
odd,  theng'=  1  (mod  p)  if  and  only  if  2""^  is  the  highest  power  of  2  dividing  m. 

Cunningham-"^"  found  the  sum  of  the  roots  of  (i/"=tl)/(?/±l)  =  0 
(mod  p). 

M.  Cipolla'"^  proved  the  existence  of  an  integer  k  such  that  k~  —  q  is  a 
quadi'atic  non-residue  of  the  prime  p  not  dividing  the  given  integer  q.     Let 

Un  =  h^q\{kWqr-{k-Vqr\, 

v^=Viik+V¥^r+{k-V¥^r\. 

By  expansion  of  the  binomials  it  is  shown  that  the  roots  of  x^=q  (mod  p) 
are  given  by  =*=W(p_i)/2  and  by  ±y(p+i)/2.     These  may  be  computed  by  use  of 

Wn=2kWn_i—qWn-2  (mod  p)  {w  =  u  or  v), 

with  the  initial  values  Uq  =  1,  Wi  =  p;  ^0  =  1?  Vi  =  k.  Although  u^,  y„  are  the 
functions  of  Lucas,  the  exposition  is  here  simple  and  independent  of  the 
theory  of  Lucas  (Ch.  XVII). 

M.  Cipolla-°^  proved  that  if  5  is  a  quadratic  residue  and  k^—q  is  a 
quadratic  non-residue  of  an  odd  prime  p,  z~^q  (mod  p^)  has  the  roots 

^lVq\{k+Vqr-{k-V~qr\, 
where  r  =  p^~\p  — 1)/2.     Other  expressions  for  the  roots  are 

^hq'{(k+V¥^r+{k-V¥^y\, 

t=ip^-2p^-'  +  l)/2,  s  =  p^-\p  +  l)/2. 

Thus  if  Zi~=q  (mod  p),  the  roots  modulo  p^  are  ^q'zi^^''^  (TonelH^^^). 
Finally,  let  n=TLpi^',  where  the  p's  are  primes  >3;  take  e,  =  =»=l  when 
Pi=^l  (mod  4).    There  exists  a  number  A  of  the  form  k^—q  such  that 

^ML'iatermddiaire  dea  math.,  8,  1901,  162. 

'^o* Assoc,  frang.  av.  sc,  31,  1902,  II,  185-201. 

»«Periodico  di  Mat.,  18,  1903,  330-5. 

»'»«Math.  Quest.  Educ.  Times,  (2),  4,  1903,  115-6;  5,  1904,  80-1. 

"'Rendiconto  Accad.  Sc.  Fis.  e  Mat.  Napoli,  (3),  9,  1903,  154-163. 

"8/6id.,  (3),  10,  1904,  144-150. 


I 


Chap.  VII]  BiNOMIAL  CONGRUENCES.  219 

(A/pi)  =  €i,. . .,  (A/pJ  =  e^,  where  the  symbols  are  Legendre's.  Call  M 
the  1.  c.  m.  of  pl^~\pi  —  ei)/2  for  {=1,...,  v.  Then  z^=q  (mod  n)  has 
the  root 

A.  Cunningham^°^  indicated  how  his  tables  may  be  used  to  solve 
directly  x''=  —1  (mod  p)  for  n  =  2,  3,  4,  6,  12.  From  p  =  a^-\-h^,  we  get  the 
roots  re  =  ±  a/6  of  a;^=— 1  (modp).  Also  p  =  a^+ 6^  =  c^+2(i^  gives  the  roots 
=i=  d{a-\-b) / (ce)  and  ±c(a±6)/(2de)  of  x^^—1  (mod  p),  where  e  =  a  or  6. 
Again,  p  =  A^+SB^  gives  the  roots  iA-B)/{2B),  {B+A)/{B-A),  and 
their  reciprocals,  of  x^=l  (mod  p). 

M.  Cipolla^°^  gave  a  report  (in  Peano's  symbolism)  on  binomial  con- 
gruences. 

M.  Cipolla^^°  proved  that  if  p  is  an  odd  prime  not  dividing  q  and  if 
z^=q  (mod  p)  is  solvable,  the  roots  are 

z=  ^2{qs,+q\+q\+  .  .  .  +5'^~'%-4+Sp-2) 
where 


s,=r+z+...+ 


m- 


Then  x^=q  (mod  p'^)  has  the  root  z^^  V,  e  =  (p^-2p^-^  +  l)/2.    For  p=l 
(mod  4),  x'^=q  (mod  p)  has  the  root 

4 1  5^S2,_i-  2  q^-%j_s+2  S  g^s^^.i  (z  =  ^)  • 

1=1  y=i  i=l  \  1     / 

M.  Cipolla^^^  extended  the  method  of  Legendre^^®  and  proved  that 

x^"'=l-\-TA  (mod  2*), 
for  A  odd  and  s^w+2,  has  a  root 

x  =  l+2^Aci-22M2^2+-  •  •  +  (-l)"'"'2"^A"c„,  n=r  _^~^  J, 

where  ^_]_  ^(2'"-l)(2-2'"-l) . .  .(n^l-2'"-l) 

are  the  coefficients  in 

il+zy^'"'  =  l+c,z-c,z'+c,z'-  . . .  -(-1)X2"+. . .. 
0.  Meissner^^^  gave  for  a  prime  p  =  Sn+5  the  known  root 

£+3  £-1 

^  =  D  »   oix^=D  (modp),  D  *  =i  (modp). 

But  if  2)^p-i)/4=  _i  (mod  p),  a  root  is  ^](p  — 1)/2)!,  since  the  square  of  the 
last  factor  is  congruent  to  (  — l)(p+^)/2  j-^y  wrjigon's  theorem. 

Tamarkine  and  Friedmann^^^  expressed  the  roots  of  z^^q  (mod  p)  by  a 
formula,  equivalent  to  Cipolla's,^^° 

^osQuadratic  Partitions,  1904,  Introd.,  xvi-xvii.     Math.  Quest.  Educ.  Times,  6,  1904,  84-5;  7, 

1905,  38-9;  8,  1905,  18-9. 
""Rendiconto  Accad.  Sc.  Fis.  e  Mat.  Napoli,  (3),  11,  1905,  13-19. 
"i/6id.,  304-9. 

'"Archiv  Math.  Phys.  (3),  9,  1905,  96. 
2"Math.  Annalen,  62,  1906,  409. 


220  History  of  the  Theory  of  Numbers.  [Chap,  vii 

(p-3)/2 

z==b2    2    q^''-''-"'s2^+,. 

m  =  0 

For,  according  as  2/^  is  or  is  not  =q  (mod  p),  we  have 

y\i-{y^-qy~'\=y  or o  (mod p). 

We  can  express  S2m+\  in  terms  of  Bemoullian  numbers. 

A.  Cunningham-^^  gave  a  tentative  method  of  solving  x'^=a  (mod  p). 
He-^^^"  noted  that  a  root  Y=2r]^  of  Y^=-l  leads  to  the  roots  of  y^=-l 
(mod  p). 

M.  Cipolla^^^  employed  an  odd  prime  p  and  a  divisor  n  of  p  — l=ni/. 
If  Ti, . . .,  r,  form  a  set  of  residues  of  p  whose  nth  powers  are  incongruent, 
and  if  ^'=1  (mod  p),  then  x''=q  (mod  p)  has  the  root 

k=0  ;=1 

Forn  =  2,  this  becomes  his^^°  earUer  formula  by  taking  1,2,...,  (p  — 1)/2  as 
the  r's.  Next,  let  p  —  l=mji,  where  m  and  /x  are  relatively  prime  and  m 
is  a  multiple  of  n.  If  7  and  8  belong  to  the  exponents  m  and  /x  modulo  p, 
the  products  7''5*  {r<m/n,  s<iJi)  may  be  taken  as  ri,. . .,  r,.  According  as 
nk=  1  or  not  (mod  /x),  we  have 

y{nk-l)m/n -j^ 

At=  -ufi — „^_i     -     or  Ak=0  (mod  p). 
7       — i 

If  n  is  a  prime  and  n"  is  its  highest  power  dividing  p  — 1,  there  exists  a 
number  co  not  an  nth  power  modulo  p  and  we  may  set  m  =  n'',  7=0)"  (mod  p). 
In  particular,  if  n  =  2,  x^^q  has  the  root 

_  1      P+2^-1  2'"-'-l 

2  «=o 

where  co  is  a  quadratic  non-residue  of  p.  If  p=5  (mod  8),  we  may  take 
CO  =  2  and  get 

M.  CipoUa^^®  considered  the  congruence,  with  p  an  odd  prime, 

x^  =a  (mod  p"*),  r<7n, 

a  necessary  condition  for  which  is  that  h  =  {a^  —  a)/p'"^'  be  an  integer. 
Determine  A  by  a^  A=h  (mod  p"*).  Then  the  given  congruence  has  the 
root  axo  if  Xq  is  a  root  of 

:r'''=l-^p'+'  (modp"*). 
This  is  proved  to  have  the  root 

'"Math.  Quest.  Educ.  Times,  (2),  13,  1908,  19-20. 

^^"^Ibid.,  10,  1906,  52-3. 

"»Math.  Annalen,  63,  1907,  54-61. 

>"Atti  R.  Accad.  Lincei,  Rendiconti,  (5),  16,  I,  1907,  603-8. 


Chap.  VII]  BiNOMIAL  CONGRUENCES.  221 

where  Ci  =  l/p*", . . .  are  given  by  the  expansion 


^1—2  ==l—CiZ  —  C2^  — 


M.  Cipolla^"  treated  x"=a  (mod  p"*)  where  n  divides  4>{p^).  We  may 
set  n  =  p'v,  where  v  divides  p  — 1.     Determine  integers  a,  j3  such  that 

Then  the  initial  congruence  has  the  root  yx^"  if  y^^=a^  (mod  p"*),  solved  as 
in  his  preceding  paper,  and  if  Xi  is  a  root  of  x''=a  (mod  p"*).  The  latter 
has  the  root 

^  A;=0  t  =  l 

where  t={p  —  l)/v,  pi^rf""'  (mod  p'"),  ri, . . .,  r^  being  integers  prime  to  p 
such  that  their  j/th  powers  are  incongruent  and  form  a  group  modulo  p"*. 

K.  A.  Posse^^^  gave  a  simplified  exposition  of  Korkine's^^^  method  of 
solving  binomial  congruences.     Cf.  Posse/^^  Schuh.^^^'^ 

F.  Stasi^^^  proved  that  we  obtain  all  solutions  of  x^=a^  (mod  n),  where 
n  is  odd  and  prime  to  a,  by  expressing  n  as  a  product  of  two  relatively 
prime  factors  P  and  Q  in  all  ways,  setting  x  —  a  =  Pz  and  finding  z  from 
Pz+2a=0  (mod  Q).  [Instead  of  his  very  long  proof,  it  may  be  shown  at 
once  that  we  may  take  x  —  a,  x+a  divisible  by  P,  Q,  respectively.] 

L.  Grosschmid^^°  gave  for  the  incongruent  roots  of  x^^r  (mod  M)  an 
expHcit  formula  obtained  by  means  of  the  ideal  factors  of  ilf  in  a  quadratic 
number-field. 

L.  Grosschmid^^^  treated  the  roots  of  quadratic  binomial  congruences. 

A.  Cunningham^^^  solved  x^=  —1  (mod  p),  where  p  =  616318177  is  a  prime 
factor  of  2^^  — 1;  by  using  various  small  moduli,  he  obtained  p  =  24561^ + 
36161 

L.  von  Schrutka^^^''  used  a  correspondence  between  the  integers  and 
certain  rational  numbers  to  treat  quadratic  congruences  without  novelty  as 
to  results.  The  method  will  be  given  under  the  topic  Fields  in  a  later 
volume  of  this  History. 

Grosschmid^^^  employed  the  products  R  and  N  of  all  the  quadratic 
residues  and  non-residues,  respectively,  ^2n  of  a  prime  p  =  4n+l.     Then 

R^={-iy+\  iv2=(_i)«  (modp). 

2"Atti  R.  Accad.  Lincei,  Rendiconti,  (5),  16,  I,  1907,  732-741. 

"'Charlkov  Soobsc.  Mat.  Obs6  (Report  Math.Soc.  Charkov),  (2),  11, 1910,  249-268  (Russian). 

"»I1  BoU.  Matematica  Gior.  Sc.-Didat.,  9,  1910,  296-300. 

«20Jour.  fur  Math.,  139,  1911,  101-5. 

""Math.  6s  Phys.  Lapok,  Budapest,  20,  1911,  47-72  (Hungarian). 

222Math.  Questions  Educat.  Times,  (2),  20,  1911,  33-4  (76). 

2««Monatshefte  Math.  Phys.,  23,  1912,  92-105. 

223Archiv  Math.  Phys.,  (3),  21,  1913,  363;  23,  1914-5,  187-8. 


222  History  of  the  Theory  of  Numbers.  [Chap,  vii 

Hence  ±7?  and  =»=iV  are  the  roots  of  x^=— 1  (mod  p)  according  as 
p  =  S7n  +  l  or  Sm  +  5.  ,;, 

U.  Concina"^  proved  the  first  result  by  Legendre.^^ 

A.  Cunningham--^  tabulated  the  roots  of  i/*=±2,  2?/"*=±l  (mod  p), 
for  each  prime  p<  1000. 

Cunningham"^  listed  the  roots  of  ?/'=  ±  1  (mod  p"),  where  l^qp", 
p  being  an  odd  prime  ^19,  p''<10^,  a  =  l  and  often  also  a  =  2,  q  a  factor 
of  p— 1. 

A.  Gcrardin  and  L.  Valroff"7  solved  2i/=l  (mod  p),  1000<p<5300. 

Cunningham-^^  announced  the  completion  of  tables  giving  all  proper 
roots  of  ?/'"=  1  (mod  p*)  for  m  odd  ^15,  and  of  'y'"=  —  1  (mod  p*)  for  m  even 
^  14.  These  tables  have  since  been  completed  up  to  p*  <  100000  and  are 
now  nearly  all  in  type. 

T.  G.  Creak^-^  announced  the  completion  of  like  tables  for  m  =  16  to 
50;  52,  54,  56,  63,  64,  72,  75,  and  10^<p'^<10^ 

H.  C.  Pocklington--^  noted  that  if  p  is  a  prime  8m+5  and  a}"'^^=-l, 
x^=a  (mod  p)  has  the  roots  =»=^(4a)'"'^^  He  showed  how  to  use  {t-\- 
u\/DY  to  solve  a;-=  —D  (mod  p=4A-+l),  and  treated  a:^=a. 

*J.  Maximoff^^°  treated  binomial  congruences  and  primitive  roots. 

*G.  Rados-^^  gave  a  new  proof  of  known  criteria  for  the  solvability  of 
x-  =  D  (mod  p).  He-^^  gave  a  new  exposition  of  the  theory  of  binomial 
congruences  without  using  indices. 

Congruences  ^''"^^l  (mod  p")  are  treated  in  Chapter  IV.  Euler**'' 
of  Ch.  XVI  solved  x-=— 1  (mod  p).  Lazzarini"^  of  Ch.  I  erred  on  the 
number  of  roots  of  2-=  —3  (mod  n).  Many  papers  in  Ch.  XX  treat  x*=a; 
(mod  10").  The  following  papers  from  the  first  part  of  Ch.  VII  treat  also 
binomial  congruences:  Euler,^  Lagrange,^  Poinsot,"  Cauchy,^^  Lebesgue," 
Epstein,i^2  Korkine."^ 

=«*Periodico  di  Mat.,  28,  1913,  212-6. 
22*Messenger  Math.,  43,  1913-4,  52-3. 
2"/Wd.,  148-163.     Cf.  Cunningham .201 
227Sphinx-0edipe,  1913,  34;  1914,  18-37,  73. 
228Messenger  Math.,  45,  1915-6,  69. 
2^Proc.  Cambridge  Phil.  Soc,  19,  1917,  57-9. 
2^Bull.  Soc.  Phys.-Math.  Kasan,  (2),  XXI. 
23iMath.  4s  Term^s  Ertesito,  33,  1915,  758-62. 
^Ibid.,  34,  1916,  641-55. 


CHAPTER  VIII. 

HIGHER  CONGRUENCES. 

A  Congruence  of  Degree  n  has  at  most  n  Roots  if  the 
Modulus  p  is  a  Prime. 

J.  L.  Lagrange^  proved  that,  if  a  is  not  divisible  by  the  prime  p,  ax'^+fex""^ 
+ .  . .  is  divisible  by  p  for  at  most  n  integers  x  between  p/2  and  —  p/2. 
For,  let  a,  jS, .  .  . ,  p,  (7  be  n+1  such  distinct  integers.     Then  the  quotient  of 

a(a:'^-a")+6(a:'^-i-a"-^)+  .  .  . 

by  a:  —  a  is  a  polynomial  aa:"~^  +  .  .  .  which  is  divisible  by  p  when  x=^,. .  .,a. 
Proceeding  as  before,  we  finally  have  a{p—<7)  divisible  by  p,  which  is 
impossible. 

L.  Euler^  noted  that  a:"  —  1  is  divisible  by  a  prime  p  for  not  more  than  n 
integers  x,  0<x<p.  For,  if  x  =  a,  is  such  an  integer,  then  x  —  a  divides 
x^  —  l—mp,  where  m  is  a  suitable  integer;  the  quotient  /  is  of  degree  n  — 1. 
If  a:  =  6  is  a  second  such  integer,  x  —  h  divides/— m'p.  Proceeding  as  in  alge- 
bra, we  obtain  the  theorem  stated.  [The  argument  is  applicable  to  any 
polynomial  of  degree  n  in  x.] 

A.  M.  Legendre^  noted  that  P^(x  —  a)Q-\-pA  has  only  one  more  root 
than  Q. 

C.  F.  Gauss'*  proved  the  theorem  by  assuming  that  there  is  a  congruence 
ox"4- . .  .  =  0  (mod  p)  with  more  than  n  roots  a, .  .  .,  and  that  every  con- 
gruence of  degree  I,  Kn,  has  at  most  I  roots.  Substituting  y+a  for  x,  we 
obtain  a  congruence  a?/"-f- ...  =0  with  more  than  n  roots,  one  of  which  is 
zero.  Removing  the  factor  y,  we  obtain  a?/"~^+.  .  .  =  0  with  more  than 
w  — 1  roots,  contrary  to  hypothesis. 

Gauss^  noted  that  if  a  is  a  root  of  ^=0  (mod  p),  then  ^  is  divisible  by 
x  —  a  modulo  p.  li  a,  b,. .  .  are  incongruent  roots,  ^  is  divisible  modulo  p 
by  the  product  (x  —  a){x  —  b)....  Hence  the  number  of  roots  does  not 
exceed  the  degree  of  ^. 

A.  Cauchy^  made  the  proof  by  use  of  X=(x  —  a)Xi  (mod  p),  identically 
in  x,  where  the  degree  of  Xi  is  one  less  than  the  degree  of  X. 

A.  L.  Crelle^  and  S.  Earnshaw^  gave  Lagrange's  proof. 

Crelle^  proved  that  if  ei, . . .,  e„  are  n  distinct  roots, ' 

ax^^-i- . .  .  =  a(x  —  ei) . .  .{x  —  e^+pN. 

iMem.  Ac.  BerUn,  24,  ann6e  1768  (1770),  p.  192;  Oeuvres,  2,  1868,  667-9. 

='Novi  Comm.  Ac.  Petrop.,  18,  1773,  p.  93;  Comm.  Arith.,  1,  519-20. 

»M^m.  Ac.  Roy.  Sc,  Paris,  1785,  466;  TWorie  des  nombres,  1798,  184. 

♦Disq.  Arith.,  1801,  Art.  43. 

^Posthumous  paper,  Werke,  2,  p.  217,  Art.  338  (p.  214,  Art.  333).     Maser's  German  translation 

of  Gauss'  Disq.  Arith.,  etc.,  1889,  p.  607  (p.  604). 
"Exercices  de  Math.,  4,  1829,  219;  Oeuvres,  (2),  9,  261;  Comptes  Rendus  Paris,  12,  1841, 

831-2;  Exercices  d'Analyse  et  de  Phys.  Math.,  2,  1841,  1-40,  Oeuvres,  (2),  12. 
'BerUn  Abhand.,  Math.,  1832,  p.  34. 
Cambridge  Math.  Jour.,  2,  1841,  79. 
•BerUn  Abhand.,  Math.,  1843,  50-54. 

223 


224  History  of  the  Theory  of  Numbers.  [Chap.viii 

'J 
L.  Poinsot^"  gave  the  proof  due  to  Crelle.'  ■ 

J.  A.  Gninert^^  proceeded  by  induction  from  n  — 1  to  n,  making  use  of 

the  first  part  of  Lagrange's  proof. 
D.  A.  da  Silva^'  gave  a  proof. 

Number  of  Roots  of  Higher  Congruences. 

G.  Libri^"  found  that /(a:,  ?/, .  .  .)  =  0  (mod  m)  has 


I 


1       b        d 

-22 

Wl  x=o  v=t 


;--!      2A-7rf     .    .    2Uj\ 

.ALi  COS \-x  sm \ 

Lfc=o        Tnn  m  \ 


sets  of  solutions  such  that  a^x^6,  c^?/^d, ....     The  total  number  of 
sets  of  solutions  is 

1  ^    ^  r,  ,         27r/,        47r/ ,  ,         Am-X)'KJ\ 

—  2   2  . .  .  O +COS  — ^+cos  — ^+  .  .  .  +COS  2^^ '-^  V 

7^1=0  v=o       y  m  m  m        \ 

V.  A.  Lebesgue^^  proved  that  if  p  is  a  prime  we  obtain  as  follows  the 
residue  modulo  p  of  the  number  S>k  of  sets  of  solutions  of  F{xx, .  .  .,  x„)  =  0 
(mod  p),  in  which  each  x,  is  chosen  from  0,  1,.  .  .,  p  — 1,  and  F  is  a  poly- 
nomial with  integral  coefficients.  Let  2A  be  the  sum  of  the  coefficients  of 
the  terms  Ax^  ■  ■  x/  of  the  expansion  of  F^~'^  in  wliich  each  of  the  exponents 
a, . . . ,  ^  is  a  multiple  >  0  of  p  - 1 .     Then  Sk=  ( - 1)  *"^^  2 A  (mod  p) . 

Henceforth,  let  p  =  hm+l.  First,  let  F  =  x"'—a.  In  F^~^  the  coefficient 
of  a;""'?-!-"^  is  (p-^)(-a)"=a"  (mod  p).  The  exponent  of  x  will  be  a  multiple 
>0  of  p  — 1  only  when  n  =  k(p  —  l)/d,  for  A:  =  0,  1,.  .  .,  d  —  1,  where  c?  is  the 
g.  c.  d.  of  m  and  p-L  Thus  51=20*^^"^^'''^  (mod  p),  while  evidently  Si<p. 
According  as  a'-^~^^^'^=l  or  not,  we  get  Si=d  or  0. 

Next,  let  F  =  x"'-ay"'-h.  Set  c  =  ay"'-\-h.  In  (x"'-c)p-^  we  omit  the 
terms  in  which  the  exponent  of  x  is  not  a  multiple  >0  of  p  — 1  and  also  the 
^rn(p-i)  jjq|.  containing  y.  Since  the  arithmetical  coefficient  is  =1  as  in  the 
first  case,  we  get 

In  this,  we  replace  c''''  by  those  terms  of  (ay"* +6)''''  in  which  the  exponents 
are  multiples  >0  of  p  — 1,  viz., 


SGB^"^") 


kh-lhUh 


Set  2/  =  1,  and  sum  for  A*  =  1, .  .  . ,  w  —  1 ;  we  get  —S2  (mod  p).     It  is  shown 
otherwise  that  *S2  is  a  multiple  <mp  of  m. 

To  these  two  cases  is  reduced  the  solution  of 

(1)  ^  =  01X1"*+. .  .-\-a^k"=a  (mod  p  =  hm-\-l). 

"Jour,  de  Math^matiques,  10,  1845,  12-15. 

"Klugel'8  Math.  Worterbuch,  5,  1831,  1069-71. 

"Proprietades . .  .Congruencias  binomias,  Lisbon,  1854.     Cf.  C.  Alasia,   Rivista  di  fisica,  mat. 

e  sc.  nat.,  4,  1903,  p.  9. 
"Mdm.  divers  Savants  Ac.  Sc.  de  I'Institut  de  France  (Math.),  5,  1838,  32  (read  1825).     Jour. 

fur  Math.,  9,  1832,  54.     To  be  considered  in  vol.  n. 
"Jour,  de  Math.,  2,  1837,  253-292.    Cf.  vol.  3,  113;  vol.  4,  366. 


Chap.  VIII]  NuMBER  OF  RoOTS  OF  CONGRUENCES.  225 

Denote  by  P  the  sum  of  the  first  /  terms  of  F  and  by  Q  the  sum  of  the  last 
k—f  terms.  Let  gr  be  a  primitive  root  of  p.  Let  P°  be  the  number  of  sets 
of  solutions  of  P=0  (mod  p);  P^'^  the  number  for  P^g^  (mod  p);  Q^  and 
Q^'^  the  corresponding  numbers  for  Q=0,  Q=g'.  Then  the  number  of 
sets  of  solution  of  P=Q  (mod  p)  is  P°Q°+/iS:=rP^*^Q^*\  Hence  we  may 
deduce  the  number  of  sets  of  solutions  of  P=0  from  the  numbers  for 
P  =  Aand  Q= -A.  For  P=  a,  we  employ  P  =  P,  Q  =  /x'"and  get  P°  =  P° 
-\-{'p  —  l)P^^\  which  determines  the  desired  P''^\ 

The  theory  is  applied  in  detail  to  (1)  for  m  =  2,  k  arbitrary,  and  for 
w  =  3,  4,  k  =  2.     Finally,  the  method  of  Libri^^  is  amplified. 

Th.  Schonemann^^  noted  that,  if  Sk  is  the  sum  of  the  A;th  powers  of  the 
roots  of  an  equation  x"+ . .  .  =0  with  integral  coefficients,  that  of  x""  being 
unity,  and  if  >S(p_i)«=n  (mod  p)  for  <=  1,  2, .  .  .,  w,  where  p  is  a  prime  >n, 
the  corresponding  congruence  a;"+  .  .  .  =  0  (mod  p)  has  n  real  roots. 

A.  L.  Cauchy^^  considered  F{x)  =  Q  (mod  M),  with  M=AB. . .,  where 
A,  B,.  .  .  are  powers  of  distinct  primes.  If  F{x)  =  0  (mod  A)  has  a  roots, 
F(x)  =  0  (mod  B)  has  /S  roots,  etc.,  the  proposed  congruence  has  a/3.  . .  roots 
in  all.  For,  if  a,  6, . .  .  are  roots  for  the  moduli  A,  B,. .  .  and  X=a  (mod  A), 
X=b  (mod  P), .  .  . ,  then  X  is  a  root  for  modulus  M. 

P.  L.  Tchebychef^°  proved  that,  if  p  is  a  prime,  a  congruence /(x)=0 
(mod  p)  of  degree  m<p  has  m  roots  if  and  only  if  the  coefficients  of  the 
remainder  obtained  by  dividing  x^—x  hj  f{x)  are  all  divisible  by  p. 

Ch.  Hermite^^  proved  the  theorem:  If  fx  and  jjl'  are  the  numbers  of 
sets  of  solutions  of  4>{x,  y)=0  for  the  respective  moduli  M  and  M',  which 
are  relatively  prime,  the  number  of  sets  of  solutions  modulo  MM'  is  /jl/j,'. 
If  0=0  is  solvable  for  a  prime  modulus  p,  it  will  be  solvable  modulo  p"  if 

have  no  common  sets  of  solutions.  In  this  case,  the  number  of  sets  of 
solutions  modulo  p"  is  p"~V  if  tt  is  the  number  for  modulus  p.  Similar 
results  are  said  to  hold  for  any  number  k  of  unknowns.  If  ikf  is  a  product 
of  powers  of  the  distinct  primes  pi, . . .,  p„,  and  if  tt,  is  the  number  of  sets 
of  solutions  of  the  congruence  modulo  Pi,  then  the  number  of  sets  for 
modulus  M  is 


M' 


k~l       TTi-.-TTn 


For  a:^+A|/^=A  (mod  M),  we  have  Xj  =  p,  — (— A/p»),  where  (a/p)  is 
=«=  1  according  as  a  is  a  quadratic  residue  or  non-residue  of  p. 

JuUus  Konig  gave  a  theorem  in  a  seminar  at  the  Technische  Hochschule 
in  Budapest  during  the  winter,  1881-2,  which  was  published  in  the  following 
paper  and  that  by  Rados.^^ 

"Jour,  fiir  Math.,  19,  1839,  293. 

"Comptes  Rendus  Paris,  25,  1847,  36;  Oeuvres,  (1),  10,  324. 
"Theorie  der  Congruenzen,  in  Russian,  1849;  in  German,  1889,  §21. 
"Jour,  fiir  Math.,  47,  1854,  351-7;  Oeuvres,  1,  243-250. 


226  History  of  the  Theory  of  Numbers.  [Chap,  viii 

G.  Raussnitz*^  proved  the  theorem,  due  to  Konig:    Let 
(2)  f{x)  =aoX^-2+OiX''-H  . .  .  +ap_2, 

where  the  a's  are  integers  and  ap_2  is  not  divisible  by  the  prime  p.    Then 
f{x)=0  (mod  p)  has  real  roots  if  and  only  if  the  cyclic  determinant 


(3)  D  = 


Oo     Oi     02     ...     ap_3     ap_2 
«!     a2     03      ...      ap_2     Oo 


Op_20o       Oi        ...        Op_4       Op.s 

is  divisible  by  p.  In  order  that  it  have  at  least  k  distinct  real  roots  it  is 
necessary  that  all  p  —  k  rowed  minors  of  D  be  divisible  by  p.  If  also  not 
all  p  —  k  —  1  rowed  minors  are  divisible  by  p,  the  congruence  has  exactly 
k  distinct  real  roots. 

The  theorem  is  applicable  to  any  congruence  not  ha\'ing  the  root  zero, 
since  we  may  then  reduce  the  degree  to  p  — 2  by  Fermat's  theorem. 

Gustav  Rados-'*  proved  Konig's  theorem,  using  the  fact  that  a  system  of 
p  —  l  linear  homogeneous  congruences  modulo  p  in  p  —  1  unknowns  has  at 
least  k  sets  of  solutions  linearly  independent  modulo  p  if  and  only  if  the 
p  —  k  rowed  minors  are  divisible  by  p. 

L.  Kronecker^^  noted  that,  if  p  is  a  prime,  the  condition  for  the  existence 
of  exactly  p—m  —  1  roots  of  (2),  distinct  from  one  another  and  from  zero, 
is  that  the  rank  of  the  system 

(3')  (a,+,)  (i,A:  =  0,  l,...,p-2) 

modulo  p  is  exactly  m,  where  Os+p_i  =  a^.  The  same  is  the  condition  for 
the  existence  of  a  {p—m  —  l)-fold  manifold  of  sets  of  solutions  of  the  system 
of  linear  congruences 

2'a,+,0,=  0  (mod  p)  ( A  =  0,  1 , . .  . ,  p  -  2) . 

fc=0 

L.  Kronecker^^  gave  a  detailed  proof  of  his  preceding  results,  noted  that 
the  rank  is  ?«  if  not  all  principal  t?? -rowed  minors  are  divisible  by  p  while 
all  VI -{-1  rowed  minors  are,  and  added  that  Co+Ci.t+ .  .  . +Cp_2a:^~^=0 
(mod  p)  has  exactly  s  roots  7^0  if  one  and  the  same  linear  homogeneous 
congruence  holds  between  every  set  of  p—s  (but  not  fewer)  successive 
terms  of  the  periodic  series  Cq,  Ci, . . . ,  Cp_2,  Cq,  Ci, . . . . 

L.  Gegenbauer^^  proved  Kronecker's  version  of  Konig's  theorem. 

Gegenbauer^^  noted  that  Kronecker's  theorems  imply  the  corollary: 

"Math,  und  Naturw.  Berichte  aus  Ungam,  1,  1882-3,  266-75. 

"Jour,  fiir  Math.,  99,  1886,  258-60;  Math.  Termea  Ertesito,  Magj'ar  Tudon  Ak.,  Budapest,  1, 

1883,  296;  3,  1885,  178. 
»Jour.  fur  Math.,  99,  1886,  363,  366. 
**Vorlesungen  liber  Zahlentheorie,  1,  1901,  389-415,  including  several  additions  by  Hensel 

(pp.  393,  399,  402-3). 
"Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  95,  II,  1887,  165-9,  610-2. 
*^Ibid.,  98,  Ila,  1889,  p.  32,  foot-note.     Cf.  Gegenbauer.« 


Chap.  VIII]  NuMBER  OF  RoOTS   OF   CONGRUENCES.  227 

There  exist  exactly  p—m—2  roots  of  (2),  distinct  from  one  another  and 
from  zero,  if  and  only  if  there  exist  exactly  p  —  m  —  2  distinct  linear  homo- 
geneous functions 

p-2 

Xak,hah  {k  =  l,. .  .,p-m-2) 

h=0 

which  remain  divisible  by  p  after  applying  all  cyclic  permutations  of  the 
ah,  so  that 

sV»a,„^0  (mod  p)  filo/iV.'.^.T™-!)- 

A  simple  proof  of  this  corollary  is  given. 

L.  Gegenbauer^^  noted  that  the  niunber  of  roots  of /(a^)=0  (mod  k)  is 

since  D{k)  =  1  or  0  according  as /(a:)  is  divisible  by  k  or  not.  Let  ki,...,ka 
be  a  series  of  increasing  positive  integers  and  g{x)  any  function.  In  the 
first  equation  take  k  =  ki,  multiply  by  g{ki)  and  sum  iorl  =  l,.  .  .,8.  Revers- 
ing the  order  of  the  summation  indices  I,  x  in  the  new  right-hand  member, 
we  get 

i  \f(x),  h\g{k)  =  S  G,  G=XD{ix)g{f,), 

1=1  1=0 

where  in  G  the  summation  index n  takes  those  of  the  values  ki,...,ki  which 
exceed  x.  Thus  G  represents  the  sum  G{f{x)]  ki,...,  k^;  x)  of  the  values 
oi  fifx)  when  ij,  ranges  over  those  of  the  numbers  ki,.  .  .,  ks  which  exceed  x 
and  are  divisors  oif{x).  In  particular,  if  ^(x)  =  1,  G  becomes  the  number  x}/ 
of  the  k's  which  exceed  x  and  divide /(a:). 

Let  f{x)=vi^nx.  Then  f{x)  =  0  (mod  k)  has  {k,  n)  roots  or  no  root 
according  as  m  is  or  is  not  divisible  by  the  g.  c.  d.  {k,  n)  of  k  and  n;  let 
{k,  71]  m)  denote  {k,  n)  or  0  in  the  respective  cases.     Then 

S  {ki,  n;  m)  g{ki)  =  s  G(m^nx;  ki,.  .  .,  k^;  x). 

'-1  x=0 

Let  G{a,  h)  denote  the  sum  of  the  values  of  g(ii)  when  n  ranges  over  all 
the  divisors  >6  of  a;  xl/{a,  h)  the  number  of  divisors  >b  of  a.  Taking 
ki  =  l  ioT  1=1,. . .,  d,  we  deduce 

S  5-1 

S  (Z,  n;  m)^(0=  S  {G{m^nx,  x)-G{m^nx,  b)\. 

1=1  z=0 

For  gil)  =  1,  this  reduces  to  Lerch's^°°  relation  (16)  in  Ch.  X.    Again, 

a  b 

2  {G{m+nx,  x  —  1)  —Gim+nx,h+x)\  =  S  {G{m—nfx,tx)  —  Giin—nfx,ti-\-a)\, 

X-l  u=0 


"SitzuDgsberichte  Ak.  Wiss.  Wien  (Math.),  98,  Ila,  1889,  28-36. 


228  History  of  the  Theory  of  Numbers.  [Chap,  viii 

which  for  g{x)  =  l  yields  the  first  formula  of  Lerch.     Next,  if  the  A;'s  are 
primes  and  g  is  a  prime  distinct  from  them, 

2  Gix'^-q]  k,,..  .,  k,;  x)=  2  (fc,-l,  n;  q)g{ki). 

x=0  /=1 

Finally,  he  treated  f{x)  of  degree  d  =  A-j  —  2,  whose  constant  term  is  prime  to 
each  A-,  and  coefficient  of  x''~'  is  divisible  by  the  prime  k^  if  i<ks  —  k^. 

Gegenbauer^"  noted  that,  if  p  —  l—n  is  the  rank  of  the  system  (3) 
modulo  p,  the  congruence,  satisfied  by  the  distinct  roots  5^0  of  (2)  and  by 
these  only,  is  given  symboUcally  by 


(-X--V 

\dai       daj 


fli+fc  I  =0  (mod  p)  {%,  k  =  0,. . .,  p-2). 


He  obtained  easily  Kronecker's"^  form  of  the  last  congruence.  He  gave 
necessary  and  sufficient  conditions,  expressed  in  terms  of  a  comphcated 
determinant  and  its  /z  —  l  successive  derivatives  with  respect  to  Op_2,  in 
order  that  (2)  and  a  second  congruence  of  degree  p  —  2  shall  have  jx  common 
roots  ?^0,  and  found  the  congruence  satisfied  by  these  ji  common  roots. 
He  deduced  determinantal  expressions  for  the  sum  o-^  of  the  rth  powers  of 
the  roots  of  (2),  and  for  the  coefficients  in  terms  of  the  cr's. 

Michael  Demeczky^^  would  employ  Euclid's  process  to  find  the  g.  c.  d. 
G{x)  modulo  p  of  (2)  and  x'^—x.  If  G{x)  =  0  (mod  p)  is  of  degree  v  it  has 
V  real  roots  and  these  give  all  the  real  roots  of  (2) .  Multiple  roots  are  then 
treated.  The  case  of  any  composite  modulus  is  known  to  reduce  to  the 
case  of  p',  p  a  prime.  If  (2)  has  X  distinct  real  roots,  not  multiple  roots,  we 
can  derive  X  real  roots  of /(a;)  =  0  (mod  p').  If  pi, .  .  . ,  p„  are  distinct  primes 
and  if /(x)  =  0  (mod  p,)  has  X,  real  roots,  then/(x)  =  0  (mod  pi.  .  .p„)  has 
Xi. .  .X„  real  roots,  and  is  satisfied  by  every  integer  x  if  the  former  are. 
Various  sets  of  necessary  and  sufficient  conditions  are  found  that  f{x)  =  0 
(mod  m=np'<)  shall  have  m  distinct  real  roots;  one  set  is  that/(x)=0 
(mod  p'<)  identically  for  each  i. 

L.  Gegenbauer^^  proved  that  a  congruence  modulo  p,  a  prime,  of  degree 
p  — 2  in  each  of  n  variables  has  a  set  of  solutions  each  ^0  if  and  only  if  p 
divides  the  determinant  of  a  cycUc  matrix 

A«       A'     .. 


A'- 

-1  ' 

A"-- 

-2 

A^ 

A'       A^     .. 

where  A"  is  itself  a  cyclic  matrix  in  B^,.  .  .,  B'^~^;  etc.,  until  we  reach 
matrices  in  the  coefficients  of  the  congruence.     An  upper  limit  is  found  for 

'"Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  98,  Ila,  1889,  652-72. 

"Math.  u.  Naturw.  Berichte  aus  Ungarn,  8,  1889-90,  50-59.     Math.  68  Term^s  Ertesito,  7, 

1889,  131-8. 
»=Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  99,  Ila,  1890,  799-813. 


Chap.  VIII]  NUMBER  OF  RoOTS   OF  CONGRUENCES.  229 

the  number  of  sets  of  solutions  each  not  divisible  by  p.     He  proved  that 

s  EzlL         n 

S  ttjXj  ^  +  S  a,+jX,+j-{-h=0  (mod  p) 
y=i  i=i 

has  p""^*"^  sets  of  solutions.     Of  these, 

have  each  x^^O,  where  r  is  the  number  of  the  2'  integers 

6  =1=01  ±02='=  ...=*=  a, 
which  are  divisible  by  p.    The  number  of  sets  of  solutions  of 

s  Pui         n 

l^QjXj  2  +i:a,+jXs+j+b=0  (modp) 

i=i  3=1 

is  expressed  in  terms  of  the  functions  used  for  quadratic  congruences. 

*E.  Snopek^^  gave  a  generalization  of  Konig's  criterion  for  the  solva- 
bility of  a  congruence  modulo  p. 

L.  Gegenbauer^^  proved  that  if  the  p  congruences 

S  Zk^x^-^-'=0  (mod  p)  (X  =  0,  1,. . .,  p-1) 

A:  =  0 

have  in  common  at  least  p—p  distinct  roots  not  divisible  by  p  then  all 
p-rowed  determinants  in  the  matrix  (^^^x)  are  divisible  by  p.  The  converse 
is  proved  when  a  certain  condition  holds.  By  specialization,  Konig's 
theorem  is  obtained. 

Gegenbauer^^  proved  that,  if  r  is  less  than  the  prime  p  and  ii  Zq,.  . .,  2r_i 
are  incongruent  and  not  divisible  by  p,  the  system  of  linear  congruences 

(4)  s'  h+,y,^0  (mod  p)  (p  =  0,  1,. . .,  p-2) 
has  all  its  sets  of  solutions  of  the  form 

(5)  2/*^'sa,2,*  (A:  =  0,  l,...,p-2) 

x=o 

or  not,  according  as  the  matrix  (bk+p),  k  =  r,  r+1, .  .  .,  p  — 2;  p  =  0, . . .,  p  — 2, 
has  a  p— r  — 1  rowed  determinant  prime  to  p  or  not.     Next,  if 

(6)  S  6;fcX^=0  (mod  p) 

A:=0 

has  exactly  r  distinct  roots  Zq,  .  .  .,  z^-i  each  not  divisible  by  p,  every  sys- 
tem of  solutions  of  (4)  is  given  by  (5),  and  conversely.  By  combining  this 
theorem  of  Kronecker's  with  the  former,  we  obtain  Kronecker's  form  of 
Konig's  theorem. 

"Prace  Mat.  Fiz.,  Warsaw,  4,  1893,  63-70  (in  PoUsh). 
'^Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  102,  Ila,  1893,  549-64. 
"Monatshefte  Math.  Phys.,  5,  1894,  230-2.     Cf.  Gegenbauer." 


230  History  of  the  Theory  of  Numbers.  [Chap,  viii 

K.  Zsigmondy^^  proved  that,  if  p  is  a  prime,  there  are  exactly 

,K«,  k)  =  p-  -  Q  p-'  +  g)  p'-'  -...+(- 1)»  g) 

congruences  x"+  .  .  .  =  0  (mod  p)  not  having  as  roots  k  given  distinct  num- 
bers.   Also, 

^(n,  k)=p4^(n-l,  k)  +  {-ir(^^,  rPin,  k  +  l)=^|^{n,  A:)-^(n-l,  k). 

If  n'^k,  \J/{n,  ^)=p"~''(p  — 1)^'.  For  n  =  k,  \p{n,  k)  is  the  number  \J/{n)  of 
congruences  of  degree  n  with  no  root.  The  number  with  exactly  i  roots  is 
{'])\l/{n  —  i).  There  are  {^V)\l/ii—r)  distinct  matrices  (3)  of  rank  i  such 
that  Qr-i  is  the  first  one  of  Qq,  ai,.  .  .  not  divisible  by  p. 

K.  Zsigmondy^^  considered  a  function  $(/)  of  a  polynomial  f{x)  such 
that  $  is  unaltered  when  the  coefficients  of  f{x)  are  increased  by  integral 
multiples  of  the  prime  p.  Let/t^'^(x),  i  =  l,.  .  .,  p'',  denote  the  polynomials 
of  degree  k  which  are  distinct  modulo  p  and  have  unity  as  the  coefficient 
of  x''.     It  is  stated  that 


p"  p- 


r,n-2 


i,  i'  J  =  1 

where  a  takes  those  values  1,  2, .  .  .,  p"  for  which /°^(x)=0  (mod  p)  does 
not  have  as  a  root  one  of  the  given  incongruent  numbers  ai, .  .  .,  a/,  while, 
in  the  outer  sums  on  the  right,  i,  i',.  .  .  range  over  the  combinations  of 
1, .  .  .,  s  without  repetitions. 

Zsigmondy^^  had  earlier  given  the  preceding  formula  for  the  case  in 
which  tti,. .  .,  a,  denote  0,  1,. . .,  p  —  \.  Then  taking  <I>(/)  =  1,  we  get  the 
number  of  congruences  of  degree  n  with  no  root  (Zsigmondy^^) .  Taking 
$(/)  =/,  we  see  that  the  sum  of  the  congruences  of  degree  n  with  no  root  is 
=  0  (mod  p),  aside  from  specified  exceptions.  Taking  $(/)=co-^,  where  co 
is  a  pth  root  of  unity,  and  n^p,  we  see  that  the  system /j;^(x)  takes  each 
of  the  values  1, .  . . ,  p  —  1  (mod  p)  equally  often. 

Zsigmondy^^  proved  his^^'^^  earher  formulas,  obtained  for  an  integral 
value  of  X  the  number  of  complete  sets  of  residues  modulo  p  into  which 
fall  the  values  of  the  fH  (^)  not  having  prescribed  roots,  and  investigated 
the  system  5„  of  the  least  positive  residues  modulo  p  of  the  left  members 
of  all  congruences  of  degree  n  having  no  root.  In  particular,  he  found  how 
often  the  system  B^  contains  each  residue,  or  non-residue,  of  a  gth  power. 
He  investigated  (pp.  19-36)  the  number  of  polynomials  in  x  which  take  k 
prescribed  residues  modulo  p  for  k  given  values  of  x. 

3«Sitzung8ber.  Ak.  Wiss.  Wien  (Math.),  103,  Ila,  1894,  135-144. 
»'Monatshefte  Math.  Phys.,  7,  1896,  192-3. 
"Jahresbericht  d.  Deutschen  Math.  Verein.,  4,  1894-5,  109-111. 
"Monatshefte  Math.  Phys.,  8,  1897,  1-42. 


Chap.  VIIIJ  NuMBER   OF  RoOTS  OF  CONGRUENCES.  231 

L.  Gegenbauer'*"  proved  that  (2)  has  as  a  root  a  quadratic  residue  or 
non-residue  of  the  prime  p  if  and  only  if  the  respective  determinant 

P  =  \  a^+i+o^+i+x  \,      N  =  \  a^+i-a^+i+,  \  (i, /i  =  0, .  .  .,  tt-I) 

be  divisible  by  p,  where  7r=  (p  — 1)/2.  From  this  it  is  proved  that  (2)  has 
exactly  ir—r  distinct  quadratic  residues  (or  non-residues)  of  p  as  roots  if 
and  only  if  P  (or  N)  and  its  tt  — 1— r  successive  derivatives  with  respect  to 
a,_i+ap_2  have  the  factor  p,  while  the  derivative  of  order  tt— r  is  prime 
to  p.    These  residues  satisfy  the  congruence 

where  K  =  P  or  N,  while  the  j^th  power  of  the  sign  of  differentiation  repre- 
sents the  ^'th  derivative.  A  second  set  of  conditions  is  obtained.  Con- 
gruence (2)  has  exactly  tt  —  I—k  distinct  quadratic  residues  as  roots  if  and 
only  if  the  determinants  of  type  P  with  now  i  =  0, .  .  .,  k,  k+1  and  fi  =  0,.  .  . , 
K,  T,  are  divisible  by  p  for  r  =  /c+l, .  .  .,  tt  — 1;  while  p  is  not  a  factor  of  the 
determinant  of  type  P  with  now  i,  fx  =  0,. . .,  k.    These  residues  are  the 

roots  of 

«■ 

S    I  a^+i+a^+i+^  I  x'~^~^=0  (mod  p), 

T=(C 

where  ^  =  0, .  . .,  k,  and  /x  =  0, . . .,  k  —  1,  t  in  the  determinants.  For  non- 
residues  We  have  only  to  use  the  differences  of  a's  in  place  of  sums. 

S.  O.  Satunovskij^^  noted  that,  for  a  prime  modulus  p,  a  congruence  of 
degree  n  (n<p)  has  n  distinct  roots  if  and  only  if  its  discriminant  is  not 
divisible  by  p  and  Sp+q=S g+i  (mod  p)  ior  q  =  l,. . .,  n  —  1,  where  Sk  is  the 
sum  of  the  kth  powers  of  the  n  roots. 

A.  Hurwitz^^  gave  an  expression  for  the  number  A^  of  real  roots  of 

f{x)=aQ-\-aiX-\-  .  .  .  -\-arX''=0  (mod  p), 
where  p  is  a  prime.    By  Fermat's  theorem, 


-1 


N=X  \l-f{xy-'\  (modp). 

a;=l 

Letf{xy-'^  =  Co-\-CiX+  ....    Then  N  is  determined  by 

Ar+l=Co+Cp_i+C2(,-i)+  . . .  (mod  p). 

Letf(xi,  X2)  be  the  homogeneous  form  of  f(x).  Let  A  be  the  number  of 
sets  of  solutions  of f{xi,  a:2)  =  0  (mod  p),  regarding  {xi,  X2)  and  (x/,  X2)  as 
the  same  solution  if  Xi=pxi,  X2=pX2  (mod  p)  for  an  integer  p.    Then 

A-l=  -0^-1-0^71+2^^:1^1  ao'^o. .  Mr^r  (mod  p), 

tto ! .  .  .  a^ ! 

"Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  110,  Ila,  1901,  140-7. 

"Kazani  Izv.  fiz.  mat.  Obsc.  (Math.  Soc.  Kasan),  (2),  12,  1902,  No.  3,  33-49.     Zap.  mat.  otd. 

Obsc,  20,  1902,  I-II. 
"Archiv  Math.  Phys.,  (3),  5,  1903,  17-27. 


232  History  of  the  Theory  of  Numbers.  [Chap,  viii 

where  the  summation  extends  over  the  sets  of  solutions  ^  0  of 

ao+ai+-  •  .+a,  =  p  — 1,  ai  +  2a2+.  .  .+ra^=0  (mod  p-1). 

The  right  member  is  an  invariant  modulo  p  oi  f{xi,  x^  with  respect  to  all 
linear  homogeneous  transformations  on  Xi,  Xo  with  integral  coefficients 
whose  determinant  is  not  divisible  by  p.  The  final  sum  in  the  expression 
for  .4  —  1  is  congruent  to  A^+1.  If  r  =  2,  p>2,  the  invariant  is  congruent 
to  the  power  (p  — 1)/2  of  the  discriminant  a^—^aQa^  of/. 

*E.  Stephan^^  investigated  the  number  of  roots  of  linear  congruences 
and  systems  of  congruences. 

H.  Kuhne^  considered  J{x)=x"'-\- . .  .-\-a„  with  no  multiple  irreducible 
factor  and  with  a^  not  a  multiple  of  the  prime  p.  For  n<m,  let  ^  =  x'*+  .  .  . 
+  6„  have  arbitrary  coefficients.  The  resultant  R{f,  g)  is  zero  modulo  p 
if  and  only  if  /  and  g  have  a  common  factor  modulo  p.  Thus  the  number 
of  all  ^'s  of  degree  n  which  have  no  common  factor  with  /  modulo  p  is  p„, 
where 

P,.^^\R{J,  gYr  (mod  p':),  co  =  p''-np-l), 

the  summation  extending  over  the  p"  possible  ^'s.  He  expressed  p„  as 
a  sum  of  binomial  coefficients.  For  any  two  binary  forms  4>,  \p  of  degrees 
w,  n,  it  is  shown  that 

is  invariant  modulo  p"  under  linear  transformations  with  integral  coeffi- 
cients of  determinant  prime  to  p ;  Ji  is  Hurwitz's^"  invariant. 

M.  Cipolla^^  used  the  method  of  Hurwitz^^  to  find  the  sum  of  the  kth 
powers  of  the  roots  of  a  congruence,  and  extended  the  method  to  show  that 
the  number  of  common  roots  of /(x)  =  0,  g{x)  =  0  (mod  p),  of  degrees  r,  s,  is 
congruent  to  —XCjKi,  where  i,  j  take  the  values  for  which 

0<i^s{p-l),  0<;^r(p-l),  i+j=0  (mod  p), 

the  Cs  being  as  with  Hurwitz,  and  similarly 

g{xy-'  =  Ko+K^x+.... 

The  number  of  roots  common  to  n  congruences  is  given  by  a  sum. 

L.  E.  Dickson^^  gave  a  two-fold  generalization  of  Hurwitz's^^  formula  for 
the  number  of  integral  roots  of  f{x)  =  0  (mod  p) .  The  first  generalization 
is  to  the  residue  modulo  p  of  the  number  of  roots  which  are  rational  in  a 
root  of  an  irreducible  congruence  of  a  given  degree.  A  further  generaliza- 
tion is  obtained  by  taking  the  coefficients  Oi  of  f{x)  to  be  elements  in  the 
Galois  field  of  order  p'*  (cf.  Galois^^  etc.).  Then  let  N  be  the  number  of 
roots  of  f{x)  =  0  which  belong  to  the  Galois  field  of  order  P  =  p"'".     Then 

"Jahresber.  Staatsoberrealsch.  Steyer,  34,  1903-4,  3-40. 
"Archiv  Math.  Phys.,  (3),  6,  1904,  174-6. 
«Periodico  di  Mat.,  22,  1907,  36-41. 
«BuU.  Amer.  Math.  Soc,  14,  1907-8,  313. 


Chap.  VIII]  NuMBER  OF  RoOTS   OF  CONGRUENCES.  233 

N=N*  (mod  p),  where  A''*+l  is  derived  from  either  of  Hurwitz's  two  sums 
for  iV+1  by  replacing  p  by  P.  The  same  replacement  in  Hurwitz's  expres- 
sion for  A  —  1  leads  to  the  invariant  A*  —  l,  where  A*  is  congruent  modulo  p 
to  the  number  of  distinct  sets  of  solutions  in  the  Galois  field  of  order  p""* 
of  the  equation /(xi,  0:2)  =0. 

G.  Rados^'^  considered  the  sets  of  solutions  of 

fix,  y)  =  i\a\i^^  x^'-^+ai''  x''-^+  .  .  .  +05,^^2)2/^"'"'=  0  (mod  p) 

for  a  prime  p.  Let  Ak  denote  the  matrix  of  D,  in  (3),  with  a^  replaced 
by  ap\  Let  C  denote  the  determinant  of  order  (p  —  iy  obtained  from  D 
by  replacing  ak  by  matrix  A^.  Then/=0  has  a  solution  other  than  x^y=0 
if  and  only  if  C  is  divisible  by  p ;  it  has  exactly  r  sets  of  solutions  other  than 
x=y=0  if  and  only  if  C  is  of  rank  (p  — 1)^— r. 

To  obtain  theorems  including  the  possible  solution  x=i/=0,  use 

cf>{x,  y)  =  S W^  x^-i+af  V-2+  .  .  .  +af.,)y''-^-'^Q  (mod  p), 


a  = 


k=0 

CL2  03 


\ap_i+aoai 


Op-3  Op-2  o,p-\  \ 

ap_2         Op.i+floO 
ap_i+ao  Oi  0 


ap_3         ap_2         0      / 

and  tt/fc  derived  from  a  by  replacing  a^  by  al^\  Let  y  be  the  determinant 
derived  from  la|  by  replacing  a^  by  matrix  a^;  and  0  by  a  matrix  whose  p"^ 
elements  are  zeros.  Then  ^=0  has  a  set  of  real  solutions  if  and  only  if 
7=0  (mod  p) ;  it  has  r  sets  of  solutions  if  and  only  if  y  is  of  rank  p^—r. 

*P.  B.  Schwacha^^  discussed  the  number  of  roots  of  congruences. 

*G.  Rados^^  treated  higher  congruences. 

Theory  of  Higher  Congruences,  Galois  Imaginaries. 

C.  F.  Gauss/''  in  a  posthumous  paper,  remarked  that  "the  solution  of 
congruences  is  only  a  part  of  a  much  higher  investigation,  viz.,  that  of  the 
factorization  of  functions  modulo  p.  Even  when  ^(x)  =  0  has  no  real  root, 
^  may  be  a  product  of  factors  of  degrees  ^2,  each  of  which  could  be  said 
to  have  imaginary  roots.  If  use  had  been  made  of  a  similar  freedom  which 
younger  mathematicians  have  permitted  themselves,  and  such  imaginary 
roots  had  been  introduced,  the  following  investigation  could  be  greatly 
condensed."     As  the  later  work  of  Serret^^  shows,  such  imaginaries  can  be 

*'Arm.  Sc.  ficole  Normale  Sup.,  (3),  27, 1910,  217-231.     Math.  6s  Term^s  firtesito  (Report  of 

Hungarian  Ac),  Budapest,  27,  1909,  255-272. 
"Ueber  die  Existenz  und  Anzahl  der  Wurzeln  der  Kongruenz  Sc<x*  =  0  (mod  w),  Progr.  Wilher- 

ing,  1911,  30  pp. 
«Math.  6s.  Term6s  Ertesito,  Budapest,  29,  1911,  810-826. 
"Werke,  2,  1863,  212-240.     Maser's  German  translation  of  Gauss'  Disq.  Arith.,  etc.,  1889, 

604-629. 


234  History  of  the  Theory  of  Numbers.  [Chap,  viii 

introduced  in  a  way  free  from  any  logical  objections.  Avoiding  their  use, 
Gauss  began  his  investigation  by  showing  that  two  polynomials  in  xwith 
integral  coefficients  have  a  greatest  common  divisor  modulo  p,  which  can 
be  found  by  Euclid's  process.  It  is  understood  throughout  that  p  is  a 
prime  (cf.  Maser,  p.  627).  Hence  if  A  and  B  are  relatively  prime  poly- 
nomials modulo  p,  there  exist  two  polynomials  P  and  Q  such  that 

PA+QB^l  {mod  p).  " 

Thus  if  A  has  no  factor  in  common  with  B  or  C  modulo  p,  we  find  by  mul- 
tiplying the  preceding  congruence  by  C  that  A  has  no  factor  in  common 
with  the  product  BC  modulo  p.  If  a  polynomial  is  divisible  by  A,  B,  C,. . ., 
no  two  of  which  have  a  common  factor  modulo  p,  it  is  divisible  by  their 
product. 

A  polynomial  is  called  prime  modulo  p  if  it  has  no  factor  of  lower  degree 
modulo  p.  Any  polynomial  is  either  prime  or  is  expressible  in  a  single 
way  as  a  product  of  prime  polynomials  modulo  p.  The  number  of  distinct 
polynomials  x''+aa:"~^+  .  . .  modulo  p  is  evidently  p".  Let  (n)  of  these  be 
prime  functions.  Then  p'^-l^d{d),  where  d  ranges  over  all  the  divisors  of 
n  (only  a  fragment  of  the  proof  is  preserved).  It  is  said  to  follow  easily 
from  this  relation  that,  if  n  is  a  product  of  powers  of  the  distinct  primes 
a,  6, ... ,  then 

n(n)=p"-2p"/''+2:p"/''^-  .... 

The  rth  powers  of  the  roots  of  an  equation  P  =  0  with  integral  coefficients 
are  the  roots  of  an  equation  Pr  =  0  of  the  same  degree  with  integral  coeffi- 
cients.    If  r  is  a  prime,  P^=P  (mod  r). 

A  prime  function  P  of  degree  m,  other  than  x  itself,  divides  x'  —  l  for 
some  value  of  vKp"".  If  v  is  the  least  such  integer,  j^  is  a  divisor  of  p*"  — 1. 
Hence  P  divides 

(1)  x^"-^-l. 

The  latter  is  congruent  modulo  p  to  the  product  of  the  prime  functions, 
other  than  x,  whose  degrees  are  the  various  divisors  of  m. 

If  P  =  x"'—Ax"'~^-{'Bx"'~^—  ...  is  a  prime  function  modulo  p,  the  re- 
mainders by  dividing  the  sum,  the  sum  of  the  products  by  twos,  etc.,  of 

^  ^p  ~p'  ^p"*~^ 

by  P  are  congruent  to  A,  B,  etc.,  respectively. 

If  V  is  not  divisible  by  p  and  if  m  is  the  least  positive  integer  for  which 
^"•=1  (mod  v),  each  prime  function  dividing  x"  —  !  modulo  p  is  a  divisor  of 
(1)  and  its  degree  is  therefore  a  divisor  of  m.  Let  6  be  a  divisor  of  m,  and 
5',  8", ...  the  divisors  <d  oi  8;  let  ai  be  the  g.  c.  d.  of  v  and  p^-1,  fx'  the 
g.  c.  d.  of  V  and  p*'  —  1, . .  .  and  set  X'  =iilii.' ,  \"  =m/m",  •  •  •  •  Then  the  num- 
ber of  prime  divisors  modulo  p  of  degree  6  of  a:"  —  1  is  iV/5,  if  A^  is  the  num- 
ber of  integers  <y.  which  are  divisible  by  no  one  of  X',  X", ....  A  method 
of  finding  all  prime  functions  dividing  j"— 1  is  based  on  periods  of  powers 
of  X  with  exponents  <  v  and  prime  to  v  (pp.  620-2). 


Chap.  VIII]  HiGHER   CONGRUENCES,   GaLOIS   ImAGINARIES.  235 

If  X  has  been  expressed  as  a  product  of  relatively  prime  factors  modulo 
p,  we  can  express  X  as  a  product  of  a  like  number  of  factors  mod  p"  con- 
gruent to  the  former  factors  modulo  p.  There  is  a  fragment  on  the  case 
of  multiple  factors. 

C.  G.  J.  Jacobi"  noted  that,  if  g  is  a  prime  6n  — 1,  x*+^=l  (mod  q)  has 
q  —  1  imaginary  roots  a +  6V— 3,  where  a  +36^=1  (modg),  besides  the  roots 
±1. 

E.  Galois®^  employed  imaginary  roots  of  any  irreducible  congruence 
F(a;)  =  0  (mod  p),  where  p  is  a  prime.  Let  i  be  one  imaginary  root  of  this 
congruence  of  degree  v.     Let  a  be  one  of  the  p"  —  1  expressions 

a-\-aii+a2i^-\- . .  .  +a^-ii''~^ 

in  which  the  a's  are  integers  <p,  not  all  zero.  Since  each  power  of  a  can 
be  expressed  as  such  a  polynomial,  we  have  a"  =  1  for  some  positive  integer 
n.  Let  n  be  a  minimum.  Then  1,  a, .  .  . ,  a"~^  are  distinct.  Multiply  them 
by  a  new  polynomial  (3  ini;  we  get  n  products  distinct  from  each  other  and 
from  the  preceding  powers  of  a.  If  2n<p''  — 1,  we  use  a  new  multiplier, 
etc.    Hence  n  divides  p"  — 1,  and 

(2)  0^"-^  =  !. 

[This  is  known  as  Galois's  generalization  of  Fermat's  theorem.]  It  follows 
that  there  exist  primitive  roots  a  such  that  a^p^l  if  e<p''  — L  Any  primi- 
tive root  satisfies  a  congruence  of  degree  v  irreducible  modulo  p. 

Every  irreducible  function  F{x)  of  degree  v  divides  x^''—x  modulo  p. 
Since  jF(x)[^"=F(xO  modulo  p,  the  roots  of  F{x)=0  are 

All  the  roots  of  x^'  =  x  are  polynomials  in  a  certain  root  ^,  which  satisfies 
an  irreducible  congruence  of  degree  v.  To  find  all  irreducible  congruences 
of  degree  v  modulo  p,  delete  from  x^''  —  x  all  factors  which  it  has  in  common 
with  x^^—x,  iJL<v.  The  resulting  congruence  is  the  product  of  the  desired 
ones;  the  factors  may  be  obtained  by  the  method  of  Gauss,  since  each  of 
their  roots  is  expressible  in  terms  of  a  single  root.  In  practice,  we  find  by 
trial  one  irreducible  congruence  of  degree  v,  and  then  a  primitive  root  of 
(2);  this  is  done  for  p  =  7,  v  =  3. 

Any  congruence  of  degree  n  has  n  real  or  imaginary  roots.  To  find 
them,  we  may  assume  that  there  is  no  multiple  root.  The  integral  roots 
are  found  from  the  g.  c.  d.  of  F{x)  and  x^~^  —  l.  The  imaginary  roots  of 
the  second  degree  are  found  from  the  g.  c.  d.  oi  F{x)  and  x^'~^  —  l;  etc. 

V.  A.  Lebesgue^^  noted  that,  if  p  is  a  prime,  the  roots  of  all  quadratic 

"Jour,  fiir  Math.,  2,  1827,  67;  Werke,  6,  235. 

"Sur  la  tMorie  des  nombres,  Bulletin  des  Sciences  Mathlmatiques  de  M.  Ferussac,  13, 1830,  428. 
Reprinted  in  Jour,  de  Math^matiques,  11,  1846,  381;  Oeuvres  Math.  d'Evariste  Galois, 
Paris,  1897,  15-23;     Abhand.  Alg.  Gleich.  Abel  u.  Galois,  Maser,  1889,  100. 

"Jour,  de  Math^matiques,  4,  1839,  9-12. 


236  History  of  the  Theory  of  Numbers.  [Chap,  viii 

congruences  modulo  p  are  of  the  form  a-}-h\/n,  where  n  is  a  fixed  quadratic 
non-residue  of  p,  while  a,  b  are  integers.  But  the  cube  root  of  a  non-cubic 
residue  is  not  reducible  to  this  form  a-\-hy/n.  The  p+1  sets  of  integral 
solutions  of  y^  —  nz'^=a  (mod  p)  yield  the  p-|-l  real  or  imaginary  roots 
x^y-[-zy/n  of  x^^=a  (mod  p).  The  latter  congruence  has  primitive  roots 
if  0  =  1. 

Th.  Schonemann^  built  a  theory  of  congruences  without  the  use  of 
Euclid's  g.  c.  d.  process.  He  began  wath  a  proof  by  induction  that  if  a 
function  is  irreducible  modulo  p  and  divides  a  product  AB  modulo  p,  it 
divides  A  or  B.  ■Much  use  is  made  of  the  concept  norm  NJ^  of  f{x)  with 
respect  to  <i>{x),  i.  e.,  the  product  /(j8i) .  .  ./(i3^),  where  i3i,. .  .,  jS^  are  the 
roots  of  4>{x)=0;  the  norm  is  thus  essentially  the  resultant  of  /  and  0. 
The  norm  of  an  irreducible  function  with  respect  to  a  function  of  lower 
degree  is  shown  by  induction  to  be  not  divisible  by  p.  Hence  if  /  is  irre- 
ducible and  Nf^=0  (mod  p),  then/  is  a  divisor  of  0  modulo  p.  A  long  dis- 
cussion shows  that  if  ai, .  .  . ,  a„  are  the  roots  of  an  algebraic  equation 
/(x)=a:"-f  .  .  .  =0  and  if /(a:)  is  irreducible  modulo  p,  then  niii]z— 0(a,)[ 
is  a  power  of  an  irreducible  function  modulo  p. 

If  a  is  a  root  of  /(x)  and  f{x)  is  irreducible  modulo  p,  and  if  4>{a) 
=^(a)-f-pi?(a),  we  write  (p^^/  (mod  p,  a);  then  (f>{x)—\p{x)  is  divisible 
by  J{x)  modulo  p.  If  the  product  of  two  functions  of  a  is  =0  (mod  p,  a), 
one  of  the  functions  is  =0. 

If /(x)  =x'*-f  ...  is  irreducible  modulo  p  and  if /(a)  =0,  then 

/(a;)  =  (x-a)(x-a'').  .  .(x-a''""'),  aP"-^=l  (mod  p,  a), 

n-l  P"~' 

x^      —  1  =  n  )x— 0,(a)|-  (mod  p,  a), 


where  4>i  is  a  polynomial  of  degree  n  —  1  in  a  with  coefficients  chosen  from 
0,  1,.  .  .,  p  — 1,  such  that  not  all  are  zero.  There  exist  (^(p"  — 1)  primitive 
roots  moduhs  p,  a,  i.  e.,  functions  of  a  belonging  to  the  exponent  p"  — 1. 

Let  F{x)  be  irreducible  modulis  p,  a,  i.  e.,  have  no  divisor  of  degree  ^  1 
modulis  p,  a.  Let  F{^)  =  0,  algebraically.  Two  functions  of  ^  with  coeffi- 
cients involving  a  are  called  congruent  modulis  p,  a,  j3  if  their  difference  is 
the  product  of  p  by  a  polynomial  in  a,  /3.     It  is  proved  that 

F{x)^ix-^){x-^n  .  ..{x-^^'"'-'"'),  /3^'""^1  (mod  p,  a,  ^). 

If  v<n,  n  being  the  degree  of  f(x),  and  if  the  function  whose  roots  are 
the  (p"— l)th  powers  of  the  roots  of /(x)  is  ^0  (mod  p)  for  x  =  l,  then /(a;) 
is  irreducible  modulo  p.  Hence  if  ???  is  a  divisor  of  p  —  1  and  if  g^  is  a  primitive 
root  of  p,  and  if  k  is  prime  to  m,  then  x"*  — ^*  is  irreducible  modulo  p. 

If  p<m,  m  being  the  degree  of  F{x),  and  if  the  function  whose  roots  are 
the  (p*^— l)th  powers  of  the  roots  of  F{x)  is  ^0  (mod  p,  a)  for  x  =  l,  then 

"Grundziige  einer  allgemeinen  Theorie  der  hohem  Congruenzen,    deren  Modul  eine  reelle 
Primzahl  ist,  Progr.,  Brandenburg,  1844.     Same  in  Jour,  fiir  Math.,  31,  1846,  269-325. 


Chap.  VIII]  HiGHER  CONGRUENCES,    GaLOIS   ImAGINARIES.  237 

F{x)  is  irreducible  modulis  p,  a.  Hence  if  m  is  a  divisor  of  p"— 1,  and  if 
^(a)  is  a  primitive  root  of 

a;^"-^=l  (mod  p,  a), 

and  if  k  is  prime  to  m,  then  x^  —  g^  is  irreducible  modulis  p,  a. 

If  F(x,  a)  is  irreducible  modulis  p,  a,  and  if  at  least  one  coefficient  satisfies 

^p'-i^j  (mod  p,  a) 
if  and  only  if  j'  is  a  multiple  of  n,  then 

1^(0^)=  n  F(a:,  a^)  (mod  p,  a) 

y=o 

has  integral  coefficients  and  is  irreducible  modulo  p. 

If  G{x)  is  of  degree  mn  and  is  irreducible  modulo  p,  and  G(a)  =  0,  alge- 
braically, and  if  ^(a)  is  a  primitive  root  of  a;^'"''=l  (mod  p,  a),  then 

X(a:)^n  (x-F),  <  =  r^     6  =  ^-^, 

y=o  p  —  1 

has  integral  coefficients  and  is  irreducible  modulo  p. 

The  last  two  theorems  enable  us  to  prove  the  existence  of  irreducible 
congruences  modulo  p  of  any  degree.    First, 

(x'"-"-'-l)/{x'"'"'-'-l) 

is  the  product  of  the  irreducible  functions  of  degree  p"  modulo  p.  To  prove 
the  existence  of  an  irreducible  function  of  degree  Zp",  where  I  is  any  integer 
prime  to  p,  assume  that  there  exists  an  irreducible  function  of  each  degree 
<Zp",  and  hence  for  the  degree  a  =  ylp",  where  A=(f){l)<l.  Let  a  be  a 
root  of  the  latter,  and  r  a  primitive  root  of  x^~^=  1  (mod  p,  a),  where  P  =  p'^. 
Since  I  divides  P  — 1  by  Euler's  generalization  of  Fermat's  theorem,  x^  —  r 
is  irreducible  modulis  p,  a.  Hence  by  the  theorem  preceding  the  last, 
JI]Zq{x^ —r'^)  is  irreducible  modulo  p.  Since  its  degree  is  Ip^'A,  the  last 
theorem  gives  an  irreducible  congruence  of  degree  Zp". 

Every  irreducible  factor  modulo  p  of  x^"~^  —  1  is  of  degree  a  divisor  of  n. 
Conversely,  every  irreducible  function  of  degree  a  divisor  of  n  is  a  factor 
of  that  binomial.  If  n  is  a  prime,  the  number  of  irreducible  functions 
modulo  p  of  degree  n"  is  (p"'— p""^"  )/n\  If  n  is  a  product  of  powers  of 
distinct  primes  A,  B,.  .  .,  say  four,  the  number  of  irreducible  congruences 
of  degree  n  modulo  p  is 

_ipABCD_pABC_  _pBCD\pABi  A.pCD_pA_  _pD] 

Tv 

where  p  =  p"/(^sc'Z))  Replacing  p  by  p"*,  we  get  the  number  of  irreducible 
congruences  of  degree  n  modulis  p,  a,  where  a  is  a  root  of  an  irreducible 
congruence  of  degree  m. 

If  n  is  a  prime  and  p  belongs  to  the  exponent  e  modulo  n,/=  {x''  —  \)/{x  —  \) 
is  congruent  modulo  p  to  the  product  of  (n  — l)/e  irreducible  functions  of 


238  History  of  the  Theory  of  Numbers.  [Chap,  viii 

degree  e  modulo  p.  Hence  if  p  is  a  primitive  root  of  n,  /  is  irreducible 
modulo  p,  and  therefore  with  respect  to  each  of  the  infinitude  of  primes 
p+7«.     Thus/  is  algebraically  irreducible. 

Schonemann^^  considered  congruences  modulo  p"*.  If  g{x)  is  not  divis- 
ible by  p,  and/=x''+  ...  is  irreducible  modulo  p"*  and  \i  A{x)  is  not  divisible 
by/  modulo  p,  then/g'=A5  (mod  p'")  implies  that  B{x)  is  divisible  by/ 
modulo  p"*.  If /=/i,  ^=6^1  (mod  p)  and  the  leading  coefficients  of  the  four 
functions  are  unity,  while  /  and  g  have  no  common  factor  modulo  p,  then 
/f/^/i^iCinod  p'")  impHes  /=/i,  g=g\  (mod  p"*).  He  proved  the  final 
theorem  of  Gauss. ^°  Next,  {x—aY-\-'pF{x)  is  irreducible  modulo  p^  if 
and  only  if  F{a)^0  (mod  p) ;  an  example  is 

^  =  {x-ir-'+pF{x),  F(l)  =  l. 

Henceforth,  let/(a;)  be  irreducible  modulo  p  and  of  degree  n.  If  f{xY-\-'pF{x) 
is  reducible  modulo  p",  then  (p.  101)  /(x)  is  a  factor  of  F(a;)  modulo  p.  If 
/(a)  =  0 and  g(a)  ^0  (mod  p, a),  then  g'^  1  (mod  p'",  a), where  e  =  p'"~Hp''- 1). 
If  the  roots  of  G{z)  are  the  (p'"~^)th  powers  of  the  roots  of /(x),  then 

G{z)^{z-^){z-n...{z-^'''-')  (mod  p-,  a). 

If  M  is  any  integer  and  if  F{x)  has  the  leading  coefficient  unity,  we  can 
find  z  and  w  such  that  {x^—iy  is  divisible  by  F(x)  modulo  M. 

A.  Cauchy^®  noted  the  uniqueness  of  the  factorization  of  a  function  f{x) 
with  integral  coefficients  into  irreducible  factors  modulo  p,  a  prime.  An  irre- 
ducible function  divides  a  product  only  when  it  divides  one  factor  modulo  p. 
A  common  divisor  of  two  functions  divides  their  g.  c.  d.  modulo  p. 

Cauchy^^  employed  an  indeterminate  quantity  or  symbol  i  and  defined 
f{i)  to  be  not  the  value  of  the  polynomial  f{x)  for  x  =  i,  but  to  be  a-\-hi  if 
a-\-hx  is  the  remainder  obtained  by  dividing /(a:)  by  x^+1.  In  particular, 
if /(x)  is  x^+1  itself,  we  have  i^+1  =0. 

Similarly,  if  w(x)  =  0  is  an  irreducible  congruence  modulo  p,  a  prime, 
let  i  denote  a  sjmiboHc  root.  Then  0(i);/'(t)  =  O  implies  either  </)(i)  =  0  or 
yp{i)  =  0  (mod  p).  At  most  n  integral  functions  of  i  satisfy /(x,  i)  =  0  (mod  p), 
if  the  degree  of  /  in  a:  is  nKp.  If  our  co(x)  divides  x"  — 1,  but  not  x"*  — 1, 
m<n,  modulo  p,  where  n  is  not  a  divisor  of  p  — 1,  call  i  a  symbolic  primi- 
tive root  of  x''=l  (mod  p).  Then  rc"-l=(.T-l)(x-i) .  .  .  (x-i"-^)  If 
s  is  a  primitive  root  of  n  and  if  n  —  l=gh,  and  p''=  1  (mod  n), 

equals  a  function  of  x  with  integral  coefficients,  while  every  factor  of  x"  —  1 
modulo  p  with  integral  coefficients  equals  such  a  product. 

«Jour.  fur  Math.,  32,  1846,  93-105. 

"Comptes  Rendus  Paris,  24,  1847,  1117;  Oeuvres,  (1).  10,  308-12. 

"Comptes  Rendus  Paris,  24,  1847,  1120;  Oeuvres,  (1),  10,  312-23. 


Chap.  VIII]  HiGHER  CONGRUENCES,   GaLOIS   ImAGINARIES.  239 

G.  Eisenstein^^  stated  that  if /(a;)  =  0  is  irreducible  modulo  p,  and  a  is  a 
root  of  the  equation /(x)  =  0  of  degree  n,  and  if  ao,  ai,.  .  .  are  any  integers, 

K  =  ao-\-aia-\-. .  .  .  +a„_ia"~-^ 
is  congruent  modulis  p,  a  to  one  and  but  one  expression 

5  =  60/3+61^^+62/3^'+  . . .  +6„_i/3^"-\ 

where  the  6's  are  integers  and  /5  is  a  suitably  chosen  function  of  a.  Hence 
the  p"  numbers  B  form  a  complete  set  of  residues  modulis  p,  a.  If  co  is  a 
primitive  nth  root  of  unity,  and  if 

(/)(X)  =a+ajV+co2V'+  .  .  .  +cu^"-i^V""\ 

the  product  0(X)0(X') ...  is  independent  of  a  if  X+X'+  ...  is  divisible  by  n. 
Th.  Schonemann^^  proved  the  last  statement  in  case  n  is  not  divisible 
by  p.  To  make  K  =  B,  raise  it  to  the  powers  p,  p^,.  .  .,  p"~^  and  reduce 
by  j8^"=/3  (mod  p,  a).  This  system  of  n  congruences  determines  /3  uniquely 
if  the  cyclic  determinant  of  order  n  with  the  elements  hi  is  not  divisible  by 
p;  in  the  contrary  case  there  may  not  exist  a  (3.  The  statement  that  the 
expressions  B  form  p"  distinct  residues  is  false  if  jS  is  a  root  of  a  congruence 
of  degree  <n  irreducible  modulo  p;  it  is  true  if  /3  is  a  root  of  such  a  con- 
gruence of  degree  n  and  if 

i8+/3^+  .  .  .  +/3^""'^0  (mod  p,  a). 

J.  A.  Serref^"  made  use  of  the  g.  c.  d.  process  to  prove  that  if  an  irre- 
ducible function  F{x)  divides  a  product  modulo  p,  a  prime,  it  divides  one 
factor  modulo  p.  Then,  following  Galois,  he  introduced  an  imaginary 
quantity  i  verifying  the  congruence  F{i)  =  0  (mod  p)  of  degree  v>l,  but 
gave  no  formal  justification  of  their  use,  such  as  he  gave  in  his  later  writings. 
However,  he  recognized  the  interpretation  that  may  be  given  to  results 
obtained  from  their  use.  For  example,  after  proving  that  any  polynomial 
a{i)  with  integral  coefficients  is  a  root  of  a^''=a  (mod  p),  he  noted  that  this 
result,  for  the  case  a  =  i,  may  be  translated  into  the  following  theorem,  free 
from  the  consideration  of  imaginaries:  If  F{x)  is  of  degree  v,  has  integral 
coefficients,  and  is  irreducible  modulo  p,  there  exist  polynomials  f{x)  and 
x(x)  with  integral  coefficients  such  that 

x''''-x=f{x)F(x)  +px{x). 

The  existence  of  an  irreducible  congruence  of  any  given  degree  and  any 
prime  modulus  is  called  the  chief  theorem  of  the  subject.  After  remarking 
that  Galois  had  given  no  satisfactory  proof,  Serret  gave  a  simple  and  ingeni- 
ous argument;  but  as  he  made  use  of  imaginary  roots  of  congruences  without 
giving  an  adequate  basis  to  their  theory,  the  proof  is  not  conclusive. 

'sjour.  fur  Math.,  39,  1850,  182. 

«'Jour.  fiir  Math.,  40,  1850,  185-7. 

"Cours  d'algdbre  sup6rieure,  ed.  2,  Paris,  1854,  343-370. 


240  History  of  the  Theory  of  Numbers.  [Chap,  viii 

R.  Dedekind"^  developed  the  subject  of  higher  congruences  by  the 
methods  of  elementary  number  theory  without  the  use  of  algebraic  prin- 
ciples. As  by  Gauss^°  he  developed  the  theory'  of  the  g.  c.  d.  of  functions 
modulo  p,  a  prime,  and  their  unique  factorization  into  prime  (or  irreducible) 
functions,  apart  from  integral  factors.  Two  functions  A  and  B  are  called 
congruent  modulis  p,  M,  \i  A—B  is  divisible  by  the  function  AI  modulo  p. 
We  may  add  or  multiply  such  congruences.  If  the  g.  c.  d.  of  A  and  B  is 
of  degree  d,  Aij=B  (mod  p,  M)  has  p'^  incongruent  roots  y{x)  modulis  p,  M. 

Let  <t){M)  denote  the  number  of  functions  which  are  prime  to  M  modulo 
p  and  are  incongruent  modulis  p,  M.  Let  ii  be  the  degree  of  M.  A  pri- 
mary function  of  degree  a  is  one  in  which  the  coefficient  of  a:"  is  =  1  (mod  p). 
If  D  ranges  over  the  incongruent  primary  divisors  of  M,  then  20(Z))=p''. 
If  M  and  A^  are  relatively  prime  modulo  p,  then  4){MN)  =4){M)<t){N).  If 
A  is  a  prime  function  of  degree  a,  0(A'')  =p*'(l  -~  Vp")-  If  Af  is  a  product 
of  powers  of  incongruent  primary  prime  functions  a,.  .  .,  p, 


*W=p-(i-l)...(i-l). 


If  F  is  prime  to  M  modulo  p,  F'^^-^^^=  1  (mod  p,  M),  which  is  the  generaliza- 
tion of  Fermat's  theorem.  Hence  if  A  is  prime  to  M,  the  above  Unear  con- 
gruence has  the  solution  y=BA'^~^. 

If  P  is  a  prime  function  of  degree  tt,  a  congruence  of  degree  n  modulis  p, 
P  has  at  most  n  incongruent  roots.     Also 

(3)  y''-'-l^Il{y-F)^(mod  p,  P), 

identically  in  y,  where  F  ranges  over  a  complete  set  of  functions  incongruent 
moduhs  p,  P  and  not  divisible  by  P.  In  particular,  l+nF=0  (mod  p,  P), 
the  generalization  of  Wilson's  theorem. 

There  are  (/)(p'— 1)  primitive  roots  modulis  p,  P.  Hence  we  may  em- 
ploy indices  in  the  usual  manner,  and  obtain  the  condition  for  solutions 
of  ?/"=A  (mod  p,  P),  where  A  is  not  divisible  by  P.  In  particular,  A 
is  a  quadratic  residue  or  non-residue  of  P  according  as 

^(p'-i)/2^_^^  or  -1  (mod  p,  P). 

His  extension  of  the  quadratic  reciprocity  law  will  be  cited  under  that  topic. 
A  function  A  belongs  to  the  exponent  p  with  respect  to  the  prime  func- 
tion P  of  degree  tt  if  p  is  the  least  positive  integer  for  which  A^''=A  (mod 
p,  P).  Evidently  p  is  a  divisor  of  tt.  Let  N{p)  be  the  number  of  incon- 
gruent functions  which  belong  to  an  exponent  p  which  divides  w.  Then 
p^='ZN'{d),  where  d  ranges  over  the  divisors  of  p.  By  the  principle  of 
inversion  (Ch.  XIX), 

isr(p)  =p''-2p''/''+2:p''/''*-2p''/'''^+ . . ., 

where  a,  6, .  .  .  are  the  distinct  primes  di\'iding  p.  Since  the  quotient  of 
this  sum  by  its  last  term  is  not  divisible  by  p,  we  have  A^(p)>0. 

"Jour,  fiir  Math.,  54,  1857,  1-26. 


■^ 


Chap.  VIII]  HiGHER  CONGRUENCES,    GaLOIS   ImAGINARIES.  241 

The  product  of  the  incongruent  primary  prime  functions  modulo  p 
whose  degree  divides  tt  is  congruent  modulo  p  to 

Then,  if  xpip)  is  the  number  of  primary  prime  functions  of  any  degree  p, 
l!id\l/{d)=p',  where  the  summation  extends  over  all  divisors  d  of  tt.  A  com- 
parison of  this  with  XN{d)=p''  above  shows  that  N{p)=p\l/{p).  Another 
proof  is  based  on  the  fact  that 

(y-A){y-A')...{y-A''-') 

is  congruent  modulis  p,  P  to  a  polynomial  in  y  with  integral  coefficients 
which  is  a  prime  function.  Moreover,  if  in  (3)  we  associate  the  linear 
factors  in  which  the  F's  belong  to  the  same  exponent,  we  obtain  a  factor 
of  the  left  member  which  is  irreducible  modulo  p. 

The  product  of  the  incongruent  primary  prime  functions  of  degree  m 
(m  being  divisible  by  no  primes  other  than  a,  6, . . . )  is  congruent  modulo  p 
to 

\m\-'n.\m/ab\ . . . 


Il\m/a\'Il\m/abc\ 


H.  J.  S.  Smith^^  gave  an  exposition  of  the  theory. 
E.  Mathieu,^^  in  his  famous  paper  on  multiply  transitive  groups,  gave 
without  proof  the  factorization  (p.  301;  for  m  =  l,  p.  275) 


h{z^"'''-z)=u\(hzy"'^''-''+ihzy'"^''-''+ .  ..+(hzy"'+hz+a}, 

a 

where  a  ranges  over  the  roots  of  a^'^^a,  while  /i^"*"=/i;  and  (p.  302;  for 
m  =  l,p.  280) 

/^(gp'""  -  z) =n(;i^  V"  -hz-^), 

where  jS  ranges  over  the  roots  of 

If  12  is  a  root  of  a  congruence  of  degree  n  whose  coefficients  are  roots  of 
z'^"'=z  and  whose  first  member  is  prime  to  z^'^  —  z,  then  (p.  303)  all  the  roots 
of  z^""'=z  are  given  by  ^o+^i^+-  •  .+^„-il2''~\  where  the  A's  satisfy 

Z^    =Z. 

J.  A.  Serret,'^^  in  contrast  to  his^°  earher  exposition,  here  avoided  at 
the  outset  the  use  of  Galois  imaginaries.  An  irreducible  function  of  degree 
V  modulo  p  divides  x^—x  modulo  p  if  and  only  if  v  divides  ^t.     A  simple 

"British  Assoc.  Reports,  1860,  120,  §§69-71;  Coll.  M.  Papers,  1,  149-155. 
"Jour,  de  Math6matiques,  (2),  6,  1861,  241-323. 

"M6m.  Ac.  Sc.  de  I'Institut  de  France,  35,  1866,  617-688.     Same  in  Cours  d'algfebre  sup6- 
rieure,  ed.  4,  vol.  2,  1879,  122-189;  ed.  5,  1885. 


242  History  of  the  Theory  of  Numbers.  [Chap,  viii 

proof  is  given  for  Dedekind's^^  final  theorem  on  the  product  of  all  irreducible 
functions  of  degree  m  modulo  p. 

A  function  F{x)  of  degree  v,  irreducible  modulo  p,  is  said  to  belong  to  the 
exponent  n  if  n  is  the  least  positive  integer  such  that  x"  —  1  is  divisible  by 
F{x)  modulo  p.  Then  n  is  a  divisor  of  p"  — 1,  and  a  proper  divisor  of  it, 
since  it  does  not  divide  p"  —  1  for  }x<v.  Let  n  be  a  product  of  powers  of  the 
distinct  primes  a,  b,.  .  ..  Then  the  product  of  all  functions  of  degree  v, 
irreducible  modulo  p,  which  belong  to  an  exponent  n  which  is  a  proper 
divisor  of  p"  — 1,  is  congruent  modulo  p  to 

n(a:"/"-l)-n(x"/"'"^-l)... 

and  their  number  is  therefore  (f>{n)/v. 

By  a  skillful  analysis,  Serret  obtained  theorems  of  practical  importance 
for  the  determination  of  irreducible  congruences  of  given  degrees.  If  we 
know  the  N  irreducible  functions  of  degree  /jl  modulo  p,  which  belong  to 
the  exponent  1=  (p"  — l)/<i,  then  if  we  replace  x  by  x^,  where  X  is  prime  to  d 
and  has  no  prime  factor  different  from  those  which  divide  p"  — 1,  we  obtain 
the  N  irreducible  functions  of  degree  Xfi  which  belong  to  the  exponent  \l, 
exception  being  made  of  the  case  when  p  is  of  the  form  4ih  — I,  fi  is  odd,  and 
X  is  divisible  by  4.  In  this  exceptional  case,  we  may  set  p  =  2H  —  l,  i'^2, 
t  odd;  X  =  2^s,  j^2,  s  odd.  Let  k  be  the  least  of  i,  j.  Then  if  we  know 
the  A^/2^~^  irreducible  functions  of  odd  degree  ju  modulo  p  which  belong  to 
the  exponent  I  and  if  we  replace  x  by  x^,  where  X  is  of  the  form  indicated, 
is  prime  to  d  and  contains  only  primes  dividing  p"  — 1,  we  obtain  N/2^~^ 
functions  of  degree  Xju  each  decomposable  into  2^~'^  irreducible  factors, 
thus  giving  A''  irreducible  functions  of  degree  \fx/2''~^  which  belong  to  the 
exponent  XL  Apply  these  theorems  to  x  —  g%  which  belongs  to  the  exponent 
(p  —  l)/d  if  ^  is  a  primitive  root  of  p  and  if  d  is  the  g.  c.  d.  of  e  and  p  —  1 ;  we 
see  that  x^—g^  is  irreducible  unless  the  exceptional  case  arises,  and  is  then 
a  product  of  2^"~^  irreducible  functions.  In  that  case,  irreducible  trinomials 
of  degree  X  are  found  by  decomposing  x" —g%  where  i'  =  2'~^X. 

If  a  is  not  divisible  by  p,  a:''  — x  — a  is  irreducible  modulo  p. 

There  is  a  development  of  Dedekind's  theory  of  functions  modulis  p  and 
F(x),  where  F{x)  is  irreducible  modulo  p.  Finally,  that  theory  is  considered 
from  the  point  of  view  of  Galois.  Just  as  in  the  theory  of  congruences  of 
integers  modulo  p  we  treat  all  multiples  of  p  as  if  they  were  zero,  so  in 
congruences  in  the  unknown  X, 

(?(X,  a:)  =  0  (mod  p,  F(a:)), 

we  operate  as  if  all  multiples  of  F{x)  vanish.  There  is  here  an  indeter- 
minate X  which  we  can  make  use  of  to  cause  the  multiples  of  F{x)  to  vanish 
if  we  agree  that  this  indeterminate  x  is  an  imaginary  root  i  of  the  irreducible 
congruence  F(a:)  =  0  (mod  p).  From  the  theorems  of  the  theory  of  func- 
tions modulis  p,  F{x),  we  may  read  off  briefer  theorems  involving  i  (cf. 
Galois^2)^ 


Chap.  VIII]  HiGHER   CONGRUENCES,    GaLOIS   ImAGINARIES.  243 

Harald  Schiitz^^  considered  a  congruence 

Z'^+aiX"-i+  .  .  .  +a„=0  (mod  Mix)) 

in  which  the  a's  and  the  coefficients  of  M  are  any  complex  integers 
(cf.  Cauchy,®^  for  real  coefficients).  Let  ai,...,  a„  be  the  roots  of  the 
corresponding  algebraic  equation.  Let  M  =  0  have  the  distinct  roots 
III,.  .  .,  jirn-  Then  the  congruence  has  n"*  distinct  roots.  For,  let  X  —  a^ 
=fi{x)  have  the  factor  x—jii,  for  i  =  1, .  .  .,  m.     Taking  i>l,  we  have 

fi{^)=fl{^)+0.p-(lpi' 

Set  X = 111.  Then  the  right  member  must  vanish.  Using  these  and  /i  (/^i)  =  0 , 
we  have  m  independent  linear  relations  for  the  coefficients  of /i(x). 

C.  Jordan^®  followed  Galois  in  employing  from  the  outset  a  symbol  for 
an  imaginary  root  of  an  irreducible  congruence,  proved  the  theorems  of 
Galois,  and  that,  if  j,  ji, .  .  .  are  roots  of  irreducible  congruences  of  degrees 
p",  q^,.  .  .  where  p,  q, .  .  .  are  distinct  primes,  their  product  jji ...  is  a  root 
of  an  irreducible  congruence  of  degree  p^q^ .... 

A.  E.  Pellet^^  stated  that,  if  t  is  a  root  of  an  irreducible  congruence  of 
degree  v  modulo  p,  a  prime,  the  number  of  irreducible  congruences  of  degree 
Vi  whose  coefficients  are  polynomials  in  i  is 

—  jp""!  —  Sp'"'i/5i+Sp'"'i/9i«2—  ...+(  —  l)"'p''V9i-  ■  -Sm  } 

if  qi,...,  qm  are  the  distinct  primes  dividing  vi.  Of  these  congruences, 
4){n)/vi  belong  to  the  exponent  n  if  n  is  a  proper  divisor  of  (p")"'  — L 

Any  irreducible  function  of  degree  ix  modulo  p  with  integral  coefficients 
is  a  product  of  5  irreducible  factors  of  degree  ix/b  with  coefficients  rational 
in  i,  where  b  is  the  g.  c.  d.  of  fx,  v. 

In  an  irreducible  function  of  degree  vi  and  belonging  to  the  exponent  n 
and  having  as  coefficients  rational  functions  of  i,  replace  x  by  x^,  where  X 
contains  only  prime  factors  dividing  n;  the  resulting  function  is  a  product  of 
2^~^D/n  irreducible  functions  of  degree  \nvi/{2^~^D)  belonging  to  the 
exponent  \n,  where  D  is  the  g.  c.  d.  of  \n  and  p""*  —  1,  and  2^~^  is  the  highest 
power  of  2  dividing  the  numerators  of  each  of  the  fractions  (p'"''+l)/2  and 
Xn/(2Z))  when  reduced  to  their  lowest  terms. 

Let  gf  be  a  rational  function  of  i,  and  m  the  number  of  distinct  values 
among  g,  g^,  g^  ,.  .  ..  If  neither  g-{-g^-{- .  .  .  +9'^"*"  nor  v/m  is  divisible  by 
p,  then  x^  —  x  —  g  is  irreducible;  in  the  contrary  case  it  is  a  product  of 
linear  functions. 

Hence  if  we  replace  a;  by  x^  — x  in  an  irreducible  function  of  degree  /x 
having  as  coefficients  rational  functions  of  i,  we  get  a  new  irreducible 
function  provided  the  coefficient  of  x''~^  in  the  given  function  is  not  zero. 

^^Untersuchungen  liber  Functionale  Congruenzen,  Diss.  Gottingen,  Frankfurt,  1867. 
^«Trait^  des  substitutions,  1870,  14-18. 
"Comptes  Rendus  Paris,  70,  1870,  328-330. 


244  History  of  the  Theory  of  Numbers.  [Chap,  viii 

[Proof  in  Pellet.®^]  In  particular,  if  p  is  a  primitive  root  of  a  prime  n, 
we  have  the  irreducible  function,  modulo  p, 

(xP-a;)"-l 
x^'-x-l 

C.   Jordan^^  listed   irreducible   functions   [errata,   Dickson,^"'^  p.   44]. 

J.  A.  Serret'^^  determined  the  product  F„  of  all  functions  of  degree  p" 
irreducible  modulo  p,  a  prime.  In  the  expansion  of  (^  —  1)"  replace  each 
power  ^^"  by  x"  ;  denote  the  resulting  polynomial  in  x  by  X^.    Then 

X.,rn^\{^-irl=ie"'-^y,  X.m^X^^'^-X    (mod    p). 

Hence  Vn  =  Xpr^/Xpn-i.     Moreover, 

X,+i  =  (^-l)''+^  =  $a-ir-(^-l)''^Z/-X,  (modp). 

Multiply  this  by  the  relations  obtained  by  replacing  /ibyju+1,-  •  .,iJi+v  —  l. 
Thus 

X,+.^Z,(X/-i-l)(X,^;J  -1) .  ..{X:+l,-l)  (mod  p). 

Take  /i  =  p"~\  /x+i'  =  p".     Hence 

F„^' "ff"   A  (mod  p),  A  =  Xjnii+x-i-l. 

X=l 

Each  /x  decomposes  into  p  — 1  factors  X—g  where  ^  =  1,...,  p  — 1.  The 
irreducible  functions  of  degree  p"  whose  product  is  A  are  said  to  belong  to 
the  Xth  class.  When  x  is  replaced  by  x^—x,  X^  is  replaced  by  X^+i  since  ^' 
is  replaced  by  ^'(^  —  1)  and  hence  (^  —  1)"  by  (^  — l)""*"^;  thus  A  is  replaced 
by  A+i)  while  the  last  factor  in  F„=nA  is  replaced  by  Xpn  —1,  which  is 
the  first  factor  in  Vn+i-  Hence  if  F{x)  is  of  degree  p"  and  is  irreducible 
modulo  p  and  belongs  to  the  Xth  class,  F{x^—x)  is  irreducible  or  the  product 
of  p  irreducible  functions  of  degree  p"  according  as  X=  or  <p'*— p"""\ 

For  n  =  l,  the  irreducible  functions  of  the  Xth  class  have  as  roots  poly- 
nomials of  degree  X  in  a  root  of  i^  —  i=l,  which  is  irreducible  modulo  p. 
Hence  if  we  eliminate  i  between  the  latter  and  f{i)  =  x,  where  f{i)  is  the 
general  polynomial  of  degree  X  in  i,  we  obtain  the  general  irreducible 
function  of  degree  p  of  the  Xth  class. 

For  any  n,  the  determination  of  the  irreducible  functions  of  degree  p"  of 
the  first  class  is  made  to  depend  upon  a  problem  of  elimination  (Algebre, 
p.  205)  and  the  relation  to  these  of  the  functions  of  the  Xth  class,  X>1,  is 
investigated. 

G.  Bellavitis'^"  tabulated  the  indices  of  Galois  imaginaries  of  order  2 
for  each  prime  modulus  p  =  4n+3^63. 

Th.  Pepin^"  proved  that  x^  —  ny^=l  (mod  p)  has  p  +  1  sets  of  solutions 

'Kllomptes  Rendus  Paris,  72,  1871,  283-290. 

"Jour,  de  Mathdmatiques,  (2),  18, 1873, 301^,  437-451.    Same  as  in  Cours  d'alg^bre  sup^rieure, 

ed.  4,  vol.  2,  1879,  190-211. 
"f-Atti  Accad.  Lincei,  Mem.  Sc.  Fis.  Mat.,  (3),  1,  1876-7,  778-800. 
««Atti  Accad.  Pont.  Nuovi  Lincei,  31,  1877-8,  43-52. 


Chap.  VIII]  HiGHER   CONGRUENCES,    GaLOIS   ImAGINARIES.  245 

X,  y  selected  from  0,  1, . .  . ,  p  —  1,  provided  n  is  a  quadratic  non-residue  of  the 
prime  p.  Then  x-\-y\/n  is  a  root  of  p+^=l  (mod  p),  which  therefore  has 
p  +  1  complex  roots,  all  a  power  of  one  root.  There  is  a  table  of  indices  for 
these  roots  when  p  =  29  and  p  =  41.     [Lebesgue.^^] 

A.  E.  Pellet^^  considered  the  product  A  of  the  squares  of  the  differences 
of  the  roots  of  a  congruence /(x)  =  0  (mod  p)  having  no  equal  roots.  Then 
A  is  a  quadratic  non-residue  of  p  if  f{x)  has  an  odd  number  of  irreducible 
factors  of  even  degree,  a  quadratic  residue  if  f{x)  has  no  irreducible  factor 
of  even  degree  or  has  an  even  number  of  them.  For,  if  5i, .  .  .,  5^  are  the 
values  of  A  for  the  various  irreducible  factors  of  f{x),  then  A=a^di.  .  .5j 
(mod  p),  where  a  is  an  integer.  Hence  it  suffices  to  consider  an  irreducible 
congruence /(a:)  =  0  (mod  p).     Let  v  be  its  degree  and  i  a  root.     In 

v-i  i-i 

!/=n  n  {x''  -x^) 

1=1  A;  =  0 

replace  x  by  the  v  roots;  we  get  two  distinct  values  if  v  is  even,  one  if  ;^  is 
odd.     In  the  respective  cases,  i/^=A  (mod  p)  is  irreducible  or  reducible. 

R.  Dedekind^^  noted  that,  if  P(x)  is  a  prime  function  of  degree  /  modulo 
p,  a  prime,  a  congruence  F{x)  =  0  (mod  p,  P)  is  equivalent  to  the  congruence 
F(a)  =  0  (mod  tt),  where  tt  is  a  prime  ideal  factor  of  p  of  norm  p^,  and  a  is 
a  root  of  P(a)  =  0  (mod  tt). 

A.  E.  Pellet^^  denoted  by/(a;)=0  the  equation  of  degree  ^(A;)  having 
as  its  roots  the  primitive  A;th  roots  of  unity,  and  by /i(i/)  =0  the  equation 
derived  by  setting  y  =  x-\-\/x.  If  p  is  a  prime  not  dividing  k,  f{x)  is  con- 
gruent modulo  p  to  a  product  of  <f>{k)/v  irreducible  factors  whose  degree  v 
is  the  least  integer  for  which  p"  —  !  is  divisible  by  k.  li  fi{y)  =  0  (mod  p) 
has  an  integral  root  a,f(x)  is  divisible  modulo  p  by  x^  —  2ax-\-l.  Either  the 
latter  has  two  real  roots  and  f{x)  and  fi{y)  have  all  their  roots  real  and 
p  —  1  is  divisible  by  k,  or  it  is  irreducible  and  f(x)  is  a  product  of  quadratic 
factors  modulo  p  and  the  roots  oifi{y)  are  all  real  and  p+1  is  divisible  by  k. 
If  k  divides  neither  p+l  nor  p  —  l,fi{y)  is  a  product  of  factors  of  equal 
degree  modulo  p.     [Cf.  Sylvester,^^  etc.,  Ch.  XVI.] 

Let  A;  be  a  divisor  f^  2  of  p  +  L  Let  X  be  an  odd  number  divisible  by  no 
prime  not  a  factor  of  k,  and  relatively  prime  to{p+l)/k.  Then  x^^  —  2ax^  + 1 
is  irreducible  modulo  p  [Serret,^^  No.  355].     Also,  if  h  is  not  divisible  by  p 

F={x-\-by''-2aix^-b^)^+ix-by^ 

is  irreducible  modulo  p;  replacing  x^  by  y,  we  obtain  a  function  of  degree  X 
irreducible  modulo  p.  If  /c  is  a  divisor  ?^2  of  p  —  1  and  if  X  is  odd,  prime  to 
{p  —  l)/k  and  divisible  by  no  prime  not  a  factor  of  k,  F  decomposes  modulo 
p  into  two  irreducible  functions  of  degree  X. 

The  function  /(x^)  is  either  irreducible  or  the  product  of  two  irreducible 
factors  of  degree  v.     In  the  respective  cases,  the  product  A  of  the  squares  of 

"Comptes  Rendus  Paris,  86,  1878,  1071-2. 

"Abhand.  K.  Gesell.  Wiss.  Gottingen,  23,  1878,  p.  25.     Dirichlet-Dedekind,  Zahlentheorie,  ed. 

4,  1894,  571-2. 
s^Comptes  Rendus  Paris,  90,  1880,  1339-41. 


246  History  of  the  Theory  of  Numbers.  [Chap,  viii 

the  differences  of  the  roots  of  /(x")  =  0  is  a  quadratic  non-residue  or  residue 
of  p  [Pellet^^j.  Let  Ai  be  the  like  product  for  J{x).  Then  A  =  (-l)'2-' 
/(0)Ai".  Hence /(ax- +6)  is  irreducible  if  (  — l)7(6)/a'' is  a  quadratic  non- 
residue  and  then /(ax  '+6)  is  irreducible  modulo  p  for  every  i  and  even  v. 

0.  H.  Mitchell^  gave  analogues  of  Fermat's  and  Wilson's  theorems 
moduhs  p  (a  prime)  and  a  function  of  x. 

A.  E.  Pellet^  considered  the  exponent  n  to  which  belongs  the  product  P 
of  the  roots  of  a  congruence  F(x)  =  0  of  degree  v  irreducible  modulo  p. 
If  g  is  a  prime  factor  of  n,  F(x')  is  irreducible  or  the  product  of  q  irreducible 
factors  of  degree  v  modulo  p  according  as  q  is  not  or  is  a  divisor  of  {p  —  \)/n. 
In  particular,  F{x^)  is  irreducible  modulo  p  if,  for  v  even,  X  contains  only- 
prime  factors  of  n  not  dividing  {p  —  \)/n;  for  v  odd,  we  can  use  the  factor  2 
in  X  only  once  if  p  =  4mH-l.  Let  i  be  a  root  of  F(x)  =  0,  I'l  a  root  of  an 
irreducible  congruence  Fi(x)  =  0  (mod  p)  of  degree  v^  prime  to  v.  Then 
ill  is  a  root  of  an  irreducible  congruence  G(x)  =  0  (mod  p)  of  degree  vvx. 
F{x)  belongs  to  the  exponent  Nn  modulo  p,  where  n  is  prime  to  (p'— 1) 
-^\{p  —  \)N\.  Let  qi  be  a  prime  factor  of  .V  not  dividing  p  —  1.  Then 
G(x'')  is  irreducible  or  decomposes  into  qi  irreducible  factors  of  degree  vvi 
according  as  qi  is  not  or  is  a  divisor  of  (p''  —  l)/N.  Thus  G(x^)  is  irreducible 
if  X  contains  onlj'  prime  factors  of  N  dividing  neither  p  —  1  nor  {p''  —  l)/N. 

0.  H.  MitchelP^  defined  the  prime  totient  of /(x)  to  mean  the  number 
of  polynomials  in  x,  incongruent  modulo  p,  of  degree  less  than  the  degree  of 
(x)  and  having  no  factor  in  conmion  with  /  modulo  p.  Those  which 
contain  S,  but  no  prime  factor  of  /  not  contained  in  S,  are  called  >S-totitives 
of/. 

C.  Dina^^  proved  known  results  on  congruences  moduhs  p  and  F{x). 

A.  E.  Pellet^^  proved  that,  if  ju  distinct  values  are  obtained  from  a 
rational  function  of  x  with  integral  coefficients  by  replacing  x  successively 
by  the  77i  roots  of  an  irreducible  congruence  modulo  p,  then  ^i  is  a  divisor 
of  m  and  these  /jl  values  are  the  roots  of  an  irreducible  congruence.  Thus 
if  A  is  a  rational  function  of  any  number  of  roots  of  congruences  irreducible 
modulo  p,  and  p  is  the  number  of  distinct  values  among  A,  A^,  A^\.  .  ., 
these  values  satisfy  an  irreducible  congruence  modulo  p.  If  A  belongs  to 
the  exponent  n  modulo  p,  then  v  is  the  least  positive  integer  for  which  p''=  1 
(mod  n).  He  proved  a  result  of  Serret's^^  stated  in  the  following  form:  If, 
in  an  irreducible  function  F{x)  modulo  p  of  degree  v  and  exponent  n,  x  is 
replaced  by  x^,  where  X  contains  only  primes  dividing  n,  then  F(x^)  is  a 
product  of  irreducible  factors  of  degree  vq  and  exponent  n\,  where  q  is  the 
least  integer  for  which  ^"^=1  (mod  n\).  He  proved  the  first  theorem  of 
Pellet^  and  the  last  one  of  Pellet." 

"Johns  Hopkins  University  Circulars,  1,  1880-1,  132. 

"Comptps  Rendus  Paris,  93,  1881,  1065-6.     Cf.  Pellet." 

"Amer.  Jour.  Math.,  4,  1881,  25-38. 

•^Giomale  di  Mat.,  21,  1883,  234-263.     For  comments  on  263-9,  see  the  chapter  on  quadratic 

reciprocity  law. 
««Bull.  Soc.  Math.  France,  17,  1888-9,  156-167. 


Chap.  VIII]  HiGHER  CONGRUENCES,    GaLOIS   ImAGINARIES.  247 

E.  H.  Moore^^  stated  that  every  finite  field  (Korper)  is,  apart  from  nota- 
tions, a  Galois  field  composed  of  the  p"  polynomials  in  a  root  of  an  irreducible 
congruence  of  degree  n  modulo  p,  a  prime. 

E.  H.  Moore^°  proved  the  last  theorem  and  others  on  finite  fields. 

K.  Zsigmondy^^  noted  that  the  number  of  congruences  of  degree  n 
modulo  p,  having  no  irreducible  factor  of  degree  i,  is 


P"-({)p"-^+(2)p"-'' 


where  /  is  the  number  of  functions  of  degree  i  irreducible  modulo  p. 

G.  Cordone^^  noted  that  if  a  function  is  prime  to  each  of  its  derivatives 
with  respect  to  each  prime  modulus  Pi,...,  Pn  and  is  irreducible  modulo 
M  =  piK  .  .pn",  it  is  irreducible  with  respect  to  at  least  one  of  Pi,...,  Pn- 
If  F(x)  is  not  identically  =0  modulo  pi,  nor  modulo  p2,  etc.,  and  if  it 
divides  a  product  modulo  M  and  is  prime  to  one  factor  according  to  each 
modulus  Pi,.  .  .,  Pn,  then  F(x)  divides  the  other  factor  modulo  M. 

Let  F{x)  be  a  function  of  degree  r  irreducible  with  respect  to  each  prime 
Pi, . .  .,  Pn,  while /(x)  is  not  divisible  by  F{x)  with  respect  to  any  one  of  the 
p's,  then  (pp.  281-8) 

l/(x)[-^^^^^l  (mod  M,  F(x)),  0,(M)=M'-(l-^,y  .  .(l-^), 

<}>r{M)  being  the  number  of  functions  Cix''~^+  .  . .  -\-Cr,  in  which  the  c's  take 
such  values  0,  1,.  .  .,  M  —  1  whose  g.  c.  d.  is  prime  to  M.  Let  A  be  the 
product  of  these  reduced  functions  modulis  M,  F{x).  Then  (pp.  316-8), 
A=  —  1  (mod  M,  F)  if  M  =  p^,  2p^  or  4,  where  p  is  an  odd  prime,  while 
A=  +  l  in  all  other  cases. 

Borel  and  Drach^^  gave  an  exposition  of  the  theory  of  Galois  imaginaries 
from  the  standpoint  of  Galois  himself. 

H.  Weber^^  considered  the  finite  field  (Congruenz  Korper)  formed  of  the 
p"  classes  of  residues  modulo  p  of  the  polynomials,  with  integral  coefficients, 
in  a  root  of  an  irreducible  equation  of  degree  n.  He  proved  the  generaliza- 
tion of  Fermat's  theorem,  the  existence  of  primitive  roots,  and  the  fact  that 
every  element  is  a  square  or  a  sum  of  the  squares  of  two  elements. 

Ivar  Damm^*  gave  known  facts  about  the  roots  of  congruences  modulis 
p,  /(x),  where /(x)  is  irreducible  modulo  p,  without  exhibiting  the  second 
modulus  and  without  making  it  clear  that  it  is  not  a  question  of  ordinary 
congruences  modulo  p.  Let  e  be  a  fixed  primitive  root  of  the  prime  p. 
Then  the  roots  of  every  irreducible  quadratic  congruence  are  of  the  form 
a±  hoi,  where  co^  =  e.     Let  k^^^  =  e,  ki  =  k^. 

89Bull.  New  York  Math.  Soc,  3,  1893-4,  73-8. 

•"Math.  Papers  Chicago  Congress  of  1893,  1896,  208-226;  University  of  Chicago  Decennial 

Publications,  (1),  9,  1904,  7-19. 
"El  Progreso  Matemdtico,  4,  1894,  265-9. 
"Introd.  th^orie  des  nombres,  1895,  42-50,  58-62,  343-350. 
•'Lehrbuch  der  Algebra,  II,  1896,  242-50,  259-261;  ed.  2,  1899,  302-10,  320-2. 
"Bidrag  till  Laran  om  Kongruenser  med  Primtalsmodyl,  Diss.,  Upsala,  1896,  86  pp. 


248  History  of  the  Theory  of  Numbers.  [Chap,  viii 

Analogous  to  the  definition  of  trigonometric  functions  in  terms  of  expo- 
nentials, he  defined  quasi  cosines  and  sines  by 

and  Tqx  as  their  quotient.  Their  relations  are  discussed.  He  defined 
pseudo  cosines  and  sines  by 

Cpx  =  Cq[{p  -  l)x]  =  e-'Cq2x,  Spx  =  -e-'Sq2x. 

For  each  prime  p<100,  he  gave  (pp.  65-86)  the  (integral)  values  of 

e^,  ind  x,  Cqx,  Sqx,  Tqx,  Cpx,  Spx 

for  x=l,  2,.  .  .,  p  +  1. 

L.  E.  Dickson^^  extended  the  results  of  Serret^*  to  the  more  general  case 
in  which  the  coefficients  of  the  functions  are  poljTiomials  in  a  given  Galois 
imaginary  (i.  e.,  are  in  a  Galois  field  of  order  p").  For  the  corresponding 
generaUzation  of  the  results  of  Serret^^  on  irreducible  congruences  modulo 
p  of  degree  a  power  of  p,  additional  developments  were  necessary.  To 
obtain  the  irreducible  functions  of  degree  p  in  the  GFlp'^']  which  are  of  the 
first  class,  we  need  the  complete  factorization,  in  the  field, 

hiz^'-z-v)  =U{h''z''-hz-^) 
where  hv  is  an  integer  and  /S  ranges  over  the  roots  of 

all  of  whose  roots  are  in  the  field.  For  the  case  v  =  0  this  factorization  is 
due  to  ]Mathieu."  Thus  K^z^—hz—^  is  irreducible  in  the  field  if  and  only 
if  B^O.  In  particular,  if  /3  is  an  integer  not  divisible  by  p,  z^  —  z—^  is 
irreducible  in  the  GF[p"]  if  and  only  if  n  is  not  divisible  by  p. 

R.  Le  Vavasseur^^  employed  Galois  imaginaries  to  express  in  brief  no- 
tation the  groups  of  isomorphisms  of  certain  tj-pes  of  groups,  for  example, 
that  of  the  abelian  group  G  generated  by  n  independent  operators  ai, . . ., 
a„,  each  of  period  a  prime  p.  If  i  is  a  root  of  an  irreducible  congruence 
of  degree  n  modulo  p,  and  if  j  =  ai-\-ia2+  .  .  .+i''~^a„,  he  defined  a^  to  be 
Oi"' . .  .  an"".  Then  the  operators  of  G  are  represented  by  the  real  and 
imaginary  powers  of  a. 

A.  Guldberg^^  considered  linear  differential  forms 

A        d^y ,       ,     dy , 
^y=^>:d^.-^----^^^di+^oy, 

^4th  integral  coefficients.  The  product  of  two  such  forms  is  defined  by 
Boole's  sjTnbolic  method  to  be 

d''  6}  d 

Ay'By={au-^-\-  •  •  •  +«o)(^/^+  •  •  •  +^^+^o)2/. 

"BuU.  Amer.  Math.  Soc.,3,  1896-7,  381-9. 
"M^m.  Ac.  Sc.  Toulouse,  (9),  9,  1897,  247-256. 
"^Comptes  Rendus  Paris,  125,  1897,  489. 


Chap.  VIII]  HiGHER   CONGRUENCES,    GaLOIS   ImAGINARIES.  249 

If  the  product  is  =Cy  (mod  p),  Ay  and  By  are  called  divisors  modulo  p 
of  Cy.  Let  A?/  be  of  order  n  and  irreducible  modulo  p.  Then  Ay  is  con- 
gruent modulis  p,  A?/  to  one  and  but  one  of  the  p"  forms 

(4)  s'c,^  (c,  =  0,  l,...,p-l). 

If  It  is  any  one  of  these  forms  (4)  and  if  e  =  'p^  —  \,  Guldberg  stated  the 
analogue  of  Fermat's  theorem 

dfu 

^='M  (mod  p,  ^y), 

but  incorrectly  gave  the  right  member  to  be  unity  [cf.  Epsteen/''^,  Dickson^"^]. 
L.  Stickelberger^^  considered  F{x)  =x^+aix''~^-\- .  .  .  with  integral  coeffi- 
cients, such  that  the  product  D  of  the  squares  of  the  differences  of  the  roots 
is  not  zero.  Let  p  be  any  prime  not  dividing  D.  Let  v  be  the  number  of 
factors  of  F{x)  which  are  irreducible  modulo  p.  He  proved  by  the  use  of 
prime  ideals  that 

(f)=(-i)»-. 

where  the  symbol  in  the  left  member  is  that  of  Legendre  [see  quadratic 
residues]. 

L.  E.  Dickson^^  proved  the  existence  of  the  Galois  field  GF[p'']  of  order 
p"  by  induction  from  r  =  n  to  r  =  qn,  by  showing  that 

(a:^"'-a:)/(x^"-x) 

is  a  product  of  factors  of  degree  q  belonging  to  and  irreducible  in  the 
(?F[p"].     Any  such  factor  defines  the  GF[p'"']. 

L.  Kronecker^°°  treated  congruences  modulis  p,  P{x)  from  the  stand- 
point of  modular  systems. 

F.  S.  Carey ^°^  gave  for  each  prime  p<  100  a  table  of  the  residues  of  the 
first  p  +  1  powers  of  a  primitive  root  a-\-hj  of  z^~^=l  (mod  p)  where /=»' 
(mod  p),  V  being  an  integral  quadratic  non-residue  of  p.  The  higher  powers 
are  readily  derived.  While  only  the  single  modulus  p  is  exhibited,  it  is 
really  a  question  of  a  double  modulus  p  and  x^—v.  Methods  of  "solving" 
2?"-!=!  are  discussed.  In  particular,  for  n  =  3,  there  is  given  a  primitive 
root  for  each  prime  p<  100. 

L.  E.  Dickson^"^  gave  a  systematic  introductory  exposition  of  the  theory, 
with  generalizations  and  extensions. 

M.  Bauer^"^  proved  that,  if /(a;)  =0  is  an  irreducible  equation  with  inte- 
gral coefficients  and  leading  coefficient  unity,  w  a  root,  D  its  discriminant, 
d  =  D/k^  that  of  the  domain  defined  by  w,  p  sl  prime  not  dividing  k,  x>l, 

«8Verhand.  I.  Internat.  Math.  Kongress,  1897,  186. 

"^BuU.  Amer.  Math.  Soc,  6,  1900,  203-4. 

looVorlesungen  iiber  Zahlentheorie,  I,  1901,  212-225  (expanded  by  Hensel,  p.  506). 
i"Proc.  London  Math.  Soc,  33,  1900-1,  294-310. 

lo'Linear  groups  with  an  exposition  of  the  Galois  field  theory,  Leipzig,  1901,  pp.  1-71. 
lo^Math.  Naturw.  Berichte  aus  Ungarn,  20, 1902,  39-42;  Math.  6s  Phyp.  Lapok,  10, 1902,  28-33. 


250  History  of  the  Theory  of  Numbers.  [Chap,  viii 

then  f{x)  is  congruent  modulo  p"  to  a  product  of  Fi(j),.  . .,  F,(a:),  each 
irreducible  modulo  p°,  such  that  F,(x)=/,(a:)'«  (mod  p),  where /(x)= 11/. (x)*' 
(mod  p),  and  /.(x)  is  irreducible  modulo  p.  There  is  an  example  of  an 
irreducible  cyclotomic  function  reducible  with  respect  to  every  prime  power 
modulus. 

P.  Bachmann^^  gave  an  exposition  of  the  general  theory. 

G.  Arnoux^"^  exhibited  in  the  form  of  tables  the  work  of  finding  a  primi- 
tive root  of  the  GF\1^]  and  of  the  GF[5^],  and  tabulated  the  reducible  and 
irreducible  congruences  of  degrees  1,  2,  3,  modulo  5. 

S.  Epsteen^°^  proved  the  result  of  Guldberg,^"  and  developed  the  theory 
of  residues  of  hnear  differential  forms  parallel  to  the  theory  of  finite  fields, 
as  presented  by  Dickson. ^°- 

L.  E.  Dickson ^°^  noted  that  the  last  mentioned  subjects  are  identical 
abstractly.     Let  the  irreducible  form  A?/  be 

d'^y  dy 

To  the  element  (4)  we  make  correspond  the  element  2c,z*  of  the  Galois 
field  of  order  p",  where  z  is  a  root  of  the  irreducible  congruence 

5„2"+...+5i2+5o=0  (mod  p). 

Since  product  relations  are  preserved  by  this  correspondence,  the  p"  resi- 
dues (4)  define  a  field  abstractly  identical  with  our  Galois  field. 

Dickson^"^''  proved  that  x'"=.t  (mod  vi  =  p")  has  p  and  only  p  roots  if  p  is 
a  prime  and  hence  does  not  define  the  Galois  field  of  order  m  as  occasionally 
stated. 

A.  Guldberg^"^''  employed  the  notation  of  finite  differences  and  wrote 

n  m  n  m 

Fy,  =  2  afi%,    Gy,  =  2  hj9%,    Fy^.Gy,  =  2  afi\  2  bfi%, 

t=0  »=0  t=0  »=0 

where  6y^  =  yjc+i,  ^'?/x  =  2/1+2.  •  •  •>  symbolically.  To  these  linear  forms  with 
integral  coefficients  taken  modulo  p,  a  prime,  we  may  apply  EucUd's  g.  c.  d. 
process  and  prove  that  factorization  is  unique.  Next,  let  6„,  be  not  di\'isible 
by  p,  so  that  Gy^  is  of  order  m.  With  respect  to  the  two  moduU  p,  Gy^,  a 
complete  set  of  p"*  residues  of  hnear  forms  is  a„,_i?/:r+m-i+  ■  •  •  +ao2/x  (^1  =  0, 
1,...,  p-1).  Amongst  these  occur  <}>{Gy^)  =p"'{l- 1 /p'"') ..  .{1-1 /p"''^) 
forms  Fy^.  prime  to  Gyj^  if  Wi, .  .  . ,  niq  are  the  orders  of  the  irreducible  factors 
of  Gyx  modulo  p,  and 

FyJ'^^'^^-^^y,  (mod  p,  Gy,) 

In  particular,  if  Gy^  is  irreducible  and  of  order  m, 

Fyr-'^yAmodp,Gy,). 


i 


i«Niedere  Zahlentheorie,  1,  1902,  363-399. 

»»Assoc.  frang.  av.  sc,  31,  1902,  II,  202-227. 

i«BuU.  Amer.  Math.  Soc,  10,  1903-4,  23-30. 

'"/bid.,  pp.  30-1. 

"'"Amer.  Math.  Monthly,  11,  1904,  39-40.  ^ 

>»"f'.\iinaU  di  Mat.,  (3),  10,  1904,  201-9.  ■ 


Chap.  VIII]  HIGHER  CONGRUENCES,    GaLOIS  ImAGINARIES.  251 

W.  H.  Bussey^°^  gave  for  each  Galois  field  of  order  <  1000  companion 
tables  showing  the  residues  of  the  successive  powers  of  a  primitive  root, 
and  the  powers  corresponding  to  the  residues  arranged  in  a  natural  order. 
These  tables  serve  the  same  purposes  in  computations  with  Galois  fields 
that  tables  of  indices  serve  in  computations  with  integers  modulo  p",  where 
p  is  a  prime. 

G.  Voronoi^°^  proved  the  theorem  of  Stickelberger.^^  Thus,  for  n  =  3, 
(D/p)  =  —  1  only  when  v  =  2.  Hence  a  cubic  congruence  has  a  single  root 
if  (D/p)  =  —1,  and  three  real  roots  or  none  if  {D/p)  =  +1. 

P.  Bachmann^^°  developed  the  general  theory  from  the  standpoint  of 
Kronecker's  modular  systems  and  considered  its  relation  to  ideals  (p.  241). 

M.  Bauer^^^  employed  a  polynomial /(z)  of  degree  n  irreducible  modulo 
p,  and  another  one  M{z)  of  degree  less  than  that  of  f{z)  and  not  divisible 
by/(2)  modulo  p.     Then  if  {t,  a)  =  l,  the  equation 

/(2)+p«M(2)=0 

is  irreducible.     The  case  a  =  1  is  due  to  Schonemann^^  (p.  101). 

G.  Arnoux,^^^  starting  with  any  prime  m  and  integer  n,  introduced  a 
symbol  i  such  that  i^^^  1  (mod  m)  and  such  that  i,  i^, .  .  .,  i^  are  distinct, 
where  s  =  m"  — 1,  without  attempting  a  logical  foundation.  If /(x)  is  irre- 
ducible modulo  m  and  of  degree  n,  there  is  only  a  finite  number  of  distinct 
residues  of  powers  of  x  modulis/(x),  m]  let  x^  andx'^'^^have  the  same  residue. 
Thus  x^  —  1  is  divisible  by/(x)  modulo  m.  It  is  stated  (p.  95)  without  proof 
that  p  divides  s.  "Call  a  a  root  of /(a:)  =  0.  To  make  a  coincide  with  the 
primitive  root  i  of  a;^=  1,  we  must  take  p  =  s,  whence  every  such  primitive 
root  is  a  root  of  an  irreducible  congruence  of  degree  n  modulo  m."  Follow- 
ing this  inadequate  basis  is  an  exposition  (pp.  117-136)  of  known  properties 
of  Galois  imaginaries. 

L.  I.  Neikirk^^^  represented  geometrically  the  elements  of  the  Galois 
field  of  order  p"  defined  by  an  irreducible  congruence 

/(x)  =rc"+aia:"-^H-  .  .  .  -fa„=0  (mod  p). 

Let  j  be  a  root  of  the  equation  f{x)  =  0  and  represent 

Ci/~^+  .  . .  +c„_ii-Fc„  (c's  integers) 

by  a  point  in  the  complex  plane.     The  p""  points  for  which  the  c's  are  chosen 
from  0,  1, .  .  .,  p  — 1  represent  the  elements  of  the  Galois  field. 

G.  A.  Miller^^^  listed  all  possible  modular  systems  p,  4>{x),  where  p  is  a 
prime  and  the  coefficient  of  the  highest  power  of  x  is  unity,  in  regard  to 
which  a  complete  set  of  prime  residues  forms  a  group  of  order  ^12.  If 
4>{x)  is  the  product  of  k  distinct  irreducible  functions  4>i, .  .  . ,  (f)^  modulo  p, 

"SBuU.  Amer.  Math.  Soc,  12,  1905,  21-38;  16,  1909-10,  188-206. 

lO'Verhand.  III.  Internat.  Math.  Kongress,  1905,  186-9. 

""AUgemeine  Arith.  d.  Zahlenkorper,  1905,  81-111. 

"iJour.  fur  Math.,  128,  1905,  87-9. 

"'Arithm^tique  Graphique,  Fonctiona  Arith.,  1906,  91-5. 

"'BuU.  Amer.  Math.  Soc,  14,  1907-8,  323-5. 

"«Archiv  Math.  Phys.,  (3),  15,  1909-10,  115-121. 


252  History  of  the  Theory  of  Numbers.  [Chap,  viii 

the  residues  prime  to  p,  (t>{x)  constitute  the  direct  product  of  the  groups 
with  respect  to  the  various  p,  0.(a:).  Not  every  abelian  group  can  be  repre- 
sented as  a  congruence  group  composed  of  a  complete  set  of  prime  residues 
with  respect  to  Fj, .  .  .,  Fx,  where  the  F's  are  functions  of  a  single  variable. 

Mildred  Sanderson^  ^^  employed  two  moduU  m  and  P{y),  the  first  being 
any  integer  and  the  second  any  polynomial  in  y  with  integral  coefficients. 
Such  a  polynomial /(?/)  is  said  to  have  an  inverse  /i  (y)  if  //i=  1  (mod  m,  P). 
If  P{y)  is  of  degree  r  and  is  irreducible  with  respect  to  each  prime  factor  of 
m,  a  function  f{y),  whose  degree  is  <r,  has  an  inverse  moduhs  m,  P{y),  if 
and  only  if  the  g.  c.  d.  of  the  coefficients  of /(?/)  is  prune  to  m.  For  such 
an/, /"  =  1  (mod  vi,  P),  where  n  is  Jordan's  function  Jr{fn)  [Jordan, ^"^ 
Ch.  V].  In  case  m  is  a  prime,  this  result  becomes  Galois'^^  generalization 
of  Fermat's  theorem.  The  product  of  the  n  distinct  residues  having 
inverses  moduHs  m,  P{y),  is  congruent  to  —1  when  m  is  a  power  of  an  odd 
prime  or  the  double  of  such  a  power  or  when  r=  1,  w  =  4;  but  congruent  to 
+  1  in  all  other  cases  — a  two-fold  generahzation  of  Wilson's  theorem. 
There  exists  a  polynomial  P{y)  of  degree  r  which  is  irreducible  with  respect 
to  each  prime  factor  of  m.  Then  if  A{y),  B{y)  are  of  degrees  <r  and  their 
coefficients  are  not  all  divisible  by  a  factor  of  m,  there  exist  polynomials 
a(i/),  ^{y),  such  that  aA+/3B=l  (mod  m,  P). 

Several  writers^^^  discussed  the  irreducible  quadratic  factors  modulo  p 
of  {x'^—\)/{x^  —  l),  where  A' =  1  or  2,  p  is  a  prime,  a  a  divisor  of  p-fl. 

G.  Tarry^^^  noted  that,  if  f=q  (mod  m),  where  5  is  a  quadratic  non- 
residue  of  the  prune  m,  the  Galois  imaginary  a+hj  is  a  primitive  root  if 
its  norm  {a-\-'bj){a  —  hj)  is  a  primitive  root  of  m  and  if  the  ratio  a:h  and  the 
analogous  ratios  of  the  coordinates  of  the  first  m  powers  of  a-\-hj  are  incon- 
gruent. 

L.  E.  Dickson^ ^^  proved  that  two  polynomials  in  two  variables  with 
integral  coefficients  have  a  unique  g.  c.  d.  modulo  p,  a  prime.  Thus  the 
unique  factorization  theorem  holds. 

G.  Tarr>'^^^  stated  that  Ap  is  a  primitive  root  of  the  GF[p^]  if  the  norm 
of  A  =  a-\-'bj  is  a  primitive  root  of  p  and  if  the  imaginar>^  p  belongs  to  the 
exponent  p+1.  The  0(p-f  1)  numbers  p  are  found  by  the  usual  process 
to  obtain  the  primitive  roots  of  a  prime. 

U.  Scarpis^-°  proved  that  an  equation  of  degree  v  irreducible  in  the 
Galois  field  of  order  p"  has  in  the  field  of  order  p"*"  either  v  roots  or  no 
root  according  as  v  is  or  is  not  a  di\dsor  of  m  [Dickson^°^,  p.  19,  lines  7-9]. 

Cubic  Congruences. 
A.  Cauchy^^°  solved  y'^-\-By-\-C={)  (mod  p)  when  it  has  three  distinct 

'"Annals  of  Math.,  (2),  13,  1911,  36-9. 

"•L'interm^diaire  dea  math.,  18,  1911,  195,  246;  19,  1912,  61-69,  95-6;  21,  1914,  158-161;  22, 

1915,  77-8.     Sphinx-Oedipe,  7.  1912,  2-3. 
"'Assoc,  fran^.  av.  sc,  40,  1911,  12-24.  "sBuU.  Amer.  Math.  Soc,  (2),  17,  1911,  293-4. 

"'Sphinx-Oedipe,  7,  1912,  43^,  49-50.  ""AnnaU  di  Mat.,  (3),  23,  1914,  45. 

""Exercices  de  Math.,  4,  1829,  279-292;  Oeuvres,  (2),  9,  326-333. 


Chap.  VIII]  CuBIC   CONGRUENCES.  253 

integral  roots  y^,  2/2,  Vs,  and  p  is  a  prime  =  1  (mod  3),  and  B^O  (mod  p) .    Set 

^Vi  =  yi+ry2-\-r%,  ^V2  =  yi+r^y2+ry3,  r^+r+ 1=0  (mod  p). 

The  roots  of  u^-\-Cu—B^/27=  0  (mod  p)  are  Ui  =  ^i^,  U2  =  ^2^.  After  finding 
Vi  from  Vi^=Ui  (mod  p),  we  get  V2=  —B/{3vi),  and  determine  the  y's  from 
'2yi=0  and  the  expressions  for  3^1,  Sv2.    Thus 

2/i=yi+z;2,  y2=r\+rv2,  ys^rvi+r^  (mod  p). 

Since  by  hypothesis  the  cubic  congruence  has  three  distinct  integral  roots, 
the  quadratic  has  two  distinct  integral  roots,  whence 

p-l  P-l  7^2       ^3 

"^"^7  '  +(~^+^7  '  ^2'  ^  '  ^^  (modp). 

Conversely,  if  the  last  two  conditions  are  satisfied,  the  cubic  congruence 
has  three  distinct  real  roots  provided  p=l  (mod  3),  B^O  (mod  p). 

G.  Oltramare^^^  found  the  conditions  that  one  or  all  of  the  roots  of 
x^+Spx+2q=0  (mod  fx)  given  by  Cardan's  formula  become  integral  modulo 
fx,  a  prime.     Set 

D  =  q^-\-p\  a=-q+VD,  T=-q-VD, 

First,  let  /x  be  a  prime  6n  —  1 .  If  D  is  a  quadratic  residue  of  /x,  there 
is  a  single  rational  root  —  2g/(p+(r^"+T^").  If  Z)  is  a  quadratic  non-residue 
of  fx,  there  are  three  rational  roots  or  no  root  according  as  the  rational  part 
M  of  the  development  of  o-^"~^  by  the  binomial  theorem  satisfies  or  does  not 
satisfy  ilfp^+g^O  (mod  //) ;  if  also  /x  =  18m+ll  and  there  are  three  rational 
roots,  they  are 

2M^,  -^(M±iW-3i)), 

if  (72"*+i  =  ikf  4-iVVD;  with  a  like  result  when  m  =  18m+5. 

Next,  let  /x  =  6n+l.  If  Z)  is  a  quadratic  non-residue  of  ^x,  there  is  one 
rational  root  or  none  according  as  the  rational  part  M  of  the  development 
of  o-^"  is  or  is  not  such  that 

(2M-l)2(ikf+l)=-2gVp3  (modM), 

and  if  a  rational  root  exists  it  is  2q/  \  p  {2M  —  1 ) } .  If  Z)  is  a  quadratic  residue 
of  IX,  there  are  three  rational  roots  or  none  according  as  cr^''=  1  (mod  /x) 
or  not.  When  there  are  three,  they  are  given  explicitly  if  ^t=18m-|-7  or 
18m  +  13,  while  if  //  =  18m  +  l  there  are  sub-cases  treated  only  partially. 

G.  T.  Woronoj^^^  (or  Voronoi)  employed  Galois  imaginaries  a -{-hi,  where 
i^—N=0  (mod  p)  is  irreducible,  p  being  an  odd  prime,  to  treat  the  solution  of 

x^—rx  —  s=0  (mod  p). 

I'lJour.  fiir  Math.,  45,  1853,  314-339. 

'"Integral  algebraic  numbers  depending  on  a  root  of  a  cubic  equation  (in  Russian),  St.  Peters- 
burg, 1894,  Ch.  I.     Cf.  Fortschritte  Math.,  25,  1893-4,  302-3.     Cf.  Voronoi."' 


254  History  of  the  Theory  of  Numbers.  [Chap,  viii 

If  4r^  — 27s^  is  a  quadratic  non-residue  of  p,  the  congruence  has  one  and 
only  one  root;  but  if  it  is  a  residue,  there  are  three  roots  or  no  root. 

G.  Cordone^^^  gave  simpler  proofs  of  Oltramare's^^^  theorem  II  on  the 
case  fjL  =  Qn  —  I,  gave  theorems  to  replace  VII  and  VIII,  and  proved  that  the 
condition  in  IX  is  sufficient  as  well  as  necessary. 

Ivar  Damm^^  found  when  Cardan's  formula  gives  three  real  roots,  one 
or  no  real  root  of  a  cubic  congruence,  and  expressed  the  roots  by  use  of 
his  quasi  sine  and  cosine  functions.  For  the  prime  modulus  p  =  3nH-l, 
f=x^-\-ax-\-b  is  irreducible  if 


c  = 


^isrea.,(-|+cf.*l. 


If  p  =  3n  — 1,  it  is  irreducible  if  c  and  (  —  6/2+0)"  are  both  imaginary. 
There  are  given  (p.  52)  explicit  expressions  for  h  such  that  /  is  irreducible. 

J.  Iwanow^^  gave  another  proof  of  the  theorem  of  Woronoj.^^^ 

Woronoj  ^^^  gave  another  proof  of  the  same  theorem  and  stated  that  the 
congruence  has  the  same  number  of  roots  for  all  primes  representable  by 
a  binary  quadratic  form  whose  determinant  equals  —  4r^+27s^. 

G.  Arnoux^^*^  gave  double-entry  tables  of  the  roots  of  the  congruences 
x''+6x*+a=0  (mod  m),  and  solved  numerical  cubic  congruences  by  in- 
terpreting Cardan's  formulas. 

G.  Arnoux^^^  treated  x^+6x+a=0  (mod  m)  by  use  of  Cardan's  formula. 
For  m  =  1 1 ,  he  gave  a  table  of  the  real  roots  for  a  ^  10,  6^  10,  and  the  residues 
of 

^-4+27 

When  R  is  a  quadratic  residue,  the  cube  roots  of  —  a/2=t  y/R  are  found  by 
use  of  a  table  for  the  Galois  field  of  order  11^  defined  by  r=2  (mod  11), 
and  the  cubic  is  seen  to  have  a  real  and  two  imaginary  roots  involving  i. 
If  jR  is  a  quadratic  non-residue,  there  are  three  real  roots  or  none.  Like 
results  are  said  to  hold  when  m  —  1  is  not  divisible  by  3.  If  w=  1  (mod  3), 
there  is  a  single  real  root  if  7?  is  a  quadratic  non-residue ;  three  real  or  three 
imaginary  roots  of  the  third  order  if  ^  is  a  residue. 

L.  E.  Dickson'^^  proved  that,  if  p  is  a  prime  >3,  x^-\-^x-\-b=0  (mod  p) 
has  no  integral  root  if  and  only  if  —4/3^  —  276^  is  a  quadratic  residue  of  p^ 
say  =81)u^,  and  if  §(  — 6+/xV  —  3)  is  not  congruent  to  the  cube  of  any 
y+zy/  —  3,  where  y  and  z  are  integers.  The  reducible  and  irreducible 
cubic  congruences  are  given  explicitly.  Necessary  and  sufficient  conditions 
for  the  irreducibility  of  a  quartic  congruence  are  proved. 

'"Rendiconti  Circolo  Mat.  di  Palermo,  9,  1895,  221-36. 

"^BuU.  Ac.  Sc.  St.  Petersburg,  5,  1896,  137-142  (in  Russian). 

'^Natural  Sc.  (Russian),  10,  1898,  329;  of.  Fortschritte  Math.,  29,  1898,  156. 

'3« Assoc,  franc;,  av.  sc,  30,  1901,  II,  31-50,  51-73;  corrections,  31,  1902,  II,  202. 

'"Assoc,  frang.  av.  sc,  33,  1904,  199-230  [182-199],  and  Amoux'",  166-202. 

'"BuU.  Amer.  Math.  Soc,  13,  1906,  1-8. 


Chap.  VIII]  CUBIC   CONGRUENCES.  255 

D.  Mirimanoff^^^  noted  that  the  results  by  Arnoux"^'^"  may  be  com- 
bined by  use  of  the  discriminant  D=  —4b^  —  27a^=  —3-Q^R  in  place  of  R, 
since  —  3  is  a  quadratic  residue  of  a  prime  p  =  Sk-{-l,  non-residue  ofp  =  Sk  —  l, 
and  we  obtain  the  result  as  stated  by  Voronoi.-^^^ 

To  find  which  of  the  values  1  or  3  is  taken  by  v  when  D  is  a  quadratic 
residue,  apply  the  theorem  that  if /(x)  =  0  (mod  p)  is  an  irreducible  con- 
gruence of  degree  n  and  if  Xq  is  one  of  its  imaginary  roots  (say  one  of  the 
roots  of  the  equation  f{x)  =  0) ,  the  roots  are 

p  pn—l 

Hence  a  function  unaltered  by  the  cycHc  substitution  (xqXi.  .  .Xn-i)  has 
an  integral  value  modulo  p.  Take  w  =  3,  D=d^,  a  a  root  5^1  of  2^=1 
(mod  p),  and  let 

M  =  (xo+aXi+a^X2)^. 

If  p=l  (mod  3),  a  is  an  integer,  and  M  is  an  integer  if  ^^  =  1,  while  M  is 
the  cube  of  an  integer  if  v  =  S.  Thus  we  have  Arnoux's  criterion  :^^^  v  =  S 
if  ilf  or-|  (  — 9a+V  — 3d)  is  a  cubic  residue  modulo  p.  If  p=  —1  (mod  3) 
J/ =  3  if  and  only  if  ilf^^l  (mod  p),  where  k  =  {p'^ -l)/3. 

For  quartic  congruences,  we  can  use  (a^o  — a:i-|-a:2  — a^s)^. 

R.  D.  von  Sterneck^"*"  noted  that  if  p  is  a  prime  >3  not  dividing  A, 
and  if  k  =  SAC  —  B^^O  (mod  p),  then  the  number  of  incongruent  values 
taken  by  Ax^+Bx^+Cx-\-D  is  i{2p+(-3/p)j ;  but,  if  k=0,  the  number 
is  p  if  p  =  Sn  —  l,  (p+2)/3  if  p  =  3n+l.     Generalization  by  Kantor.^^^ 

C.  Cailler^^^  treated  x^+px-\-q=0  (mod  I),  where  I  is  a  prime  >3.  By 
the  algebraic  method  leading  to  Cardan's  formula,  we  write  the  congruence 
in  the  form 

(1)  x^-Sabx+abia+b)^0  (mod  /), 

where  a,  h  are  the  roots  of  z^-{-Sqz/p—p/S=0  (mod  T),  whence 

z={xQ-\-aXi+ a^X2) V (9p) ,  a^ + a  + 1  =  0  (mod  I) . 

Let  A  =  4p^+27g^.  If  3A  is  a  quadratic  residue  of  i,  a  and  b  are  distinct 
and  real.  If  3A  is  a  non-residue,  a  and  b  are  Galois  imaginaries  r±s\/iV^ 
where  N  is  any  non-residue.     For  a  root  a:  of  (1), 

Use  is  made  of  a  recurring  series  S  with  the  scale  of  relation  [a +6,  —ab] 
to  get  2/0,  Vi,.. ..     Write  Q  =  (3A/Z).     If  ;  =  3m-l,  Q  =  l,  then 

If  l  =  3m-\-l,  Q  =  l,  the  congruence  is  possible  only  when  the  real  number 
a/b  is  a  cubic  residue,  i.  e.,  if  2/^=0  in  S;  let  a/b  belong  to  the  exponent 
3m=f1  modulo  I,  whence 

'"L'enseignement  math.,  9,  1907,  381-4. 

"oSitzungsber.  Ak.  Wiss.  Wien  (Math.),  116,  1907,  Ila,  895-904. 

"'L'enseignement  math.,  10,  1908,  474r-487. 


256  History  of  the  Theory  of  Numbers.  [Chap,  viii 

w=  I  7  I     J  x= or > 

\o/  2/2^-1         2/m 

according  as  the  upper  or  lower  sign  holds.     If  l==Sm-\-l,  Q=  —1,  then 
2/3m+2=0,  (rl        =-,  realx=-; . 

\0/  0  y2m+l 

li  l  =  Sm  —  l,  Q=  —1,  there  are  three  real  roots  if  and  only  if  a/b  is  a  cubic 
residue  of  I,  viz.,  2/^=0;  when  real,  the  roots  may  be  found  as  in  the  second 
case. 

Cailler^^^  noted  that  a  cubic  equation  X  =  0  has  its  roots  expressible 
rationally  in  one  root  and  VA,  where  A  is  the  discriminant  (Serret's 
Algebre,  ed.  5,  vol.  2,  466-8).  Hence,  if  p  is  a  prime,  X=0  (mod  p)  has 
three  real  roots  if  one,  when  and  only  when  A  is  a  quadratic  residue  of  p. 
If  p  =  9m^l,  his^^^  test  shows  that  x^  — 3xH-l  =  0  (mod  p)  has  three  real 
roots,  but  no  real  root  for  other  prime  moduli  5^3.  The  function 
F{x)  =x^-{-x-  —  2x  —  l  for  the  three  periods  of  the  seventh  roots  of  unity  is 
divisible  by  the  primes  7m=*=  1  (then  3  real  roots,  Gauss®",  p.  624)  and  7, 
but  by  no  other  primes. 

E.  B.  Escott"^  noted  that  the  equation  F(x)  =0  last  mentioned  has  the 
roots  a,  ^  =  a^—2,  7=/3^— 2,  so  that  F{x)  =  0  (mod  p)  has  three  real  roots 
if  one  real  root.  To  find  the  most  general  irreducible  cubic  equation  with 
roots  a,  (3,  y  such  that 

^=/(a),     y=m,     a=f{y), 

we  may  assume  that/(x)  is  of  degree  2.     For /(a)  =  a^—n,  we  get 

(2)  x^-\-ax'^-{a'-2a-\-3)x-ia^-2a^-\-da-l)=0, 

with  ^  =  a^  —  c,  y=j3^—c,  a=y^—c,  c  =  o^  — a+2.  The  corresponding  con- 
gruence has  three  real  roots  if  one.  To  treat/(a)=a^+^a+Z,  add  k/2  to 
each  root.  For  the  new  roots,  jS'  =  a'^  —  n,  as  in  the  former  case.  To  treat 
/(a)  =  ta^-\-ga-\-h,  the  products  of  the  roots  by  t  satisfy  the  preceding  relation. 
L.  E.  Dickson^^  determined  the  values  of  a  for  which  the  congruence 
corresponding  to  (2)  has  three  integral  roots.     Replace  x  by  z—a;  we  get 

z^-2az^+{2a-S)z-\-l  =  0  (mod  p). 

If  one  root  is  z,  the  others  are  1  —  1/z  and  1/(1—2).  Evidently  a  is  rational 
in  z.  If  —3  is  a  quadratic  non-residue  of  p,  there  are  exactly  (p  —  2)/S 
values  of  a  for  which  the  congruence  has  three  distinct  integral  roots.  If 
—  3  is  a  residue,  the  number  is  (pH-2)/3.  A  second  method,  yielding  an 
explicit  congruence  for  these  values  of  a,  is  a  direct  application  of  his^^® 
general  criteria  for  the  nature  of  the  roots  of  a  cubic  congruence. 

T.  Hayashi^^^  treated  cyclotomic  cubic  equations  with  three  real  roots 
by  use  of  Escott's^'*^  results. 

"«L'interm^diaire  des  math.,  16,  1909,  185-7.  '"/bid.,  (2),  12,  1910-11,  149-152. 

'"Annals  of  Math.,  (2),  11,  1909-10,  86-92.  >«/6id.,  189-192. 


Chap.  VIII]  MISCELLANEOUS  RESULTS   ON   CONGRUENCES.  257 

Miscellaneous  Results  on  Congruences. 

Linear  congruences  will  be  treated  in  Vol.  2  under  linear  diophantine 
equations,  quadratic  congruences  in  two  or  more  variables,  under  sums  of 
four  squares;  ax''+hy''-}-cz''=0,  under  Fermat's  last  theorem. 

Fermat^^^  stated  that  not  every  prime  p  divides  one  of  the  numbers 
a+1,  a^+l,a^+l,.  .  ..  For,  if  /c  is  the  least  value  for  which  a^'  —  1  is  divis- 
ible by  p  and  if  k  is  odd,  no  term  a^-{-l  is  divisible  by  p.  But  if  k  is  even, 
^fc/2_|_2  jg  divisible  by  p. 

Fermat^^^  stated  that  no  prime  12n±l  divides  S'^+l,  every  prime 
12n=t5  divides  certain  S'^+l,  no  prime  10n±l  divides  5""+!,  every  prime 
lOn^S  divides  certain  5"^+!,  and  intimated  that  he  possessed  a  rule  relating 
to  all  primes.     See  Lipschitz.^^*^ 

A.  M.  Legendre^^°  obtained  from  a  given  congruence  x"=ax'*~^+-  ■  • 
(mod  p),  p  SiTi  odd  prime,  one  having  the  same  roots,  but  with  no  double 
roots.  Express  x^^"^'^^  in  terms  of  the  powers  of  a;  with  exponents  <n,  and 
equate  the  result  to  +1  and  to  —1  in  turn.  The  g.  c.  d.  of  each  and  the 
given  congruence  is  the  required  congruence.  An  exception  arises  if  the 
proposed  congruence  is  satisfied  by  0,  1, .  .  .,  p  — 1. 

Hoen4  de  Wronski^^^  developed  {ni-\- .  .  .-\-nJ"',  replaced  each  multi- 
nomial coefficient  by  unity,  and  denoted  the  result  by  A[ni+  .  .  .+nj\"*. 
Thus  A[ni+n2f  =  ni^+nin2+n2^.     SetiV„  =  ni+  .  .  .  +n„.     Then  (pp.65-9), 

(1)  A[N^-nX-A[N^-nX={n,-n,)A[NT~'=0  (mod  n,-n,). 

Let  (ni.  .  .nS)m  be  the  sum  of  the  products  of  ni,.  .  .,  n„  taken  m  at  a 
time.     Then  (p.  143),  if  A[Nj'  =  l, 

(2)  A[NJ={n,..  .nJ,A[N^Y-'-in,..  .nJ^AWJ-' 

+  in,..  .n^)sA[NJ-'-  .  .  .+{-iy+\n,.  .  .nJ),A[Nj. 

He  discussed  (pp.  146-151)  in  an  obscure  manner  the  solution  of  Xi=X2 
(mod  X),  where  the  X's  are  polynomials  in  ^  of  degree  v.  Set  N^  =  ni-\-  .  .  . 
H-n„_2+np+ng.  Let  the  negatives  of  Ui,...,  n„_2,  Up  be  the  roots  of 
P  =  Po-\-PiX+  .  .  .+P^^2^"~^-\-x"~^  =  0;  the  negatives  of  ni,...,  n^-2,  "riq 
the  roots  of  Q  =  Qo+ .  .  .+x"~^  =  0.  We  may  add  fiX  and  ^2^  to  the 
members  of  our  congruence.  It  is  stated  that  the  new  first  member  may 
be  taken  to  be  A[A^„— nj"",  whence  by  (2) 

X,-\-^,X  =  P^_2A[N^-nr-'-P.-3A[N^-nX-^+ .  . ., 

and  the  A's  may  be  expressed  in  terms  of  the  P's  by  (2).  Similarly, 
^2+^2^  niay  be  expressed  in  terms  of  the  Q's.  By  (1),  X  =  nq—np  =  Q^_2 
—  P„_2.  Since  P  =  0,  Q  =  0  have  co  — 2  roots  in  common,  we  have  further 
conditions  on  the  coefficients  Pi,  Qi.     It  is  argued  that  w  — 3  of  the  latter 

"^Oeuvres,  2,  209,  letter  to  Frenicle,  Oct.  18,  1640, 
i^Oeuvres,  2,  220,  letter  to  Mersenne,  June  15,  1641. 
i"M6m._Ac.  Sc.  Paris,  1785,  483. 

"^Introduction  a  la  Philosophic  des  Math^matiques  et  Technie  de  I'Argorithmie,  Paris,  1811. 
He  used  the  Hebrew  aleph  for  the  A  of  this  report.    Cf.  Wronski^^'  of  Ch.  VII. 


258  History  of  the  Theory  of  Numbers.  [Chap,  viil 

remain  arbitrary,  and  that  ^  is  a  function  of  them  and  one  of  the  n's,  which 
has  an  arbitrary  rational  value. 

A.  Cauchy^"  noted  that  if  /  and  F  are  polynomials  in  x,  Lagrange's 
interpolation  formula  leads  to  polynomials  u  and  v  such  that  uJ-\-vF  =  R, 
where  i?  is  a  constant  [provided  /  and  F  have  no  common  factor].  If  the 
coefficients  are  all  integers,  R  is  an  integer.  Hence  R  is  the  greatest  of  the 
integers  di\-iding  both  /  and  F.  For  /=  x^—x,  we  may  express  i2  as  a  prod- 
uct of  trigonometric  functions.  If  also  F{x)=  (x"+l)/(a:+l),  where  n  and 
p  are  primes,  R=0  or  ±2  according  as  p  is  or  is  not  of  the  form  nx-\-\. 
Hence  the  latter  primes  are  the  only  ones  dividing  x^+l,  but  not  x-\-\. 

Cauchy^^^  proved  that  a  congruence /(x)  =  0  (mod  p)  of  degree  m<p  is 
equivalent  to  (x— r)'</)(x)  =  0,  where  4>  is  of  degree  m—i,  if  and  only  if 

/(r)^0,  /'(r)  =  0,.  .  .,  r-'\r)^Q  (mod  p), 

where  p  is  a  prime.  The  theorem  fails  if  m'^p.  He  gave  the  method  of 
Libri  (M^moires,  I)  for  solving  the  problem:  Given"/(a:)  =  0  (mod  p)  of 
degree  m^p  and  with  exactly  m  roots,  and/i(x)  of  degree  l^m,  to  find  a 
polynomial  </)(a;),  also  with  integral  coeflficients,  whose  roots  are  the  roots 
common  to/  and /i.  He  gave  the  usual  theorem  on  the  number  of  roots  of 
a  binomial  congruence  and  noted  conditions  that  a  quartic  congruence  have 
four  roots. 

Cauchy^^  stated  that  if  7  is  an  arbitrary  modulus  and  if  ri, .  . .,  r„,  are 
roots  of /(x)=0  (mod  7)  such  that  each  difference  Vi—Tj  is  prime  to  7,  then 

f{x)={x-ri) .  .  .(x-rJQ(x)  (mod  7). 

If  in  addition,  m  exceeds  the  degree  of /(x),  then/(x)  =  0  (mod  7)  for  every  x. 
A  congruence  of  degree  n  modulo  p^,  where  p  is  a  prime,  has  at  most  n 
roots  unless  every  integer  is  a  root.  If /(r)  =  0  (mod  7)  and  if  in  the  irre- 
ducible fraction  equal  to 

_/(r) 

the  denominator  is  prime  to  7,  then  r— r7  is  a  root  of /(a:)=0  (mod  P). 

V.  A.  Lebesgue^^^  wrote  a/b=c  (mod  p)  if  h  is  prime  to  p  and  a=bc 
(mod  p),  and  a/b=c/d  (mod  p)  \i  h,  d  are  prime  to  p  and  ad=bc  (mod  p). 

J.  A.  Serret^^^  stated  and  A.  Genocchi  proved  that,  if  p  is  a  prime,  the 
sum  of  the  mth.  powers  of  the  p"  polynomials  in  x,  of  degree  n  —  1  and  with 
integral  coefficients  <p,  is  a  multiple  of  p  if  m<p''  — 1,  but  not  if  m  =  p''  — 1. 

J.  A.  Serret^^^  noted  that  all  the  real  roots  of  a  congruence  f{x)  =  0 
(mod  p),  where  p  is  a  prime,  satisfy  \j/{x)^0,  where  \f/  is  the  g.  c.  d.  of  f{x) 
andxP~^-l. 

'"Exercices  de  Math.,  1,  1826,  160-6;  Bull.  Soc.  Philomatique;  Oeuvres,  (2),  6,  202-8. 

^"Exercices  de  Math.,  4,  1829,  253-279;  Oeuvres,  (2),  9,  298-326.  j 

i"Compte8  Rendus  Paris,  25,  1847,  37;  Oeuvres,  (1),  10,  324-30.  ~\ 

'"Nouv.  Ann.  Math.,  9,  1850,  436. 

i^Nouv.  Ann.  Math.,  13.  1854,  314;  14,  1855,  241-5 

"'Cours  d'alg^bre  sup6rieure,  ed.  2,  1854,  321-3. 


Chap.  VIII]  MISCELLANEOUS  RESULTS   ON   CONGRUENCES.  259 

N.  H.  AbeP^^  proved  that  we  can  solve  by  radicals  any  abelian  equation, 
i.  e.,  one  whose  roots  are  r,  0(r),  0^(r)  =  <t)[<i){r)], .  .  .,  where  </>  is  a  rational 
function.  H.  J.  S.  Smith^^^  concluded  that  when  the  roots  of  a  congru- 
ence can  be  similarly  expressed  modulo  p,  its  solution  can  evidently  be 
reduced  to  the  solution  of  binomial  congruences,  and  the  expressions  for 
the  roots  of  the  corresponding  equation  may  be  interpreted  as  the  roots 
of  the  congruence.  For  the  special  case  a:"=l,  this  was  done  by  Poinsot 
in  1813-20  in  papers  discussed  in  the  chapter  on  primitive  roots. 

M.  Jenkins^^^"  noted  that  all  solutions  of  a^=l(mod  x)  are  x=  Un=UiU2 
.  .  .Un,  where  Ui  is  any  divisor  of  any  power  of  a  — 1;  u^  any  divisor  prime 
to  a  — 1,  of  any  power  of  a'"  — 1;.  .  .;  u,,  any  divisor,  prime  to  a^"-2  — 1, 
of  any  power  of  a^^-'^  —  l.  For  a*+l  =  0  (mod  x),  modify  the  preceding 
by  taking  odd  factors  of  a+1  instead  of  factors  of  a  — 1. 

J.  J.  Sylvester^®°  proved  that  if  p  is  a  prime  and  the  congruence /(a:)  =  0 
(mod  p)  of  degree  n  has  n  real  roots  and  if  the  resultant  of  f{x)  and  g{x) 
is  divisible  by  p,  then  g{x)^0  has  at  least  one  root  in  common  with /(a;)  =  0. 
There  are  exactly  p  —  1  real  roots  of  x^~^=l  (mod  p^). 

A.  S.  Hathaway^^^  noted  the  known  similarity  between  equations  and 
congruences  for  a  prime  modulus.  He^^^  made  abstruse  remarks  on  higher 
congruences. 

G.  Frattini^^^  proved  that  x^ — Dy'^=\  and  x'^  —  Dy^=\  are  each  solvable 
when  the  modulus  is  a  prime  p>5  and  Dp^O.  If  d  =  B^—AC^O,  then 
Ax'^-i-2Bx^y+Cy"=\  (mod  p)  is  solvable  since  dx'^+XC  can  be  made  con- 
gruent to  a  square  and  hence  to  {Cy-{-Bx^y.     Likewise  for  ax'^-\-2bx-\-c=y'^. 

A.  Hurwitz^^^  discussed  the  congruence  of  fractions  and  the  theory  of 
the  congruence  of  infinite  series.  If  (/)(x)  =ro-\-riX-{- .  .  .  +r„a:V^-+  •  •  •  and 
if  yp{x)  is  a  similar  series  with  the  coefficients  s^,  then  0  and  \l/  are  called 
congruent  modulo  m  if  and  only  if  Vn^s^  (mod  m)  for  n  =  1,  2, .  .  . . 

G.  Cordone^^^  treated  the  general  quartic  congruence  for  a  prime 
modulus  ji  by  means  of  a  cubic  resolvent.  The  method  is  similar  to  Euler's 
solution  of  a  quartic  equation  as  presented  by  Giudice  in  Peano's  Rivista 
di  Matematica,  vol.  2.  For  the  special  case  x'^-\-%Hx^-\-K=0  (mod  /x), 
set  t  =  {ix  —  l)/2,  r^  =  9H^  —  K;  then  if  K  is  a  quadratic  residue  of  /jl,  there 
are  four  rational  roots  or  none  according  as  (  — 3/f+r)'=+l  or  not;  but 
if  K  is  a  non-residue,  there  are  two  rational  roots  or  none  according  as  one 
of  the  congruences 

(-3H+r)'=4-l,  i-ZH-ry^-l 

is  satisfied  or  not. 

i^sjour.  fur  Math.,  4,  1829,  131;  Oeuvres,  1,  114. 

"sReport  British  Assoc.  1860,  120  seq.,  §66:  Coll.  M.  Papers,  1,  141-5. 

"9aMath.  Quest.  Educ.  Times,  6,  1866,  91-3. 

""Amer.  Jour.  Math.,  2,  1879,  360-1;  Johns  Hopkins  University  Circulars,  1,  1881,  131.     Coll. 

Papers,  3,  320-1. 
i"Johns  Hopkins  Univ.  Circulars,  1,  1881,  97.  "^Amer.  Jour.  Math.,  6,  1884,  316-330. 

"^Rendiconti  Reale  Accad.  Lincei,  Rome,  (4),  1,  1885,  140-2. 
i"Acta  Mathematica,  19,  1895,  356. 
«6Rendiconti  Circolo  Mat.  di  Palermo  9,  1895,  209-243. 


260  History  of  the  Theory  of  Numbers.  [Chap,  viii 

R.  Lipschitz^^^  examined  Fermat's^^^  statement  and  proved  that  the 
primes  p  for  which  a' +1  =  0  (mod  p)  is  impossible  are  those  and  only  those 
for  which  a  solution  u  of  w"  =a  (mod  p)  is  a  quadratic  non-residue  of  p 
and  for  which  X^A.',  where  2^  is  the  highest  power  of  2  dividing  p  —  1. 
Cases  when  a''+l  =  0  is  impossible  and  not  embraced  by  Fermat's  rule  are 
a  =  2,  p  =  89,  337;  a  =  S,  p  =  13;  a=-2,  p  =  281;  etc. 

L.  Kronecker^^^  called /(.r)  an  invariant  of  the  congruence  k=k'  (mod  m), 
if  the  latter  congruence  implies  the  equality /(/v)  =/(//).  If  also,  conversely, 
the  equality  implies  the  congruence,  f{x)  is  called  a  proper  (or  characteristic) 
invariant,  an  example  being  the  least  positive  residue  of  an  integer  modulo 
m.  It  is  shown  that  every  invariant  of  k=k'  (mod  m)  can  be  represented 
as  a  symmetric  function  of  all  the  integers  congruent  to  k  modulo  m. 

G.  Wertheim^^^  proved  that  a^+l  =  0  (mod  p)  is  impossible  if  a  belongs 
to  an  odd  exponent  modulo  p  [Fermat^'^^]. 

E.  L.  Bunitzky^*^^  (Bunickij)  noted  that,  for  any  integer  M,  the  con- 
gruences 

f(a+kh)=rk  (mod  M)  (A:  =  0,  1,.  .  .,  n) 

hold  if  and  only  if  the  coefficients  Ak  of /(x)  satisfy  the  conditions 
k\h''Ak=A%  (mod  M)  {k  =  1, .  .  . ,  n). 

If  k  is  the  least  value  of  x  for  which  xlh""  is  divisible  by  M,  and  if  the 
g.  c.  d.  of  M  and  h  is  k<m,  where  m  is  a  divisor  of  M,  then  if /(a;)=0  (mod 
M)  has  the  roots  a,  a-\-h,...,  a-\-{k  —  l)h,  it  has  also  the  roots  a-\-jh 
{j  =  k,k-\-l,...,m-l).^ 

G.  Biase^^''  called  a  similar  to  h  in  the  ratio  m:n  modulo  k  if  the  remainders 
on  dividing  a  and  h  by  k  are  in  the  ratio  m:n.  Two  numbers  similar  to  a 
third  in  two  given  ratios  modulo  k  are  similar  to  each  other  modulo  k  in 
a  ratio  equal  to  the  quotient  of  the  given  ratios. 

The  problem^ '^^  to  find  n  numbers  whose  n^  —  n  differences  are  incon- 
gruent  modulo  n^  — n+1  is  possible  for  n  =  6,  but  not  for  n  =  7. 

R.  D.  von  Sterneck^^°  proved  that,  if  A  is  not  divisible  by  the  odd 
prime  p,  Ax'^+Bx^+C  takes  \p{2AB,  p)  incongruent  values  (when  x  ranges 
over  the  set  0,  1, . .  .,  p  — 1)  if  5  is  not  divisible  by  p,  while  if  B  is  divisible 
by  p,  it  takes  (p+3)/4  or  (p-fl)/2  values  according  as  p  =  4n-l-l  or  p  = 
471  —  1.     In  terms  of  Legendre's  symbol, 

>««Bull.  des  Sc.  Math.,  (2),  22,  I,  1898,  123-8.     Extract  in  Oeuvres  de  Fermat,  4,  196-7. 

>"Vorlesungen  iiber  Zahlcntheorie,  I,  1901,  131-142. 

"^AnfangsRTunde  der  Zahlenlehre,  1902,  265. 

''"Zap.  mat.  otd.  Obsc.  (Soc.  of  natur.),  Odessa,  20,  1902,  III- VIII  (in  Russian);  cf.  Fortschr. 

Math.,  33,  1902,  p.  205. 
•^"Il  Boll.  Matematica  Gior.  Sc.  Didat.,  Bologna,  4,  1905,  96. 
i"L'interm6diaire  des  math.,  1906,  141;  1908,  64;  19,  1912,  130-1.     Amcr.  Math.  Monthly,  13, 

1906,  215;  14,  1907,  107-8. 


Chap.  VIII]  MISCELLANEOUS   RESULTS   ON   CONGEUENCES.  261 

E.  Landau^'^  proved  that,  if /(x)  =0  is  an  equation  with  integral  coeffi- 
cients and  at  least  one  root  of  odd  multiplicity,  there  exist  an  infinitude  of 
primes  p  =  4:n  —  l  such  that /(a;)  =  0  (mod  p)  has  a  root. 

R.  D.  von  Sterneck^^^  found  the  number  of  combinations  of  the  ith.  class 
(with  or  without  repetition)  of  the  numbers  prime  to  p  of  a  complete  set  of 
residues  modulo  p^  whose  sum  is  congruent  to  a  given  integer  modulo  p^, 
p  being  a  prime. 

E.  Piccioli^^'*  gave  known  theorems  on  adding  and  multiplying  con- 
gruences. 

C.  Jordan^^^  found  the  number  of  sets  of  integers  aik  for  which  the 
determinant  |aa-|  of  order  n  is  congruent  to  a  given  integer  modulo  M. 

C.  Krediet^^^  gave  theorems  on  congruences  of  degree  n  for  a  prime 
modulus  analogous  to  those  for  an  algebraic  equation  of  degree  n,  including 
the  question  of  multiple  roots.  The  determination  of  roots  is  often  sim- 
plified by  seeking  first  the  roots  which  are  quadratic  residues  and  then 
those  which  are  non-residues.     The  exposition  is  not  clear  or  simple. 

G.  Rados"^  proved  that,  if  p  is  a  prime, 

fix)  =  ao^^-^+  •  •  ■  +«p-2=  0,  g{x)  =  hx^'-^-i- . . .  +bp_2=  0  (mod  p) 

have  a  common  root  if  and  only  if  each  Ri=0  (mod  p),  where 

*(w)  =Rou''-^+Riu''-^-\- . . .  +Rp-i 

aou+bo  aiU+bi       ...       ap_2W+6p_2 

aiu+bi  a2U-\-b2       . .  .       aoU-{-bo 

ap_2U+bp-2  aou+bo       ...       ttp-s^+^p-s 

For  ^=/',  let  ^(u)  become  DoU^~^-\- .  .  .  +i)p_2;  thus/(a:)^0  (mod  p)  has  a 
multiple  root  if  and  only  if  each  A=0  (mod  p).  Each  of  these  theorems 
is  extended  to  three  congruences.  Finally,  if  f(x)  and  f'(x)  are  relatively 
prime  algebraically,  there  is  only  a  finite  number  of  primes  p  for  which  the 
number  of  roots  of  /=  0  (mod  p'')  exceeds  the  degree  of  /. 

G.  Frattini^^^  proved  that  if  p  and  q  are  primes,  q  a  divisor  of  p  —  1, 
every  homogeneous  symmetric  congruence  in  q  variables  is  solvable  modulo 
p  by  values  of  the  variables  distinct  from  each  other  and  from  zero  except 
when  the  degree  of  the  congruence  is  divisible  by  q. 

C.  Grotzsch^'^  noted  that  if  a  is  a  root  of  a^'^^a  (mod  p),  where  a  is  prime 
to  p,  then  x=a  (mod  p^  —  p)  is  a  root,  and  proved  that  if  d  is  the  g.  c.  d.  of 
ind  a  and  p  —  1  and  if  ind  a>0,  it  has  exactly 

^''^Handbuch .  .  .Verteilung  der  Primzahlen,  1,  1909,  440. 
^"Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  118,  1909,  Ila,  119-132. 
'^"11  Pitagora,  Palermo,  16,  1909-10,  125-7. 
"5Jour.  de  Math.,  (6),  7,  1911,  409-416. 
'^^Wiskundig  Tijdschrift,  Haarlem,  7,  1911,  193-202  (Dutch). 
i"Ami.  sc.  ecole  norm,  sup.,  (3),  30,  1913,  395-412. 
"speriodico  di  Mat.,  29,  1913,  49-53. 
"'Archiv  Math.  Phys.,  (3),  22,  1914,  49-53. 


262  History  of  the  Theory  of  Numbers.  [Chap,  viii 


iV  =  0(p-l)+2^^) 


roots  incongruent  modulo  p(p  —  1),  where  5  ranges  over  all  divisors  >  1  of  ^. 
If  ind  a  =  0,  the  number  of  such  roots  is  p  —  l-\-N,  where  now  5  ranges  over 
the  di\asors  >1  of  p  — 1. 

A.  Chdtelet^^°  noted  that  divergences  between  congruences  and  equa- 
tions are  removed  by  not  Umiting  attention  to  the  given  congruence  fix)  =  0 
of  degree  n,  but  considering  simultaneously  all  the  polynomials  g{x)  derived 
from  f{x)  by  a  Tschirnhausen  transformation  ky  —  (i>{x),  where  k  is  an 
integer  and  </>  has  integral  coefficients  and  is  of  degree  n  — 1. 

*M.  Tihanyi^^""  proved  a  simple  congruence. 

R.  Kantor^^^  discussed  the  number  of  incongruent  values  modulo  m 

taken  by  a  polynomial  in  n  variables,  and  especially  for  ax^-\- . .  .+d 

modulo  p',  generalizing  von  Sterneck.^"*" 

The  solvabiUty  of  x^+9a;+6=0  and  x^-\-y{y-\-l)  =  0  (mod  p)  has  been 
treated.^«2 

A.  Cunningham^^  announced  the  completion,  in  conjunction  with 
Woodall  and  Creak,  of  tables  of  least  solutions  {x,  a)  of  the  congruences 

T^=^y%    rV==*=l  (mod p*<  10000),     r  =  2, 10; ?/  =  3,  5,  7, 11. 

T.  A.  Pierce^^^  gave  two  proofs  that /(a;)  =  0  (mod  p)  has  a  real  root  if 
and  only  if  the  odd  prime  p  divides  11(1— a^^"^),  where  a^  ranges  over  the 
roots  of  the  equation  f{x)  =  0. 

Christie^^  stated  that  P(F+1)  =  1  (mod  p)  if  t=  2  sin  18°  and  p  is  any 
odd  prime.     Cunningham  gave  a  proof  and  a  generalization. 

*G.  Rados^^^  found  the  congruence  of  degree  r  having  as  its  roots  the 
r  distinct  roots  ?^0  of  a  given  congruence  of  degree  p  —  2  modulo  p,  a 
prime. 

i8'€omptes  Rendus  Paris,  158,  1914,  250-3. 

"""Math,  es  Phys.  Lapok,  Budapest,  23,  1914,  57-60. 

i8iMonatshefte  Math.  Phys.,  26,  1915,  24-39.  J 

'»2Wiskundige  Opgaven,  12,  1915,  211-2,  215-7. 

'"Messenger  Math.,  45,  1915-6,  69. 

'"Annals  of  Math.,  (2),  18,  1916,  53-64. 

i«*Math.  Quest.  Educ.  Times,  71,  1899,  82-3. 

'»Math.  is  Term6s  firtesito,  33,  1915,  702-10. 


CHAPTER  IX. 

DIVISIBILITY  OF  FACTORIALS  AND  MULTINOMIAL  COEFFICIENTS. 
Highest  Power  of  a  Prime  Dividing  ml 

Genty^  noted  that  the  highest  power  of  2  dividing  (2'*) !  is  2^""^  and  the 
quotient  is  3"-'(5-7)"-2(9-lM3-15)"-^(17. .  .31)""^  .  .(2"-l).  In  general 
if  P  =  2"'+2"'+. .  .+2%  where  the  n's  decrease,  the  highest  power  of  2 
dividing  P!  is  2^-^ 

A.  M.  Legendre^  proved  that  if  p"  is  the  highest  power  of  the  prime  p 
which  divides  m !,  and  if  [x]  denotes  the  greatest  integer  ^  x, 

where  s  =  ao+  •  •  •  +««  is  the  sum  of  the  digits  of  m  to  the  base  p: 

Th.  Bertram^  stated  Legendre's  result  in  an  equivalent  form. 
H.  Anton*  proved  that,  U  n  =  vp+a,  a<p,  v<p,  and  p  is  a  prime, 

=  (p  — l)'a!y!  (mod  p), 


n! 


P 

while,  if  v  =  vp-\-a,   a'<p,  v'<p, 

■^,=  {p-iy+^'a\a\v\v\  (modp). 

D.  Andr^^  stated  that  the  highest  power  p"  of  the  prime  p  dividing  n! 
is  given  expHcitly  by  }i=^lZi[n/p^]  and  claimed  that  merely  the  method  of 
finding  ii  had  been  given  earHer.  He  appHed  this  result  to  prove  that  the 
product  of  n  consecutive  integers  is  divisible  by  n!. 

J.  Neuberg*'  determined  the  least  integer  m  such  that  m\  is  divisible  by 
a  given  power  of  a  prime,  but  overlooked  exceptional  cases. 

L.  Stickelberger^  and  K.  HenseP  gave  the  formula  [cf.  Anton*]. 

(2)  ^^(-irao!ai!...aj(modp). 

F.  de  Brun^  wrote  g[u]  for  the  exponent  of  the  highest  power  of  the 
prime  p  dividing  u.     He  gave  expressions  for 

■^  rP{n;k)=Uf\  g[rP(n;k)] 

3  =  1 

in  terms  of  the  functions  h{a;  k)  =  l*+2*-f  .  .  .  +a^.     A  special  case  gives  (1). 

^Hist.  et  M6m.  Ac.  R.  Sc.  Inscript.  et  Belles  Lettres  de  Toulouse,  3,  1788,  97-101  (read  Dec.  4, 

1783). 
''Th^orie  des  nombres,  ed.  2,  1808,  p.  8;  ed.  3,  1830,  I,  p.  10. 
'Einige  Satze  aus  der  Zahlenlehre,  Progr.  Coin,  Berlin,  1849,  18  pp. 
*Archiv  Math.  Phys.,  49,  1869,  298-9. 
"Nouv.  Ann.  Math.,  (2),  13,  1874,  185. 

^Mathesis,  7,  1887,  68-69.     Cf.  A.  J.  Kempner,  Amer.  Math,  Monthly,  25,  1918,  204-10. 
'Math.  Annalen,  37,  1890,  321. 
sArchiv  Math.  Phys.,  (3),  2,  1902,  294. 
»Arkiv  for  Matematik,  Astr.,  Fysik,  5,  1904,  No.  25  (French). 

263 


264  History  of  the  Theory  of  Numbers.  [Chap,  ix 

R.  D.  CarmichaeP"  treated  the  problem  to  find  m,  given  the  prime  p 
and  s  =  ^ai,  in  Legendre's  formula;  a  given  solution  m-,  leads  to  an  infinitude 
of  solutions  m-zv'',  k  arbitrary.  Again,  to  find  771  such  that  p"*~'  is  the  highest 
power  of  p>2  dividing  m\,  we  have  m  —  t={m  —  s)/{p  —  l),  and  see  that  m 
has  a  hmited  number  of  values;  there  is  always  at  least  one  solution  m. 

Carmichael"  used  the  notation  H\y\  for  the  index  of  the  highest  power 
of  the  prime  p  dividing  y,  and  evaluated 

;i=i/{n(xa+c)|, 

where  a,  c  are  relatively  prime  positive  integers.  Set  Co  =  c  and  let  %  be 
the  least  integer  such  that  iVa+Cr-i  is  divisible  by  p,  the  quotient  being  Cr. 
Let 


ei 


=  [^4^']'       ^'=[^]'  ^>i- 


t-i 
Then  /i= 2(6^+1),  where  t  is  the  least  subscript  for  which 

r=l 

Ct{a-\-Ct){2a-^Ct) . . .  (cta+Ct) 
is  not  divisible  by  p.    It  follows  that 

where  R  is  the  index  of  the  highest  power  of  p  not  exceeding  7i  —  1 .  If  n  is  a 
power  of  p,  /i  =  (n  — l)/(p  — 1).  But  if  n  =  8kp''-\- .  .  .-\-dip-\-8o,  8k9^0,  and 
at  least  one  further  8  is  not   zero, 

^SftSfc+^.  <r  =  6.+  ...+6o. 

P  — 1  p  — 1 

In  case  the  first  x  for  which  xa-\-c  is  divisible  by  p  gives  c  as  the  quotient, 
all  the  Cr  are  equal  and  hence  all  the  v;  then 

,  _  rn  —  1—i+pl  .  rn  —  l—i  —  ip-\-p^l  ,  rn  —  l—i  —  ip  —  ip'^-i-p^l  . 

L      p      J"^L        p'        J"^L  w  J"^'" 

The  case  a  =  c  =  l  yields  Legendre's^  result.     The  case  a  =  2,  c  =  1,  gives 

Hll.3.5. .  .(2„-l)(  =  [?^^]  +  [?^^^>. . .. 

E.  Stridsberg^^  wrote  H^  for  (1)  and  considered 

Trt  =  a{a-\-m) .  .  .{a-{-'mt), 

where  a  is  any  integer  not  divisible  by  the  positive  integer  m.  Let  p  be  a 
prime  not  dividing  m.  Write  a^  for  the  residue  of  aj  modulo  m.  He  noted 
that,  if  pj=l  (mod  m), 

"BuU.  Amer.  Math.  Soc,  14,  1907-8,  74-77;  Amer.  Math.  Monthly,  15,  1908,  15-17. 

''Ibid.,  15,  1908-9,  217. 

"Arkiv  for  Matematik,  Astr.,  Fysik,  6,  1911,  No.  34. 


1 


Chap.  IX]  DIVISIBILITY  OF  FACTORIALS,  MULTINOMIAL  COEFFICIENTS.         265 

is  an  integer,  and  wrote  L^  for  its  residue  modulo  p^+\     Set 


:.=  P 


He  proved  that  tt^  is  divisible  by  p*,  where  s  =  Hi+Sj;io2^.     If  t„  is  the 
first  one  of  the  numbers  tq,  ti,  .  .  .  which  is  <p  — 1,  tt^  is  divisible  by  p", 

k 

A.  Cunningham^^  proved  that  if  /  is  the  highest  power  of  the  prime  z 
dividing  p,  the  number  of  times  p  is  a  factor  of  p"!  is  the  least  of  the 
numbers  ^„   2^«-^+i-l 

for  the  various  primes  z  dividing  p. 

W.  Janichen^^  stated  and  G.  Szego  proved  that 

i:tx{n/d)v{d)=ci>{n)/{v-l), 

summed  for  the  divisors  d  of  n,  where  v{d)  is  the  exponent  of  the  highest 
power  of  p  (a  prime  factor  of  n)  which  divides  d\,  for  /x  as  in  Ch.  XIX. 

Integral  Quotients  Involving  Factorials. 

Th.  Schonemann^^  proved,  by  use  of  symmetric  functions  of  pth  roots 
of  unity,  that  if  b  is  the  g.  c.  d.  of  fjL,v,'..., 
8-{m-l)l 


.ii/l 


=  integer,  {m—fx+v-\- . .  .). 


fJLlVl 

He  gave  (p.  289)  an  arithmetical  proof  by  showing  that  the  fractions 
obtained  by  replacing  8  by  fi,  v, ..  .  are  integers. 
A.  Cauchy^^  proved  the  last  theorem  and  that 

^ -^ ^  =  integer,     {m  =  a-\- .  .  .+k). 

a\.  .  .k\ 

D.  Andre^*^  noted  that,  except  when  n  =  l,  a  =  4,  n(n  +  l).  .  .(na  — 1)  is 
not  or  is  divisible  by  a"  according  as  a  is  a  prime  or  not. 

E.  Catalan^^  found  by  use  of  elliptic  functions  that 

{m-\-n-\)\  (2m) !  (2n) ! 

m!n!  m\n\{m-\-n)\ 

are  integers,  provided  m,  n  are  relatively  prime  in  the  first  fraction. 

^^L'intermediaire  des  math.,  19,  1912,  283-5.     Text  modified  at  suggestion  of  E.  Maillet. 

"Archiv  Math.  Phys.,  (3),  13,  1908,  361;  24,  1916,  86-7. 

»8Jour.  fur  Math.,  19,  1839,  231-243. 

"Comptes  Rendus  Paris,  12,  1841,  705-7;  Oeuvres,  (1),  6,  109. 

20N0UV.  Ann.  Math.,  (2),  11,  1872,  314. 

"/fttd.,  (2),  13,  1874,  207,  253.     Arith.  proofs,  Amer.  Math.  Monthly,  18,  1911,  41-3. 


266  History  of  the  Theory  of  Numbers.  [Chap,  ix 

P.  Bachmann^-  gave  arithmetical  proofs  of  Catalan's  results. 

D.  Andr^-^  proved  that,  if  aj, .  .  . ,  a„  have  the  sum  N  and  if  k  of  the  a's 
are  not  di\isible  by  the  integer  >1  which  divides  the  greatest  number  of 
the  a's,  then  (iV— A;)!  is  di\'isible  by  ai!.  .  .a„!. 

J.  Bourguet^^  proved  that,  if  k^2, 

{kmi)l  ikmo)\...{knh)\ 

— ,  .  ^    , ; r:  =  mteger. 

Wi!.  .  .nikl  (wi+.  .  .+mk)l 

M.  WeilP^  proved  that  the  multinomial  coefficient  (tq) !  -r-  {q\y  is  divisible 
by  tl 

WeilP^  stated  that  the  following  expression  is  an  integer : 

(a+i8+  ■  •  ■  +pg+Pigi+  ■  ■  •  +rst) ! 
am.  .  .{piyq\{p,\y^q,\.  .  .{r\ns\yt\' 

WeilP^  stated  the  special  case  that  {a-\-^-\-pq-\-rs)\  is  divisible  by 
a\^\{q\rp\{s\yrl 

D.  Andr^-^  proved  that  (tq) !  -r-  (g!)'  is  divisible  by  (<!)*  if  for  every  prime 
p  the  sum  of  the  digits  of  q  to  base  p  is  ^k. 

Ch.  Hermite'^^  proved  that  n!  divides 

m{7n+k){m+2k) .  .  .  lm+{n-l)k]k''-\ 

C.  de  PoUgnac^"  gave  a  simple  proof  of  the  theorem  by  WeilP^  and 
expressed  the  generalization  by  Andr^^^  in  another  and  more  general  form. 

E.  Catalan^^  noted  that,  if  s  is  the  number  of  powers  of  2  having  the  sum 

^+^'  (2a)!  (26)! 

a!6!(a+6)! 
is  an  even  integer  and  the  product  of  2'  by  an  odd  number. 
E.  Catalan^^  noted  that,  if  n  =  a+6+  .  .  .  -{-t, 

n\{n+t) 
a\h\...tl 

is  divisible  by  a+t,  h+t,. .  .,  a-\-h-\-t,. .  .,  a-\-b+c+t,. . .. 

E.  Ces^ro^^  stated  and  Neuberg  proved  that  (p)  is  divisible  by  n(n  — 1) 
if  p  is  prime  to  n(n  — 1),  and  p  —  l  prime  to  n  — 1;  and  divisible  by  (p  +  1) 
X(p+2)  if  p-\-l  is  prime  to  n+1,  and  p+2  is  prime  to  (n  +  l)(n+2). 

"Zeitschrift  Math.  Phys.,  20,  1875,  161-3.     Die  Elemente  der  Zahlentheorie,  1892,  37-39. 
«Bull.  Soc.  Math.  France,  1,  1875,  84. 

"Nouv.  Ann.  Math.,  (2),  14,  1875,  89;  he  wrote  r(n)  incorrectly  for  n!;  see  p.  179. 
'MDomptes  Rendus  Paris,  93,  1881,  1066;  Mathesis,  2,  1882,  48;  4,  1884,  20;  Lucas,  Th^orie 

des  nombres,  1891,  365,  ex.  3.     Proof  by  induction,  Amer.  M.  Monthly,  17,  1910,  147. 
»«Bull.  Soc.  Math.  France,  9,  1880-1,  172.     Special  case,  Amer.  M.  Monthly,  23,  1916,  352-3. 
"Mathesis,  2,  1882,  48;  proof  by  Li6nard,  4,  1884,  20-23. 
"Comptes  Rendus  Paris,  94,  1882,  426. 
"Faculty  des  Sc.  de  Paris,  Cours  de  Hermite,  1882,  138;  ed.  3,  1887,  175;  ed.  4,  1891,  196. 

Cf.  Catalan,  M6m.  Soc.  Sc.  de  Li6ge,  (2),  13,  1886,  262-^  (  =  Melanges  Math.);  Heine."" 
»"Comptes  Rendus  Paris,  96, 1883, 485-7.    Cf .  Bachmann,  Niedere  Zahlentheorie,  1, 1902.  59-62. 
«Atti  Accad.  Pont.  Nouvi  Lincei,  37,  1883-4,  110-3. 
"Mathesis,  3,  1883,  48;  proof  by  Cesiro,  p.  118. 
"Ibid.,  5,  1885,  84. 


Chap.  IX]  DIVISIBILITY  OF  FACTORIALS,  MULTINOMIAL  COEFFICIENTS.         267 

E.  Catalan^^  noted  that 

e::r)'(?)"*e)-«" 

F.  Gomes  Teixeira^^  discussed  the  result  due  to  Weill. ^® 
De  Presle^®  proved  that 

{k-\-l)(k+2)...{k+hl)     .  ^ 

nw ^  ^^  ^^^'' 

being  the  product  of  an  evident  integer  by  {hl)\/{U{h\y}.  , 

E.  Catalan^^  noted  that,  if  n  is  prime  to  6, 

(2n-4)! 


nl{n-2)\ 
H.  W.  Lloyd  Tanner^^  proved  that 


=  integer. 


=  integer. 


{\,\...\,\ng\y 

L.  Gegenbauer  stated  and  J.  A.  Gmeiner^^  proved  arithmetically  that, 
if  n=Sjrja_,iaj2.  •  Oys,  the  product 

m{m+k)(m-h2k) . . .  {m+{n-l)k}k''-' 
is  divisible  by 

where  m,  k,  n,  an,-  ■  ■,  o,rs  are  positive  integers.  This  gives  Hermite's'^' 
result  by  taking  r  =  s  =  l.  The  case  m  =  A;  =  l,s  =  2,  is  included  in  the  result 
by  Weill.26 

Heine^^"  and  A.  Thue^°  proved  that  a  fraction,  whose  denominator  is  k\ 
and  whose  numerator  is  a  product  of  k  consecutive  terms  of  an  arithmetical 
progression,  can  always  be  reduced  until  the  new  denominator  contains  only 
such  primes  as  divide  the  difference  of  the  progression  [a  part  of  Her- 
mite's^^  result]. 

F.  Rogel'*^  noted  that,  if  P  be  the  product  of  the  primes  between  (p  — 1)/2 
and  p  +  1,  while  n  is  any  integer  not  divisible  by  the  prime  p, 

(n-l)(n-2).  ..{n-p-^l)P/p=0  (mod  P). 
S.  Pincherle^^  noted  that,  if  n  is  a  prime, 

P={x+l){x+2) .  ..(x+n-l) 
is  divisible  by  n  and,  if  x  is  not  divisible  by  n,  by  n !.    If  n  =  Up",  P  is  divisible 

="Nouv.  Ann.  Math.,  (3),  4,  1885,  487.     Proof  by  Landau,  (4),  1,  1901,  282. 

35Archiv  Math.  Phys.,  (2),  2  1885,  265-8.  ^eBuU.  Soc.  Math.  France,  16,  1887-8,  159. 

"M6m.  Soc.  Roy.  Sc.  Li^ge,  (2),  15,  1888,  111  (Melanges  Math.  III).     Mathesis,  9,  1889,  170. 

"Proc.  London  Math.  Soc,  20,  1888-9,  287.        ^QMonatshefte  Math.  Phys.,  1,  1890,  159-162. 

"a Jour,  fur  Math.,  45,  1853,  287-8.     Cf.  Math.  Quest.  Educ.  Times,  56,  1892,  62-63. 

"Archiv  for  Math,  og  Natur.,  Kristiania,  14,  1890,  247-250. 

"Archiv  Math.  Phys.,  (2),  10,  1891,  93. 

"Rendiconto  Sess.  Accad.  Sc.  Istituto  di  Bologna,  1892-3,  17. 


268  History  of  the  Theory  of  Numbers.  [Chap,  ix 

by  n !  if  and  only  if  divisible  by  IIp"'''^,  where  /3  is  the  exponent  of  the  power 
of  p  dividing  (n  — 1)!. 

G.  Bauer^^  proved  that  the  multinomial  coefficient  (n+ni+n2+.  .  .)' 
-7-  {7i!7?i! . .  . }  is  an  integer,  and  is  even  if  two  or  more  n's  are  equal. 

E.  Landau^^  generalized  most  of  the  preceding  results.  For  integers 
Qij,  bij,  each  ^  0,  and  positive  integers  Xj,  set 

Then  /  is  an  integer  if  and  only  if 

m  n 

t=i       t=i 
for  all  real  values  of  the  Xj  for  which  O^Xj^l.     A  new  example  is 

(4m)!(4n)!  a 

m!n!(2m+n)!(m+2n)!~^^^^^^'  | 

P.  A.  MacMahon^^  treated  the  problem  to  find  all  a's  for  which 

is  an  integer  for  all  values  of  n;  in  particular,  to  find  those  "ground  forms" 
from  which  all  the  forms  may  be  generated  by  multiplication.  For  m  =  2, 
the  ground  forms  have  (ai,  a2)  =  (1,  0)  or  (1,  1).  For  m  =  3,  the  additional 
ground  forms  are  (1,  1,  1),  (1,  2,  1),  (1,  3,  1).  For  ?7i  =  4,  there  are  3  new 
ground  forms;  for  m  =  5,  13  new. 

J.  W.  L.  Glaisher^®  noted  that,  if  Bp{x)  is  Bernoulli's  function,  i.  e.,  the 
polynomial  expression  in  x  for  F~^+2^"^+ .  .  .  +  (x  — 1)^"^  [Bernoulli^^"'' 
of  Ch.  V], 

x{x-\-l) .  .  .{x-\-p  —  l)/p=Bp{x)—x  (mod  p). 

He  gave  (ibid.,  33,  1901,  29)  related  congruences  involving  the  left  member 
and  Bp_i{x). 

Glaisher^^  noted  that,  if  r  is  not  divisible  by  the  odd  prime  p,  and 
l  =  kp+t,  0^t<p, 

l{r+l){2r+l)  .  .  .  {(p-l)r+i)/p^-|[^]^+A:}     (mod  p), 

where  [t/p]r  denotes  the  least  positive  root  of  px=t  (mod  r).  The  residues 
mod  p^  of  the  same  product  l{r-\-l) . . .  are  found  to  be  complicated. 

E.  Maillet*^  gave  a  group  of  order  t\{q\y  contained  in  the  sjrmmetric 
group  on  tq  letters,  whence  follows  Weill's^^  result. 

«SitzunKsber.  Ak.  Wiss.  Miinchen  (Math.),  24,  1894,  34&-8. 

"Nouv.  Ann.  Math.,  (3),  19,  1900,  344-362,  576;  (4),  1,  1901,  282;  Archiv  Math.  Phys.,  (3),  1, 

1901,  138.     Correction,  Landau." 
«Trans  Cambr.  Phil.  Soc,  18,  1900,  12-34. 
"Proc.  London  Math.  Soc,  32,  1900,  172. 
^'Messenger  Math.,  30,  1900-1,  71-92. 
*»Mem.  Pr6s.  Ac.  Sc.  Paris,  (2),  32,  1902,  No.  8,  p.  19. 


Chap.  IX]    DIVISIBILITY  OF  FACTORIALS,  MULTINOMIAL  COEFFICIENTS.        269 

M.  Jenkins^^"  counted  in  two  ways  the  arrangements  of  n  =  4>f-\-'yg-\- . .  . 
elements  in  4>  cycles  of  /  letters  each,  7  cycles  of  g  letters,  . .  . ,  where/,  g,  ... 
are  distinct  integers  >  1,  and  obtained  the  result 

fct>\g'y\...         \2\     Sr4\     '  '  '  ^^     ^  n\)' 
C.  de  Polignac*^  investigated  at  length  the  highest  power  of  n!  dividing 
(nx)  \/{x\y.     Let  rip  be  the  sum  of  the  digits  of  n  to  base  p.     Then 

{x-\-n)p  =  Xp+np-k{p  -1) ,  {xn)p  =  Xp-np-k'{p-l), 

where  k  is  the  number  of  units  "carried"  in  making  the  addition  x-\-n,  and  k' 
the  corresponding  number  for  the  multiplication  x-n. 

E.  Sch6nbaum^°  gave  a  simplified  exposition  of  Landau's  first  paper.^^ 

S.  K.  Maitra^i  proved  that  (n  - 1)  (2n  - 1) .  .  .  { (n  -  2)n  - 1 }  is  divisible  by 
(n  —  1) !  if  and  only  if  n  is  a  prime. 

E.  Stridsberg^^  gave  a  very  elementary  proof  of  Hermite's^^  result. 

E.  Landau^^  corrected  an  error  in  his^'*  proof  of  the  result  in  No.  Ill  of  his 
paper,  no  use  of  which  had  been  made  elsewhere. 

Birkeland^^  of  Ch.  XI  noted  that  a  product  of  2^k  consecutive  odd  in- 
tegers is^l  (mod  2^). 

Among  the  proofs  that  binomial  coefficients  are  integers  may  be  cited 
those  by: 

G.  W.  Leibniz,  Math.  Schriften,  pub.  by  C.  I.  Gerhardt,  7,  1863,  102. 

B.  Pascal,  Oeuvres,  3,  1908,  278-282. 

Gioachino  Pessuti,  Memorie  di  Mat.  Soc.  Italiana,  11,  1804,  446. 

W.  H.  Miller,  Jour,  fiir  Math.,  13,  1835,  257. 

S.  S.  Greatheed,  Cambr.  Math.  Jour.,  1,  1839,  102,  112. 

Proofs  that  multinomial  coefficients  are  integers  were  given  by: 

C.  F.  Gauss,  Disq.  Arith.,  1801,  art.  41. 

Lionnet,  Complement  des  elements  d'arith.,  Paris,  1857,  52. 
V.  A.  Lebesgue,  Nouv.  Ann.  Math.,  (2),  1,  1862,  219,  254. 

Factorials  Dividing  the  Product  of  Differences  of  r  Integers. 

H.  W.  Segar^"  noted  that  the  product  of  the  differences  of  any  r  distinct 
integers  is  divisible  by  (r  — l)!(r  — 2)!.  .  .2!.  For  the  special  case  of  the 
integers  1,  2, .  .  .,  n,  r+1,  the  theorem  shows  that  the  product  of  any  n 
consecutive  integers  is  divisible  by  n!. 

A.  Cayley®^  used  Segar's  theorem  to  prove  that 

m{m  —  n) .  .  .{m  —  r  —  ln)-rf 

is  divisible  by  r!  if  m,  n  are  relatively  prime  [a  part  of  Hermite's-^  result]. 
Segar®^  gave  another  proof  of  his  theorem.     Applying  it  to  the  set 

^8aQuar.  Jour.  Math.,  33,  1902,  174-9.     "Bull.  Soc.  Math.  France,  32,  1904,  5-43. 
"Casopis,  Pras,  34,  1905,  265-300  (Bohemian). 
"Math.  Quest.  Educat.  Times,  (2),  12,  1907,  84-5. 

^^Acta  Math.,  33,  1910,  243.  "Nouv.  Ann.  Math.,  (4),  13,  1913,  353-5. 

soMessenger  Math.,  22,  1892-3,  59.  "Messenger  Math.  22,  1892-3,  p.  186.     Cf.  Hermite." 

^Hbid.,  23, 1893-4,  31.     Results  cited  in  I'interm^diaire  des  math.,  2,  1895, 132-3,  200;  5,  1898, 
197;  8,  1901,  145. 


270  History  of  the  Theory  of  Numbers.  [Chap,  ix 

a,  a-\-N,. . .,  a+A^",  we  conclude  that  the  product  of  their  differences  is 
divisible  by  n!(n  — 1)!. .  .21  =  p.    But  the  product  equals 

p=iN-ir-'  (ir--ir-\ .  .{N^-'^-iyiN"-'-!), 

multiplied  by  a  power  of  A^.  Hence,  if  N  is  prime  to  n!,  P  is  divisible  by  v; 
in  any  case  a  least  number  X  is  found  such  that  N^P  is  divisible  by  ;'.  It  is 
shown  that  the  product  of  the  differences  of  mi,.  .  .,  m^  is  divisible  by 
k\{k  —  l)\.  .  .2!  if  there  be  any  integer  p  such  that  Wi+p, .  .  .,  nik+p  are 
relatively  prime  to  each  of  1,  2, . . . ,  A;.  It  is  proved  that  the  product  of  any 
n  distinct  integers  multiplied  by  the  product  of  all  their  differences  is  a 
multiple  of  n!(n-l)!.  .  .2!. 

E.  de  Jonquieres^^  and  F.  J.  Studnicka^  proved  the  last  theorem. 

E,  B.  Elliott^^  proved  Segar's  theorem  in  the  form:  The  product  of  the 
differences  of  n  distinct  numbers  is  di\'isible  by  the  product  of  the  differences 
of  0,  1,...,  n  — 1.  He  added  the  new  theorems:  The  product  of  the 
differences  of  n  distinct  squares  is  divisible  by  the  product  of  the  differences 
of  0",  1",...,  (n  — 1)";  that  for  the  squares  of  n  distinct  odd  numbers, 
multiplied  by  the  product  of  the  n  numbers,  is  divisible  by  the  product  of 
the  differences  of  the  squares  of  the  first  n  odd  numbers,  multiplied  by  their 
product. 

Residues  of  Multinomial  Coefficients. 

Leibniz^' '^  of  Ch.  Ill  noted  that  the  coefficients  in  (ZaY—Za^  are 
di\'isible  by  p. 

Ch.  Babbage^^  proved  that,  if  n  is  a  prime,  (^n-/)  — 1  is  divisible  by 
n^,  while  ("p")  —  1  is  divisible  by  p  if  and  only  if  p  is  a  prime. 

G.  Libri^''  noted  that,  if  m  =  6p-hl  is  a  prime, 

2^p-^-^ep-l-(^^P~^y+(^P~^y'-  ..=0  (mod  m). 

E.  Kummer^^  determined  the  highest  power  p^  of  a  prime  p  dividing 

^;  ^1  >         A  =  ao+aip-{-.  .  .-{-aip\     B=ho+hip-{- .  .  .+bip\ 

where  the  a,  and  6,  belong  to  the  set  0,  1,.  .  .,  p  — 1.  We  may  determine 
Ci  in  this  set  and  e,  =  0  or  1  such  that 

(3)  ao+6o  =  €oP+Co,  €o+ai+6i  =  eip+Ci,  ei +a2  +  ?>2  =  €2^4-^2, •■  •• 

Multiply  the  first  equation  by  1,  the  second  by  p,  the  third  by  p^,  etc.,  and 
add.     Thus 

A+B  =  Co-\-Cip+  .  .  .+Cip'+e,p'+\ 

"Comptes  Rendus  Paris,  120,  1895.  408-10.  534-7. 

"Vpstnik  Ceske  Ak.,  7,  1898,  No.  3,  165  (Bohemian). 

«Messinger  Math.,  27,  1897-8,  12-15. 

"Edinburgh  Phil.  Jour.,  1,  1819,  46. 

"Jour,  fur  Math.,  9, 1832,  73.     Proofs  by  Stern,  12, 1834,  288. 

"/6ui.,  44,  1852,  115-6.     Cayley,  Math.  Quest.  Educ.  Times,  10,  1868,  88-9. 


Chap.  IX]  DIVISIBILITY  OF  FACTORIALS,  MULTINOMIAL  COEFFICIENTS.        271 

Hence,  by  Legendre's  formula  (1), 

ip-l)N  =  A-\-B-y-ei-{A-a)-{B-P),      a=Sa„      ^  =  26^,      7=Sc,. 
Insert  the  value  of  a+/3  obtained  by  adding  equations  (3).    Thus 

A.  Genocchi'^^  proved  that,  if  m  is  the  sum  of  n  integers  a,h,...,k,  each 
divisible  by  p  —  l,  and  if  m<p"  — 1,  then  m\-i-  {a\bl.  .  .k\}  is  divisible  by  the 
prime  p. 

J.  Wolstenholme^^  proved  that  f"ll)  =  l(mod  n^)  if  n  is  a  prime > 3. 

H.  Anton^  (303-6)  proved  that  if  n  =  vp+a,  r  =  wp+h,  where  a,  h,  v,  w 
are  all  less  than  the  prime  p, 

according  as  a^  6  or  a < 6. 

M.  Jenkins^^"  considered  for  an  odd  prime  p  the  sum 


"^^    ^\mr+k{p-l)J' 


extended  over  all  the  integers  k  between  nr/(p  —  l)  and  —mr/{p  —  l),  in- 
clusive, and  proved  that  (Tr=o'p  (mod  p)  if  the  g.  c.  d.  of  r,  p  — 1  equals  that 
of  p,  p-1. 

E.  Catalan^^  noted  that  C'^_})  =  l(mod  p),  if  p  is  a  prime. 

Ch.  Hermite^^  proved  by  use  of  roots  of  unity  that  the  odd  prime  p  divides 

/2n+l\  ,  /2n+l\  ,  /2n+l\  , 
{p-l)  +  [2p-2r[sp-3r-' 

E.  Lucas'^^  noted  that,  if  m  =  pmi-\-ii,  n  =  pni+v,  ii<p,  v<p,  and  p  is 
a  prime. 

In  general,  if  fxi,  fJL2,  ■  ■  ■  denote  the  residues  of  m  and  the  integers  contained 
in  the  fractions  m/p,  m/p^, . .  . ,  while  the  v's  are  the  residues  of  n,  [n/p], .  . . , 

e)-t;)t)-  '-'^^'- 

E.  Lucas'^'^  proved  the  preceding  results  and 

0-0.  f  ;>(-!)".  (^:>0(modp), 

according  as  n  is  between  0  and  p,  0  and  p  —  l,  or  1  and  p. 

"Nouv.  Ann.  Math.,  14,  1855,  241-3. 

"Quar.  Jour.  Math.,  5,  1862,  35-9.     For  mod.  w^  Math.  Quest.  Educ.  Times,  (2),  3,  1903,  33. 
"'^Math.  Quest.  Educ.  Times,  12,  1869,  29.  ^■'Nouv.  Corresp.  Math.,  1,  1874-5,  76. 

'^Jour.  fur  Math.,  81,  1876.  94.  '^Bull.  Soc.  Math.  France,  6,  1877-8,  52. 

"Amer.  Jour.  Math.,  1,  1878,  229,  230.     For  the  second,  anon.«  of  Ch.  Ill  (in  1830). 


272  History  of  the  Theory  of  Numbers.  [Chap,  ix 

J.  Wolstenholme"^  noted  that  the  highest  power  of  2  dividing  i^""^^) 
isq  —  p  —  l,  where  q  is  the  sum  of  the  digits  of  2m  — I  to  base  2,  and  2"  is  the 
highest  power  of  2  dividing  ?«. 

J.  Petersen"^  proved  by  Legendre's  formula  that  C^'')  equals  the 
product  of  the  powers  of  all  primes  p,  the  exponent  of  p  being  (ta+tb  —  ta+b) 
-^(p  —  1),  where  ta  is  the  sum  of  the  digits  of  a  to  base  p. 

E.  Cesaro^°  treated  Kummer's^^  problem.  He  stated  (Ex.  295)  and 
Van  den  Broeek^^  proved  that  the  exponent  of  the  highest  power  of  the 
prime  p  dividing  (-„")  is  the  number  of  odd  integers  among  [2n/p],  [2n/p^], 
[2n/p'],.... 

O.  Schlomilch^^"  stated  in  effect  that  („  +  i)  is  divisible  by  n. 

E.  Catalan^'-  proved  that  if  n  is  odd, 


p:)+io(t-^) 


=  0  (modn+2). 


W.  J.  C.  Sharp^^"  noted  that  {p-\-n)\  —  p\n\  is  divisible  by  p^,  if  p  is  a 
prime  >n.     This  follows  also  from  (''t")  — 1  (mod  p)  [Dickson^"]. 

L.  Gegenbauer^^  noted  that,  if  a  is  any  integer,  r  one  of  the  form  6s  or 
3s  according  as  n  is  odd  or  even, 

The  case  n  odd,  a  =  2,  r  =  3,  gives  Catalan's  result. 

E.  Catalan^  proved  Hermite's'^^  theorem. 

Ch.  Hermite^^  stated  that  (Z)  is  divisible  by  m— n-f-1  if  w  is  divisible 
by  n;  by  (m— n+l)/€  if  e  is  the  g.  c.  d.  of  m+1  and  n;  by  m/8,  if  5  is 
the  g.  c.  d.  of  m,  n. 

E.  Lucas^^  noted  that,  ifn^p  — 1,  p  —  2,  p  — 3,  respectively, 

(^;3)-(-ir(^^±lM)(:nodp), 

if  p  is  a  prime,  and  proved  Hermite's'^  result  (p.  506). 

F.  RogeP^  proved  Hermite's"^  theorem  by  use  of  Fermat's. 

^*Jour.  de  math.  6\6m.  et  spec.,  1877-81,  ex.  360. 

^»Tidsskrift  for  Math.,  (4),  6,  1882,  138-143. 

soMathesis,  4,  1884,  109-110. 

8'7feid.,  6,  1886,  179. 

«>"Zeitschrift  Math.  Naturw.  Unterricht,  17,  1886,  281. 

<«M6m.  Soc.  Roy.  Sc.  de  Li6ge,  (2),  13,  1886, 237-241  ( =  Melanges  Math.).     Mathesis,  10,  1890. 

257-8. 
82aMath.  Quest.  Educ.  Times,  49,  1888,  74. 
s^Sitzunpsber.  Ak.  Wiss.  Wien  (Math.),  98,  1889,  Ila,  672. 
wM6m.  Soc.  Sc.  Li^KC,  (2),  15,  1888,  253-4  (Melanges  Math.  III). 
«*Jour.  de  math,  sp^ciales,  problems  257-8.     Proofs  by  Catalan,  ibid.,  1889,  19-22;  1891,  70; 

by  G.  B.  Mathews,  Math.  Quest.  Educ.  Times,  52,  1890,  63;  by  H.  J.  Woodall,  57, 

1892,  91. 
"Th^orie  des  nombres,  1891,  420.  "Archiv  Math.  Phys.,  (2),  11,  1892,  81-3. 


Chap.  IX]    DIVISIBILITY  OF  FACTORIALS,  MULTINOMIAL  COEFFICIENTS.       273 

C.  Szily^^  noted  that  no  prime  >2a  divides 

?M)  • 

and  specified  the  intervals  in  which  its  prime  factors  occur. 

F.  Morley^^  proved  that,  if  p  =  2n+l  is  a  prime,  (2„")-(-l)"2*"  is 
divisible  by  p^  ii  p>S.  That  it  is  divisible  by  p^  was  stated  as  an  exercise 
in  Mathews'  Theory  of  Numbers,  1892,  p.  318,  Ex.  16. 

L.  E.  Dickson^°  extended  Rummer's''^  results  to  a  multinomial  coefficient 
M  and  noted  the  useful  corollary  that  it  is  not  divisible  by  a  given  prime  p 
if  and  only  if  the  partition  of  m  into  nii,...,  nit  arises  by  the  separate 
partition  of  each  digit  of  m  written  to  the  base  p  into  the  corresponding 
digits  oi  TUi, .  .  . ,  rrit.     In  this  case  he  proved  that 

^=  n      .1),''"     .,,  (mod  p),  m,  =  ao''Y-{- . .  .  +a,^''K 

This  also  follows  from  (2)  and  from 

(xi+ . . .  +xtr=  {x,+  ...  +x,)"»(xiP+ . . .  +x,o"»-i . . .  (0^1^''+ . . .  +x/'ro 

(mod  p). 

F.  Mertens^^  considered  a  prime  p^n,  the  highest  powers  p"  and  2"  of 
p  and  2  which  are  ^n,  and  set  n„  =  [n/2"].  Then  nl-^  {niln2l. .  .nj}  is 
divisible  by  Up'",  where  p  ranges  over  all  the  primes  p. 

J.  W.  L.  Glaisher^^  gave  Dickson's^"  result  for  the  case  of  binomial 
coefficients.  He  considered  (349-60)  their  residues  modulo  p"',  and  proved 
(pp.  361-6)  that  if  {n)r  denotes  the  number  of  combinations  of  n  things  r 
at  a  time,  'Z{n)r^(j)k  (mod  p),  where  p  is  any  prime,  n  any  integer  =j 
(mod  p  —  l),  while  the  summation  extends  over  all  positive  integers  r, 
f"^n,  r=k  (mod  p  —  l),  and  j,  k  are  any  of  the  integers  1,. .  .,  p  —  l.  He 
evaluated  S[(?^)r-^p]  when  r  is  any  number  divisible  by  p  —  l,  and  (n)^  is 
divisible  by  p,  distinguishing  three  cases  to  obtain  simple  results. 

Dickson^^  generalized  Glaisher's^^  theorem  to  multinomial  coefficients: 
Let  k  be  that  one  of  the  numbers  1,  2, .  .  .,  p  —  l  to  which  m  is  congruent 
modulo  p  —  l,  and  let  ki,...,  kt  be  fixed  numbers  of  that  set  such  that 
ki-\-  ■  ■  ■  -\-kt=k  (mod  p  —  l).     Then  if  p  is  a  prime, 

where                                                   ,      .         ,      m 
(mi, . . . ,  nit)  = J ; 

The  second  of  the  two  proofs  given  is  much  the  simpler. 

ssNouv.  Ann.  Math.,  (3),  12,  1893,  Exercices,  p.  52.*     Proof,  (4),  16,  1916,  39-42. 

s^Annals  of  Math.,  9,  1895,  168-170. 

^"Ibid.,  (1),  11,  1896-7.  75-6:  Quart.  Jour.  Math.,  33,  1902,  378-384. 

siSitzungsber.  Ak.  Wiss.  Wien  (Math.),  106,  lla,  1897,  255-6. 

'2Quar.  Jour.  Math.,  30,  1899,  150-6,  349-366. 

o^Ibid.,  33,  1902,  381-4. 


274  History  of  the  Theory  of  Numbers.  [Chap,  ix 

Glaisher^^  discussed  the  residues  modulo  p^  of  binomial  coefficients. 
T.  Hayashi^^  proved  that  if  p  is  a  prime  and  fjL+v  =  p, 

(nsr>(-i)'C).».i(-<ip)- 

according  as  0<s^v,  v<s<p,  or  s  =  0. 

T.  Hayashi^^  proved  that,  if  Iq  is  the  least  positive  residue  of  I  modulo  p, 
and  if  v  =  p—ii, 

modulo  p.     Special  cases  of  the  first  result  had  been  given  by  Lucas. *^ 

A.  Cunningham^^  proved  that,  if  p  is  a  prime, 

(^;^)^(-ir  (modp),  ^(p^)^!  (modp^p>3). 

B.  Ram^^  noted  that,  if  (^),  m  =  l,. . .,  n  — 1,  have  a  common  factor 
o>l,  then  a  is  a  prime  and  n  =  a''.  There  is  at  most  one  prime  <n  which 
does  not  di\dde  n(^)  for  m  =  l,. .  .,  n  — 2,  and  then  only  when  n+l=?a^ 
where  a  is  a  prime  and  q<a.  For  m  =  0,  1, . . .,  n,  the  number  of  odd  (^) 
is  always  a  power  of  2. 

P.  Bachmann^^  proved  that,  if  h{p  —  l)  is  the  greatest  multiple  <A;  of 

p-i, 

(,!i)+(2(pii))+-+(M/-i>''("^°'^^>' 

the  case  k  odd  being  due  to  Hermite.'^ 

G.  Fonten^  stated  and  L.  Grosschniid^°°  proved  that 

(p(pil))^(-l)'  (^odp),  P  =  p\  a^O. 

A.  Fleck^^i  proved  that,  if  0^p<p,  aH-6=0  (mod  p), 

N.  Nielsen^"^  proved  Bachmann's^^  result  by  use  of  Bernoulli  numbers. 

wQuar.  Jour.  Math.,  31,  1900,  110-124. 

"Jour,  of  the  Physics  School  in  Tokio,  10,  1901,  391-2;  Abh.  Geschichte  Math.  Wias.,  28,  1910, 
26-28. 

"Archiv  Math.  Phys.,  (3),  5,  1903,  67-9. 

•'Math.  Quest.  Educat.  Times,  (2),  12,  1907,  94-5. 

"Jour,  of  the  Indian  Math.  Club,  Madras,  1,  1909,  39-43. 

"Niedere  Zahlentheorie,  II,  1910,  46. 
""•Xouv.  Ann.  Math.,  (4),  13,  1913,  521-4. 
"'Sitzungs.  BerUn  Math.  Gesell.,  13,  1913-4,  2-6.     Cf.  H.  Kapferer,  Archiv   Math.   Phys. 

(3),  23,  1915,  122. 
"«Annali  di  mat.,  (3),  22,  1914,  253. 


Chap.  IX]    DIVISIBILITY  OF  FACTORIALS,  MULTINOMIAL  COEFFICIENTS.       275 

A.  Fleck"^  proved  that 

if  and  only  if  p  is  a  prime.     The  case  a  =  1  is  Wilson's  theorem. 

Gu^rin^'^  asked  if  Wolstenholme's^^  result  is  new  and  added  that 

(iLij  — ^~1  (modp^),    p  prime  >3. 

The  Congruence  1-2-3. .  .(p— 1)/2=  ±1  (mod  p). 
J.  L.  Lagrange^^"  noted  that  p  — 1,  p— 2, ...,  (p+l)/2  are  congruent 
modulo  p  to    —1,    —2,...,    —  (p  — 1)/2,    respectively,    so  that  Wilson's 
theorem  gives 

(4)  (l-2.3. .  .P=iy^(-1)¥  (mod  p). 
For  p  a  prime  of  the  form  4n+3,  he  noted  that 

(5)  l-2-3...^==tl  (modp). 

E.  Waring^^^  and  an  anonymous  writer^^^  derived  (4)  in  the  same  manner. 

G.  L.  Dirichlet^^^  noted  that,  since  —1  is  a  non-residue  of  p  =  4n+3,  the 
sign  in  (5)  is  -f  or  — ,  according  as  the  left  member  is  a  quadratic  residue  or 
non-residue  of  p.  Hence  if  m  is  the  number  of  quadratic  non-residues 
<p/2ofp, 

l-2.3...^=(-ir  (modp). 

C.  G.  J.  Jacobi^^^  observed  that,  for  p>3,  m  is  of  the  same  parity  as  N, 
where  2N—l  =  {Q—P)/p,  P  being  the  sum  of  the  least  positive  quadratic 
residues  of  p,  and  Q  that  of  the  non-residues.  Writing  the  quadratic 
residues  in  the  form  ^k,  l^A;^|(p  — 1),  let  m  be  the  number  of  negative 
terms  —k,  and  —  T  their  sum.  Since  —  1  is  a  non-residue,  m  is  the  number 
of  non-residues  <  ^p  and 

^{=i=k)=Sp,  P=2{+k)-{-X{p-k)=mp+Sp, 

Since  p  =  4n+3,  N  =  n-\-l—m—S.    But  7i-|-l  and  S  are  of  the  same  parity 
since 

p>S+2r  =  l+2+...+Kp-l)=iy-l)  =  (2n-H)(n+l). 

lo^Sitzungs.  BerUn  Math.  GeseU.,  15,  1915,  7-8. 

lo^L'intermgdiaire  des  math.,  23,  1916,  174. 

""Nouv.  M6m.  Ac.  BerHn,  2,  1773,  ann6e  1771,  125;  Oeuvres,  3,  432. 

"iMeditat.  Algebr.,  1770,  218;  ed.  3,  1782,  380. 

i"Jour.  fur  Math.,  6,  1830,  105. 

"'/bid.,  3,  1828,  407-8;  Werke,  1,  107.     Cf.  Lucas,  Th^orie  des  nombres,  438;  rinterm^diaire 

des  math.,  7,  1900,  347. 
i"/6id.,  9,  1832,  189-92;  Werke,  6,  240-4. 


276  History  of  the  Theory  of  Numbers.  [Chap,  ix 

He  stated  empirically  that  N  is  the  number  of  reduced  forms  ay^+hyz-\-cz^, 
4:ac  —  b~  =  p  for  b  odd,  ac  —  \b~  =  p  for  b  even,  where  b<a,  b<c. 

C.  F.  Arndt^^^  proved  in  two  ways  that  the  product  of  all  integers 
relatively  prime  to  M  =  %r  or  2/)",  and  not  exceeding  (ilf  — 1)/2,  is  =±1 
(mod  M),  when  p  is  a  prime  4fe+3,  the  sign  being  +  or  —  according  as  the 
number  of  residues  >M/2  of  M  is  even  or  odd.     Again, 
{l-3-5-7...(p-2)}2=±l  (modp), 

the  sign  being  +  or  —  according  as  the  prime  p  is  of  the  form  4n+3  or 
472+1.     In  the  first  case,  1-3.  .  .(p  — 2)=±1  (mod  p). 

L.  Kronecker^^^  obtained,  for  Dirichlet's^^^  exponent  m,  the  result  m=j/ 
(mod  2),  where  v  is  the  number  of  positive  integers  of  the  form  q^^^'^r^  in 
the  set  p  — 2^,  p  — 4",  p  — 6", .  . .,  and  g  is  a  prime  not  dividing  r.  Liou- 
\'ille  (p.  267)  gave  m=k-\-v"  (mod  2),  when  p  =  8^+3  and  v"  is  the  number 
of  positive  integers  of  the  form  g'^'+V^  in  the  set  p— 4^,  p  — 8^,  p  — 12^, . . .. 

J.  Liou\'ille^"  gave  the  result  ?n=cr+r  (mod  2),  for  the  case  p  =  8^'+3, 
where  r  is  the  number  of  positive  integers  of  the  form  2g^'"^^  r^  {q  a  prime  not 
dividing  r)  in  the  set  p  —  1^,  p— 3^,  p  — 5^, .  . . ,  and  a  is  the  number  of  equal 
or  distinct  primes  4gr+l  di\'iding  b,  where  p  =  a^+26"  (uniquely). 

A.  Korkine^^^  stated  that,  if  [x]  is  the  greatest  integer  ^x, 


_p-3 


(p-3)/4 


S     [Vp^l  (modp). 


4 
J.  Franel"^  proved  the  last  result  by  use  of  Legendre's  symbol  and 

(-i)""'TG>    ©=(-^)''   "-TBI    (-°'^2)- 

M.  Lerch^^°  obtained  Jacobi's"*  result. 

H.  S.  Vandiver^^""  proved  Dirichlet's"^  result  and  that 

(p-i)/2r.-2-i 
m=    S    Y~\     (mod  2). 

R.  D.  CarmichaeP^^  noted  that  (4)  holds  if  and  only  if  p  is  a  prime. 

E.  Malo^^^  considered  the  residue  ±r  of  1-2.  .  .(p  — 1)/2  modulo  p, 
where  p  is  a  prime  4m+l,  and  0<r<p/2.  Thus  r^=  —1.  The  numbers 
2,  3, .  .  .,  (p  — 1)/2,  with  r  excluded,  may  be  paired  so  that  the  product 
of  the  two  of  a  pair  is  =  =•=  1  (mod  p) .  If  this  sign  is  minus  for  k  pairs, 
1-2.  .  .(p-l)/2=(-l)V  (mod  p). 

*J.  Ouspensky  gave  a  rule  to  find  the  sign  in  (5). 

Other  Congruences  Involving  Factorials. 

V.  Bouniakowskyi29  noted  that  (p-l)!  =  PP',  P±P'=0  (mod  p)  accord- 
ing as  p  =  4/j=f1.     For,  if  p  is  a  primitive  root  of  p,  we  may  set  P  =  pp^ 

I'SArchiv  Math.  Phys.,  2,  1842,  32,  34-35.  i^o^Amer.  Math.  Monthly,  11,  1904,  51-6. 

"«Jour.  de  Math.,  (2),  5,  1860,  127.  ^^'IMd.,  12,  1905,  106-8. 

^^Ubid.,  128.  '22i;interm6diaire  des  math.,  13,  1906,  131-2 

»«L'interm6diaire  des  math.,  1,  1894,  95.  »23Bu11.  Soc.  Phys.  Math.  Kasan,  (2),  21. 

"»/6id.,  2,  1895,  35-37.  i"M6m.  Ac.  Sc.  St.  P6tersbourg,  (6),  1,  1831,  564. 

'"Prag  Sitzungsber.  (Math.).  1898,  No.  2. 


Chap.  IX]       DiVISIBILTY  OF  FACTORIALS,  MULTINOMIAL  COEFFICIENTS.     277 

...p\  P'  =  p'+^..p^-'  with  t  =  {v-l)/2,  when  p  =  4A;-l;  but  P=pp''-^ 
pV-^  .  .,  P'  =  pY-''  pV"".  •  .,  when  p  =  4A;  +  l. 

G.  01tramare^^°  gave  several  algebraic  series  for  the  reciprocal  of  the 
binomial  coefficient  C^)  and  concluded  that,  if  the  moduli  are  primes, 

!+(-')=  -2{(i)%(||)%(i|^J+  . . .}  (mod  4^+1), 

2=+(-') = -K(iy+(riT+(wy+ ■  •  •}  ^^'^  *-+3)- 

V.  Bouniakowsky^^^  considered  the  integers  qi,.  .  .,  Qs,  each  <N  and 
prime  to  N,  arranged  in  ascending  order  of  magnitude.  If  X  is  any  chosen 
integer  ^s,  multiply 

q,  =  N-qi,  qs-i  =  N-q2,...,  g,_x+i  =  iV-gx 

together  and  multiply  the   resulting   equation  by  qi.  .  . q^^x-     Apply  the 
generalized  Wilson  theorem  qi.  .  .g^+(  — 1)^=0  (mod  A'").     Hence 

9i?2-  •  •5x-gi?2. .  .g.-x-f(-l)'+^=0  (mod  N). 
For  N  a  prime,  we  have  s  =  N—l  and 

X!(iV-l-X)!+(-l)'=0  (mod  N)  (l^XSN-l). 

C.  A.  Laisant  and  E.  Beaujeux^^^  gave  the  last  result  and 


{'-.'} 


(-If  (mod  p),  ^  =  ^- 


F.  G.  Teixeira^^^  proved  that  if  a=-2^''-^p-a,  a<2p-l, 

a{a+l) . .  .{a+2p-l)=3^-5\  .  .{2p-iyp 

(mod  a+a+l+a+2+...+a+2p-l). 
Thus,  for  p  =  3,  a  =  1 ,  a  =  95, 

95-96-97-98-99-100=32-52-3  (mod  585  =  95+ ••  .+100). 

M.  Vecchi^^^  noted  that  the  final  formula  by  Bouniakowsky^^^  follows 
by  induction.  Taking  X  =  (iV— 1)/2,  we  get  Lagrange's  formula  (4). 
From  the  latter,  we  get 

{3.5-7. . .  (22/-l)}2|  (^^^^)  \f/2'^^{-l)'^  (mod  p). 

The  case  y={p  —  l)/2  gives  Arndt's"^  result 

(6)  {3-5-7...(p-2)P=(-l)~  (modp). 

Vecchi^^^  proved  that,  if  v  is  the  number  of  odd  quadratic  non-residues 
of  a  prime  p  =  4n+3,  then  1-3-5.  .  .(p  — 2)  =  (  — 1)"  (mod  p).  If  n  is  the 
number  of  non-residues  <p/2,  1-3-5.  .  .{p-2)={-iy+^2^''-^^^^  (mod  p). 

""M^m.  de  I'lnstitut  Nat.  Genevois,  4,  1856,  "sjomal  de  Sciencias  Math,  e  Astr.,  3,  1881, 

33-6.  105-115. 

"iBull.  Ac.  Sc.  St.  P^tersbourg,  15,  1857, 202-5.  i^^Periodico  di  Mat.,  16,  1901,  22-4. 

"2Nouv.  Corresp.  Math.,  5,  1879,  156  (177).  '^Hbid.,  22,  1907,  285-8. 


278  History  of  the  Theory  of  Numbers.  [Chap.  DC 

R.  D.  Carmichael^^^  proved  that,  if  a+1  and  2a +  1  are  both  primes, 
(a!)*  — 1  is  di^'isible  by  (a  +  l)(2a  +  l),  and  conversely. 

A.  Ar^valo^^^  proved  (6)  and  Lucas'"  residues  of  binomial  coefficients. 
N.  G.  W.  H.  Beeger"^  proved  that  [if  p  is  a  prime] 

(p_l)!+l  =  s-p+l  (rnodp^),        s  =  l+2^-'-\-  .  .  .+{p-iy-'  =  pK.„ 

where  h  is  a.  Bernoulli  number  defined  by  the  symbolical  equation  (/i  +  l)" 
=  /i",  hi  =  l/2.     By  use  of  Adams'^""  table  of  /i„  2<114,  it  was  verified 
that  p  =  5,  p  =  13  are  the  only  p<114  for  which  (p  — 1)!+1=0  (mod  p^). 
T.  E.  Mason^^^  and  J.  M.  Child^'^  noted  that,  if  p  is  a  prime  >3, 

inp)\  =  nl(piy  (modp"+^). 

N.  Nielsen^^''  proved  that,  if  p  =  2n+l,  P=l-3-5.  .  .  (2n-l), 

(-l)'^2np2=22'».3.5.  .  .  (4n-l)     (mod  IGn^). 

If  p  is  a  prime  >3,  P=(-l)"2^"n!     (mod  p^).     He  gave  the  last  result 
also  elsewhere.  ^"^^ 

C.  I.  Marks"-  found  the  smallest  integer  x  such  that  2-4. .  .{2n)x  is  di- 
visible by  3-5 ...  (2n- 1). 

i»«Revista  de  la  Sociedad  Mat.  Espanola,  2,  "»Math.  Quest.  Educat.  Times,  26,  1914,  19. 

1913,  130-1.  ""Annali  di  mat.,  (3),  22,  1914,  81-2. 

"'Messenger  Math.,  43,  1913-4,  83-4.  »«K.  Danske  Vidensk.  Selsk.  Skrifter,  (7),  10 
""a Jour,  fiir  Math.,  85,  1878,  269-72.  1913,  353. 

"STohoku  Math.  Jour.,  5,  1914,  137.  "'Math.  Quest.  Educ.  Times,  21,  1912,  84-6. 


I 


CHAPTER  X. 

SUM  AND  NUMBER  OF  DIVISORS. 

The  sum  of  the  A;th  powers  of  the  divisors  of  n  will  be  designated  crk(n) 
Often  (r(n)  will  be  used  for  (7i(n),and  T{n)  for  the  number  ao{n)oi  the  divisors 
of  n;  also, 

!r(7i)=T(l)+T(2)+...+r(n). 

The  early  papers  in  which  occur  the  formulas  for  T{n)  and  a{n)  were  cited 
in  Chapter  II. 

L.  Euler^'^'^  applied  to  the  theory  of  partitions  the  formula 

(1)  p{x)='n.{l-x'')=s^l-x-x'+z^+x^-x'^-.... 

fc=l 

Euler''  verified  for  n<300  that 

(2)  a{n)=(T{n-l)+(7{n-2)-a{n-b)-(j{n-7)+(7in-12)+..  ., 

in  which  two  successive  plus  signs  alternate  with  two  successive  minus 
signs,  while  the  differences  of  1,  2,  5,  7,  12, .  .  .  are  1,  3,  2,  5,  3,  7, .  .  .,  the 
alternate  ones  being  1,  2,  3,  4, .  .  .  and  the  others  being  the  successive  odd 
numbers.     He  stated  that  (2)  can  be  derived  from  (1). 

Euler^  noted  that  the  numbers  subtracted  from  n  in  (2)  are  pentagonal 
numbers  (3a;^— a:)/2  for  positive  and  negative  integers  x,  and  that  if  a(n—n) 
occurs  it  is  to  be  replaced  by  n.  He  was  led  to  the  law  of  the  series  s  by 
multipljdng  together  the  earlier  factors  of  p{x),  but  had  no  proof  at  that 
time  that  p  =  s.  Comparing  the  derivatives  of  the  logarithms  of  p  and  s, 
he  found  for  —xdp/{pdx)  the  two  expressions  equated  in 

,„.  «     nx""      x+2x^-bx^-1x^+l2x^^+ .  .  . 

{o)  2j ^= 

n=l    \—X  S 

He  verified  for  a  few  terms  that  the  expansion  of  the  left  member  is 
(4)  I  a;V(n). 

n=l 

Multiplying  the  latter  by  the  series  s  and  equating  the  product  to  the  numer- 
ator of  the  right  member  of  (3),  he  obtained  (2)  from  the  coefficients  of  x". 
Euler®  proved  (1)  by  induction.  To  prove  (2),  multiply  the  left  member 
of  (3)  by  —dx/x  and  integrate.  He  obtained  log  p{x)  and  hence  log  s, 
and  then  (3)  by  differentiation. 

^Letter  to  D.  Bernoulli,  Jan.  28,  1741,  Corresp.  Math.  Phys.  (ed.  Fuss),  II,  1843,  467. 

''Euler,  Introductio  in  Analysin  Infinitorum,  1748,  I,  ch.  16. 

^Novi  Comm.  Ac.  Petrop.,  3,  1750-1,  125;  Comm.  Arith.,  1,  91. 

^Letter  to  Goldbach,  Apr.  1,  1747,  Corresp.  Math.  Phys.  (ed.  Fuss),  I,  1843,  407. 

»Posth.  paper  of  1747,  Comm.  Arith.,  2,  639;  Opera  postuma,  1, 1862,  76-84.  Novi  Comm.  Ac. 
Petrop.,  5,  ad  annos  1754-5,  59-74;  Comm.  Arith.,  1,  146-154. 

"Letter  to  Goldbach,  June  9,  1750,  Corresp.  Math.  Phys.  (ed.  Fuss),  I,  1843,  521-4.  Novi 
Comm.  Ac.  Petrop.,  5,  1754-5,  75-83;  Acta  Ac.  Petrop.,  41,  1780,  47,  56;  Comm.  Arith., 
1,  234-8;  2,  105.     Cf.  Bachmann,  Die  Analytische  Zahlentheorie,  1894,  13-29. 

279 


280  History  of  the  Theory  of  Numbers.  [Chap,  x 

Material  on  (1)  will  be  given  in  the  chapter  on  partitions  in  Vol.  II. 
J.  H.  Lambert/  by  expanding  the  terms  by  simple  division,  obtained 

n  =  l   1—X 

in  which  the  coefficient  of  x"*  is  T{n).  Similarly,  he  obtained  (4)  from  the  left 
member  of  (3). 

E.  Waring^  reproduced  Euler's^  proof  of  (2). 

E.  Waring^  employed  the  identity 

n  {x''-l)=x''-x'-'-x'-^+x'-^+x'-''-  -...=A, 

k=l 

the  coefficient  of  x^'",  for  v^n,  being  (  —  1)^  if  v={3z^^z)/2  and  zero  if  v  is 
not  of  that  form.  If  m^n,  the  sum  of  the  mth  powers  of  the  roots  of 
A=0  is  a{m).  Thus  (2)  follows  from  Newton's  identities  between  the 
coefficients  and  sums  of  powers  of  the  roots.     He  deduced 

m(m-l)    ,o^  ,  m{m-l)im-2)            m{m-l){m-2){m-S) 
\o)   I (t{2)-\ a{S) o-(4) 

,  ?n(m-l)(m-2)(m-3)(        .^ 
+  ■■■+ ^ {o-(2)j^-...=  c-ml, 

where  c=  =•=  1  or  0  is  the  coefficient  of  x^~"*  in  series  A.    Let 

U{x''-l)=x''-x'''-'-x''-^+x''-''-\-x''-^-  . . .  =A', 

where  p  ranges  over  the  primes  1,  2,  3,  5, .  .  .,  n.  If  m^n,  the  sum  of  the 
mth  powers  of  the  roots  of  A'  =  0  equals  the  sum  a'{m)  of  the  prime  divisors 
of  m.    Thus 

ff'(m)  =(r'(w- 1) +o-'(m-2) -t7'(m-4) -o-'(w-8) +(r'(m- 10) +o-'(m- 11) 
-(T'{m-12)-a'{m-lQ)+.  .  .. 

We  obtain  (5)  with  a  replaced  by  a',  and  c  by  the  coefficient  of  ic^'"*"  in  series 
A'.     Consider 

n  {x^^-l)=x^-x^-^  -x^-2'+x^-^'+  ...  =5, 

with  coefficients  as  in  series  A.  The  sum  of  the  (Zm)th  powers  of  the  roots 
of  B  =  0  equals  the  sum  (T^^\m)  of  those  divisors  of  m  which  are  multiples  of  I. 
Thus  ^ 

(T'^'\m)=(T'^'^{m-l)W\m-2l)-a^'\m-U)-  .  .  ., 
with  the  same  laws  as  (2) .  The  sum  of  those  divisors  of  m  which  are  divisible 

'Anlage  ziir  Architectonic,  oder  Theorie  des  Ersten  und  des  Einfachen  in  der  phil.  und  math. 

Erkenntniss,  Riga,  1771,  507.     Quoted  by  Glaisher.'* 
^Meditationes  Algebraicse,  ed.  3,  1782,  345. 
•Phil.  Trans.  Roy.  Soc.  London,  78,  1788,  388-394. 


Chap.  X]  SuM   AND   NuMBER  OF   DiVISORS.  281 

by  the  relatively  prime  numbers  a,h,  c,. .  .  is 

Waring  noted  that  o-(a|8)  =  ao-(/3)  +  (sum  of  those  divisors  of  jS  which  are  not 
divisible  by  a) .     Similarly, 

<T(a^y .  .  . )  =  aai^y .  .  . )  +  (sum  of  divisors  of  187 .  .  .  not  divisible  by  a) 
=  a(3a{yd .  .  .)  +  (sum  of  divisors  of  JS7.  .  .  not  divisible  by  a) 
+a(sum  of  divisors  of  76 .  .  .  not  divisible  by  jS), 

etc.    Again,  (r^'^(a/3)=ao-^"(i8)  +  (sum  of  divisors  of  ^  divisible  by  I  but  not 
by  a).     The  generalization  is  similar  to  that  just  given  for  a. 
C.  G.  J.  Jacobi^^  proved  for  the  series  s  in  (1)  that 

00 

s^  =  l-3x-\-5x^-7x^+...=  S   (-l)"(2n+l)  a;"'"+^)/2. 

n=0 

Jacobi^^  considered  the  excess  E{n)  of  the  number  of  divisors  of  the 
form  4w  +  l  of  n  over  the  number  of  divisors  of  the  form  4m +3  of  n.  If 
n  =  2^uv,  where  each  prime  factor  of  u  is  of  the  form  4m +  1  and  each  prime 
factor  of  V  is  of  the  form  4m+3,  he  stated  that  E{n)  =0  unless  y  is  a  square, 
and  then  E{n)  =t{u). 

Jacobi^^  proved  the  identity 

(6)     {l+x-{-x^-{- . . .  +a;'(^+i)/24- . .  .y  =  l-\-a(3)x+  . . .  +(r(2n+lK+  .... 

A.  M.  Legendre^^  proved  (1). 

G.  L.  Dirichlet^^  noted  that  the  mean  (mittlerer  Werth)  of  (T{n)  is  x^n/6 
—  1/2,  that  of  T{n)  is  log  n+2C,  where  C  isEuler's  constant  0.57721.  .  .  . 
He  stated  the  approximations  to  T{n)  and  \pin),  proved  later^'^,  without  ob- 
taining the  order  of  magnitude  of  the  error. 

Dirichlet^^  expressed  m  in  all  ways  as  a  product  of  a  square  by  a  com- 
plementary factor  e,  denoted  by  v  the  number  of  distinct  primes  dividing  e, 
and  proved  that  22"  =  T(m). 

Stern^^"  proved  (2)  by  expanding  the  logarithm  of  (1).  If  C"„  is  the 
number  of  all  combinations  with  repetitions  with  the  sum  n, 

(T(n)=nCn-C\ain-l)-C'2(T{n-2)- . . .. 

Let  S{n)  be  the  sum  of  the  even  divisors  of  n.    Then,  by  (1), 

S{2n)=Si2n-2)-\-S{2n-4:)-S{2n-10)-Si2n-U)-\- .  .  .,      S{0)=2n. 

"Fundamenta  Nova,  1829,  §  66,  (7);  Werke,  1,  237.  Jour,  fiir  Math.,  21,  1840,  13;  French 
transl.,  Jour,  de  Math,  7,  1842,  85;  Werke,  6,  281.     Cf.  Bachmann,«  pp.  31-7. 

"Zfeid.,  §40;  Werke,  1,  1881,  163. 

i^Attributed  to  Jacobi  by  Bouniakowsky"  without  reference.  See  Legendre  (1828)  and 
Plana  (1863)  in  the  chapter  on  polygonal  numbers,  vol.  2. 

"Th^orie  des  nombres,  ed.  3,  1830,  vol.  2,  128. 

"Jour,  fiir  Math.,  18,  1838,  273;  Bericht  Berlin  Ak.,  1838,  13-15;  Werke,  1,  373,  351-6. 

^'Ibid.,  21,  1840,  4.     Zahlentheorie,  §  124. 

i5»76id.,  177-192. 


282  History  of  the  Theory  of  Numbers.  [Chap,  x 

Let  S'(n)  be  the  sum  of  the  odd  di\'isors  of  n,  and  C„  be  the  number  of  all 
combinations  without  repetitions  with  the  sum  n,  so  that  C7  =  5.     Then 

S'in)=nCn-S'{n-l)Ci-S\n-2)C2+ . . ., 

Z)(n)  =  -D(n-l)-D(n-3)-D(n-6)-...,    D{n)=S'{n)-S{n). 

A  complicated  recursion  formula  for  T(n)is  derived  from 

\og{{l-x){l-x^y{l-3^)r  .  .}  =  -  I  ^-Tin)x\ 

n=in 

Complicated  recursion  fonnulas  are  found  for  the  number  of  integers 
<m  not  factors  of  m,  and  for  the  sum  of  these  integers.  A  recursion 
formula  for  the  sum  Sr{n)  of  the  di\'isors  ^r  of  n  is  obtained  by  expanding 

log  {l-x)(l-x2)...(l-a:'-)l  =  -  S  -Sr(n)x". 

n=in 

Jacobi^®  proved  (1). 

Dirichlet^^  obtained  approximations  to  T(n).  An  integer  s^n  occurs 
in  as  many  terms  of  this  sum  as  there  are  multiples  of  s  among  1,  2, .  .  . ,  n. 
The  number  of  these  multiples  is  [n/s],  the  greatest  integer  ^n/s.     Hence 


''(")=iG] 


This  sum  is  approximately  the  product  of  n  by 

£i  =  logn+C+i+.... 

Hence  T{n)  is  of  the  same  order  of  magnitude  as  n  log  n. 

Let  ju  be  the  least  integer  ^  y/n  and  set  v  =  [n/ii].    Then  if  g{x)  is  any 
function  and  G{x)=g{l)-\-g{2)-\- . . .  +^(x), 

2  r^i^(s)= -.GGu)+s  pi^(s)+s  Gjr^ii- 

»=:LsJ  «=iLsJ  «=i    LLsJJ 

In  particular,  if  ^(x)  =  1, 

«=iLsJ      «=iLsJ 
Giving  to  [n/s]  the  approximation  n/s,  we  see  that 
(7)  T(n)=n  log,n+(2C-l)n+e, 

where  €  is  of  the  same  order  of  magnitude  as  Vn. 

Let  pin)  be  the  number  of  distinct  prime  factors  >1  of  ti.     Then  2"^"^  is 
the  number  of  ways  of  factoring  n  into  two  relatively  prime  factors,  taking 

"Jour,  fur  Math.,  32,  1846,  164;  37,  1848,  67,  73. 

"Abhand.  Ak.  Wiss.  Berlin,  1849,  Math.,  69-83;  Werke,  2,  49-66.     French  transl.,  Jour,  de 
Math.,  (2),  1,  1856,  353-370. 


4 


Chap.  X]  SuM  AND  NuMBER  OF  DiVISORS.  283 

account  of  the  order  of  the  factors.  The  number  of  pairs  of  relatively  prime 
integers  ^,  17  for  which  ^r^^n  is  therefore 

y=i 
For  the  preceding  C  and  r(n),  it  is  proved  that 

r(n)=S^,/.[P],    .        t  =  [V^], 
^in)=^(log.n+'-^+2C-l)+m,  "  C-  I  H^, 

IT  IT  8=2      S 

where  m  is  of  the  order  of  magnitude  of  n\  8>y/2,  while  7  is  determined  by 
2)s~^  =  l  (s  =  2  to  00).  Moreover,  T(n)  is  the  number  of  pairs  of  integers 
X,  y  for  which  xy^n.    He  noted  that 


(7(l)+(r(2)  +  ...+(r(n)=Ssr^l 

8=1  Ls-i 


and  that  the  difference  between  this  sum  and  ir^n^/12  is  of  an  order  of  magni- 
tude not  exceeding  n  loge  n. 

G.  H.  Burhenne^^  proved  by  use  of  infinite  series  that 

r(n)=i2)/"K0),  fix)^-  "^^ 


and  then  expressed  the  result  as  a  trigonometric  series. 

V.  Bouniakowsky^^  changed  x  into  x^  in  (6),  multiplied  the  result  by  x'^ 
and  obtained 

(x^ +x' +x^  +  . .  .)*  =  x*+(r(3)x'2_^  . .  .  +(r(2w+l)a:^"'+H  .... 

Thus  every  number  8m+4  is  a  sum  of  four  odd  squares  in  (r(2w+l)  ways. 
By  comparing  coefficients  in  the  logarithmic  derivative,  we  get 


(8)     (l2-2m+l)(r(2m+l)  +  (3^-2m-l)(7(2m-l)  +  (52-2m-5)(r(2m-5) 

+  ...=0, 

in  which  the  successive  differences  of  the  arguments  of  <r  are  2,  4,  6,  8, ... . 
For  any  integer  N, 


(9)  {l^-N)a{N)  +  {S^-N-h2)(r{N-l-2)  +  i5''-N-2-3)a{N-2'3) 

+  -..=0, 

where  o-(O) ,  if  it  occurs,  means  A^/6.     It  is  proved  (p.  269)  by  use  of  Jacobi's^° 
result  for  s^  that 

l+x+x'+x'+  . . .  =P^=  (i+x)a+x'){l+x') .  .  . 
{l-x')(l-x'){l-x')..., 

"Archiv  Math.  Phys.,  19,  1852,  442-9. 

»M6m.  Ac.  Sc.  St.  P^tersbourg  (Sc.  Math.  Phys.),  (6),  4,  1850,  259-295  (presented,  1848). 
Extract  in  Bulletin,  7,  170  and  15,  1857,  267-9. 


284  History  of  the  Theory  of  Numbers.  [Chap,  x 

where  the  exponents  in  the  series  are  triangular  numbers.  Hence  if  we 
count  the  number  of  ways  in  which  n  can  be  formed  as  a  sum  of  different 
terms  from  1,  2,  3, .  .  .  together  w^ith  different  terms  from  2,  4,  6, .  .  .,  first 
taking  an  even  number  of  the  latter  and  second  an  odd  number,  the  differ- 
ence of  the  counts  is  1  or  0  according  as  n  is  a  triangular  number  or  not. 
It  is  proved  that 

(10)  <r(n)  +  {(T(2)-4o-(l))(r(n-2)+(r(3)(T(n-4)  +  {(r(4)-4(r(2))(r(n-6) 
+(r(5)(7(n-8)  +  {(r(6)-4(r(3))tr(n-10)+.  .  .  =^(7(n+2). 

The  fact  that  the  second  member  must  be  an  integer  is  generaUzed  as 
follows:  for  n  odd,  (T(n)  is  even  or  odd  according  as  n  is  not  or  is  a  square; 
for  n  even,  (T{n)  is  even  if  n  is  not  a  square  or  the  double  of  a  square,  odd  in  the 
contrary  case.  Hence  squares  and  their  doubles  are  the  only  integers  whose 
sums  of  divisors  are  odd. 

V.  Bouniakowsky-'^  proved  that  (r(A^)  =  2  (mod  4)  only  when  N  =  kc^  or 
2kc^,  where  A:  is  a  prime  4Z+1  [corrected  by  Liouville^°]. 

V.  A.  Lebesgue-^  denoted  by  l-{-AiX+A2X^-\- .  . .  the  expansion  of  the 
mth  power  of  p{x),  given  by  (1),  and  proved,  by  the  method  used  by  Euler 
for  the  case  m  =  1,  that 

a{n)+A,a{n-l)-\-A2<T{n-2)+  .  .  .+Ar,_,a{l)-\-nAjm  =  0. 
This  recursion  formula  gives 

.      m(m— 3)       .       —m(m  —  l)(m  —  S) 
A,=  -m,    A,  =  — ^^2— '     ^^  = 1:2:3 •••• 

The  expression  for  Aj,  was  not  found. 

E.  MeisseP2  proved  that  (c/.  Dirichlet^^) 

(11)  T{n)  =  i^[jj  =^i:[j]  -''        (^  =  [V^])- 

J.  Liouville^^  noted  that  by  taking  the  derivative  of  the  logarithm  of 
each  member  of  (6)  we  get  the  formula,  equivalent  to  (8) : 


J       5m(m+l)1    /o     ,  1        2        N     n 
S^n ^ Ya{2n+l—m—m)=0, 


summed  for  m  =  0,  1, .  .  .,  the  argument  of  a  remaining  ^0. 
J.  Liouville^^  stated  that  it  is  easily  shown  that 


Sd<T(d)=s(|y(r(d), 


20M6m.  Ac.  Sc.  St.  P^tersbourg,  (6),  5,  1853,  303-322. 
"Nouv.  Ann.  Math.,  12,  1853,  232-4. 
"Jour,  fur  Math.,  48,  1854,  306. 
"Jour,  de  Math.,  (2),  1,  1856,  349-350  (2,  1857,  412). 

^Ibid.,  (2),  2,  1857,  56;  Nouv.  Ann.  Math.,  16,  1857,  181;  proof  by  J.  J.  Hemming,  ibid.,  (2),  4, 
1865,  547. 


Chap.  X]  SuM  AND   NuMBER   OF  DiVISORS.  285 

where  d  ranges  over  the  divisors  of  m.     He  proved  (p.  411)  that 

S(-l)'"/'^fi  =  2(T(m/2)-(7(m). 

J.  Liouville^^  stated  without  proof  the  following  formulas,  in  which  d 
ranges  over  all  the  divisors  of  m,  while  5  =  m/d : 

Xaid)  =2:5r(d),  S0(d)r(5)  =(r(m),  20(d)r(6)  =  [rim)}^, 

XcT{d)cr{8)  =SdT(d)r(5),  Sr(d)r(5)  =2:|t(^)}' 

where  (}){d)  is  the  number  of  integers  <  d  and  prime  to  d,  6{d)  is  the  number  of 
decompositions  of  d  into  two  relatively  prime  factors,  and  the  accent  on  S 
denotes  that  the  summation  extends  only  over  the  square  divisors  D^  of  m. 
He  gave  (p.  184) 


S0(cf)=r(m2),  ^'e{^)i=r{m), 


the  latter  being  implied  in  a  result  due  to  Dirichlet.^^ 

Liouville"*'  gave  the  formulas,  numbered  (a),.  .  .,  {k)  by  him,  in  which 
X(m)  =  +1  or  —1,  according  as  the  total  number  of  equal  or  distinct  prime 
factors  of  m  is  even  or  odd: 

Sr(d2'')=T(m)r(m''),     2r(d2'')T(5)  =ST(d)rOT,     S(^(5)(7((i)  =mT(m), 
S5(7(d)  =SdT(d),  SX(c^)  =  1  or  0,  ^\{d)d{d)r{b)  =  1  or  0, 

according  as  m  is  or  is  not  a  square; 

i:\{d)d{d)r{h^)  =  l,  X\{d)e{d)=\im),    SX(d)0(5)  =  l, 

l^X{d)d{d)did)=0,  SX(5)o-(ci)  =mS'-^. 

The  number  of  square  divisors  D^  of  m  is  '2\(d)T{8). 

Liouville^^  gave  the  formulas,  numbered  I-XVIII  by  him: 

ST(52)(/)(d)  =S5^(d),  Sdr(52)  =S^(5)(r(d), 

ST(52)X(d)  =T(m),  2  {T{8)}Md)d{d)  =tW, 

S0(d)T(5)r(5'')  =SdT(62''),  ^e{b)T{d)r{d'')  =Sr(52)T(d-''), 

ST(52'')(7(d)  =25r(d)T(d''),  S'0(i))T(^)  =S'Z)  ^(^) , 

SX(5)T(d)TW  =2't(^)  .  ^\id)(T{d)  =mX(m)S'- 


2)2 


'^Jour.  de  Mathematiques,  (2),  2, 1857, 141-4.    "Sur  quelques  fonctiona  num^riques,"  1st  article. 

Here  Sa6c  denotes  S(a6c). 
^^Ihid.,  244-8,  second  article  of  his  series. 
"76id.,  377-384,  third  article  of  his  seriea 


286  History  of  the  Theory  of  Numbers.  [Chap,  x 

S'X(i))r(^2)  =^"d(^^'  2{W!''  =  T(m^), 

Sr(0^(5)  =S{^(d))''T(52),         2r(OX(5)  =2|0(^)  j^ 

where,  in  2",  e  ranges  over  the  biquadrate  divisors  of  m. 
Liouville^^  gave  the  formula 

X{T{d)V={Xr{d)]', 

which  implies  that  if  2m  (m  odd)  has  no  factor  of  the  form  4)u+3  and  if  we 
find  the  number  of  decompositions  of  each  of  its  even  factors  as  a  sum  of 
two  odd  squares,  the  sum  of  the  cubes  of  the  numbers  of  decompositions 
found  will  equal  thesquare  of  their  sum.     Thus,  for  m  =  25, 

50=l2+72  =  72+l2  =  52+5^  10  =  32+12  =  12+32,  2  =  1  +  1, 

whence  3H2Hl'  =  62. 

Liouville2^  stated  that,  if  a,  6, . .  .  are  relatively  prime  in  pairs, 

a^iah.  .  .)=o'n(oVn(?>)-  •  •, 
while  if  p,  9, .  . .  are  distinct  primes, 

He  stated  the  formulas 

2(r^((i)</)(5)  =m<T,_,{m),  2ciV,(5)  =2d''(T^(5), 

2X(d)r(d2)(r^(5)  =2d''r(5)X(5),  2dV^(6)  =2c/''r(d), 

2dX(d)  =252v,(d),  2dV3,(5)  =2c^V2,(d), 

2dX+,(d)(7,(5)  =2(iV,+,(d)(r,(5),    2X(d)(T,(5)  =S'(^)' 

2T(d2'')(r,(5)  =2^^(5)7(5"),  2{^(rf)}  V,(5)  =2d''r(52'), 

and  various  special  cases  of  them.  To  the  seventh  of  these  Liouville^"  later 
gave  several  forms,  one  being  the  case  p  =  0  of 

2d''-V.+X^)(r,+,(5)=2d''-X+,((i)(r,+,(5), 

and  proved  (p.  84)  the  known  theorem  that  a{m)  is  odd  if  and  only  if  m  is  a 
square  or  the  double  of  a  square  [cf.  Bouniakowsky,^^  end].  He  proved  that 
(t{N)  =  2  (mod  4)  if  and  only  if  N  is  the  product  of  a  prime  4X  +  1,  raised  to 
the  power  4Z  +  1  (Z^O),  by  a  square  or  by  the  double  of  a  square  not  divis- 

"Jour.  de  Math6matiques,  (2),  2,  1857,  393-6;  Comptes  Rendus  Paris,  44,  1857,  753, 
^^Ibid.,  425-432,  fourth  article  of  his  series, 
"/bid.,  (2),  3,  1858,  63. 


Chap.  X]  SuM   AND   NuMBER   OF   DiVISORS.  287 

ible  by  the  prime  4X+1.  The  condition  given  by  Bouniakowsky^°  is  neces- 
sary, but  not  sufficient.    Also, 

o-3(m)  =  S  <T{2j-l)a{2m-2j+l)  {m  odd). 

J.  Liouville's  series  of  18  articles,  "Sur  quelques  formules . .  .utiles  dans  la 
th^orie  des  nombres,"  in  Jour,  de  Math.,  1858-1865,  involve  the  function 
(r„,  but  will  be  reported  on  in  volume  II  of  this  History  in  connection  with 
sums  of  squares.  A  paper  of  1860  by  Kronecker  will  be  considered  in 
connection  with  one  by  Hermite.'^'^ 

C.  Traub^^  investigated  the  number  {N;  M,  t)  of  divisors  T  oi  N  which 
are  =  t  (mod  M) ,  where  M  is  prime  to  t  and  N.  Let  a,h,.  .  .,lhe  the  integers 
<  M  and  prime  to  M ;  let  them  belong  modulo  M  to  the  respective  exponents 
a',  h',. . .,  V;  let  m  be  a  common  multiple  of  the  latter.  Since  any  prime 
factor  of  N  is  of  the  form  Mx+k,  where  k  =  a,. .  .,1,  any  T  is  congruent  to 

a^6^. .  .Z^=«  (mod  M),  O^A<a',. . .,  O^KV. 

Let  A',. . .,  L'  he  one  of  the  n  sets  of  exponents  satisfying  these  conditions. 
If  P  is  a  primitive  mth  root  of  unity,  the  function 

1/'  =  ^7-^SP^  e  =  {A-A')am/a'+  . .  .+{L-L')\m/V, 

summed  for  all  sets  0^a<a', . .  .,  O^X<r,  has  the  property  that  i^  =  l  if 
A  =  A'(mod  a'),.  .  .,  L=L'(mod  V)  simultaneously,  while  i/'  =  0  in  all  other 
cases.  Thus  {N]  M,  t)  =SSt/',  where  one  summation  refers  to  the  n  sets 
mentioned,  while  the  other  refers  to  the  various  divisors  T  of  N.  This 
double  sum  is  simplified. 

[The  properties  found  (pp.  278-294)  for  the  set  of  residues  modulo  M 
of  the  products  of  powers  oi  a,.  .  .,  I  may  be  deduced  more  simply  from  the 
modern  theory  of  commutative  groups.] 

V.  Bouniakowsky^^  considered  the  series 

n=l'c.  n=l    "' 

By  forming  the  product  of  xl/ix)""'^  by  \{/{x) ,  he  proved  that  z„,  2  is  the  number 
No{n)=T{n)  of  the  divisors  of  n,  and  Zn,m  equals 

where  (and  below)  d  ranges  over  the  divisors  of  n.    Also, 

\p(x)\l/{x-l)=  2)  ——• 

n=l     '«' 

From  \l/{xYxl/(x-iy  for  (i,  j)  =  (2,  1),  (2,  2),  (1,  2),  he  derived  the  first  and 
fourth  formulas  of  Liouville's^^  first  article  and  the  fourth  of  his^^  second 
article.     He  extended  these  three  formulas  to  sums  of  powers  of  the  divisors 

^lArchiv  Math.  Phys.,  37,  1861,  277-345. 

32M^m.  Ac.  Sc.  St.  P^tersbourg,  (7),  4,  1862,  No.  2,  35  pp. 


288  History  of  the  Theory  of  Numbers.  [Chap,  x 

and  proved  the  second  formula  in  Liouville's  first  article  and  the  first  two 
summation  formulas  of  Liouville.-^    He  proved 

i.(2.-l)=2.-l+z[^-^],  .=  [^], 

where  77  =  1  or  0  according  as  2o-—  1  is  divisible  by  3  or  not.  The  last  two 
were  later  generaUzed  by  Gegenbauer.^^ 

E.  Lionnet^  proved  the  first  two  formulas  of  Liouville.^^ 

J.  Liou^ille^  noted  that,  if  q  is  divisible  by  the  prime  a, 

(r,(a5)+a''a-M^|j  =  (a''  +  l)(r,(g). 

C.  Sardi^^  denoted  by  A„  the  coefficient  of  x"  in  Jacobi's^''  series  for  s^, 
so  that  An  =  0  unless  n  is  a  triangular  number.     From  that  series  he  got 

S(-l)P(2p+l)(7{7i-p(p  +  l)/2)=(-l)^'+^'/W3orO       {t  =  Vl+8n), 
p 
according  as  n  is  or  is  not  a  triangular  number,  and 

|.4„+A„_,cr(l)+...+Ai(r(n-l)+Ao(r(n)=0. 

This  recursion  formula  determines  A„  in  terms  of  the  c's,  or  (T{n)  in  terms  of 
the  A's.  In  each  case  the  values  are  expressed  by  means  of  determinants  of 
order  n. 

IM.  A.  Andreievsk}^^  wrote  N^h^^i  for  the  number  of  the  divisors  of  the 
form  4/i±  1  of  n  =  a'^h^ .  .  . ,  where  a,  b,. .  .  are  distinct  primes.     We  have 

where  d  ranges  over  all  the  di\'isors  of  n  and  the  symbols  are  Legendre's. 
Evidently  „  /-i\a' 

S  (— ^)  =  a  +  l    if  a  =  4Z  +  l, 

a'=o\  a  } 

=  0  or  1  if  a  =  4Z-l, 

according  as  a  is  odd  or  even.  Hence,  if  any  prime  factor  4Z  —  1  of  n  occurs 
to  an  odd  power,  we  have  iV4A+i=iV4A_i.     Next,  let  *| 

where  each  p,  is  a  prime  of  the  form  4Z + 1 ,  each  g,  of  the  form  4?  —  1 .     Then 

iV4A+i-iV4.-i  =  (ai  +  l)(ao  +  l).  .  .  =r(^),  D  =  q,\^\  .  .. 

^'Souv.  Ann.  Math.,  (2),  7,  1868,  68-72. 

"Jour,  de  math.,  (2),  14,  1869,  263-4. 

"Giomale  di  Mat.,  7,  1869,  112-5. 

3%Iat.  Sbomik  (Math.  Soc.  Moscow),  6,  1872-3,  97-106  (Russian). 


Chap.  X]  SUM  AND   NUMBER  OF   DiVISORS.  289 

The  sum  of  the  N's  is  T{n)  ^riD^Mn/D^).     Hence 

N^^^rm  +  l 

which  is  never  an  integer  other  than  1  or  2  when  n  is  odd.  If  it  be  2,  t(D^)  =  3 
requires  that  D  be  a  prime.     Similarly,  for  Legendre's  symbol  (2/a), 

is  zero  if  any  prime  factor  8^±  3  of  n  occurs  to  an  odd  power,  but  is  11  (a^H- 1) 
if  in  n  each  p,  is  a  prime  8Z±1  and  each  Qi  a  prime  8Z±3.  For  n  odd, 
Ngh^i/Nsh^s  can  not  be  an  integer  other  than  1  or  2;  if  2,  D  is  a  prime. 

F.  Mertens"  proved  (11).  He  considered  the  number  v{n)  of  divisors 
of  n  which  are  not  divisible  by  a  square  >  1.  Evidently  v{n)  =2",  where  p 
is  the  number  of  distinct  prime  factors  of  n.  If  ju(n)  is  zero  when  n  has  a 
square  factor  >  1  and  is  + 1  or  —  1  according  as  n  is  a  product  of  an  even  or 
odd  number  of  distinct  primes,  v{n)  =XiJL^{d),  where  d  ranges  over  the  divisors 
of  n.    Also, 

fc=i  k=i  \/c  / 

He  obtained  Dirichlet's^'^  expression  \l/{n)  for  this  sum,  finding  for  m  a  limit 
depending  on  C  and  n,  of  the  order  of  magnitude  of  \/n  log^  n. 

E.  Catalan^'^"  noted  that  So-(i)o-(i)  =80-3(72)  where  i-{-j  =  4n.  Also,  if  i  is 
odd,  €r{i)  equals  the  sum  of  the  products  two  at  a  time  of  the  E's  of  the  odd 
numbers  whose  sum  is  2i,  where  E  denotes  the  excess  of  the  number  of 
divisors  4/i+l  over  the  number  of  divisors  4/^  — 1. 

H.  J.  S.  Smith^^  proved  that,  if  m  =  pi''ip2"2. .  ., 

..W-2..(^)+S.,(^)-. ..=.•- 

For,  if  P=l+p'+...+p",     P'  =  l+p'+...+p'"-"',  then 

c.(m)  =  P.P. ... ,  <..  (^)  =  P/P, .  . . ,  a.  (^J  =  P/P/Pa .... 

and  the  initial  sum  equals  (Pi  —  Pi){P2  —  P2)  ■  .  .=m\ 

J.  W.  L.  Glaisher^^  stated  that  the  excess  of  the  sum  of  the  reciprocals  of 
the  odd  divisors  of  a  number  over  that  for  the  even  divisors  is  equal  to  the 
sum  of  the  reciprocals  of  the  divisors  whose  complementary  divisors  are 
odd.  The  excess  of  the  sum  of  the  divisors  whose  complementary  divisors 
are  odd  over  that  when  they  are  even  equals  the  sum  of  the  odd  divisors. 

G.  Halphen^°  obtained  the  recursion  formula 

(T(n)=3(r(n-l)-5(r(n-3)+.  .  . -(-l)"(2a;+l)Jn-^^^^|+.  . ., 

"Jour,  flir  Math.,  77,  1874,  291-4. 

^'"Recherches  sur  quelques  produits  indefinis,  M^m.  Ac.  Roy.  Belgique,  40,  1873,  61-191. 

Extract  in  Nouv.  Ann.  Math.,  (2),  13,  1874,  518-523. 
asProc.  London  Math.  Soc,  7,  1875-6,  211. 
'^Messenger  Math.,  5,  1876,  52. 
^oBuU.  Soc.  Math.  France,  5,  1877,  158. 


290 


History  of  the  Theory  of  Numbers. 


[Chap.X 


where,  if  n  is  of  the  form  x{x+l)/2,  (t(0)  is  to  be  taken  to  be  n/3  [Glai- 
sher^"].  The  proof  follows  from  the  logarithmic  derivative  of  Jacobi's^" 
expression  for  s^,  as  in  Euler's^  proof  of  (2). 

Halphen"*^  formed  for  an  odd  function /(z)  the  sum  of  s.i 


pc! 


(-1)7 


(t*«)' 


n- 


■  +^n-l  —  "~o-^n> 


X  taking  all  integral  values  between  the  two  square  roots  of  a,  and  y  ranging 
over  all  positive  odd  divisors  of  a—x^.     This  sum  is 

if  a  is  a  square,  zero  if  a  is  not  a  square.  Taking /(s)  =z,  we  get  a  recursion 
formula  for  the  sum  of  those  di\dsors  d  oi  x  for  which  x/d  is  odd  [see  the 
topic  Sums  of  Squares  in  Vol.  II  of  this  History].  Taking  f{z)=a^  —oT', 
we  get  a  recursion  formula  for  the  number  of  odd  di\dsors  <a/m  of  a. 
A  generalization  of  (2)  gives  a  recursion  formula  for  the  sima  of  the  divisors 
of  the  forms  2nk,  n{2k-\-\)^m,  wdth  fixed  n,  m. 

E.  Catalan^'-  denoted  the  square  of  (1)  by  l+LiX+ .  .  . +L„x'*H- . .  .. 
Thus 

o-(n) +IiO-(7i- 1) +L20-(n-2)  + 

I'n-I>n-l-I>.-2+I^n-54-Z>„_7-  .  .  .  =  0  Or   (2X  +  1)(-1)\ 

according  as  n  is  not  or  is  of  the  form  X(X  +  l)/2.  In  \dew  of  the  equality 
of  (3)  and  (4)  and  the  fact  that  l/p=2;/'(n)x",  where  yp{n)  is  the  number  of 
partitions  of  n  into  equal  or  distinct  positive  integers,  he  concluded  that 

(7(n)=;//(n-l)+2,/'(n-2)-5;/'(n-5)-7i/^(n-7)  +  12^(n-12)+.  .  .". 

J.  W.  L.  Glaisher^^  noted  that,  if  B{n)  is  the  excess  of  the  sum  of  the  odd 
di\'isors  of  n  over  the  sum  of  the  even  dii'isors, 

e{n)  +<9(n  - 1)  +d{n  -  3)  +d{n  -  6)  -f  .  .  .  =  0, 

where  1,  3,  6, .  . .  are  the  triangular  numbers,  and  B{n—n)  =  —n. 

E.  Cesaro^  denoted  bj^  s„  the  sum  of  the  residues  obtained  by  dividing  n 
by  each  integer  <n,  and  stated  that 

s„+(7(l)+(7(2)+...+(7(n)=n2. 

E.  Catalan^^  proved  the  equivalent  result  that  the  sum  of  the  divisors  of 
1, .  .  . ,  n  equals  the  sum  of  the  greatest  multiples,  not  >/?,  of  these  numbers. 
Catalan'*^  stated  that,  if  <^(a,  n)  is  the  greatest  multiple  ^^  of  a, 

n 

a{n)=  2  {</)(a,  n)—4>{a,  n  — 1)). 


I 


"Bull.  Soc.  Math.  France,  6,  1877-S,  119-120,  173-188. 

"Assoc,  frang.  avanc.  sc,  6,  1877,  127-8.     Cf.  Catalan.^^" 

«Messenger  Math.,  7,  1877-8,  66-7. 

"Nouv.  Corresp.  Math.,  4,  1878,  329;  5,  1879,  22;  Nouv.  Ann.  Math.,  (3),  2,  1883,  289;  4,  1885, 

473. 
«/Wd.,  5,  1879,  296-8;  stated,  4,  1879,  ex.  447. 
*mid.,  6,  1880,  192. 


Chap.  X]  SuM   AND   NuMBER   OF   DiVISORS.  291 

Radicke  (p.  280)  gave  an  easy  proof  and  noted  that  if  we  take  n  =  1, . .  . ,  m 
and  add,  we  get  the  result  by  E.  Lucas^^ 

o-(l)  +  .  .  .  +o-(m)  =0(1,  m)  +  .  .  .  +0(m,  m). 

J.  W.  L.  Glaisher^^  stated  that  if  f{n)  is  the  sum  of  the  odd  divisors  of  n 
and  if  g{n)  is  the  sum  of  the  even  divisors  of  n,  and  /(O)  =0,  g{0)  =n,  then 
/(n)+/(n-l)+/(n-3)+/(n-6)+/(n-10)  +  ... 

=  g{n)+g{n-l)+gin-S)+ .  .  .. 
Chr.Zeller^^  proved  (11). 

R.  Lipschitz^o  wrote  G(t)  for  <t{1)-\- .  .  .-\-a{t),  D{t)  for  {t''+t)/2,  and 
$(0  for  (^(1)  +  .  .  .  +(f>{t),  using  Euler's  (f>{t).  Then  if  2,  3,  5,  6, . . .  are  the 
integers  not  divisible  by  a  square  >  1 , 


^w--[l]--[l]--G] 


+  . .  .  =n, 


G(n)-2G 
D(n)-D 


the  sign  depending  on  the  number  of  prime  factors  of  the  denominator.  He 
discussed  (pp.  985-7)  Dirichlet's^'^  results  on  the  mean  of  T{n),  o-(n),  (f>{n). 

A.  Berger^^  proved  by  use  of  gamma  functions  that  the  mean  of  the  sum 
of  the  divisors  d  of  n  is  ir^n/Q,  that  of  S  d/2'^  is  1,  that  of  Sl/d!  is  ir^/Q, 

G.  Cantor^^"  gave  the  second  formula  of  Liouville^^  and  his^^  third. 

A.  Piltz^^  considered  the, number  Tk{n)  of  sets  of  positive  integral  solu- 
tions of  Ui.  .  .Uk  =  n,  where  differently  arranged  u's  give  different  sets. 
Thus  T'i(n)  =  1,  T'2(n)  =T{n).  If  a-  is  the  real  part  of  the  complex  number  s, 
and  n*  denotes  e^  '°^ "  for  the  real  value  of  the  logarithm,  he  proved  that 

n=l      "'  m  =  0 

where  l  =  \—(T  —  \/k,  and  the  6's  are  constants,  6^  =  0 for  s ?^  1 ;  while 0(/)  is^'^ 
of  the  order  of  magnitude  of  /.  Taking  s  =  0,  we  obtain  the  number  'SiTkin) 
of  sets  of  positive  integral  solutions  of  t^i .  .  .u^'^x. 

H.  Ahlborn^Hreated  (11). 

E.  Cesaro^^  noted  that  the  mean  of  the  difference  between  the  number 
of  odd  and  number  of  even  divisors  of  any  integer  is  log  2 ;  the  limit  for 

^^Nouv.  Corresp.  Math.,  5,  1879,  296. 

48NOUV.  Corresp.  Math.,  5,  1879,  176. 

"Gottingen  Nachrichten,  1879,  265. 

^oComptes  Rendus  Paris,  89,  1879,  948-50.     Cf.  Bachmann^"  of  Ch.  XIX. 

"Nova  Acta  Soc.  Sc.  Upsal.,  (3),  11,  1883,  No.  1  (1880).   Extract  by  Catalan  in  Nouv.  Corresp. 

Math.,  6,  1880,  551-2.     Cf.  Gram.«^« 
"«G6ttmgen  Nachr.,  1880,  161;  Math.  Ann.,  16,  1880,  586. 
"Ueber  das  Gesetz,  nach  welchem  die  mittlere  Darstellbarkeit   der  natiirlichen  Zahlen   ala 

Produkte  einer  gegebenen  Anzahl  Faktoren  mit  der  Grosse  der  Zahlen  wachst.    Diss., 

Berlin,  1881. 
"Progr.,  Hamburg,  1881. 
"Mathesis,  1,  1881,  99-102.     Nouv.  Ann.  Math.,  (3),  1,  1882,  240;  2,  1883,  239,  240.     Also 

Ces^ro,"  113-123,  133. 


292  History  of  the  Theory  of  Numbers.  [Chap,  x 

7i=  00  of  r(r2)/(n  log  r?)  is  l;cf.  (7);  themean  of  2(d+p)-'  is  (1  +  1/2+... 
+  l/p)/P-  -'^s  generalizations  of  Berger's^^  results,  the  mean  of  H.d/'p^  is 
l/(p  — 1);  the  mean  of  the  sum  of  the  rth  powers  of  the  divisors  of  n  is 
^r  ^(/--f  1)  and  that  of  the  inverses  of  their  rth  powers  is  f(r+l),  where 

(12)  f(s)=ilM 

n=l 

J.  W.  L.  Glaisher^^  proved  the  last  formula  of  Catalan^^  and 

(r(n)-(r(n-4)-(r(n-8)+(T(n-20)+o-(n-28)-... 

=  Q(n-l)+3Q(n-3)-6Q(n-6)-10Q(n-10)+..., 

where  Q{n)  is  the  number  of  partitions  of  n  without  repetitions,  and  4,  8, 
20, . . .  are  the  quadruples  of  the  pentagonal  numbers.  He  gave  another 
formula  of  the  latter  tj-pe. 

R.  Lipschitz,^^  using  his  notations,^"  proved  that 

7'(n)-.r0+.r[^]-...=n+z[|], 
G(n)-SaGg]+2«6G[^]  -  . . .  =n+Sp[|], 

D(.)  -SD  [2]  +XD    [^]  -  . . .  =$(n)  +2*[|] , 

where  P  ranges  over  those  numbers  ^  n  which  are  composed  exclusivelj''  of 
primes  other  than  given  primes  a,h,.  .  .,  each  ^ n. 

Ch.  Hermite"^  proved  (11)  very  simply. 

R.  Lipschitz^^  considered  the  number  T^it)  of  those  divisors  of  t  which  are 
exact  sth  powers  of  integers  and  proved  that 

where  p'  is  the  largest  sth  power  ^rz,  and  v  =  [n/ii'].  The  last  expression, 
found  by  taking  /i  =  [n^^"^*^"  ],  gives  a  generahzation  of  (11). 

T.  J.  Stieltjes^^  proved  (7)  by  use  of  definite  integrals. 

E.  Cesaro^°  proved  (7)  arithmetically  and  (11). 

E.  Cesaro^^  proved  that,  if  d  ranges  over  the  divisors  of  n,  and  5  over 
those  of  X, 

(13)  2G(d)/Q)=2^(d)FQ),  F{x)^i:fi8),  G{x)^Xg{8). 
Taking  g(x)  =  l,f{x)  =x,  4>{x),  1/x,  we  get  the  first  two  formulas  of  Liouville^^ 

"Messenger  Math.,  12,  1882-3,  16&-170. 

"Comptes  Rendus  Paris,  96,  1883,  327-9. 

"Acta  Math.,  2,  1883,  299-300. 

"/&id.,  301-4. 

5»Cdmpte9  Rendus  Paris,  96,  1883,  764-6. 

^Hhid.,  1029. 

"Mdm.  Soc.  Sc.  Li^e,  (2),  10,  1883,  Mem.  6,  pp.  26-34. 


Chap.  X]  SuM   AND   NuMBER   OF   DiVISORS.  293 

and  the  fourth  of  Liouville.^^    Taking  g  =  x,  f=(t>,  we  get  the  third  for- 
mula of  Liouville.^^    For  g=l/x,f=(j),  we  get 


S#(d)(7  0) 


^    =Sdl 


For  g=(f)  or  x'',f=x%  we  get  the  first  two  of  Liouville's^^  summation  formulas. 
If  ir(x)  is  the  product  of  the  negatives  of  the  prime  factors  5^  1  of  a:, 

Sx(d)*(d)<70)i  =  T(»),  ST(d)<#,(d)J3  =  ^,2#(d). 

Further  specializations  of  (13)  and  of  the  generalization  (p.  47) 

2G(d)/(^)  =i:F(d)g(^fj,        F{x)^l:^|^{^)f(^^y        G(a;)  ^2,^(5)^  (^), 

led  Cesaro  (pp.  36-59)  to  various  formulas  of  Liouville^^"^^  and  many- 
similar  ones.     It  is  shown  (p.  64)  that 

n=l    fl'  n=l    "• 

for  f  and  F  as  in  (12),  (13).  For /(n)  =(^(n),  we  have  the  result  quoted 
under  Cesaro^^  in  Ch.  V.     For/(n)  =  1  and  n'',  m  —  k>l, 

^—^=r{m),  S-— -  =  ^(m)f(m-A;). 

n=l    '«'  n     '«' 

If  (n,  j)  is  the  g.  c.  d.  of  n,  j,  then  (pp.  77-86) 

.ST^-jr  =  2S(7(d)-l,  nT(n)=i:a{n,j),  <j{n)=^T{n,  3), 

S  (Tk{n,  j)=n(Tk-i(n),  ^j(T{n,  j)  =-^lnT{n)-\-a{n)}. 

y=i  ^ 

If  in  the  second  formula  of  Liouville^^  we  take  m  =  l,. .  .,n  and  add,  we  get 

s0(i)rr?i=s<T(i). 

Similarly  (pp.  97-112)  we  may  derive  a  relation  in  [x]  from  any  given  relation 
involving  all  the  divisors  of  x,  or  any  set  of  numbers  defined  by  x,  such  as 
the  numbers  a,  h,.  .  .  for  which  x  —  a^,  x  —  W,.  .  .  are  all  squares.  Formula 
(7)  is  proved  (pp.  124-8).  It  is  shown  (pp.  135-143)  that  the  mean  of  the 
sum  of  the  inverses  of  divisors  of  n  which  are  multiples  of  k  is  7rV(6A;^) ;  the 
excess  of  the  number  of  divisors  4)U+1  over  the  number  of  divisors  4^i+3 
is  in  mean  7r/4,  and  that  for  4/x+2  and  4ju  is  ^  log  2;  the  mean  of  the  sum  of 
the  inverses  of  the  odd  divisors  of  any  integer  is  ttYS  ;  the  mean  is  found  of 
various  functions  of  the  divisors.  The  mean  (p.  172)  of  the  number  of 
divisors  of  an  integer  which  are  mth  powers  is  f(^)j  and  hence  is  7rV6  if 


294  History  of  the  Theory  of  Numbers.  [Chap,  x 

m  =  2.     The  mean  (pp.  216-9)  of  the  number  of  divisors  of  the  form  aix+r 
of  n  is,  for  r>0, 


i+ijlog«/a+2C-/;ij^dx} 


(cf.  pp.  341-2  and,  for  a  =  4,  6,  pp.  136-8),  while  several  proofs  (also,  p.  134) 
are  given  of  the  known  result  that  the  number  of  divisors  of  n  which  are 
multiples  of  a  is  in  mean 

-(log7i/a+2C). 
a 

If  (pp.  291-2)  a  ranges  over  the  integers  for  which  [2n/d]  is  odd,  the 
number  (sum)  of  the  a's  is  the  excess  of  the  number  (sum)  of  the  divisors  of 
n  +  1,  n+2, .  .  . ,  2n  over  that  of  1, .  .  . ,  n;  the  means  are  n  log  4  and  7rW/6. 
If  (pp.  294-9)  k  ranges  over  the  integers  for  which  [n/k]  is  odd,  the  number 
of  the  A:'s  is  the  excess  of  the  number  of  odd  divisors  of  1, .  .  .,  n  over  the 
number  of  their  even  divisors,  and  the  sum  of  the  A;'s  is  the  sum  of  the  odd 
divisors  of  1, .  .  . ,  w;  also 


S*W=9^  9=[^]' 


Several  asymptotic  evaluations  by  Cesaro  are  erroneous.  For  instance, 
for  the  functions  \{n)  and  At(n),  defined  by  Liouville^^  and  Mertens,^^ 
Cesaro  (p.  307,  p.  157)  gave  as  the  mean  values  6/7r^  and  36/7r*,  whereas 
each  is  zero.^^ 

J.  W.  L.  Glaisher^^  considered  the  sum  A(n)  of  the  odd  divisors  of  n. 
If  n  =  2^m  {m  odd),  A(n)  =(j{m).  The  following  theorems  were  proved  by 
use  of  series  for  elliptic  functions : 

A(l)A(2n-l)+A(3)A(2n-3)+A(5)A(2n-5)+...+A(2n-l)A(l) 

equals  the  sum  of  the  cubes  of  those  divisors  of  n  whose  complementary 
divisors  are  odd.     The  sum  of  the  cubes  of  all  divisors  of  2n+l  is 

A(2n+l)  +  12{A(l)A(2n)+A(2)A(2n-l)+.  .  . +A(2n)A(l)). 

If  A,  £,  C  are  the  sums  of  the  cubes  of  those  divisors  of  2n  which  are  respec- 
tively even,  odd,  with  odd  complementary  divisor, 

2A(2n)+24JA(2)A(2n-2)+A(4)A(2n-4)+.  .  .+A(2n-2)A(2)) 

=  i(2A-2J5-C)=i(3-23^-10)5 
o  7 

if  2n  =  2'"m  (ttz  odd).     Halphen's  formula^*'  is  stated  on  p.  220.     Next, 

n(r(2n+l)  +  (n-5)(r(2n-l)  +  (n-15)(T(2n-5) 

+  (n-30)(7(2n-ll)+.  .  .  =0, 

"H.  V.  Mangoldt,  Sitzungsber.  Ak.  Wiss.  Berlin,  1897,  849,  852;  E.  Landau,  Sitzungsber,  Ak. 

Wiss.  Wien,  112,  II  a,  1903,  537. 
«Quar.  Jour.  Math.,  19,  1883,  216-223. 


Chap.  X]  SuM  AND   NuMBER  OF   DiVISORS.  295 

in  which  the  differences  between  the  arguments  of  a  in  the  successive  terms 
are  2,  4,  6,  8, ... ,  and  those  between  the  coefficients  are  5,  10,  15, ... ,  while 
o-(O)  =0.     Finally,  there  is  a  similar  recursion  formula  for  A(n). 

Glaisher^^  proved  his^^  recursion  formula  for  Q{n),  gave  a  more  compli- 
cated one  and  the  following  for  (j{n) : 

o-(n)-2{o-(n-l)+o-(n-2))  +3{(7(n-3)+(r(n-4)+(7(n-5)!  -  . . . 
+  (-irV{...+(r(l))  =  (-l)V-s)/6, 

where  s  =  r  unless  ra-(l)  is  the  last  term  of  a  group,  in  which  case,  s  =  r+l. 
He  proved  Jacobi's^^  statement  and  concluded  from  the  same  proof  that 
E{n)  =JlE{ni)  if  n=nn„  the  n's  being  relatively  prime.  It  is  evident  that 
E{p')=r-\-l  if  p  is  a  prime  4m+l,  while  £'(pO  =  l  or  0  if  p  is  a  prime 
4m+3,  according  as  r  is  even  or  odd.  Also  £'(2'')  =  1.  Hence  we  can  at 
once  evaluate  E{n).  He  gave  a  table  of  the  values  oi  E{n),n  =  \,.  .  .,  1000. 
By  use  of  elliptic  functions  he  found  the  recursion  formulae 

E{n)-2E{n-^)+2E{n-\io)-2E{n-m)+  .  .  .  =0  or  (-l)'^-^^/^^, 
for  n  odd,  according  as  n  is  not  or  is  a  square;  for  any  n. 
E{n)-E{n-l)-E{n-S)+E{n-6)-{-E{n-10)-  .  . . 

=  0  or  (-ir{(-l)('-i)/2^-l}/4,         ^^  Vs^^fl, 

according  as  n  is  not  or  is  a  triangular  number  1,  3,  6,  10, . . ..  He  gave 
recursion  formulae  for 

S{2n)  =E(2)+E(4)+  .  .  .  +E{2n), 
S{2n-l)=E{l)-\-E{S)+ .  .  .+Ei2n-1). 

The  functions  E,  S,  6,  a  are  expressed  as  determinants. 
J.  P.  Gram^^"  deduced  results  of  Berger^^  and  Cesaro.^'* 
Ch.  Hermite^^  expressed  (T(l)+<r (3)  +  .  •  -  +o-(2n-l),       (t(3) +o-(7)  +  . . . 

+o-(4n  — 1)  and  o-(1)+<j(5)+  .  . .  +(7(4n+l)  as  sums  of  functions 

E,{x)=^{[xf-^[x\]/2. 

Chr.  Zeller^^  gave  the  final  formula  of  Catalan.^^ 

J.  W.  L.  Glaisher®^  noted  that,  if  in  Halphen's^"  formula,  n  is  a  triangular 
number,  (T{n—n)  is  to  be  given  the  value  n/3;  if,  however,  we  suppress  the 
undefined  term  (7(0),  the  formula  is 

(T(n)-3(j(n-l)+5(7(n-3)-  .  .  .  =0  or  {-lY-\l''+2''+ .  .  .+r''), 

according  as  n  is  not  a  triangular  number  or  is  the  triangular  number 
r(r+ 1)/2.  He  reproduced  two  of  his^^'^"*'^^  own  recursion  formulas  for 
<T{n)  (with  yp  for  <j  in  two)  and  added 

o-(n)-{(7(n-2)+o-(n-3)+(r(n-4)j  +  !(7(n-7)-f(r(n-8)+(7(n-9) 

+o-(n-10)+(7(n-ll)[-{(T(n-15)+...)  +  ...=A-B, 

«^Proc.  London  Math.  Soc,  15,  1883-4,  104-122. 

"°Det  K.  Danske  Vidensk.  Selskabs  Skrifter,  (6),  2   1881-6  (1884),  215-220  296. 

"^Amer.  Jour.  Math.,  6,  1884,  173-4. 

6«Acta  Math.,  4,  1884,  415-6. 

6Troc.  Cambr.  Phil.  Soc,  5,  1884,  108-120. 


296  History  of  the  Theory  of  Numbers.  [Chap,  x 

where  A  and  B  denote  the  number  of  positive  and  negative  terms  respec- 
tively, not  counting  cr{0)  =n  as  a  term; 

n<T(n)+2{(n-2)(r(n-2)  +  (n-4)(r(n-4)l 

+3{(n-6)(r(n-6)  +  (n-8)(7(n-8)  +  (n-10)tr(n-10)}  +  ... 
=  a{n)  +  {V-+3')\a{n-2)+a{n-4:)] 

+  (l-+3-  +  5') {(r(n-6)+(r(n-8)+<r(n-10)l  +  ...  (n  odd). 

He  reproduced  his^  formulas  for  d{n)  and  E{n).  He  announced  {ibid.,  p.  86) 
the  completion  of  tables  of  the  values  of  ^(n),  T{n),  (T{n)  up  to  n  =  3000,  and 
inverse  tables. 

Mobius^^  obtained  certain  results  on  the  reversion  of  series  which  were 
combined  by  J.  W.  L.  Glaisher^^  into  the  general  theorem:  Let  a,h,  ... 
be  distinct  primes;  in  terms  of  the  undefined  quantities  e^,  %,...,  let  e„ 
=  e^V/ ...  if  n  =  a'^lP .  .  . ,  and  let  ei  =  1.     Then,  if 

F(a:)=SeJ(a:"), 
where  n  ranges  over  all  products  of  powers  of  a,  6, . .  . ,  we  have 

/(x)=S(-l)^6^(a;0, 

where  v  ranges  over  the  numbers  wdthout  square  factors  and  divisible  by 
no  prime  other  than  a,  6, ... ,  while  r  is  the  number  of  the  prime  factors  of  v. 
Taking 

Glaisher  obtained  the  formula  of  H.  J.  S.  Smith^^  and 

aM  -2aV,(^)  +2a'fcV.(^^^  -  . . .  =  1. 

Using  the  same/,  but  taking  €2  =  0,  Cp  =  p\  when  p  is  an  odd  prime,  he  proved 
that,  if  Ar{n)  is  the  sum  of  the  rth  powers  of  the  odd  divisors  of  n, 


A,(„)-.A,©-f.A,(|) 


=  0  or  rf, 


according  as  n  is  even  or  odd.     In  the  latter  case,  it  reduces  to  Smith's. 

If  A'r(^)  is  the  sum  of  the  rth  powers  of  those  di\'isors  of  n  whose  com- 
plementary divisors  are  odd,  while  Er{n)  [or  E'r{n)]  is  the  excess  of  the  sum 
of  the  rth  powers  of  those  divisors  of  n  which  [whose  complementary  divisors] 
are  of  the  form  47?i  + 1  over  the  sum  of  the  rth  powers  of  those  divisors  which 
[whose  complementary  divisors]  are  of  the  form  4772+3, 

A',(n)-2a'A',(^)+2a'6'A',(^^)  -  , . ,  =;,  =  ijl-(-l)-), 
A',(„)-2A',Q+2A-,(^)-...=n-, 

"Jour,  fiir  Math.,  9,  1832,  105-123;  Werke,  4,  591. 
"London,  Ed.  Dublin  Phil.  Mag.,  (5),  18,  1884,  518-540. 


Chap.  X]  SuM  AND   NUMBER   OF   DiVISORS.  297 

EM -s^.(^)  +s^.(£)  - . . .  =  (-i)^"-^>/v., 

E'Xn)  -^a^E\(^^  +^a%^E',{^^  -...  =^{-\r-'^'\ 
E\{n)-U-ir-'"'E\{^  +S(-l)(^«-i)/2^;(^JL^  -  . .  .  =n^ 

where  A,B,.  .  .  are  the  odd  prime  factors  of  n.  Note  that  ^  =  0  or  1  according 
as  n  is  even  or  odd.  By  means  of  these  equations,  each  of  the  five  functions 
(Tr{i^),-  ■  -J  E'r{n)  is  expressed  in  two  or  more  ways  as  a  determinant  of 
order  n. 

Ch.  Hermite^°  quoted  five  formulas  obtained  by  L.  Kronecker'^^  from  the 
expansions  of  elHptic  functions  and  involving  as  coefficients  the  functions 
$(n)=o-(n),  the  sum  Z(n)_of  the  odd  divisors  of  n,  the  excess  ^(n)  of  the 
sum  of  the  divisors  >\/n  of  n  over  the  sum  of  those  <\/n,  the  excess 
$'(n)  of  the  sum  of  the  divisors  of  the  form  8k=^l  of  n  over  the  sum  of  the 
divisors  of  the  form  8k^S,  and  the  excess  ^'(n)  of  the  sum  of  the  divisors 
8A;±  1  exceeding  Vn  and  the  divisors  8A;±  3  less  than  y/n  over  the  sum  of  the 
divisors  Sk=i=l  less  than  \/n  and  the  divisors  8A;±  3  exceeding  \/n.  Hermite 
found  the  expansions  into  series  of  the  right-hand  members  of  the  five 
formulas,  employing  the  notations 

Ei{x)  =  [a:+i]  -  [x],  E^ix)  =  [x\[x+\]/2, 

a  =  l,  3,  5,...;  6  =  2,4,6,...;  c  =  l,  2,  3,..., 

and  A  for  a  number  of  type  a,  etc.     He  obtained 

Z(l)+X(3)+. .  .+X(A)=SE2(^), 
(r(l)  +(7(2)  +  .  . .  +  (7  (C)  =SE2(C/c), 

^(l)+^(2)+.  .  .+^(0=2^2 (^'), 
X(2)+Z(4)+  . . .  +Z(B)  =is|a[^]  +&^i[|]| 
«l>'(l)+*'(3)+. .  .+$'(A)=S(-l)^"^-^)/«a[^], 
^'(l)+^'(3)+  . . .  +^'{A)  =S(-l)('^^+^>/«a|  l^^+^^-^'j 

"BuU.  Ac.  Sc.  St.  Petersbourg,  29,  1884,  340-3;  Acta  Math.,  5,  1884-5,  315-9. 
"Jour,  fiir  Math.,  57,  1860,  bottom  p.  252  and  top  p.  253. 


298  History  of  the  Theory  of  Numbers.  [Chap,  x 

The  first  three  had  been  found  and  proved  purely  arithmetically  by  Lipschitz 
and  communicated  to  Hermite. 

Hermite  proved  (11)  by  use  of  series.     Also, 


i  F{a)=i  r^l/(a),         F(n)^2/(d), 

a  =  l  a  =  l  LuJ 


where  d  ranges  over  the  di\'isors  of  n.  When  f{d)  =  I,  F{n)  becomes  T{n) 
and  the  formula  becomes  the  first  one  by  Dirichlet.^^ 

L.  Gegenbauer"'-  considered  the  sum  p^. ,  (n)  of  the  A:th  powers  of  those 
divisors  d,  of  n  whose  complementary  divisors  are  exact  ^th  powers,  as  well 
as  Jordan's  function  Jk{n)  [see  Ch.  V].  By  means  of  the  f -function,  (12),  he 
proved  that 

2  (Tk{m)po,  2(n)  =2po.  2t{d)pk,  t K j ' 

where  d  ranges  over  the  divisors  of  r,  and  7n,  n  over  all  pairs  of  integers  for 
which  mv}  =  r', 

2  Jtk{n)p,,  t{m)  =  rV^jt.  <(r) ,  2 <T,_k{m)T{n)m'' ='Epk,  t{d)p,,  t  \j) ' 

the  latter  for  t  =  l  being  Liouville's^®  seventh  formula  ioT  v  =  0; 

2dV,(^)  =2dV..(0,  2/.(d)dV.(0  =P.+.<(r), 

the  latter  f or  ^  =  t-  =  1 ,  A-  =  0,  being  the  second  formula  of  Liouville"^,  while  for 
<  =  1  it  is  the  final  formula  by  Cesaro^^°  of  Ch.  V; 

2X(d)dV.,2.(0  =2X(d)p,.,((i)p*.,^0  =0  or  P2k,t{\^), 
according  as  r  is  not  or  is  a  square; 

2X(n)p,.  M  =p,,2t{r),  2X(d)T(d2)  =Hr)T{r), 

^r\d)J,Q  =r^^,  2dV(d2)(7,(0  =2dV(d), 

2  \l\r{x')  =2  At),  2  r^1x(x)(7,(x)  =2  p^r). 

2=iLa;J  r=l  i=lL3;J  r=l 

By  changing  the  sign  of  the  first  subscript  of  p,  we  obtain  formulas  for  the 
sum  Pk,i{n)=n''p_i,j{7i)  of  the  A:th  powers  of  those  divisors  of  7i  which  are 
tth  powers.  By  taking  the  second  subscript  of  p  to  be  unity,  we  get  formulas 
for  (Tkin).  There  are  given  many  formulas  invohnng  also  the  number 
fain)  of  solutions  of  nin2.  .  .71^  =  71,  and  the  number  co(«)  of  ways  n  can  be 
expressed  as  a  product  of  two  relatively  prime  factors.  Two  special  cases 
[(107), (128)]  of  these  are  the  first  formula  of  Liou\ille-^  and  the  ninth 
summation  formula  of  Liouville,^^  a  fact  not  observed  by  Gegenbauer.  He 
proved  that,  if  p^n, 

2   B{x)  =  -    2    Cix)-\-Bn-Ap, 

i=p+l  x  =  A  +  l 

"Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  89,  II,  1884,  47-73,  76-79. 


Chap.  X]                       SuM  AND  Number  of  Divisors.  299 

where  

and  B  =  B{n),  A=B{p+l);  also  that 

2   D{x)=     S    F{x)+Dn-Ep, 

z=p+l  x=D+l 

where  

and  D  =  D(n),  E  =  D(p-{-l).  It  is  stated  that  special  cases  of  these  two 
formulas  (here  reported  with  greater  compactness)  were  given  by  Dirichlet, 
Zeller,  Berger  and  Cesaro.  In  the  second,  take  t  =  1,  p  =  0,  and  choose  the 
integers  a,  j8,  b,  n  so  that 

hn''^-^>a>h{n-iy-\-^, 

whence  D  =  0.  If  Xr  is  the  number  of  divisors  of  r  which  are  of  the  form 
bx'+jS,  we  get 

Change  n  to  n  +  1  and  set  i3  =  0,  6  =  o-  =  l,  whence  a  =  n  [also  set  p  =  [Vn]]; 
we  get  Meissel's^^  formula  (11).  Other  speciaHzations  give  the  last  one 
of  the  formulas  by  Lipschitz,^^  and 

where  v  =  [-\/n\,  k{r)  is  the  number  of  odd  divisors  of  r,  while  ^  =  0  or  1 
according  as  [n/v  —  ^]>v  —  \  or  =v  —  \. 

L.  Gegenbauer'^^  proved  by  use  of  ^-functions  many  formulas  involving 
his'^^  functions  p,  /  and  divisors  d<.  Among  the  simplest  formulas,  special 
cases  of  the  more  general  ones,  are 

2(r,(d)d^=S(r,+x(0rf'=2(r^(0d^+\  Xn\d,)  =SX(A), 

mh)tx\d,)  =Xfx\h),  i:r{h')}x\d,)  =Zd{h),  2m'(^)^(0  ='r(r'), 

Xrid^Uh)  =e{r),  Xi/{d)J,(^^  ^^dMh), 

summed  for  d,  c?2,  d^,  where  h  =  Vr/dg.  Other  special  cases  are  the  fourth 
and  sixth  formulas  of  Liouville,^^  the  first,  third  and  last  of  Liouville.^^ 
Beginning  with  p.  414,  the  formulas  involve  also 

oi,{n)^n'Ti  {l  +  \/vt),  n=ilp:\ 

1=1  1=1 

"Sitzungsber.  Ak.  Wien  (Math.),  90,  II,  1884,  395-459.     The  functions  used  are  not  defined 
in  the  paper.     For  his  \pf,,  ^,  u,  we  write  0-^,  r,  e,  where  e  is  the  notation  of  Liouville." 


n" 


300  History  of  the  Theory  of  Numbers.  [Chap,  x 

Beginning  with  p.  425  and  p.  430  there  enter  the  two  functions 

■.?,(i)-{© -'■'-}• 

in  which  (A/p)  is  Legendre's  symbol,  with  the  value  1  or  —1. 

J.  W.  L.  Glaisher^'*  investigated  the  excess  tr{n)  of  the  sum  of  the  rth 
powers  of  the  odd  di\'isors  of  n  over  the  sum  of  the  rth  powers  of  the  even 
divisors,  the  sum  A'r(n)  of  the  rth  powers  of  those  divisors  of  n  whose 
complementary  divisors  are  odd,  wrote  f  for  f  i,  and  A'  for  A'l,  and  proved 

A'3(n)=nA'(n)+4A'(l)A'(n-l)+4A'(2)A'(n-2)+.  .  .+4A'(n-l)A'(l), 
r3(n)  =  (2n-l)r(n)-4r(l)r(n-l)-4r(2)r(n-2)-...-4r(n-l)f(l), 
nA'(n)=A'(l)A'(2n-l)-A'(2)A'(2n-2)  +  . .  .+A'(2n-1)A'(1), 

(-l)''-H(n)=A'(n)+8r(l)A'(n-2)+8f(2)A'(n-4)  +  . . ., 
A'3(n)=7zA'(n)+A'(2)A'(2n-2)+A'(4)A'(2n-4)  +  .  .  .+A'(2n-2)A'(2), 
-f3(7i)=3A(n)+4{A(l)A(n-l)+9A(2)A(n-2)+A(3)A(n-3) 

+9A(4)A(n-4)+ . . .  +A(n-1)A(1))  {n  even), 

2^-W2.+i(n)_[l,2r-l]      [3,  2r-3]  [2r-l,  1] 

(2r)!  l!(2r-l)!"^3!(2r-3)!'^""'^(2r-l)!l!' 

where 

b,?]=(7p(lK(2n-l)+(rp(3K(2n-3)  +  .  .  .+(Tp(2n-lK(l). 

For  n  odd,  f  (n)  =A'(n)  =a{n)  and  the  fourth  formula  gives 

(/i-lM7i)=8{(7(lMn-2)H-r(2M7i-4)+(7(3Mn-6)+r(4Mn-8)  +  .  . .). 

Glaisher'''^  proved  that 

5o'3(n)  —  6w(7(n)  -\-(j{n) 

=  12{(7(l)(7(7i-l)+o-(2)(r(n-2)+...+(7(n-l)(T(l)), 
(r(l)(r(2n-l)+(r(3)(r(2n-3)  +  .  .  .+or(2n-l)(T(l) 

=A'3(n)=|{cr3(2ii)-(T3(n)}. 

The  latter  includes  the  first  theorem  in  his^^  earUer  paper. 
Glaisher'^^  proved  for  Jacobi's^^  E{n)  that 

cr(2m  +  l)=E(l)E(4m  +  l)+£;(5)^(4w-3)+E(9)E(4w-7)  +  . . . 
+£(4m  +  l)E(l), 
E(0-2^(^-4)+2E(<-16)-2E(«-36)  +  . . .  =0  («  =  8n+5), 

(7(y)-2(r(y-4)+2(7(y-16)-2(7(y-36)  +  .  .  .  =0  (y  =  8n+7), 

<t(w) +(t(?/- 8)  H-(r(i/- 24) +(r(w- 48) +  .  .  .  =4{(r(w)+2(T(m-4) 
+2(r(rn-16)+2(7(m-36)+.  .  .)  (m  =  2n  +  l,   w  =  8n+3), 

and  three  formulas  analogous  to  the  last  (pp.  125,   129).     He  repeated 
(p.  158)  his^'*  expressions  for  A'3(n). 

^^Messenger  Math.,  14,  1884r-5,  102-8. 

"•Hbid.,  156-163. 

'•Quar.  Jour.  Math.,  20,  1885,  109,  116,  121,  118. 


r,,.,(n)=SM.(^|);'' 


Chap.  X]  SuM  AND   NuMBER  OF  DiVISORS.  301 

L.  Gegenbauer^^  considered  the  number  Ti(k)  of  the  divisors  ^[\/n]  of  ^ 
and  the  number  T2ik)  of  the  remaining  divisors  and  proved  that 

STi(/c)=5(log,n+2C)+0(V^), 

^r^ik)  =|(log,n+2C-2)+0(V^), 

0(s)  being  ^°  of  the  order  of  magnitude  of  s.  He  proved  (p.  55)  that  the  mean 
of  the  sum  of  the  reciprocals  of  the  square  divisors  of  any  integer  is  7rV90; 
that  (p.  64)  of  the  reciprocals  of  the  odd  divisors  is  ttVS;  the  mean  (p.  65) 
of  the  cubes  of  the  reciprocals  of  the  odd  divisors  of  any  integer  is  7r^/96, 
that  of  their  fifth  powers  is  7r^/960.  The  mean  (p.  68)  of  Jacobi's^^  E{n)  is 
7r/4. 

G.  L.  Dirichlet^^  noted  that  in  (7),  p.  282  above,  we  may  take  e  to  be  of 
lower  [unstated]  order  of  magnitude  than  his  former  -\/n- 

L.  Gegenbauer^^  considered  the  sum  r^  k,s  (n)  of  the  kth  powers  of  those 
divisors  of  n  which  are  rth  powers  and  'are  divisible  by  no  (sr)  th  power 
except  1 ;  also  the  number  Qa{b)  of  integers  ^  b  which  are  divisible  by  no  ath 
power  except  1.  It  follows  at  once  that,  if  /Xg(^)  =0  if  m  is  divisible  by  an 
sth  power  >1,  but  =1  otherwise, 

where  the  summation  extends  over  all  the  divisors  dr  of  n  whose  com- 
plementary divisors  are  rth  powers,  and  that 

(14)  ST..,,(a;)=  S    -;  hr^V.(^),  v  =  Wn]. 

From.the  known  formula  Qr{n)  =S[n/a;'^]/z(x),  x  =  1, . . .,  j^,  is  deduced 

the  right  member  reducing  to  n  for  A;  =  0  and  thus  giving  a  result  due  to 
Bougaief.    From  this  special  result  and  (14)  is  derived 

From  these  results  he  derived  various  expressions  for  the  mean  value  of 
Tr,-k,s{^)  and  of  the  sum  t^.aj.X^)  of  the  A;th  powers  of  those  divisors  of  n 
which  are  rth  powers  and  are  divisible  by  at  least  one  (sr)  th  power  other 
than  1.  He  obtained  theorems  of  the  type:  The  mean  value  of  the  number 
of  square  divisors  not  divisible  by  a  biquadrate  is  15/x^;  the  mean  value  of 
the  excess  of  the  number  of  divisors  of  one  of  the  forms  4rjLi+y(j  =  l,  3, . . ., 
2r  — 1)  over  the  number  of  the  remaining  odd  divisors  is 

1  i  cot(2LdV. 
4ri^i  4r 

"Denkschr.  Akad.  Wien  (Math.),  49,  I,  1885,  24. 

78G6ttingen  Nachrichten,  1885,  379;  Werke,  2,  407;  letter  to  Kronecker,  July  23,  1858. 

"Sitzungsberichte  Ak.  Wiss.  Wien  (Math.),  91,  II,  1885,  600-^21. 


302  History  of  the  Theory  of  Numbers.  [Chap,  x 

L.  Gegenbauer^"  considered  the  number  Ao{a)  of  those  di\'isors  of  a 
which  are  congruent  modulo  k  and  have  a  complementary  divisor  =1 
(mod  k).     He  proved  that,  if  p<k, 

If  we  replace  (t  by  <t  —  1  and  subtract,  we  obtain  expressions  for  Ao{k(T—p). 
The  above  formulas  give,  for  k  =  2,  p  =  l, 

and  formulas  of  Bouniakowsky.^^    The  same  developments  show  that  an 
odd  number  a  is  a  prime  if 


L2(2a:+1)  ^2j      L2(2x+l)^2j 


for  x^[(a  —  Z)/2];  likewise  for  a  =  6fc=*=l  if  the  same  equality  holds  when 
x^[(a  —  5)/Q],  with  similar  tests  for  a  =  3n  — 1,  or  4n  — 1. 

C.  Runge^^  proved  that  T{n)/n*  has  the  limit  zero  as  n  increases  indefi- 
nitely, for  every  e>0. 

E.  Catalan^-  noted  that,  if  x^p  is  the  number  of  ways  of  decomposing  a 
product  of  n  distinct  primes  into  p  factors  >1,  order  being  immaterial, 

x„p  =  px„_ip+x„_i,_i  =  jp'-^-(PTi)(p-ir-^+(^-^)(p-2r-^-...±i} 

-^{(p-l!). 
E.  Cesaro^  considered  the  number  F„  (x)  of  integers  ^x  which  are  not 
divisible  by  mth  powers,  and  the  number  T^  (x)  of  those  di\'isors  of  x  which 
are  mth.  powers,  evaluated  sums  involving  these  and  other  functions,  and 
determined  mean  values  and  probabilities  relating  to  the  greatest  square 
divisor  of  an  arbitrary  integer. 

R.  Lipschitz^  considered  the  sum  k{m)  of  the  odd  di\'isors  of  m  increased 
by  half  the  sum  of  the  even  di\'isors,  and  the  function  l(m)  obtained  by 
interchanging  the  words  "even,"  "odd."     He  proved  that 

k{m)-2k{m-l)+2k{m-9)-  .  .  .  =(-1)"-'^  or  0, 
according  as  m  is  a  square  or  is  not; 

l{m)+l{m-l)+l{77i-S)+l{m-6)+.  .  .  = -m  or  0, 
according  as  m  is  a  triangular  number  or  is  not ; 

XW=A'(1)+A'(2)+...+A-W  =  H+[|]+3[|]+2[^]  +  ...+m[^], 
L(m)  =  /(l)+Z(2)+  .  .  .  +l{m)  =  -[m]-^2[f\  -^[f\+^[f\  -■■■> 

"Sitzunpsberichte  Ak.  Wien.  (Math.),  91,  II,  1885,  1194-1201.       "Acta  Math.,  7,  1885,  181-3. 
"M^m.  80C.  roy.  sc.  Lifege,  (2),  12,  1885,  18-20;  Melanges  Math.,  1868,  18. 
"Annali  di  Mat.,  (2),  13,  1885,  251-268.     Reprint  "Excursions  arith.  k  Tinfini,"  17-34. 
"Comptes  Rendus  Paris,  100,  1885,  845.     Cf.  Glaisher"«,  also  Fergola"  of  Ch.  XI,  Vol.  II. 


Chap.  X]  SuM  AND   NuMBER  OF  DiVISORS.  303 

where  fx  =  m  or  m/2  according  as  m  is  odd  or  even.     Cf .  Hacks.^^ 

M.  A.  Stern^^  noted  that  Zeller's^^  formula  follows  from  B=pA,  where 

1  00  n  00 


=  A=  Si/'(n)a;",      -^=S=  S  o-(n)a:'*-\        p  =  l+2a:-5x*-7x^+ 


pix)  n=o  V{x) 

where  p(rc)  is  defined  by  (1),  yp{n)  is  the  number  of  partitions  of  n,  and 
the  second  equation  follows  from  the  equality  of  (3)  and  (4)  after  remov- 
ing the  factor  x.  Next,  if  N{n)  denotes  the  number  of  combinations  of 
1,  2, .  . .,  n  without  repetitions  producing  the  sum  n, 


X  N(n)x''=  {l+x){l+x^) . . .  = 


2,  (l-x')il-x') 


rZi  ^  '        '    '    ''    '^■■-      {l-x){l-x')...' 
then  by  the  second  equation  above, 

B{\-x^-x'^x^''+x^^-  . .  .)=pSA^(n)x~, 

(r(n)-o-(n-2)-o'(7i-4)+(r(n-10)+(7(n-14)- . . . 
=i\r(n-l)+2i\r(n-2)-5iV(n-5)-7iV(n-7)+..., 

where  (T{n  —  n)  =0,  N{n—n)  =  1. 

S.  Roberts^^  noted  that  Euler's^  formula  (2)  is  identical  with  Newton's 
relation  S^n  =  S-n+i+S^n+2—  ■  ■  ■  for  obtaining  the  sum  aS_„  of  the  (— n)th 
powers  of  the  roots  of  s  =  0,  where  s  and  p  are  defined  by  (2).  In  p,  the  sum 
of  the  ( —  n)th  powers  of  the  roots  of  1  —a:^  =  0  is  A;  or  0  according  as  k  is  or  is 
not  a  divisor  of  n.  Hence  the  like  sum  for  p  is  (r{n).  [Cf.  Waring^.]  The 
process  can  be  applied  to  products  of  factors  1  —f(k)x^.  His  further  results 
may  be  given  the  following  simpler  form.  Let  0„  be  the  sum  of  the  even 
divisors  of  n,  and  xpn  the  sum  of  the  odd  divisors,  and  set  s„=0„+2i^„  if  n 
is  even,  s„=  —  2i^„  if  n  is  odd.     By  elliptic  function  expansions, 

S2n  +  8{s2n_itAi+3S2H-2'/'2+S2n-3'A3  +  3S2n-4^4+  •  •  •  -hSiXf/ 2n-l]  +12ni/^2n  =  0, 
S2n+l+8{s2„lAi+3S2„_l^2+  ■  •  ■  +3SilA2n)  +(4n  +  2)l/'2n+l=0» 

the  coefficients  being  1  and  3  alternately.     He  indicated  a  process  for  finding 
a  recursion  formula  involving  the  sums  of  the  cubes  of  the  even  divisors  and 
the  sums  of  the  cubes  of  the  odd  divisors,  but  did  not  give  the  formula. 
N.  V.  Bougaief^®"  obtained,  as  special  cases  of  a  summation  formula, 
^{Sx+5-5i2u-iy}(Ti2x  +  l-u''  +  u)  =  0,   S{n -3(7(t/)}P{n -o-(w)}  =  0, 

where  P{n)  is  the  number  of  solutions  u,  v  of  a{u)  +(t(v)  =n. 

L.  Gegenbauer^®''  proved  that  the  number  of  odd  divisors  of  1,  2, . . .,  n 
equals  the  sum  of  the  greatest  integers  in  (n+l)/2,  (n+2)/4,  (nH-3)/6, .  .  ., 
(2n)/(2n).  The  number  of  divisors  of  the  form  Bx—'yoil,...,nis  ex- 
pressed as  a  sum  of  greatest  integers;  etc. 

J.  W.  L.  Glaisher^^  considered  the  sum  A^n)  of  the  sth  powers  of  the  odd 
divisors  of  n,  the  Hke  sum  Dsin)  for  the  even  divisors,  the  sum  D',{n)  of  the 

s^Acta  Mathematica,  6,  1885,  327-8. 
8«Quar.  Jour.  Math.,  20,  1885,  370-8. 
8««Comptes  Rendus  Paris,  100,  1885,  1125,  1160. 
sebDenkschr.  Akad.  Wiss  Wien  (Math.),  49,  II,  1885,  111. 
8'Messenger  Math.,  15,  1885-6,  1-20. 


304  History  of  the  Theory  of  Numbers.  [Chap,  x 

sth  powers  of  the  divisors  of  n  whose  complementary  divisors  are  even, 
the  excess  f ',(«)  of  the  sum  of  the  sth  powers  of  the  divisors  whose  com- 
plementary'' di\'isors  are  odd  over  that  when  they  are  even,  and  the  similar 
functions'^  A'„  f „  a,.  The  seven  functions  can  be  expressed  in  terms  of  any 
two: 

where  the  arguments  are  all  n.  Since  D\{2k)  =(T,(k),  we  may  express  all  the 
functions  in  terms  of  a^n)  and  (TXn/2),  provided  the  latter  be  defined  to  be 
zero  when  n  is  odd.        Employ  the  abbreviation  'EfF='S,Ff  for 

/(l)F(n-l)+/(2)F(n-2)+/(3)F(n-3)  +  . .  .+/(n-l)F(l). 

This  sum  is  evaluated  when/  and  F  are  any  two  of  the  above  seven  functions 
w^th  s  =  1  (the  subscript  1  is  dropped) .     If 

f{n)=aa(n)+^D'in),  F(n)  =aV(n)+/3'Z)'(n), 

then 

2/F  =  aa'2(T(r+(ai8'+a'/3)S(Ti)'H-/3iS'2i)'Z)'. 

By  using  the  first  formula  in  each  of  two  earUer  papers,'^'  '^  we  get 

12'Z(ja  =  5(T2{n)  —6na{n)  -\-a{n) , 

122D'D'  =  5Ds'(n)-dnD'{n)+D'in), 

242o-D'  =  2(r3(n)  +  (l-3n)(7(7i)  +  (l-6n)D'(n)+8Z)3'(n). 

Hence  all  21  functions  can  now  be  expressed  at  once  linearly  in  terms  of 
0-3,  Ds',  (T  and  D'.  The  resulting  expressions  are  tabulated;  they  give  the 
coefficients  in  the  products  of  any  two  of  the  series  2f/(n)x",  where/ is  any 
one  of  our  seven  functions  ■vsithout  subscript. 

Glaisher^^  gave  the  values  of  l^a^ai  for  2  =  3,  5,  9  and  ^<r^(TT,  where  the 
notation  is  that  of  the  preceding  paper.     Also,  if  p  =  7i—r, 

12  S  rpa{f)a{fi)  ^n^a^in)  -nV(n),  S  rj{r)F{p)  =^/F. 

r=l  r=l  A 

L.  Gegenbauer^^  gave  purely  arithmetical  proofs  of  generalizations  of 
theorems  obtained  by  Hermite'"  by  use  of  elliptic  function  expansions.     Let 

5,(r)  =2/,  (7=J^5,([^])  -v&,{y),  v^\yM' 

Then  (p.  1059), 

The  left  member  is  knowTi  to  equal  the  sum  of  the  ^th  powers  of  all  the 
divisors  of  1,  2, .  .  .,  n.  The  first  sum  on  the  right  is  the  sum  of  the  A-th 
powers  of  the  divisors  ^  y/n  of  1, .  .  . ,  n.     Hence  if  A^fx)  is  the  excess  of  the 

"Messenp;er  Math.,  15,  188.5-6,  p.  36. 

'•Sitzungsberichte  Ak.  Wien  (Math.),  92,  II,  1886,  1055-78. 


Chap.  X]  SuM  AND   NuMBER  OF   DiVISORS.  305 

sum  t/'fc'(a;)  of  the  kth  powers  of  the  divisors  >  \/x  of  x  over  the  sum  of  the 
A;th  powers  of  the  remaining  divisors,  it  follows  at  once  that 


Also 


n  V     ['fi~\ 

x=l  x=lL-CJ 

^J,\x)  =£'S^([3)  +^''+M  -  (^+i)5,(.), 


with  a  similar  formula  for  ^^^(a;),  where  "^k(^)  is  the  excess  of  xpki^)  over  the 
sum  of  the  A;th  powers  of  the  divisors  <  y/x  of  x.  For  k  —  l,  the  last  formula 
reduces  to  the  third  one  of  Hermite's. 

Let  Xk{^)  be  the  sum  of  the  kth  powers  of  the  odd  divisors  of  x;  Xk\^) 
that  for  the  odd  divisors  >  \/x;  Xk"{x)  the  excess  of  the  latter  sum  over  the 
sum  of  the  A;th  powers  of  the  odd  divisors  <  ^/x  of  x;  Xk"'{'^)  the  excess  of  the 
sum  of  the  kih.  powers  of  the  divisors  8s±l>\/x  of  x  over  the  sum  of  the 
kih.  powers  of  the  divisors  8s='=3<\/^  of  ^-  For  y  =  2x  and  y  =  2x  —  l,  the 
sum  from  x  =  l  to  a:  =  n  of  Xkiv),  Xk'iv),  X/iy)  and  Xk"{y)  are  expressed  as 
complicated  sums  involving  the  functions  Sk  and  [x\. 

E.  Pfeiffer^°  attempted  to  prove  a  formula  like  (7)  of  Dirichlet/^  where 
now  e  is  0{'n}^^^^)  for  every  k>0.  Here  Og{T)  means  a  function  whose 
quotient  hy  g{T)  remains  numerically  less  than  a  fixed  finite  value  for  all 
sufficiently  large  values  of  7".  E.  Landau^ ^  noted  that  the  final  step  in 
the  proof  fails  from  lack  of  uniform  convergence  and  reconstructed  the 
proof  to  secure  this  convergence. 

L.  Gegenbauer,^^  in  continuation  of  his^°  paper,  gave  similar  but  longer 
expressions  for 

S  r{y),  S  (Tkiy)  (2/  =  4a;+l,  6a;+l,  8a:+3,  8a:+5,  8a:+7) 

and  deduced  similar  tests  for  the  primality  of  y. 

Gegenbauer^^"  found  the  mean  of  the  number  of  divisors  \x-\-a  of  a 
number  of  s  digits  with  a  complementary  divisor  iiy-\-^;  also  for  divisors 
ax'^+hy'^. 

Gegenbauer^^''  evaluated  A(l)  +  . . .  +A(n)  where  A{x)  is  the  sum  of  the 
pth  powers  of  the  crth  roots  of  those  divisors  d  oi  x  which  are  exact  o-th 
powers  and  whose  complementary  divisors  exceed  kdJ/'^.  A  special  case 
gives  (11),  p.  284  above. 

Gegenbauer^^"  gave  a  formula  involving  the  sum  of  the  A;th  powers  of 
those  divisors  of  1, . . . ,  m  whose  complementary  divisors  are  divisible  by  no 
rth  power  >1. 

'"Ueber  die  Periodicitat  in  der  Teilbarkeit .  .  . ,  Jahresbericht  der  Pf eiffer'schen  Lehr-  und  Erzieh- 

ungs-Anstalt  zu  Jena,  1885-6,  1-21. 
"Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  121,  Ila,  1912,  2195-2332;  124,  Ila,  1915,  469-550. 

Landau.^^^ 
»276id.,  93,  II,  1886,  447-454. 

"-^Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  93,  1886,  II,  90-105. 
92676id.,  94,  1886,  II,  35-40. 
"^c/feid.,  757-762. 


306  History  of  the  Theory  of  Numbers.  [Chap,  x    1  C^' 

Ch.  Hermite^'  proved  that  if  F{N)  is  the  number  of  odd  divisors  of  N, 

n  =  l 

and  then  that 

F(l)+F(2)+  . .  .  +F{N)  =^N  log  iV+  (c-^N, 

$(l)+$(2)+  . . .  +$(iV)  =^N  log  N/k+  (c-^N, 

asATnptotically,  where  ^(N)  is  the  number  of  decompositions  of  N  into  two 
factors  d,  d',  such  that  d'>kd. 

E.  Catalan^^"  noted  that,  if  n  =  i+i'  =  2i"d, 

'La{i)a{i')=2(P,  i:{aii)a{2n-i)}  =Si:{<j{i)<x{n-i)]. 

E.  Ces^ro^  proved  Lambert's^  result  that  T{n)  is  the  coefficient  of  x"  in 
2xV(l  — a;*).    Let  T,{n)  be  the  number  of  sets  of  positive  integral  solutions  of 

and  s,{n)  the  sum  of  the  values  taken  by  ^,.    Then 

sM  =  TM  +  TXn-v)  +  TXn  -2v)+..., 
T{n)  =Si(n)  -Szin)  +83(71) 

Let  aa^)=S(-l)'*+^r,(x-n), 

summed  for  the  divisors  d  of  n.    Then 

Tin)=tM+t2{n)+  . .  .+TM-T2{n)  +  Ts{n)-  . . .. 

E.  Busche^^  employed  two  complementary  di\isors  5^  and  8  J  of  m, 
an  arbitrary  function/,  and  a  function  y=^(x)  increasing  with  x  whose 
inverse  function  is  x  =  ?/)/' (7/).     Then 

2  limx)],  X)  -/(O,  x) }  =2 {/(5'^,  5  J  -/(5'^- 1,  5 J ) , 

x=l 

where  in  the  second  member  the  summation  extends  over  all  divisors  of  all 
positive  integers,  and  $(w)^6;„^a.     In  particular, 

2  /(x)[tA(x)]  =2/(5J,  2  [rA(x)]  =  number  of  5„, 

1=1  2=1 

subject  to  the  same  inequalities.     In   the  last   equation   take  \l/(x)=x, 
a  =  [\/n]',  we  get  (11). 

J.  Hacks^®  proved  that,  if  7?i  is  odd, 

^W^T(l)+r(3)+T(5)  +  .  .  .  +r(7n)  =2[^], 

wjour.  fur  Math.,  99,  1886,  324-8. 

•"^Mdm.  .Soc.  R.  Sc.  Li^ge,  (2),  13,  1886,  318  (Melanges  Math.,  II). 

•*Jomal  de  sciencias  math,  e  astr.,  7,  1886,  3-6. 

«Jour.  fiir  Math.,  100,  1887,  459-464.     Cf.  Busche.»" 

"Acta  Math.,  9,  1887,  177-181.     Corrections,  Hacks, »^  p.  6,  footnote. 


Chap.  X]  SuM  AND  NUMBER  OF  DiVISORS.  307 

@(m)=or(l)+(7(3)+(r(5)  +  .  .  .+(7(m)=S^[^^], 

where  t  ranges  over  the  odd  integers  ^  m.  For  the  K  and  L  of  Lipschitz^ 
and  G{m)  =(r(l)+o-(2)  +  .  .  .  -\-<T(m),  it  is  shown  that 

LW^(?(m)^[Vm]  +  [^|],  !r(m)^[v^]   (mod  2). 

J.  Hacks^^  gave  a  geometrical  proof  of  (11)  and  of  Dirichlet's^^  expression 
for  T{n),  just  preceding  (7).  He  proved  that  the  smn  of  all  the  divisors, 
which  are  exact  ath  powers,  of  1,  2, . .  . ,  m  is 

m 

S{1^+2«+...+[a/^H. 

3  =  1 

He  gave  (pp.  13-15)  several  expressions  for  his^^  i^M,  &(m),  K{m). 
L.  Gegenbauer^^"  gave  simple  proofs  of  the  congruences  of  Hacks. ^^ 
M.  Lerch^^  considered  the  number  \p{a,  h)  of  divisors  >6  of  a  and  proved 

that 

[n/2]  n 

(15)  X  \}/{n—p,  p)  =71,  Si/'(n4-p,  p)=2n. 

p=0  p=0 

A.  Strnad^^  considered  the  same  formulas  (15). 

M.  Lerch^°°  considered  the  number  x(«j  b)  of  the  divisors  ^6  of  a  and 
proved  that 

[(m-l)/a] 

S      {\J/{m—aa,k+a)—xi'm—aa,a)] 

<7  =  0 

k 

+  S  {\J/{m+\a,  X-l)-x(m+Xa,  a))  =0. 
x=i 

This  reduces  to  his  (15)  for  a  =  l,  k  =  l  orm  +  1.  Let  (k,  n;  m)  denote  the 
g.  c.  d.  {k,  n)  of  k,  n  or  zero,  according  as  {k,  n)  is  or  is  not  a  divisor  of  m. 
Then 

a— 1  a 

(16)  S  {i/'(m4-an,  a)— ;/'(m+an,  a)]  =  S  (A;,  n;  m). 

In  case  m  and  n  are  relatively  prime,  the  right  member  equals  the  number 
0(a,  n)  of  integers^  a  which  are  prime  to  n.     Finally,  it  is  stated  that 

(17)  S  1/^(772  — an,  a)  =  S  x(^  —  ctn,  n),  c=    . 

a  =  0  a  =  0  L       n       J 

Gegenbauer,^^  Ch.  VIII,  proved  (16)  and  the  formula  preceding  it. 

"Acta  Math.,  10,  1887,  9-11. 

"oSitzungsber.  Ak.  Wiss.  Wien  (Math.),  95,  1887,  II,  297-8. 
"Prag  Sitzungsberichte  (Math.),  1887,  683-8. 
"Casopis  mat.  fys.,  18,  1888,  204. 
""Compt.  Rend.  Paris,  106, 1888, 186.     Bull,  des  sc.  math,  et  astr.,  (2),  12, 1, 1888, 100-108, 121-6. 


308  History  of  the  Theory  of  Numbers.  [Chap,  x 

C.  A.  Laisant^"^  considered  the  number  nk{N)  of  ways  N  can  be  expressed 
as  a  product  of  k  factors  (including  factors  unity),  counting  PQ .  .  .  and  QP .  . . 
as  distinct  decompositions.     Then 

n,{N)  =n,_,{N)u(l-\-^y  N=Upr. 

E.  Ces^ro^"^  proved  Gauss'  result  that  the  number  of  di\'isors,  not 
squares,  of  n  is  asymptotic  to  Gtt"^  logn.  Hence  T{n~)  is  asymptotic  to 
Stt"^  log^n.  The  number  of  decompositions  of  n  into  two  factors  whose 
g.  c.  d.  has  a  certain  property  is  asymptotic  to  the  product  of  log  n  by  the 
probability  that  the  g.  c.  d.  of  two  numbers  taken  at  random  has  the  same 
property. 

E.  Busche^''^  gave  a  geometric  proof  of  his^^  formula.  But  if  we  take 
$(x)  to  be  a  continuous  function  decreasing  as  x  increases,  with  $(0)>0, 
then  the  number  of  positive  divisors  of  y  which  are  ^4^iy)  is  S[$(a:)/x], 
summed  for  x  =  1,  2, .  . . ,  with  $(x)  ^  0.  This  result  is  extended  to  give  the 
number  of  non-associated  di\'isors  of  y+zi  whose  absolute  value  is  ^4>{y,  z). 

J.  W.  L.  Glaisher^^  considered  the  excess  H{n)  of  the  number  of  divisors 
=  1  (mod  3)  of  n  over  the  number  of  divisors  =2  (mod  3),  proved  that 
H{pq)  =H{p)H{q)  if  p,  q  are  relatively  prime,  and  discussed  the  relation  of 
H{n)  to  Jacobi's^i  E{n). 

Glaisher^^^  gave  recursion  formulae  for  H{n)  and  a  table  of  its  values  for 
n  =  l,...,  100. 

L.  Gegenbauer^°^  found  the  mean  value  of  the  number  of  divisors  of  an 
integer  which  are  relatively  prime  to  given  primes  pi, . .  . ,  Pa,  and  are 
also  (pr)  th  powers  and  have  a  complementary  divisor  which  is  di\'isible 
by  no  rth  powers.  Also  the  mean  of  the  sum  of  the  reciprocals  of  the  A;th 
powers  of  those  di\'isors  of  an  integer  which  are  prime  to  pi, .  .  . ,  p„  and  are 
rth  powers.     Also  many  similar  theorems. 

Gegenbauer^o^"  expressed  S^=S  i^(4.T+l)  and  SF(4a;+3)  in  terms  of 
Jacobi's  symbols  (A/?/)  and  greatest  integers  [ij]  when  F{x)  is  the  sum  of  the 
A;th  powers  of  those  divisors  ^  -x/x  of  x  which  are  prime  to  D,  or  are  divisible 
by  no  rih.  power  >  1,  etc.;  and  gave  asymptotic  evaluations  of  these  sums. 

J.  P.  Gram^°^  considered  the  number  D„(m)  of  di\dsors  ^m  of  n,  the 
number  iV"2, 3  ..(^)  of  integers  ^n  which  are  products  of  powers  of  the 
primes  2,  3, .  . ,  and  the  sum  -Lo. 3. . . (n)  of  the  values  of  \(k)  whose  arguments 
k  are  the  preceding  N  numbers,  where  X(2"3^ .  .  . )  =  ( ~  1)"+'^+    •. 

If  p  =  Pi°'P2°'.  .  .,  where  the  p,  are  distinct  primes, 

Dj,{n)  =  Nin)  -'LN(n/p,''^+')  +SiV(n/pi"'+ ^2'^'^')  -  •  •  •  • 

""Bull.  Soc.  Math.  France,  16,  1888,  150. 
i^Atti  R.  Accad.  Lincei,  Rendiconti,  4,  1888,  I,  452-7. 
>MJour.  fur  Math.,  104,  1889,  32-37. 

"xProc.  London  Math.  Soc,  21,  1889-90,  198-201,  209.  ^°Hbid.,  395-402.     See  Glaisher.'" 

'»«Denkschi-iften  Ak.  Wiss.  Wien  (Math.),  57,  1890,  497-530. 
'""'Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  99,  1890,  Ila,  390-9. 

'"Det  K.  Danske  Videnskab.  Selskaba  Skrifter  (natur.  og  math.),  (6),  7,  1890,  1-28,  with  r6- 
8um6  in  French,  29-34. 


i 


Chap.  X] 


Sum  and  Number  of  Divisors. 


309 


In  particular,  if  the  pi  include  all  the  primes  in  order,  we  may  replace 
N{x)  by  [x],  the  greatest  integer  ^x.  Since  there  are  as  many  divisors 
>a  of  n  as  there  are  divisors  <n/a, 


D»+i),(^^^=€+n(a,+l), 


where  €  =  1  or  0  according  as  n  is  or  is  not  a  divisor  of  p.  These  two  formulas 
serve  as  recursion  formulas  for  the  computation  of  N{n).  For  the  case  of 
two  primes  pi  =  2,  p2  =  3, 

The  functions  L  satisfy  similar  formulas  and  are  computed  similarly. 

J.   W.    L.   Glaisher^"^   stated  a  theorem,  which  reduces  for  m  =  l  to 
Halphen's,4° 

/S=o-^(n)-3o-^(n-l)+5o-^(n-3)-7(r„(n-6)+9(7^(n-10)-.  . . 
=  2s(^'^^{cr^_,(n-l)-(l^+2V^_,(n-3)+(lH2H3V^_,(n-6)-. . .  j 

provided  m  is  odd,  where  k  ranges  over  the  even  numbers  2,  4, .  . .,  ?n  — 1, 
while  6  =  0  or  6  =  1  according  as  n  is  not  or  is  of  the  form  g'(^  +  l)/2.  As  in 
Glaisher^^  for  m  =  l,  the  series  are  stopped  before  any  term  (Ti{n  —  n)  is 
reached;  but,  if  we  retain  such  terms,  we  must  set  6  =  0  for  every  n  and 
define  o-i(O)  by 


m+2 
m 


©''^^'^^l^-^'+i^^'K 


<r(0)  = 

©.;(o)= 


m+2 


B. 


where  Bi,  B2, .  .  .  are  the  Bernoullian  numbers. 

Glaisher^°^  stated  the  simpler  generalization  of  Halphen'*": 

'S+S 2Hk^(!k)  f^-'^^'')  -3^+V^_,(n-l)+5^+V_,(n-3)  -  . . . } 

where  the  summation  index  k  ranges  over  the  even  numbers  2,  4, ... ,  m  —  1, 
and  m  is  odd.  If  we  include  the  terms  <T2,_i(0)  =  (  —  !)' 5r/(4r)  in  the  left 
member,  the  right  member  is  to  be  replaced  by 


5{-iy 


2'"+2(m+2) 


"^Messenger  Math.,  20,  1890-1,  129-135. 
^o^Ibid.,  177-181. 


310  History  of  the  Theory  of  Numbers.  [Chap,  x 

Glaisher""  considered  the  set  Gn[\l/(d),  x(^>-  •• )  of  the  values  of  \l/(d), 
x(cO,  •  •  •  when  d  ranges  over  all  the  di\isors  of  n,  and  wrote  —G{\J/,  x,-) 
for  G{—\f/,  —  X,  •  •)•  By  use  of  the  ^-function  (12),  he  proved  (p.  377)  that 
the  numbers  given  by 

GM-Gn-i(d,  d=^l)+Gn-z{d,  d^l,  d=^2)-G,.M  d^l,  ^±2,  ^±3)  + . . . 

all  cancel  if  n  is  not  a  triangular  number,  but  reduce  to  one  1,  two  2's, 
three  3's, .  . .,  g  g's,  each  taken  with  the  sign  (  — )*~\  if  n  is  the  ^th  tri- 
angular number  ^(gr+l)/2.     For  example,  if  n  =  6,  whence  g  =  S, 

{1,  2,  3,  6)  -  {1,  5;  2,  6;  0,  4}  +  {l,  3;  2,  4;  0,  2;  3,  5;  -1,  1} 

=  {1,2,2,3,3,3). 

Let  \l/{d)  be  an  odd  function,  so  that  \f/{  —  d)=—\l/{d),  and  let  2r/(c?) 
denote  the  sum  of  the  values  of  f(d)  when  d  ranges  over  the  divisors  of  r. 
Then  the  above  theorem  implies  that 

2„iA(d)  -2„-i  iHd)  +4^{d=^  1) )  +2„_i_2  [rPid)  +rl^{d^  1)  -{-^p{d^  2) ) 

-2„_i_2-3  {rpid)  +yp{d^  1)  +iA(d±  2)  +^(d±  3)  1  + . . . 

=  5( - 1)''-^  {,^(1)  +2^(2)  +3V^(3)  +  . . .  +g^l^{g) } , 

where  5  =  0  or  1  according  as  n  is  not  or  is  of  the  form  ^(^+l)/2,  and  where 
\l/{d^i)  is  to  be  replaced  by  \}/{d+i)-{-yl/{d  —  i).  Taking  \l/{d)  =d"',  where  m 
is  odd,  we  obtain  Glaisher's^°^  recursion  formula  for  (Tm{n),  other  forms  of 
which  are  derived.     For  the  function^^  ^3,  we  derive 

Un)+Un-1)+Un-S)  + . .  .+Q{nn-l)-{l'-2'mn-3) 

+  (l2-22+3^)r(n-6)-...) 
=  (-l)''-^(l^-2H3'-  . . .  +(-l)'-y)  or  0, 

according  as  n  is  of  the  form  ^(^+l)/2  or  not. 

Next  he  proved  a  companion  theorem  to  the  first : 


2d-\-7   \, 
-[2d-7];+ 


^/    2d+l    \     ^     /    2d+3    V^     /    2d+5   \     r     f 

all  cancel  if  n  is  not  a  triangular  number,  but  reduce  to  1,  3,  5,. ..,  2^  —  1, 
each  taken  with  the  sign  (  — )",  together  with  (  — l)''+^(2^+l)  taken  g 
times,  if  n  is  the  gth  triangular  number  ^(^+l)/2.     For  example,  if  n  =  6, 

{-tt  J-;?}-{-t"M-3;-"}={-'-'-- ^- ^- ^}- 

Hence  if  x(c^)  be  any  even  function,  so  that  x(—d)  =  x{d), 
2„{x(2d+l)-x(2d-l)l-2„_i{x(2ci+3)-x(2d-3)l+S„_3-.... 

=  5(-l)''-M^x(2^  +  l)-x(l)-x(3)- .  .  . -x(2^-l)l. 
Taking  ^(A;)  =  fc'""*"\  where  k  and  m  are  odd,  we  get  Glaisher's^°^  formula. 

"oProc.  London  Math.  Soc,  22,  1890-1,  359-410.     Results  stated  in  London,  Edinb.,  Dublin 
Phil.  Mag.,  (5),  33,  1892,  54-61. 


I 


Chap.  X]  SUM  AND   NuMBER  OF   DiVISORS.  311 

He  proved  two  theorems  relating  to  the  divisors  of  1,  2, . . .,  w: 

+  (G^n-3  +  ^n-4  +  G^n-5)(   _y_3l  )  —  •  •  • 

all  cancel  with  the  exception  of  —2,  —  4, . . .,  —  (p— 2),  each  taken  twice, 
p  taken  p  —  1  times  and  —  0,  if  p  be  even;  but  with  the  exception  of  1,  3, ... , 
p  —  2,  each  taken  twice,  and  —p  taken  p  —  1  times,  if  p  be  odd,  where 
p(p+l)/2  is  the  triangular  number  next  >n; 

all  cancel  with  the  exception  of  k  taken  k  times,  for  A;  =  1,  3,  5, . . . ,  p  —  1,  if  p 
be  even;  and  of  —A;  taken  k  times,  for  A;  =  2,  4,  6, . . . ,  p  —  1,  if  p  be  odd;  here 
zeros  are  ignored. 

The  last  two  theorems  yield  (as  before)  corresponding  relations  for  any- 
even  function  x  aiid  any  odd  function  xp.  Applying  them  to  x{d+l) 
=  (d+l)"'  and  \l/{d)=d"*,  where  m  is  odd,  and  in  the  first  case  dividing 
by  2(m+l),  and  modifying  the  right  members,  we  get  for 

r=o-Jn)-2{(T^(n-l)+(r^(n-2)}+3{Mn-3) 

+(rm{n-4:)+(Tm{n-5)]-. .. 
the  respective  relations 

+3*+^  (next  three)  -  . . . } 

-^-^M2(m^"^^^  +3  V2J^P      -5V4/3"^      +...=^2      -p^, 
where  s  =  (m+l)/2  and  0-^(0)  terms  are  suppressed; 
!r=S2(^^{(r^_,(n-l)+(7^_fc(n-2)-2^  (next  three) +  ( 1^3') (next  four) 

_(2^+4*^) (next  five)  +  (lH3*+5'=) (next  six)  - (2H4H6') (next  seven)  + . . . } 
r     ^m+i_^^m+i_^^m+i_^       _{_  (^  _  l)-+i^  if  p  bg  eveu, 

i-|_ 2^+1  _ 4^+1  _gm+i_       _(p_i)m+i^  if  p  be  odd, 

where,  in  each,  A:  takes  the  values  2,  4, . . . ,  m  —  1.  These  sums  of  like  powers 
of  odd  or  even  numbers  are  expressed  by  the  same  function  of  Bernoullian 
numbers.  For  m  =  l,  the  first  formula  becomes  that  by  Glaisher,^^  repub- 
Ushed.^^  Three  further  (t„  formulas  are  given,  but  not  applied  to  o-„. 
J.  Hammond"^  wrote  (n;  m)  =  l  or  0  according  as  n/m  is  integral  or 
fractional,  also  t{x)  =a{x)=0  if  x  is  fractional,  and  stated  that 

00  00 

T(n/m)  =  S  (n;  jm),            a  (n/m)  =  2  j(n;  jm). 
y=i }=i 

"^Messenger  Math.,  20,  1890-1,  158-163. 


312  History  of  the  Theory  of  Numbers.  [Chap,  x 

From  the  sum  of  Euler's  (f}(d)  for  the  di\'isors  d  of  n,  he  obtained 
a{n)=iT(jj<t>iJ),  nr{n)=i<T(tjct>{j). 

E.  Lucas^^-  proved  the  last  formulas,  the  result  of  Cesaro/*  and  the 
related  one  o-(n)+s„  =  s„_i+2n  — 1. 

A.  Berger"^  considered  the  mean  of  the  number  of  decompositions  of 
1,2,. . .,  X  into  three  or  more  factors,  and  gave  long  expressions  for  \l/{l)-\- 
..  .+^(7J),  where  i/'(A-)  =2d'c?i*',  summed  for  the  solutions  of  ddi  =  k.  He 
gave  (pp.  116-125)  complicated  results  on  the  mean  value  of  (Tk{n). 

D.  N.  Sokolov  and  D.  T.  Egorov^^^"  proved,  by  use  of  Bougaief's  formu- 
las for  sums  extending  over  all  the  divisors  of  a  number,  the  formulas  in 
Liou\'ille's-°"'-^  series  of  four  articles. 

J.  W.  L.  Glaisher^^"*  gave  Zeller's^^  formula  and 

P(n  - 1)  +2-P(n  -  2)  -  5-P(n  -  5)  -  7'P{n  -  7)  + . . . 

=  ^|5(r3(n)-(18n-lMn)l, 

where  1,  2,  5, . . .  are  pentagonal  numbers  (3r=i=r)/2  and  P(0)  =  1. 
Glaisher"^  proved  formulae  which  are  greatly  shortened  by  setting 

a./n)=n''(rXn)-3(n-l)V/n-l)+5(n-3)VXn-3)-7(n-6)'(T,(n-6)+.... 

Write  Qij  for  ay(n).     Besides  the  formula  [of  Halphen^°]  aoi  =  0,  he  gave 

40 
ao3-2an  =  0,  ao5-10ai3+ya2i  =  0, 

126       ,756         ^_         . 

007 5~^15H H~^23  —  105031  =  0, 

Oo9  -  50017+720025  -  336O033 +336O041  =  0, 

with  the  agreement  that  o-(O)  =n/3  and 

—  t--\-l  f  —  1  —  ^'*+l  t^—1 

^3(0)  =  -24o-'  <^M  =  -^'          ^7(0)— ^^>          <^9i0)=-^, 

where  t  =  Sn+l,  but  did  not  find  the  general  formula  of  this  type.  Next, 
he  gave  five  formulas  of  another  set,  the  first  one  being  that  of  his  earlier 
paper,^  the  second  involving  the  same  function  of  0-3  with  added  terms  in 
ra{r).     Finally,  denoting  Euler's  formula  (2)  by  Ea{n)  =0,  it  is  shown  that 

5Eas{n)-lSE{n(T{n)\  =0. 

Glaisher^^^  showed  that  his"^  third  formula  holds  for  all  odd  numbers  v 
not  expressible  as  a  sum  of  three  squares  and  hence  in  particular  for  the 

"T'htorie  des  nombres,  1891,  403-6,  374,  388. 

'"Nova  Acta  Soc.  Upsal.,  (3),  14,  1891  (1886),  No.  2,  p.  63. 

>"^Math.  Soc.  Moscow,  16,  1891,  89-112,  236-255. 

"♦Messenger  Math.,  21,  1891-92,  47-8. 

^''Ibid.,  4^-69. 

^*Ibid.,  122,  126.    The  further  results  are  quoted  in  the  chapter  on  sums  of  three  squares. 


Chap.  X]  SUM  AND   NuMBER  OF   DiVISORS.  313 

former  case  v=7  (mod  8).    Also  the  left  member  of  the  third  formula 
equals 

4jE(y-l)-3^(y-9)+5^(y-25)- . . .) 

when  V  is  odd,  provided  E{0)  =  1/4.     If  A'(n)  denotes  the  sum  of  those  divi- 
sors of  n  whose  complementary  divisors  are  odd, 

A'(7i)-2A'(n-l)+2A'(n-4)-2A'(n-9)+.  .  .  =0  or  (-l)'^-^ 

according  as  n  is  not  or  is  a  square.     [Cf.  Lipschitz.^^]     Since  A'(n)  =a{n) 
for  n  odd,  we  deduce  a  formula  involving  c's  and  A"s. 
M.  Lerch"''  proved  (11)  and 

if  F{n)  =2/(d),  d  ranging  over  the  divisors  of  n. 

K.  Th.  Vahlen^^^  proved  Liouville's^^  formula  and  Jacobi's^"  result. 

A.  P.  Minin"^  proved  that  2,  8,  9,  12,  18,  8g  and  12p  (where  g  is  a  prime 
>2,  p  a  prime  >3)  are  the  only  numbers  such  that  each  is  divisible  by  the 
number  of  its  divisors  and  the  quotient  is  a  prime.  Minin^^°  found  that 
1,  3,  8,  10,  18,  24  and  30  are  the  only  numbers  N  for  which  the  number  of 
divisors  equals  the  number  of  integers  <  A^  and  prime  to  A^. 

M.  Lerch^^^  considered  the  number  x{a,  b)  and  sum  X{a,  h)  of  the  divisors 
^b  oi  a,  proved  his^°°  final  formula  (17)  and 

c  c 

a,  X{m  —  a.n,  a)  =  2a{x(m  — an,  n)—yp{m  —  an,  a)}, 

o=l  a=l 

(18)  S  \l/[m-an,  ^)  =  2  xim-an,  rn),       c=\  ^^-—   • 

a=o    \  ^/      a=o  L    n    J 

If  6  ranges  over  the  divisors  of  n, 

i  sV{(a-am,  n)}  =2^^^^,  |  sV|(a-am,  n)\  =2(5,  m)  a), 

't  a=0  0  n  a=0 

S  (am,  n)  =nS^-(5,  (m,  n)). 

a=l  0 

TO— 1 

He  quoted  (p.  8)  from  a  letter  to  him  from  Chr.  Zeller  that  2  a\p{m  —  a,  a) 

a=l 

equals  the  sum  of  the  remainders  obtained  on  dividing  m  by  the  integers 
<m. 

M.  Lerch^^^  proved  that 

'Zxf/im+p—crn,  a)  =2x(w+p— o-n,  n)  — 2 p  \, 

i:\pim-p-pn,  a)=Xx{m-p-(rn,  ^) -2[^:j:YjLn+fJ' 

i^Casopis,  Prag,  21,  1892,  90-95,  185-190  (in  Bohemian).     Cf.  Jahrbuch  Fortschritte  Math., 

24,  1892,  186-7. 
"sjour.  fur  Math.,  112,  1893,  29. 
i^Math.  Soc.  Moscow,  17,  1893,  240-253. 
i2o/6id.,  17,  1894,  537-544. 
"iPrag  Sitzungsberichte  (Math.),  1894,  No.  11. 
^Ubid.,  No.  32. 


314  History  of  the  Theory  of  Numbers.  [Chap,  x 

summed  for  p,  o-  =  0,  1, .  .  .,  with  p^<r.     Also, 

"s  (-l)V(w-a,a)=2  2  (-l)"e'(/n-a)  +  (-l)'"m, 

a-O  a=0 

m— 1  m  PtwH 

2^'(7n-a,a)=2(-l)*-Mf    , 

a-O  t  =  l  L^J 

2  Um-a,  2a)  =m+^-ii- ^+2  2  (-1)'  "^   , 

«=o  ^  v=i  L    if    J 

where  6' (A")  is  the  number  of  odd  divisors  of  k;  yp'{n,  a)  is  the  number  of 
div'isors  >a  of  n  whose  complementary  di\isors  are  odd;  while  \po{k,  n)  is  the 
number  of  even  di\'isors  >/i  of  k. 

In  No.  33,  he  expressed  in  terms  of  greatest  integer  functions 

X{\p{7n—p—an,  k-jr(T)—xi'm—p  —  (7n,n)}} 
2{^(w  — a,  k-\-a)  —  {k-]-a)\l/{m  —  a,  k-\-a)}- 

a 

E.  Busche^^^  gave  a  geometrical  proof  of  Meissel's^^  (11). 
J.  Schroder^^  obtained  (11)  and  the  first  relation  (15)  of  Lerch^^  as 
special  cases  of  the  theorem  that 

0,1,2,...  m  m 

2     \J/r,+sin-ri:  ipi,  2p,) 

Pi.--.Pm=  »  =  1  »  =  1 

equals  the  coefficient  of  x"  in  the  expansion  of 

m-l 

1-  n  (l-a:"+0 

1=0 

li::ri ' 

(l-x^"*)  n(l-x"+0 

t=0 

where  ypr,+>{o-j  ^)  is  the  number  of  di\isors  of  a  which  are  >^  and  have  a 
complementary  divisor  of  the  form  rv-{-s{v  =  0,l,. .  .).     He  obtained 

2    \f/r,+i{n-rp,  p)  =  y J • 

Schroder^^  determined  the  mentioned  coefficient  of  x". 
Schroder ^^^  proved  the  generaUzation  of  (11): 

p=iLpJ       p=iLpJ       p=2  LpJ 

For  (j{\)-\- . . .  +(r(n),  Dirichlet,"  end,  he  gave  the  value 

E.  Busche^^^  proved  that  if  X  =  4)(m)  is  an  increasing  (or  decreasing) 
function  whose  inverse  function  is  m=<l>(X),  the  divisors  of  the  natural 

i»Mittheilungen  Math.  GeseU.  Hamburg,  3,  1894,  167-172. 
"*Ibid.,  177-188. 
^Ibid.,  3,  1897,  302-8. 
"•/feid.,  3,  1895,  219-223. 
"Ubid.,  3,  1896,  239-40. 


Chap.  X]  SuM  AND  NuMBEK  OF   DiVISORS.  315 

numbers  between  <^(m)  and  a,  including  the  limits,  are  the  numbers  x  from 
1  to  o  (or  those  ^a)  each  taken  ^=  [$(x)/x]  times,  and  the  numbers  within 
the  limits  which  are  multiples  of  x  are  x,2x,. .  .,  ^x.  For  example,  if  a  =  3, 
</)(m)=900/m^,  then  <I>(a;)=30/Vx  and  it  is  a  question  of  the  divisors  of 
3. . .,  17;  for  x  =  'd,  ^  =  5  and  3  is  a  divisor  of  3,  6,  9,  12,  15.  For«I>(x)=n, 
a  =  l,  the  theorem  states  that  among  the  divisors  of  1,.  .  .,  n  any  one  x 
occurs  [n/x]  times  and  that  these  divisors  are  l,...,7i;l,...,  [n/2];  1,.  .  ., 
[w/3];  etc.    Hence  the  sum  of  the  divisors  of  1, . . . ,  n  is 

and  their  product  is 

n3.[n/x]=n[nA]!. 

x=l  x=\ 

He  proved  (pp.  244-6)  that  the  number  of  divisors  =r  (mod  s)  of  1,  2, . . . ,  n 
equals  A-\-B,  where  A  is  the  number  of  integers  [n/x]  for  x  =  \,.  .  .,  n 
which  have  one  of  the  residues  r,  r+l,...,s  —  1  (mod  s),  and  B  is  the  number 
of  all  divisors  of  1,  2, .  .  . ,  [n/s\.     The  number  of  the  divisors  b  of  m,  such  that 


\  n 


n 

and  such  that  5"  divides  m/8,  equals  the  number  of  divisors  of  1,  2, . . .,  n. 
The  number  of  primes  among  n,  [n/2], .  .  .,  [n/n]  equals  the  number  of  those 
divisors  of  1, . . .,  n  which  are  primes  decreased  by  the  number  of  divisors 
which  exceed  by  unity  a  prime. 

P.  Bachmann^^^  gave  an  exposition  of  the  work  of  Dirichlet,^^'  ^^  Mer- 
tens,^'^  Hermite,^^  Lipschitz,^^  Ces^ro,^''  Gegenbauer,'^'^  Busche,^^^'  ^" 
Schr6der.i24.  126 

N.  V.  Bougaiefi29  stated  that 

where  d  ranges  over  the  divisors  >  1  of  n,  and  v  =  [Vn] ; 


-i^HMM' 


where  d  ranges  over  the  divisors  of  n  for  which  d  <n.    If  6  is  any  function, 

nZ-^d{d)=  Xi:d{d), 

a  y=i  d 

where,  on  the  left,  d  ranges  over  all  the  divisors  of  n;  on  the  right,  only  over 
those  ^  [nVj].     For  6{d)  =  l,  this  gives 


„a(n)  =  Sx(n,[y]). 


"8Die  Analytische  Zahlentheorie,  1894,  401-422,  431-6,  490-3. 

i^'Comptes  Rendus  Paris,  120,  1895,  432-4.     He  used  $  (a,  6),  ^(a,  b)  with  the  same  meaning 
as  xib,  a),  X{b,  a)  of  Lerch,"^  and  fi(n)  for  (r{n). 


316  History  of  the  Theory  of  Numbers.  [Chap,  x  i>^ 

M.  Lerch"°  proved  relations  of  the  type 

The  number  of  solutions  of  [n/ x]  =  [n/ (x-\-l)],  x<n,  is 

2i/^(n-r,  r)+2  x{n-p,  p)  (A:= -§  + Vn  +  1/4  ). 

F.  NachtikaP^^  gave  an  elementary  proof  of  (15). 
M.  Lerch^^'  proved  that 


2   \yp{vi—aa,-)-]r'4^{m—(Ta,ra)\ 


remains  unaltered  if  we  interchange  r  and  s.     He  proved  (18)  and  showed 
that  it  also  follows  from  the  special  case  (17) .     From  (17)  f  orn  =  2  he  derived 

L.  Gegenbauer^^^"  proved  a  formula  which  includes  as  special  cases  four 
of  the  five  general  formulas  by  Bougaief  .^^^  When  x  ranges  over  a  given  set  S 
of  n  positive  integers,  the  sum2/(x)[x(a:)]  is  expressed  as  sums  of  expressions 
$(p)  and  <J>i(p),  where  p  takes  values  depending  upon  x,  while  $(2)  is  the  sum 
of  the  values  of /(x)  for  x  in  *S  and  x^z,  and  $1(2)  is  the  analogous  sum  with 

X'^Z. 

F.  RogeP^^  differentiated  repeatedly  the  relation 

|x|<l, 


00 

n(i- 

-xr''-'=e-^, 

then  set 

a:  =  0  and  found  that 

22  - 

(-1)'' 

\<T^{2)\^       Jcr 

.(r)l 

r'=ss( 

-!)■/ 

the  summations  extending  over  all  sets  of  a's  for  which 

CI1+CI2+  .  .  .  +ar  =  i,  ai+2a2+  .  .  .  -\-ra.r  =  r. 

Starting  with  the  reciprocals  of  the  members  of  the  initial  relation,  he 
obtained  similarly  a  second  formula;  subtracting  it  from  the  former  result, 
he  obtained 

.„(r)=."+|22{nf^+^;^-i)-(-i)'np} 

ai!.  .  .a,_3!;=2l    j    J 

""Casopis,  Prag,  24,  1895,  25-34,  118-124;  25,  1896,  228-30. 

"i/Wd.,  25,  1896,  344-6. 

»«Jornal  de  Scienciaa  Math,  e  Astr.  (TeLxeira),  12,  1896,  129-136. 

"^'Monatshefte  Math.  Phys.,  7,  1896,  26. 

i»Sitzung8ber.  Geaell.  Wiss.  (Math.),  Prag,  1897,  No.  7,  9  pp. 


'y-'.J 


Chap.  X]  SuM  AND   NuMBER   OF  DiVISORS.  317 

where  i  =  3,  5,  7,.  .  .  in  S',  while  the  a's  range  over  the  solutions  of 
ai+.  .  .+ar-3  =  i,  ai+2a2+.  .  . +(r-3)a,_3  =  r. 

The  case  n  =  0  leads  to  relations  for  T{r). 

J.  de  Vries^^^"  proved  the  first  formula  of  Lereh's.^^'^ 

A.  Berger"^  considered  the  excess  \l/{k)  of  the  sum  of  the  odd  divisors  of  k 
over  the  sum  of  the  even  divisors  and  proved  that 

xl/{n)+x{/in-l)+\l/{n-3)+x(/{n-Q)+xl/in-10)+  .  .  .=0  orn, 

according  as  n  is  not  or  is  a  triangular  number;  also  Euler's  (2). 
J.  FraneP^^  employed  two  arbitrary  functions  /,  g  and  set 

d{n)  ^Xmg( fj,  F{n)  =  ifU),  G{n)  =  S  gU), 

where  d  ranges  over  the  divisors  of  n.    Then 

id{j)  =  2 /(r)G[^]  +^j(r)F\j]  -F{v)G{v), 

where  v  =  ['\^n].  The  case  f{x)=g(x)  =  l  gives  Meissel's^"  (11).  Next,  he 
evaluated  St?(j),  where  ??(n)  =Xf{x)g{y)h(z),  sunmied  for  the  sets  of  positive 
integral  solutions  of  xyz  =  n.  In  particular,  ■d(n)  is  the  number  of  such  sets 
if  f=g  =  h  =  l.    Using  Dirichlet's  series,  it  is  shown  (p.  386)  that 

2^(i)  =  Si(logn+3C-l)2-3C2+6Ci+l}+e, 

where  e  is  of  the  order  of  magnitude  of  nP^^  log  n,  C  is  Euler's  constant  and 
Ci  =  0.0728 . .  .     [Piltz,^2  Landau^"]. 
FraneP^^  proved  that 

2^  =  1  log'  P+2C  log  p+e+Ao, 
r=i    r 

where  Aq  is  a  coefficient  in  a  certain  expansion,  and  ep^^^  remains  in  absolute 
value  inferior  to  a  fixed  number  for  every  p. 

E.  Landau ^^'^  gave  an  immediate  proof  of  (11)  and  of 


S7^3(^)=2T(.)r-l 

y  =  l  u=l  Li'J 


where  T^iv)  is  the  number  of  decompositions  of  v  into  three  factors.     He 
obtained  by  elementary  methods  a  formula  yielding  the  final  result  of 

R.  D.  von  Sterneck^^^"  proved  Jacobi's^^  formula  for  s^. 

i33aK.  Akad.  Wetenschappen  te  Amsterdam,  Verslagen,  5,  1897,  223. 

i3*Nova  Acta  Soc.  Sc.  UpsaUensis,  (3),  17,  1898,  No.  3,  p.  26. 

"^Math.  Annalen,  51,  1899,  369-387. 

i3676id.,  52,  1899,  536-8. 

i"/6ici.,  54,  1901,  592-601. 

i^^Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  109,  Ila,  1900,  31-33. 


318  History  of  the  Theory  of  Numbers.  [Chap,  x 

J.  Franel"^  stated  that,  if /(n)  is  the  number  of  positive  integral  solutions 
of  x''y^  =  n,  where  a,  h  are  distinct  positive  integers, 


2/(r)  =^i^l)n^+^{^n'+o{n'-^ 


+6J, 


where'"  0{s)  is  of  the  order  of  magnitude  of  s.  Taking  a  =  l,  6  =  2,  we  see 
that  /(n)  is  the  number  of  di\isors  of  q,  where  <f  is  the  greatest  square  divid- 
ing n,  and  that  the  mean  of  /(n)  is  tt"/  6. 

E.  Landau^^^  proved  the  preceding  formula  of  Franel's. 

EUiott^^  of  Ch,  V  gave  formulas  involving  a{n)  and  rin). 

L.  Kronecker^'*'^  proved  that  the  sum  of  the  odd  divisors  of  a  number  equals 
the  algebraic  sum  of  all  its  di\isors  taken  positive  or  negative  according  as 
the  complementary^  di\'isor  is  odd  or  even  (attributed  to  Euler^);  proved 
(pp.  267-8)  the  result  of  Dirichlet'^  and  (p.  345)  proved  (7)  and  found  the 
median  value  (Mittelwert)  of  T{n)  to  be  log^  n+2C  with  an  error  of  the  order 
of  magnitude  of  n~^^^when  the  number  of  values  employed  is  of  the  order  of 
n^''^.  Calling  a  di\isor  of  n  a  smaller  or  greater  di\dsor  according  as  it  is  less 
than  or  greater  than  \/n,  he  found  (pp.  343-369)  the  mean  and  median  value 
of  the  sum  of  all  smaller  (or  greater)  di\'isors  of  1,  2, . .  . ,  A^  [cf.  Gegenbauer^^], 
the  sum  of  their  reciprocals,  and  the  sum  of  their  logarithms.  The  mean  of 
Jacobi's^^  E{n)  is  x/4  (p.  374). 

J.  W.  L.  Glaisher^^^  tabulated  for  n  =  l,...,  1000  the  values  of  the 
function^'^  H{n)  and  of  the  excess  J{n)  of  the  number  of  divisors  of  n  which 
are  of  the  form  8A-  +  1  or  8A;+3  over  the  number  of  divisors  of  the  form 
8A:+5  or  8A:+7.  WTien  n  is  odd,  2J{n)  is  the  number  of  representations  of  n 
by  x'-\-2f. 

J.  W.  L.  Glaisher"^  derived  from  Dirichlet's^^  formula,  and  also  inde- 
pendently, the  simpler  formula 

2  0.(s)  =  -pG(p)+f[5],(.)+£(?{g]}, 

where  p  =  {\/n].  The  case  g{s)  =  l  gives  Meissel's^-  formula  (11),  which  is 
applied  to  find  asymptotic  formulae  involving  n/s  —  [n/s].  The  error  of  the 
approximation  (7)  is  discussed  at  length  (pp.  38-75,  180-2).  The  first 
formula  above  is  applied  (pp.  183-229)  to  find  exact  and  asymptotic  formu- 
las for  2/(s),  when/(n)  is  Jacobi's"  E{n),  Glaisher's^^^  H{n)  or  J{n),  or  the 
excess  D{n)  of  the  number  of  odd  divisors  of  n  over  the  number  of  even 
di\'isors,  or  more  general  functions  (p.  215,  p.  223)  involving  the  number  of 
di\isors  with  specified  residues  modulo  r. 

G.  Voronoi^^^  proved  a  formula  like  Dirichlet's^^  (7),  but  with  e  now  of  the 
same  order  of  magnitude  as  -^/n  log^  n. 

•"L'intermddiaire  des  math.,  6,  1899,  53;  18,  1911,  52-3. 

"»/6ui.,  20,  1913,  155. 

•"Vorlesungen  iiber  Zahlentheorie,  I,  1901,  54-55. 

"'Messenger  Math.,  31,  1901-2,  64-72,  82-91. 

>«Quar.  Jour.  Math.,  33,  1902,  1-75,  180-229. 

'«Jour.  fiir  Math.,  126,  1903,  241-282. 


Chap.  X]  SuM  AND   NUMBER  OF   DiVISORS.  319 

H.  Mellin"^  obtained  asymptotic  expressions  for2T(n),  So-(n). 
I.  Giulini^^^  noted  that,  if  m  and  h  are  given  integers,  and  /3(r)  is  the  sum 
of  the  divisors  d  =  mk-\-h  of  r,  then 

i8(l)  +  . .  .+/3(n)=2:4n/d],  A;  =  0,  1,.  . .,  [{n-h)/m], 

k 

The  number  and  sum  of  the  divisors  d='mk-\-h  oil,. . .,  n  are 

[(n-w«]r^-i  [n/im+h)]      /^-hA  ,  ,%'^\-n-sh  ,  ,-| 

S     hj    '  ^     2     Eal ]+hi:\ +1  h 

fc=o    LttJ  r=i         \  mr  /        s=iL  ms         J 

respectively,  where  ^2(2^)  =  W[a^+l]/2. 

G.  Voronoi^^^"  gave  for  T{x)  the  precise  analytic  expression 

x(logx+2(7-l)+i+Ma^)-2|  g{t)dt^      {g{-x+ti)-g(-x-U)}idt, 

and  (p.  515)  approximations  to  these  integrals,  where 

,(x)=  -i  log  .-iC-!2i^+2^  i^x(»)  ,og5(^+-i-). 

He  discussed  at  length  the  function  g{x)  and  (pp.  467,  480-514)  the  asymp- 
totic value  of  Sr(n)(x  — 7i)V^!- 

J.  Schroder^^®  proved  that  the  sum  of  the  I'th  powers  of  1, . . . ,  n  is 

S  pf-1  =n(T._i(n)+  "S  p'+  r  pT ^1, 
p=i     LpJ  p=t+i  P=i      LPJ 

where  t  =  [n/2],  and  the  accent  on  the  last  S  denotes  that  the  summation 
extends  only  over  the  values  ^  ^  of  p  which  are  not  divisors  of  n. 

E.  Busche^*^  proved  that,  if  we  multiply  each  divisor  of  m  by  each  divisor 
of  n,  the  number  of  times  we  obtain  a  given  divisor  a  of  mn  is  Tiixv/a),  where 
jjL  is  the  g.  c.  d.  of  m,a,  and  v  is  that  of  n,  a.  A  like  theorem  is  proved  for 
th^  divisors  of  mnp ....    He  stated  (p.  233;  cf.  Bachmann^^^)  that 


(Th{m)(Th{n)  =SdV;,f  ^j, 


where  d  ranges  over  the  common  divisors  of  m,  n. 

C.  Hansen ^^^  denoted  by  Ti{n)  and  T^{n)  the  number  of  divisors  of  n 
of  the  respective  forms  4/c  — 1  and  4/^  —  3,  and  set 

A„=r3(4n-3)-ri(4n-3). 

By  use  of  Jacobi's  B^{v,  s)  for  ^  =  1/4,  he  proved  that 

„=i     '^^       ~Zx^      ^      l-s'^"-^     l-2s^+2s^«+... 

i«Acta  Math.,  28,  1904,  49. 

i«Giornale  di  mat.,  42,  1904,  103-8. 

i^'^Annales  sc.  I'ecole  norm,  sup.,  (3),  21,  1904,  213-6,  245-9,  258-267,  472-480.     Cf.  Hardy.i^" 

"«Mitt.  Math.  GeseU.  Hamburg,  4,  1906,  256-8. 

"V6id.,  4,  1906,  229. 

"^Oversigt  K.  Danske  Videnskabemes  Selskabs  Forhandlinger,  1906,  19-30  (in  French). 


320  History  of  the  Theory  of  Numbers.  [Chap,  x 

and  hence  deduced  the  law  of  a  recursion  formula  for  A„.  The  law  of  a 
recursion  formula  for  B„  =  4{T2{7i)  —  Ti{n)\  is  obtained  from 

S  ^y  S  s''"+'^'  cos(2n+l)^=  i;  (2n+l)s<2n+i)'sijj(2n+l)7, 

n=0  n=0  4        n=0  4 

with  Bo=l,  which  was  found  by  use  of  Jacobi's  d{\,  s).     Next, 

is  shown  to  satisfy  the  functional  equation 

$(ts)=lj$(s)-$(-s)}-$(s2)+2$(s*). 

If  a  convergent  series  liens'"  is  a  solution  $(s)  of  the  latter,  the  coefficients  are 
uniquely  determined  by  the  C4i_3(/j  =  1,  2, .  .  . ),  which  are  arbitrary.  Hence 
the  function  5„  is  determined  for  all  values  of  n  by  its  values  forn  =  4/c— 3 

(A:  =  l,  2,...). 

S.  Wigert^^^  proved  that,  for  sufficiently  large  values  of  n,  r(n)<2', 
where  f  =  (l  +  e)  log  n^-log  log  n,  for  every  e  >0;  while  there  exist  certain 
values  of  n  above  any  limit  for  which  riri)  >2',  s  =  (1  —  e)  log  n  -i-log  log  n. 

J.  V.  Pexider^^°  proved  that,  if  a,  n  are  positive,  a  an  integer, 

by  the  method  used,  for  the  case  in  which  n  is  an  integral  multiple  of  a, 
by  E.  Cesaro.^°  Taking  a  =  [Vn],  we  have  the  second  equation  (11).  Proof 
is  given  of  the  first  equation  (11)  and 

S.[g=2.W,  2[2][!^]=S<?-.W, 

where  d  ranges  over  the  divisors  of  [n]. 

0.  Meissner^"  noted  that,  if  m  =pi".  .  .p/",  where  pi  is  the  least  of  the 
distinct  primes  pi, .  .  . ,  7?„,  then 

,=iPj  — 1       m       ,=2Pt  — 1  w  log  m 

where  G  is  finite  and  independent  of  m.     If  /v>  1,  (Tk{m)/rn!'  is  bounded. 

W.  Sierpinski^^^  proved  that  the  mean  of  the  number  of  integers  whose 
squares  divide  n,  of  their  sum,  and  of  the  greatest  of  them,  are 

x^  1,         .3^  3  ,         ,  9C  ,  36  -  logs 

— ,  -logn+^C,  -:2lognH — rA — -^Z  ^^, 

respectively,  where  C  is  Euler's  constant. 

J.  W.  L.  Glaisher^^^  derived  formulas  differing  from  his^^°  earlier  ones 
only  in  the  replacement  of  d  by  {  —  lY~^d,  i.  e.,  by  changing  the  sign  of  each 

"»Arkiv  for  mat.,  ast.,  fys.,  3,  1906-7,  No.  18,  9  pp. 

»"Rendiconti  Circolo  Mat.  Palermo,  24,  1907,  58-63. 

"'Archiv  Math.  Phys.,  (3),  12,  1907,  199. 

"^prawozdania  Tow.  Nank.  (Proc.  Sc.  Soc.  Warsaw),  1,  1908,  215-226  (Polish). 

i"Proc.  London  Math.  Soc,  (2),  6,  1908,  424-^67. 


Chap.  X]  SuM  AND   NuMBER  OF   DiVISORS.  321 

even  divisor  d.  In  the  case  of  the  theorems  on  the  cancellation  of  actual 
divisors,  the  results  follow  at  once  from  the  earlier  ones.  But  the  recursion 
formulae  for  o-„  and  f „  are  new  and  too  numerous  to  quote.  Cancellation 
formulas  (pp.  449-467)  are  proved  for  the  divisors  whose  complementary- 
divisors  are  odd,  and  applied  to  obtain  recursion  formulae  for  the  related 
function  A/(n)  of  Glaisher.''^' " 

E.  Landau ^^^  proved  that  log  2  is  the  superior  limit  for  x=  oo  of 
log  T(x)-log  log  a;-^log  X. 

M.  Fekete^^^  employed  the  determinant  RkX  obtained  by  deleting  the 
last  t  rows  and  last  t  columns  of  Sylvester's  eliminant  of  x'^— 1  =  0  and 
a;"-l  =  0.     Set,  for  A;^n, 

Then  6„(A;)  =  1  or  0  according  as  k  is  or  is  not  a  divisor  of  n;  while  c„(i,  k)  =  l 
if  ik  =  n  and  i  is  relatively  prime  to  k,  but  =  0  in  the  contrary  cases.     Thus 

T(n)=S6„(fc),  (T{n)=ikK{k), 

k  =  \  A:  =  l 

while  the  number  and  sum  of  those  divisors  d  of  n,  which  are  relatively  prime 
to  the  complementary  divisors  n/d,  equal,  respectively, 

n  1       " 

S   Cn{i,  k),  -  i:  (i+k)  c^{i,  k). 

i,  k  =  \  ^  i,k  =  l 

J.  Schroder^"  deduced  from  his^^"*  final  equation  the  results 

The  final  sum  equals  XIZI  i//(s,  [s/{r+l)]). 

P.  Bachmann^^^  gave  an  exposition  of  the  work  of  Euler,^'^  Glaisher,^^' ^^ 
Zeller,66  stern,^^  Glaisher,iio  Liouville.^^ 

E.  Landau ^^^  proved  that  the  number  of  positive  integers^  a;  which  have 
exactly  n  positive  integral  divisors  is  asymptotic  to 

Aa;^/^^-^^(log  log  rr)'"-Vlog  x, 

where  p  is  the  least  prime  factor  of  n,  and  p  occurs  exactly  w  times  in  n, 
while  A  depends  only  on  n. 

K.  Knopp^^"  obtained,  by  enumerations  of  lattice  points, 

n  n  w  w 

i:Mq,k)=  i:hik,q)=  i:f,iq,k)+i:f2{k,q)-F{w,w), 

k=l  k=l  k=l  k=l 

where  q  =  [n/k]  and 

/i(r,  k)=k  fij,  k),           h{k,  s)=i  f{k,  j),           F(r,  s)=i  Mr,  j). 
y=i y=i i=i 

i66Handbuch.  .  .Verteilung  der  Primzahlen,  I,  1909,  219-222. 

i^^Math.  6s  Phys.  Lapok  (Math.  phys.  soc),  Budapest,  18,  1909,  349-370.     German  transl., 

Math.  Naturwiss.  Berichte  aus  Ungarn,  26,  1913  (1908),  196-211, 
>"Mitt.  Math.  Gesell.  Hamburg,  4,  1910,  467-470. 
issNiedere  Zahlentheorie,  II,  1910,  268-273,  284-304,  375. 
i^Annaes  So.  Acad.  Polyt.  do  Porto,  Coimbra,  6,  1911,  129-137. 
""Sitzungsber.  Berlin  Math.  Gesell.,  11,  1912,  32-9;  with  Archiv  Math.  Phys. 


322  History  of  the  Theory  of  Numbers.  [Chap,  x 

Taking  /(/j,  k)  =  l,  we  obtain  Meissel's"  (11),   a  direct  proof  of  which 
is  also  given.     Taking /(/i,  k)=f{h)g{hk),  we  get 


S    S/0>(j/c)=S/(/c)S^(iA:), 


-m-       I 


special  cases  of  which  yield  niany  known  formulas  involving  Mobius's  func- 
tion ju(n)  or  Euler's  function  (f>{n). 

E.  Landau^^^  proved  the  result  due  to  PfeifTer^°,  and  a  theorem  more 
effective  than  that  by  Piltz^^,  having  the  0  terms  replaced  by  0{x°-),  where, 
for  every  e>0, 

k-1        . 

E.  Landau^^-  extended  the  theorem  of  Piltz^^  to  an  arbitrary  algebraic 
domain,  defining  Tk{n)  to  be  the  number  of  representations  of  n  as  the  norm 
of  a  product  of  k  ideals  of  the  domain. 

J.  W.  L.  Glaisher^^,  generalizing  his^^^  formula,  proved  that 

Sf[^]^(s)=Sf[^]^(s)  +  2g[^]/(s)-F(p)G(p), 

where F(s)  =/(!)+  .  .  .  +/(s),  G{s)=g{l)+. .  . -\-g{s),  p  =  [v^].  A  similar 
generalization  of  another  formula  by  Dirichlet^^  is  proved,  also  analogous 
theorems  involving  only  odd  arguments. 

Glaisher^^  applied  the  formulas  just  mentioned  to  obtain  theorems  on 
the  number  and  sum  of  powers  of  divisors,  which  include  all  or  only  the 
even  or  only  the  odd  divisors.  Among  the  results  are  (11)  and  those  of 
Hacks.^®'^^  The  larger  part  of  the  paper  relates  to  asymptotic  formulas 
for  the  functions  mentioned,  and  the  theorems  are  too  numerous  to  be 
cited  here. 

E.  Landau^^  gave  another  proof  of  the  result  by  Voronoi^^^.  He  proved 
(p.  2223)  that  T(n)< 471^/^ 

J.  W.  L.  Glaisher^^^  stated  again  many  of  his^^  results,  but  without 
determining  the  limits  of  the  errors  of  the  asymptotic  formulas. 

S.  Minetola^^^  proved  that  the  number  of  ways  a  product  of  m  distinct 
primes  can  be  expressed  as  a  product  of  n  factors  is 

iy{»"-G)("-')"+(2)(»-2)"--(„:ii)4 

T.  H.  GronwalP^^  noted  that  the  superior  limits  for  a:=  oo  of 
aM/x"     (a>l),  (r{x)/{x\oglogx) 

are  the  zeta  function  f  (a)  and  e^,  respectively,  C  being  Euler's  constant. 

"'Gottingen  Nachrichten,  1912,  687-690,  716-731. 

"»Tran8.  Amer.  Math.  Soc,  13,  1912,  1-21. 

»«Quar.  Jour.  Math.,  43,  1912,  123-132. 

^**Ibid.,  315-377.     Summary  in  Glaisher.i« 

'"Messenger  Math.,  42,  1912-13,  1-12. 

i««Il  Boll,  di  Matematica  Gior.  Sc.-Didat.,  Roma,  11,  1912,  43-46;  cf.  Giomale  di  Mat.,  45, 1907, 

344-5;  47,  1909,  173,  §1,  No.  7. 
"Trans.  Amer.  Math.  Soc,  14,  1913,  113-122. 


Chap.  X]  SuM  AND   NuMBEE  OF  DiVISORS.  323 

P.  Bachmann^^^  proved  the  final  formula  of  Busche.^^' 
K.  Knopp^^^  studied  the  convergence  of  26„a:V(l  — 2:"),  including  the 
series  of  Lambert^,  and  proved  that  the  function  defined  in  the  unit  circle 
by  Euler's^  product  (1)  can  not  be  continued  beyond  that  circle. 

E.  T.  BelP'°  proved  that,  if  P  is  the  product  of  all  the  distinct  prime 
factors  of  m,  and  X  is  their  number,  and  d  ranges  over  all  divisors  of  m, 


6^Sr(d)T(^^  =r{m)T{Pm)T{P''m). 


J.  F.  Steffensen^'^^  proved  that,^°  if  Ix  denotes  log  x, 

S.  Wigert^^^  proved,  for  the  sum  n's{n)  of  the  divisors  of  n, 


(1  —  €)e^  log  log  n< s{n) <  {l-\-e)e^  log  log  n, 
^s{n)  =  '^x-^l^{x),  rP{x)  =  x^    1  +  2  Ip(^), 

for  €>  0  and  p{x)=x  —  [x].    For  x  sufficiently  large, 

(i-e)  log  x<xP{x)<(l+e)  log  x. 
Besides  results  on  Ss(a^)(x— n)*,  lls{n)  log  x/n,  he  proved  that 

X  ns{n)=^+xlh\ogx-rP{x)}+0{x). 

E.  Landau^^^  gave  corrections  and  simplifications  in  the  proofs  by 
Wigert."2 

E.  T.  Bell^^^  introduced  a  function  including  as  special  cases  the  functions 
treated  by  Liouville,^^"-^  restated  his  theorems  and  gave  others. 
J.  G.  van  der  Corput^^^  proved,  for  ix(d)  as  in  Chapter  XIX, 

Sd'')u(d)So-„(A;)=x. 
S.  Ramanujan^'^®  proved  that  t{N)  is  always  less  than  2*  and  2',  where^" 

^  =  lWV+«  {  (IsSpf  '=^*-('°^  ^)+^f'°«  iVe— .-!, 

for  Li(x)  as  in  Ch.  XVIII,  and  for  a  a  constant.  Also,  t(N)  exceeds  2*'and 
2'  for  an  infinitude  of  values  of  N.  A  highly  composite  number  N  is  one 
for  which  TiN)>T{n)  when  N>n',  if  Ar  =  2''^3"». .  .p"p,  then  aa^as^ag^ 

"SArchiv  Math.  Phys.,  (3),  21,  1913,  91. 

"9Jour.  flir  Math.,  142,  1913,  283-315;  minor  errata,  143,  1913,  50. 

""Amer.  Math.  Monthly,  21,  1914,  130-1. 

i"Acta  Math.,  37,  1914,  107.     Extract  from  his  Danish  Diss.,  "Analytiske  Studier  med  Anven- 

delser  paa  Taltheorien,"  Kopenhagen,  1912. 
"HUd.,  113-140. 

"^Gottingsche  gelehrte  Anzeigen,  177,  1915,  377-414. 
"<Univ.  of  Washington  PubUcations  Math.  Phys.,  1,  1915,  6-8,  38-44. 
'"Wiskundige  Opgaven,  12,  1915,  182-4. 
"oProc.  London  Math.  Soc,  (2),  14, 1915,  347-409. 


324 


History  of  the  Theory  of  Numbers. 


[Chap.  X 


.  .  .  ^flp,  while  ap=  1  except  when  A'' =  4  or  36.  The  value  of  X  for  which 
a2>ai>  .  .  .  >ax  is  investigated  at  length.  The  ratio  of  two  consecutive 
highly  composite  numbers  A''  tends  to  unity.  There  is  a  table  of  A"s  up 
to  t{N)  =  10080.  An  N  is  called  a  superior  highly  composite  number  k 
there  exists  a  positive  number  e  such  that 


N.^ 


N'   =  Ni^ 


for  all  values  of  A^  and  No  such  that  A'2>  A'>  A^i.     Properties  of  t{N)  are 
found  for  (superior)  highly  composite  numbers. 

Ramanujan^"  gave  for  the  zeta  function  (12)  the  formula 

and  found  asjTnptotic  formulae  for 


j=i 


S  r'(j), 
=1 


Sr(jr+c), 


S(7„(j>i(i), 


A(n), 


for  a  =  0  or  1 ,  where 


A(n)  =  S^r(i.)  =SM(d)r  (0Z),(^), 
summed  for  the  di\isors  d  of  v.     If  5  is  a  common  di\isor  of  u,  v, 
xM=iMW.g)rg)=2.W.@x(0. 


E.  Landau^'^^''  gave  another  asjinptotic  formula  for  the  number  of  de- 
compositions of  the  numbers  ^  x  into  k  factors,  A'  ^  2. 
Ramanujan^'*  wrote  c^O)  =^^("5)  and  proved  that 


2,,(,>.(n-,)  ^^Pm^  ■  ^^l^lV 


J=0 


r(r+s  +  2)  f(r+s+2)    ^^+*+^ 

r(i-r)+r(i-^) 


(n) 


/Z(r,+,_i(n)+0(n2'^+«+^^/^), 


for  positive  odd  integers  r,  s.     Also  that  there  is  no  error  term  in  the  right 
member  if  r=l,  s  =  1,  3,  5,  7,  11;  r  =  3,  s=3,  5,  9;  r  =  5,  s  =  7. 

J.  G.  van  der  Corput^"^  wrote  s  for  the  g.  c.  d.  of  the  exponents  ai,  a-z,... 
in  m='n.pi''i  and  expressed  in  terms  of  zeta  function  f(i),  i=2, .  .  .,  k-\-l, 

2  {a,{s)-l]/m 

m=2 

if  A'  >  1 ;  the  sum  being  1  —  CifA=— 1,  where  C  is  Euler's  constant. 

"'Messenger  Math.,  45,  1915-6,  81-84. 
"'"Sitzungsber.  Ak.  Wiss.  Miinchen,  1915,  317-28. 
i^Trans.  Cambr.  Phil.  Soc,  22,  1916,  159-173. 
"•Wiskundige  Opgaven,  12,  1916,  116-7. 


Chap.  X]  SuM  AND   NUMBEK   OF   DiVISORS.  325 

G.  H.  Hardy^^°  proved  that  for  Dirichlet's^^  formula  (7)  there  exists  a 
constant  K  such  that  e  >  Kn^^'^,  e  <  —  Kn^^'^,  for  an  infinitude  of  values  of  n 
surpassing  all  limit.     In  Piltz's^^  formula 

S  Tk{n)  =  x{akii\og  xy-'-\-  .  .  .  -^-a^k}  +e„ 

n=l 

ek>Kx\  ik<—Kx\  where  t={k  —  \)/{2k).     He  gave  two  proofs  of  an 
equivalent  to  Voronoi's^^^''  explicit  expression  for  T{x). 

Hardy^^^  wrote  A(n)  for  Dirichlet's  e  in  (7)  and  proved  that,^°  for  every 
positive  e,  /l{n)  =  0{n'^^''^)  on  the  average,  i.  e., 

iJiA(0M«=O(n'+"*). 

G.  H.  Hardy  and  S.  Ramanujan^^^  employed  the  phrase  "almost  all 
numbers  have  a  specified  property"  to  mean  that  the  number  of  the  num- 
bers ^  X  having  this  property  is  asymptotic  to  a:  as  a;  increases  indefinitely, 
and  proved  that  if  /  is  a  function  of  n  which  tends  steadily  to  infinity  with  n, 
then  almost  all  numbers  have  between  a  — 6  and  a-\-h  different  prime  factors, 
where  a  =  log  log  n,  h=f-\/d.  The  same  result  holds  also  for  the  total 
number  of  prime  factors,  not  necessarily  distinct.  Also  a  is  the  normal 
order  of  the  number  of  distinct  prime  factors  of  n  or  of  the  total  number 
of  its  prime  factors,  where  the  normal  order  of  g{n)  is  defined  to  mean  f{n) 
if ,  for  every  positive  e,  (1— €)/(n)<gr(n)<(l+e)/(n)  for  almost  all  values 
of  n. 

S.  Wigert^^^  gave  an  asymptotic  representation  for  l!,n^j:r{n){x  —  n)^. 

E.  T.  BelP^  gave  results  bearing  on  this  chapter. 

F.  RogeP^^  expressed  the  sum  of  the  rth  powers  of  the  divisors  ^g*  of 
m  as  an  infinite  series  involving  Bernoullian  functions. 

A.  Cunningham^^^  found  the  primes  p<  lO'*  (or  10^)  for  which  the  number 
of  divisors  of  p  — 1  is  a  maximum  64  (or  120). 

Hammond^^  of  Ch.  XI  and  RogeP^^  of  Ch.  XVIII  gave  formulas  involv- 
ing (J  and  r.  Bougaief^^' ^^  of  Ch.  XIX  treated  the  number  of  divisors 
^  m  of  n.  Gegenbauer^°  of  Ch.  XIX  treated  the  sum  of  the  pth  powers  of 
the  divisors  ^  m  of  n. 

i^oProc.  London  Math.  Soc,  (2),  15,  1916,  1-25. 

i"/&id.,  192-213. 

i82Quar.  Jour.  Math.,  48,  1917,  76-92. 

i83Acta  Math.,  41,  1917,  197-218. 

is^Annals  of  Math.,  19,  1918,  210-6. 

i85Math.  Quest.  Educ.  Times,  72,  1900,  125-6. 

i86Math.  Quest,  and  Solutions,  3,  1917,  65. 


I 


CHAPTER  XL 

MISCELLANEOUS  THEOREMS  ON  DIVISIBILITY,  GREATEST 
COMMON  DIVISOR.  LEAST  COMMON  MULTIPLE. 

Theorems  on  Divisibility. 

An  anonymous  author^  noted  that  for  n  a  prime  the  sum  of  1 ,  2, . . . ,  n  —  1 
taken  by  twos  (as  1+2,  1+3,. . .),  by  fours,  by  sixes,  etc.,  when  divided 
by  n  give  equally  often  the  residues  1,  2,...,  n  — 1,  and  once  oftener  the 
residue  0.  The  sum  by  threes,  fives, . . . ,  give  equally  often  the  residues 
1,. .  .,  n  — 1  and  once  fewer  the  residue  0. 

J.  Dienger^  noted  that  if  w''-+'±l  and  (m^'-+2_  1)7(^2 _i)  are  divisible 
by  the  prime  p,  then  the  sum  of  any  2r+l  consecutive  terms  of  the  set 
1,  m^",  m^'^",  m^'^", . . .  is  divisible  by  p.  The  case  m  =  2,  r  =  l,  p  =  3  or  7 
was  noted  by  Stifel  (Arith.  Integra). 

G.  L.  Dirichlet^  proved  that  when  n  is  divided  by  1,  2, .  .  .,  n  in  turn 
the  number  of  cases  in  which  the  remainder  is  less  than  half  the  divisor 
bears  to  n  a  ratio  which,  as  n  increases,  has  the  limit  2  — log  4  =  0.6137 
.  .  . ;  the  sum  of  the  quotients  of  the  n  remainders  by  the  corresponding 
divisors  bears  to  n  a  ratio  with  the  limit  0.423 .... 

Dirichlet^  generalized  his  preceding  result.  The  number  h  of  those 
divisors  1,2,.  .  . ,  p  (p^  ti),  which  yield  a  remainder  whose  ratio  to  the  divisor 
is  less  than  a  given  proper  fraction  a,  is 


-liH-B-"]} 


Assuming  that  pVn  increases  indefinitely  with  n,  the  limit  of  /i/p  is  a 
if  n/p  increases  indefinitely  with  n,  but  if  n/p  remains  finite  is 

J.  J.  Sylvester^  noted  that  2"""^^  is  a  factor  of  the  integral  part  of  /c^"*"*"^ 
and  of  the  integer  just  exceeding  h^"^,  where  ^  =  l  +  \/3- 

N.  V.  Bougaief^  called  a  number  primitive  if  divisible  by  no  square  >1, 
secondary  if  divisible  by  no  cube.     The  number  of  primitive  numbers  ^  n  is 

H,{n)=i:q{u)+iq{u)+..  .,  <i  =  [VnA'], 

1  1  ' 

where  q{u)  is  zero  if  u  is  not  primitive,  but  is  +1  or  —1  for  a  primitive  u, 
according  as  ?/  is  a  product  of  an  even  or  odd  number  of  prime  factors. 

iJour.  fiir  Math.,  6,  1830,  100-4.  ^Archiv  Math.  Phys.,  12,  1849,  425-9. 

3Abh.  Ak.  Wiss.  BerMn,  1849,  75-6;  Werke,  2,  57-58.     Cf.  Sylvester,  Amer.  Jour.   Math.,  5, 

1882,  298-303;  CoU.  Math.  Papers,  IV,  49-54. 
<Jour.  fur.  Math.,  47,  1854,  151-4.     Berlin  Berichte,  1851,  20-25;  Werke,  2,  97'-104;  French 

transl.  by  O.  Terquem,  Nouv.  Ann.  Math.,  13,  1854,  396. 
^uar.  Joum.  Math.,  1,  1857,  185.     Lady's  and  Gentleman's  Diary,  London,  1857,  60-1. 
"Comptes  Rendus  Paris,  74,  1872,  449-450.     BuU.  Sc.  Math.  Astr.,  10,  I,  1876,  24.     Math. 

Sbornik  (Math.  Soc.  Moscow),  6,  1872-3,  I,  317-9,  323-331. 

327 


328  History  of  the  Theory  of  Numbers.  [Chap,  xi 

To  obtain  the  number  Hoin)  of  secondary  numbers  ^n,  replace  square  roots 
by  cube  roots  in  the  /,.     We  have 

ffi(n)+Hi([|:,])+/fi([|])  +  . . .  =n,         H2{n)+Ho{]^^  + .  . .  =n, 

and  similarly  for  Hi,_i{n)  given  by  (2)  below. 

J.  Grolous"  considered  the  probability  R^  that  a  number  be  divisible 
by  at  least  one  of  the  integers  Qi,.  . .,  Qk,  relatively  prime  by  twos,  and 
showed  that 

Chr.  Zeller''"  modified  Dirichlet's^  expression  for  h.    The  sums 

,=iLs        J  ,=iLs  +  aJ 

are  equal.  The  sum  of  the  terms  of  the  second  with  s>fjL  =  [\/p]  equals 
the  excess  of  the  sum  of  the  first  n  terms  of  the  first  over  fx^  or  ju^  —  1 ,  the 
latter  in  the  case  of  numbers  between  fi^  and  m^+m-  Hence  we  may  abbre- 
\iate  the  computation  of  h. 

E.  Cesaro^  obtained  Dirichlet's^'^  results  and  similar  ones.  The  mean 
(p.  174)  of  the  number  of  decompositions  of  A^  into  two  factors  having  p  as 
their  g.  c.  d.  is  6(log  N)/(p~Tr^).  The  mean  (p.  230)  of  the  number  of 
di\isors  common  to  two  positive  integers  n,  n'  is  7rV6,  that  of  the  sum  of 
their  common  di\isors  is 

ilog,  nn'+2C-Y^+i, 

where  C  =  0.57721 ....  The  sum  of  the  inverses  of  the  nth  powers  of  two  posi- 
tive integers  is  in  mean  ^^+2)  where  s"  is  defined  by  (12)  of  Ch.  X. 

E.  Cesaro^  proved  the  preceding  results  on  mean  values;  showed  that 
the  number  of  couples  of  integers  whose  1.  c.  m.  is  n  is  the  number  of  divisors 
of  n",  if  (a,  b)  and  (6,  a)  are  both  counted  when  a^^b;  found  the  mean  of 
the  1.  c.  m.  of  two  numbers;  found  the  probability  that  in  a  random  division 
the  quotient  is  odd,  and  the  mean  of  the  first  or  last  digit  of  the  quotient; 
the  probability  that  the  g.  c.d.  of  several  numbers  shall  have  specified 
properties. 

Cesaro^"  noted  that  the  probability  that  an  integer  has  no  divisor  >  1 
which  is  an  exact  rth  power  is  l/f(r). 

L.  Gegenbauer^°  proved  that  the  number  of  integers  ^  x  and  divisible 
by  no  square  is  asymptotic  to  Gx/tt",  with  an  error  of  order  inferior  to 
\/x-     He  proved  the  final  formulas  of  Bougaief.^ 

'Bull.  Sc.  Soc.  Philomatique  de  Paris,  1872,  11(>-128. 
'"Nachrichten  Gesell.  Wiss.  Gottingen,  1879,  265-8. 

8M6m.  Soc.  R.  Sc.  de  Li^ge,  (2),  10,  1883,  No.  6,  175-191,  219-220  (corrections,  p.  343). 
•Annali  di  mat.,  (2),  13,  1885,  235-351,  "Excursions  arith.  4  I'lnfini." 
•"Nouv.  Ann.  Math.,  (3),  4,  1885,  421. 

I'Denkschr.  Akad.  Wien  (Math.),  49, 1,  1885,  47-8.  Sitzungsber.  Akad.  Wien,  112,  II  a,  1903, 
562;  115,  II  a,  1906,  589.  Cf.  A.  Berger,  Nova  Acta  Soc.  Upsal.,  (3),  14,  1891,  M6m.  2, 
p.  110;  E.  Landau,  Bull.  Soc.  Math.  France,  33,  1905,  241.     See  Gegenbauer,".'"  Ch.  X. 


Chap.  XI]  MISCELLANEOUS   THEOREMS   ON   DIVISIBILITY.  329 

Gegenbauer^°"  proved  that  the  arithmetical  mean  of  the  greatest  integers 
contained  in  k  times  the  remainders  on  the  division  of  n  by  1,  2, . . .,  n 
approaches 

k—l 

k\ogk-{-k-l-ki:i/x 

as  n  increases.    The  case  A;  =  2  is  due  to  Dirichlet. 

Gegenbauer^^  gave  formulas  involving  the  greatest  divisor  t^n),  not 
divisible  by  a,  of  the  integer  n.  In  particular,  he  gave  the  mean  value  of 
the  greatest  divisor  not  divisible  by  an  ath  power. 

L.  Gegenbauer,^^  employing  Merten's  function  ix  (Ch.  XIX)  and 
R{a)=a  —  \a\,  gave  the  three  general  formulas 


2  sV(^V(2/)  =  sV(A;)  -  i:  m  -  i  m, 
Xi  j/=i  \y /        k=i        A=i        it=i 


where  X2  ranges  over  the  divisors  >n  of  (r  — l)n+l,  (r  — l)n+2, .  .  .,  rn, 
while  Xi  ranges  over  all  positive  integers  for  which 

r-\-n    ~       g  r  n  \  '  g'  '       '  gj 

where  g  is  the  g.  c.  d.  of  r,  n.  Take  f{x)  =  1  or  0  according  as  x  is  an  sth 
power  or  not.     Then  the  functions 

(1)  2 /(A;),  2/x(^)/(2/) 

k  =  \  y  =  \      \y/ 

become  [-^m]  and  \{^),  with  the  value  0  if  the  exponent  of  any  prime 
factor  of  X  is  ^0,  1  (mod  s),  otherwise  the  value  (  —  1)",  where  a  is  the 
number  of  primes  occurring  in  x  to  the  power  /cs+1.     Thus 

2x,(x2) = \y^  -  \</V^r?^  -  [i/i\  • 

If  j{x)  =  0  or  1  according  as  x  is  divisible  by  an  sth  power  or  not,  the  func- 
tions (1)  become  Qs(w)  and  ix{\/x)j  the  former  being  the  number  of  integers 
^  m  divisible  by  no  sth  power.  If  J{x)  =  1  or  0  according  as  x  is  prime  or 
not,  the  functions  (1)  become  the  number  of  primes  ^m  and  a  simple  func- 
tion a(x) ;  then  the  third  formula  shows  that  the  mean  density  of  the  primes 

loiDenkschr.  Akad.  Wien  (Math.),  49,  II,  1885,  108. 
"Sitzungsber.  Akad.  Wiss.  Wien  (Math.),  94,  1886,  II,  714. 
i276id.,  97,  1888,  Ila,  420-6. 


330  History  of  the  Theory  of  Numbers.  [Chap,  xi 

If /(x)=log  X,  the  second  function  (1)  becomes  v{x),  ha\ing  the  value* 
log  p  when  x  is  a  power  of  the  prime  p,  otherwise  the  value  0.  Besides  the 
resulting  formulas,  others  are  found  by  taking  J{x)  =  v{x),  Jacobi's  symbol 
(A/x)  in  the  theor>'  of  quadratic  residues,  and  finally  the  number  of  repre- 
sentations of  X  by  the  system  of  quadratic  forms  of  discriminant  A. 

L.  Saint-Loup^^  represented  graphically  the  divisors  of  a  number. 
Write  the  first  300  odd  numbers  in  a  horizontal  line;  the  300  following 
numbers  are  represented  by  points  above  the  first,  etc.  Take  any  prime  as 
17  and  mark  all  its  multiples;  we  get  a  rectilinear  distribution  of  these  mul- 
tiples, which  are  at  the  points  of  intersection  of  two  sets  of  parallel  lines. 

J.  Hacks^^  proved  that  the  number  of  integers  ^m  which  are  divisible 
by  an  nth  power  >1  is 

p„(m)  =S  g„]  -2  [^„]  +S  [jtTi^.]  -  . . . , 

where  the  A;'s  range  over  the  primes  >1  [Bougaief^].  Then  yp2{fn)  = 
m—p2{'m)  is  the  number  of  integers  ^m  not  divisible  by  a  square  >1,  and 


^.w+^.(f)+^.(f)+...+^.([-^.)  = 


m. 


A  like  formula  holds  for  \p3  =  7n  —  p3(m),  using  quotients  of  m  by  cubes. 

L.  Gegenbauer""  found  the  mean  of  the  sum  of  the  reciprocals  of  the 
A:th  powers  of  those  divisors  of  a  term  of  an  unlimited  arithmetical  progres- 
sion which  are  rth  powers ;  also  the  probabiUty  that  a  term  be  divisible  by  no 
rth  power;  and  many  such  results. 

L.  Gegenbauer^^  noted  that  the  number  of  integers  1, .  .  . ,  n  not  divisible 
by  a  Xth  power  is 

(2)  Qx(n)=  S^[5J/x(x). 

Ch.  de  la  Valine  Poussin^®  proved  that,  if  x  is  divided  by  each  positive 
number  ky-\-b^x,  the  mean  of  the  fractional  parts  of  the  quotients  has  for 
x=  00  the  limit  1  — C;  if  x  is  divided  by  the  primes  ^x,  the  mean  of  the 
fractional  parts  of  the  quotients  has  for  x  =  co  the  limit  1  —  C.  Here  C  is 
Euler's  constant.^ 

L.  Gegenbauer^^  proved,  concerning  Dirichlet's^  quotients  Q  of  the 
remainders  (found  on  di\'iding  n  by  1 ,  2, .  .  . ,  n  in  turn)  by  the  corresponding 
divisors,  that  the  number  of  Q's  between  0  and  1/3  exceeds  the  number  of 
Q's  between  2/3  and  1  by  approximately  0.179n,  and  similar  theorems. 

♦Cf.  Bougaief  1"  of  Ch.  XIX. 

"Comptes  Rendus  Paris,  107.  1888,  24;  ficole  Norm.  Sup.,  7,  1890,  89. 

"Acta  Math.,  14,  1890-1,  329-336. 

""Sitzungsber.  Ak.  Wien  (Math.),  100,  Ila,  1891,  1018-1053. 

^Ibid.,  100,  1891,  Ila,  1054.     Denkschr.  Akad.  Wien  (Math.),  49  I,  II,  1885;  50  I,  1885.     Cf. 

Gegenbauer"  of  Ch.  X. 
"Annale.^  de  la  soc.  ac.  Bruxellea,  22,  1898,  84-90. 
"Sitzungsberichte  Ak.  Wiaa.  Wien  (Math.),  110,  1901,  Ila,  148-161. 


Chap.  XI]  MISCELLANEOUS  ThEOEEMS   ON  DIVISIBILITY.  331 

He  investigated  the  related  problem  of  Dirichlet.*  Finally,  he  used  as 
divisors  all  the  sth  powers  ^  n  and  found  the  ratio  of  the  number  of  remain- 
ders less  than  half  of  the  corresponding  divisors  to  the  number  of  the  others. 

L.  E.  Dickson^^"  and  H.  S.  Vandiver  proved  that  2">2(7i+l)(n'  +  l) .  .  ., 
if  1,  n,  n', .  •  •  are  the  divisors  of  an  odd  number  n>  3. 

R.  Birkeland^^  considered  the  sum  Sg  of  the  qth.  powers  of  the  roots 
Oi, . . .,  flm  of  z'^+Aiz"'~^-\- . . .  +^m  =  0.  If  Si, . . . ,  s^  are  divisible  by  the 
power  a^  of  a  prime  a,  then  A^  is  divisible  by  a"  unless  q  is  divisible  by 
a.  If  g  is  divisible  by  a,  and  a^'  is  the  highest  power  of  a  dividing  q,  then 
Afi  is  divisible  by  a^~^\  Then  (n+aai) . .  .  (n+aa^)  —n""  is  divisible  by  a^. 
In  particular,  the  product  of  m  consecutive  odd  integers  is  of  the  form 
1+2^^  if  m  is  divisible  by  2". 

E.  Landau^^  reproduced  Poussin's^^  proof  of  the  final  theorem  and  added 
a  simplification.  He  then  proved  a  theorem  which  includes  as  special  cases 
the  two  of  Poussin  and  the  final  one  by  Dirichlet^.  Given  an  infinite  class 
of  positive  numbers  q  without  a  finite  limit  point  and  such  that  the  number 
of  g's  ^a;  is  asymptotic  to  x/w{x),  where  w{x)  is  a  non-decreasing  posi- 
tive function  having 

x=oo    w{x) 

then  if  x  is  divided  by  all  the  q's  ^  x,  the  mean  of  the  fractional  parts  of  the 
quotients  has  for  x  =  «>  the  limit  1  —  C. 

St.  GuzeP°  wrote  5(n)  for  the  greatest  odd  divisor  of  n  and  proved  in 
an  elementary  way  the  asymptotic  formulas 

[X]  rfi  \X\    U^\ 

S  5(n)  =|-+0(x),  S  ^^  =f:r+0(l), 

n=l  O  n=l     n 

for  0  as  in  Pfeiffer^",  Ch.  X. 

A.  Axer^^  considered  the  x'''''(^)  decompositions  of  n  into  such  a  pair 
of  factors  that  always  the  first  factor  is  not  divisible  by  a  Xth  power  and 
the  second  factor  not  by  a  z^th  power,  X^2,  v'^2.  Then  S^iix'"'"  (n)  is 
given  asymptotically  by  a  compHcated  formula  involving  the  zeta  function. 

F.  RogeP^  wrote  Rx,n  for  the  algebraic  sum  of  the  partial  remainders 
<— [i]  in  (2),  with  n  replaced  by  2,  and  obtained 

Qx(2)=2P,,„-|-i2x.n,  Px.n=  n   (l-:;^x)' 

where  p„  is  the  nth  prime  and  Pn''^  2<p„+i.  He  gave  relations  between  the 
values  of  Qx{z)  for  various  2's  and  treated  sums  of  such  values,  and  tabu- 
lated the  values  of  ^2(2)  and  jB2,n  for  2^288.     He^^"  gave  many  relations 

I'^Amer.  Math.  Monthly,  10,  1903,  272;  11,  1904,  38-9. 

"Archiv  Math,  og  Natur.,  Kristiania,  26,  1904,  No.  10. 

"Bull.  Acad.  Roy.  Belgique,  1911,  443-472. 

"Wiadomoaci  mat.,  Warsaw,  14,  1910,  171-180. 

"Prace  mat.  fiz.,  22,  1911,  73-99  (Polish),  99-102  (German).    Review  in  Bull,  des  sc.  math., 

(2),  38,  II,  1914,  11-13. 
^Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  121,  Ila,  1912,  2419-52. 
"«/6id.,  122,  Ila,  1913,  669-700.     See  RogeP«  of  Ch.  XVIII. 


332  History  of  the  Theory  of  Numbers.  [Chap,  xi 

between  the  QAz),  relations  involving  the  number  A{z)  of  primes  ^z,  and 
relations  involving  both  Q's  and  A's. 

A.  Rothe'*^^  called  b  a  maximal  divisor  of  a  if  no  larger  divisor  of  a  con- 
tains 6  as  a  factor.  Then  a/b  is  called  the  index  of  b  with  respect  to  a.  1| 
If  also  c  is  a  maximal  divisor  of  b,  etc.,  a,b,  c, .  .  .,  I  are  said  to  form  a  series 
of  composition  of  a.  In  all  series  of  composition  of  a,  the  sets  of  indices 
are  the  same  apart  from  order  [a  corollary  of  Jordan's  theorem  on  finite 
groups  applied  to  the  case  of  a  cyclic  group  of  order  a]. 

*Weitbrecht-^  noted  tricks  on  the  divisibility  of  numbers. 

*E.  Moschietti-^  discussed  the  product  of  the  divisors  of  a  number. 

Each-^  of  the  consecutive  numbers  242,  243,  244,  245  has  a  square  factor 
>  1 ;  likewise  for  the  sets  of  three  consecutive  numbers  beginning  with  48 
or  98  or  124. 

C.  Avery  and  N.  Verson^^  noted  that  the  consecutive  numbers  1375, 
1376,  1377  are  divisible  by  5^  2\  3^  respectively. 

J.  G.  van  derCorput^^ evaluated  the  sum  of  thenth  powersof  all  integers, 
not  divisible  by  a  square  >1,  which  are  ^x  and  are  formed  of  r  prime 
factors  of  m. 

Greatest  Common  Divisor,  Least  Common  Multiple. 

On  the  number  of  divisions  in  finding  the  g.  c.  d.  of  two  integers,  see 
Lame^^  et  seq.  in  Ch.  XVII;  also  Binet^^  and  Dupre^. 

V.A.Lebesgue^^notedthatthel.c.m.of  a, . .  .,A;is(p]P3P5. .  ■)/{v2ViP&-  ■  ■) 
if  pi  is  the  product  of  a, ... ,  k,  while  p2  is  the  product  of  their  g.  c.  d.'s  two 
at  a  time,  and  ps  the  product  of  their  g.  c.  d.'s  three  at  a  time,  etc.  If  a,  6,  c 
have  no  common  divisor,  there  exist  an  infinitude  of  numbers  ax-^b  rela- 
tively prime  to  c. 

V.  Bouniakowsky^^  determined  the  g.  c.  d.  N  of  all  integers  represented 
by  a  polynomial /(x)  with  integral  coefficients  without  a  common  factor. 
Since  A^  divides  the  constant  term  of  f{x),  it  remains  to  find  the  highest 
power  p"  of  a  prime  p  which  divides  J{x)  identically,  i.  e.,  for  x  =  1,  2, .  .  . ,  p". 
Divide /(x)  by  Xp={x  —  1).  .  .{x  —  p)  and  call  the  quotient  Q  and  remain- 
der R.  Then  must  R^O  (mod  p")  for  x  =  l,.  .  .,  p,  so  that  each  coefficient 
of  R  is  divisible  by  p",  and  iu  =  Mu  vvhere  p"'  is  the  highest  power  of  p  divid- 
ing the  coefficients  of  i?.     If /ii  =  l,  wehaveju=  1.     Next,  let /ii>l.     Divide 

^^Zeitschrift  Math.-Xaturw.  Unterricht,  44,  1913,  317-320. 

"Vom  Zahlenkunststiick  zur  Zahlentheorie,  Korrcspondenz-Blatt  d.  Schulen  Wiirttembergs, 

Stuttgart,  20,  1913,  200-6. 
"Suppl.  al  Periodico  di  Mat.,  17,  1914,  115-6. 
i^Math.  Quest.  Educ.  Times,  36,  1881,  48. 
2'Math.  Miscellany,  Flushing,  N.  Y.,  1,  1836,  370-1. 
"Nieuw  Archief  voor  Wi.skunde,  (2),  12,  1918,  213-27. 
"Jour,  de  Math.,  (1),  6,  1841,  453. 
»Ibid.,  (1),  11,  1846,  41. 
'*Nouv.  Ann.  Math.,  8,  1849,  350;  Introduction  k  la  th6orie  des  nombres,  1862,  51-53;  Exercises 

d'analyse  num^rique,  1859,  31-32,  118-9. 
"M<5m.  acad.  sc.  St.  P^tersbourg,  (6),  ac.  math,  et  phys.  6  (so.  math.  phys.  et  nat.  8),  1857 

305-329  (read  1854);  extract  in  Bulletin.  13,  149. 


Chap.  XI]  GREATEST  COMMON   DiVISOR.  333 

Q  by  (x  —  p  —  l)...  {x  —  2p)  and  call  the  quotient  Q'  and  remainder  R'. 
Then  must  X^R'+XopQ'^O  and  hence  XpR'=0  (mod  p").  Thus  if  1^2  is 
the  exponent  of  the  highest  power  of  p  dividing  the  coefficients  of  R',  we 
have  At^M2  +  l-  In  general,  if  [j.^  and  X^-i  are  the  exponents  of  the  highest 
powers  of  p  dividing  the  coefficients  of  the  remainder  i?'^~^^  and  X^k-Dp 
identically,  then  fi^ iJLk+\k-i'  Finally,  if  l  =  [m/p],  /x^X^.  The  extension 
to  several  variables  is  said  to  present  difficulties.  [For  simpler  methods, 
see  Hensel^^  and  Borel.^^]     It  is  noted  (p.  323)  that 

are  identically  divisible  by  p".  It  is  conjectured  (p.  328)  that/(a;)/iV  repre- 
sents an  infinitude  of  primes  when  f{x)  is  irreducible. 

E.  Cesaro"  and  J.  J.  Sylvester^^  proved  that  the  probabihty  that  two 
numbers  taken  at  random  from  1 , .  .  . ,  n  be  relatively  prime  is  Q/tt^  asymp- 
totically. 

L.  Gegenbauer^^  gave  16  sums  involving  the  g.  c.  d.  of  several  integers 
and  deduced  37  asymptotic  theorems  such  as  the  fact  that  the  square  of 
the  g.  c.  d.  of  four  integers  has  the  mean  value  IS/tt^.  He  gave  the  mean 
of  the  kth.  power  of  the  g.  c.  d.  of  r  integers. 

J.  Neuberg^^"  noted  that,  if  two  numbers  be  selected  at  random  from 
1, .  .  .,N,  the  probability  that  their  sum  is  prime  to  N  is  k=cf){N)  0Tk/{N—l) 
according  as  N  is  odd  or  even. 

T.  J.  Stieltjes,^^  starting  with  a  set  of  n  integers,  replaced  two  of  them 
by  their  g.  c.  d.  and  1.  c.  m.,  repeated  the  same  operation  on  the  new  set, 
etc.  Finally,  we  get  a  set  such  that  one  number  of  every  pair  divides  the 
other.  Such  a  reduced  set  is  unique.  The  1.  c.  m.  of  a, .  .  . ,  ?  can  be 
expressed  (pp.  14-16)  as  a  product  a'.  .  J'  of  relatively  prime  factors  divi- 
ding a,...,l,  respectively.  The  1.  c.  m.  (or  g.  c.  d.)  oi  a,h,.  .  .,1  equals  the 
quotient  oi  P  =  ab.  .  .Ihy  the  g.  c.  d.  (or  1.  c.  m.)  of  P/a,  P/b, .  .  . ,  P/l. 

E.  Lucas^^  gave  theorems  on  g.  c.  d.  and  1.  c.  m. 

L.  Gegenbauer^^"  considered  in  connection  with  the  theory  of  primes, 
the  g.  c.  d.  of  r  numbers  with  specified  properties. 

J.  Hacks^^  expressed  the  g.  c.  d.  of  m  and  n  in  the  forms 

ql^]-.n^.^n,         2'|;[f]+2'}:g]-2[|][|]-^. 

where  €  =  0  or  1  according  as  m,  n  are  both  or  not  both  even. 

J.  Hammond^^  considered  arbitrary  functions  /  and  F  oi  p  and  a,  such 

"Mathesis,  1,  1881,  184;  Johns  Hopkins  Univ.  Circ,  2,  1882-3,  85. 

"Johns  Hopkins  Univ.  Circ,  2,  1883,  45;  Comptes  Rendus  Paris,  96,  1883,  409;  Coll.  Papers,  3, 

675;  4,  86. 
"Sitzungsberichte  Ak.  Wiss.  Wien  (Math.)   92,  1885,  II,  1290-1306. 
39«Math.  Quest.  Educ.  Times,  50,  1889,  113-4. 

^''Sur  la  theorie  des  nombres,  Annales  de  la  fac.  des  sciences  de  Toulouse,  4,  1890,  final  paper. 
^iTheorie  des  nombres,  1891,  345-6;  369,  exs.  4,  5. 
""Monatshefte  Math.  Phys.,  3,  1892,  319-335. 
*2Acta  Math.,  17,  1893,  208. 
^Messenger  Math.  24   1894^5   17-19. 


334  History  of  the  Theory  of  Numbers.  [Chap,  xi 


that /(p,  0)  =  1 ,  F{p,  0)=0,  and  any  two  integers  m=np",  n='n.p^,  where 
the  p's  are  distinct  primes  and,  for  any  p,  a  ^  0,  /3  ^  0.     Set 

rP{7n)=Uf{p,a),  $=2F(p,  a). 

By  the  usual  proof  that  mn  equals  the  product  of  the  g.  c.  d.  M  of  m  and  n 
by  their  1.  c.  m.  n,  we  get 

yPim)4/{n)=^P{M)xl/(jx),  <I>(w)+$(n)  =$(M)+$(iu). 

In  particular,  if  m  and  n  are  relatively  prime, 

yl/{m)\p{n)  =\p{vin),  $(w)+<l>(n)  =<l>(mn). 

These  hold  if  i/'  is  Euler's  0-function,  the  sum  o-(m)  of  the  divisors  of 
m  or  the  number  T{m)  of  divisors  of  ?n ;  also,  if  ^{m)  is  the  number  of  prime 
factors  of  vi  or  the  sum  of  the  exponents  a  in  m  =  Iip''. 

K.  Hensel^  proved  that  the  g.  c.  d.  of  all  numbers  represented  by  a 
polynomial  F{u)  of  degree  n  with  integral  coefficients  equals  the  g.  c.  d.  of 
the  values  of  F{u)  for  any  n+1  consecutive  arguments.  For  a  polynomial 
of  degree  ni  in  Ui,  712  in  ^2,  •  ■  •  we  have  only  to  use  ni  +  1  consecutive 
values  of  ui,  712+ 1  consecutive  values  of  U2,  etc. 

F.  Klein"*^  discussed  geometrically  Euclid's  g.  c.  d.  process. 

F.  ^Vlertens^^  calls  a  set  of  numbers  primitive  if  their  g.  c.  d.  is  unity. 
If  7719^0,  k>\,  and  ai,. . .,  o^,  m  is  a  primitive  set,  we  can  find  integers 
Xi,.  .  .,  Xk  so  that  ai-\-mxx,.  .  .,  ak+mxk  is  a  primitive  set.  Let  d  be  the 
g.  c.  d.  of  fli, .  .  .,  Oi-  and  find  5,  ji  so  that  db-\-vi^  =  \.  Take  integral  solu- 
tions a  of  OittiH-.  .  .+akak  =  d  and  primitive  solutions  ^i  not  all  zero  of 
aij3i+ .  .  . +aii3/;  =  0.  Then  7i=/3,+6a,('i  =  l,.  .  .,  k)  is  a  primitive  set. 
Determine  integers  ^  so  that  71^1+.  .  .+7*^^  =  1  and  set  a:,=/i^<.  Then 
Ci+TTix,  form  a  primitive  set. 

R.  Dedekind^^  employed  the  g.  c.  d.  d  oi  a,h,  c;  the  g.  c.  d.  (6,  c)  =Oi, 
(c,  a)  =  61,  (a,  h)  =  Ci.  Then  a'  =  ajd,  h'  =  hi/d,  c'  =  C]/d  are  relatively  prime 
in  pairs.  Then  cf6'c' is  the  1.  c.  m.  of  61,  Ci,  and  hence  is  a  divisor  of  a.  Thus 
a  =  dh'c'a",  h  =  dc'a'b",  c  =  da'h'c".  The  7  numbers  a', .  .  .,a" ,. .  .,d  are  called 
the  " Kerne"  of  a,  h,  c.  The  generalization  from  3  to  n  numbers  is  given. 

E.  Borel'*^  considered  the  highest  power  of  a  prime  p  which  di\'ides  a 
polynomial  P{x,  y,.  .  .)  with  integral  coefficients  for  all  integral  values  of 
X,  y,.  .  ..  If  each  exponent  is  less  than  p,  we  have  only  to  find  the  highest 
power  of  p  dividing  all  the  coefficients.  In  the  contrary  case,  reduce  all 
exponents  below  p  by  use  of  x^  =  x-\-pxi,Xi'  =  Xi  -\-px2,.  .  .  and  proceed  as 
above  with  the  new  polynomial  in  x,  Xi,  X2,...,y,yi,....  Then  to  find  all 
arithmetical  divisors  of  a  polynomial  P,  take  as  p  in  turn  each  prime  less 
than  the  highest  exponent  appearing  in  P. 

L.  Kronecker^^  found  the  number  of  pairs  of  integers  i,  k  having  t  as 
their  g.  c.  d.,  where  l^i^m,  l^k^n.     The  quotient  of  this  number  by 

«Jour.  fur  Math.,  116,  1896,  350-6. 

"Ausgewahlte  Kapitel  der  Zahlentheorie,  I,  1896. 

*«Sitzung8berichte  Ak.  Wiss.  Wien  (Math.),  106,  1897,  II  a,  132-3. 

*^Ueber  Zerlegungen  von  Zahlen  durch  d.  grossten  gemeinsamen  Teller,  Braunschweig,  1897. 

"BuU.  Sc.  Math.  Astr.,  (2),  24  I,  1900,  75-80.     Cf.  Borel  and  Drach'^  of  Ch.  III. 

"Vorlesungen  uber  Zahlentheorie,  I,  1901,  306-312. 


i 


Chap.  XI]  GREATEST  COMMON   DiVISOR.  335 

mn  is  the  mean.  When  m  and  n  increase  indefinitely,  the  mean  becomes 
Q/iirH"^).  The  case  ^=1  gives  the  probability  that  two  arbitrarily  chosen 
integers  are  relatively  prime;  the  proof  in  Dirichlet's  Zahlentheorie  fails  to 
establish  the  existence  of  the  probability. 

E.  DintzP°  proved  that  the  g.  c.  d.  A(a, .  . .,  e)  is  a  linear  function  of 
a, .  .  . ,  e,  and  reproduced  the  proof  of  Lebesgue's^^  formula  as  given  in 
Merten's  Vorlesungen  iiber  Zahlentheorie  and  by  de  Jough.^^ 

A.  Pichler,^°"  given  the  1.  c.  m.  or  g.  c.  d.  of  two  numbers  and  one  of  them, 
found  values  of  the  other  number. 

J.  C.  Kluyver^^  constructed  several  functions  z  (involving  infinite  series 
or  definite  integrals)  which  for  positive  integral  values  of  the  two  real 
variables  equals  their  g.  c.  d.  He  gave  to  Stern's^^  function  the  somewhat 
different  form  [;r]     /     \ 

W.  Sierpinski^^  stated  that  the  probability  that  two  integers  ^n  are 
relatively  prime  is  .   „         p  -,2 

contrary  to  Bachmann,  Analyt.  Zahlentheorie,  1894,  430. 
G.  Darbi^^  noted  that  if  a  =  (a,  N)  is  the  g.  c.  d.  of  a,  N, 


(iV,abc...)=a(6,^)(c,^(^_^/J 


and  gave  a  method  of  finding  the  g.  c.  d.  and  1.  c.  m.  of  rational  fractions 
without  bringing  them  to  a  common  denominator. 

E.  Gelin^®  noted  that  the  product  of  n  numbers  equals  ah,  where  a  is 
the  1.  c.  m.  of  their  products  r  at  a  time,  and  h  is  the  g.  c.  d  of  their  products 
n  —  r  at  a  time. 

B.  F.  Yanney^'^  considered  the  greatest  common  divisors  Di,  D2, ...  of 
tti, .  .  . ,  a„  in  sets  of  k,  and  their  1.  c.  m.'s  Li,  L2, .  .  . .     Then 

HA  Lt'  ^  (ai .  .  .  «n)^  ^  n  D  ^-^L„  5  =  (^y  c  =  (^~  I) . 

The  limits  coincide  ii  k  =  2.    The  products  have  a  single  term  iik  =  n. 

P.  Bachmann^^  showed  how  to  find  the  number  N  obtained  by  ridding 
a  given  number  n  of  its  multiple  prime  factors.  Let  d  be  the  g.  c.  d.  of  n 
and  0(n).  If  d  =  n/d  occurs  to  the  rth  power,  but  not  to  the  (r+l)th  power 
in  n,  set  ni  =  n/5^     From  rii  build  di  as  before,  etc.     Then  N  =  86182 .... 

"Zeitschrift  fiir  das  Realschulwesen,  Wien,  27.  1902,  654-9,  722. 

6o«76id.,  26,  1901,  331-8. 

"Nieuw  Archief  voor  Wiskunde,  (2),  5,  1901,  262-7. 

^^K.  Ak.  Wetenschappen  Amsterdam,  Proceedings  of  the  Section  of  Sciences,  5,  II   1903,  658- 

662.     (Versl.  Ak.  Wet.,  11,  1903,  782-6.) 
"Jour,  flir  Math.,  102,  1888,  9-19. 
"Wiadomosci  Mat.,  Warsaw,  11,  1907,  77-80. 
^^Giornale  di  Mat.,  46,  1908,  20-30. 
S6I1  Pitagora,  Palermo,  16,  1909-10,  26-27. 
"Amer.  Math.  Monthly,  19,  1912,  4-6. 
6«Archiv  Math.  Phys.,  (3),  19,  1912,  283-5. 


336 


History  of  the  Theory  of  Numbers. 


[Chap.  XI 


Erroneous  remarks^^  have  been  made  on  the  g.  c.  d.  of  2""  — 1,  3""  — 1. 

]\I.  Lecat^°  noted  that,  if  a,j  is  the  1.  c.  m.  of  i  and  j,  the  determinant 
loyl  was  evaluated  by  L.  Gegenbauer,^^  who,  however,  used  a  law  of  multi- 
pHcation  of  determinants  valid  only  when  the  factors  are  both  of  odd  class. 

J.  Barinaga®^"  proved  that,  if  5  is  prime  to  iV  =  nk,  the  sum  of  those  terms 
of  the  progression  A'',  N-\-d,  iV+25,  .  .  . ,  which  are  between  nk  and  n{k-\-hd) 
and  which  have  with  n  =  mp  the  g.  c.  d.  p,  is  ^n<l){n/p){2k-\-hd)h. 

R.  P.  Willaert®-  noted  that,  if  P{n)  is  a  polynomial  in  n  of  degree  p  with 
integral  coefficients,  f{n)=aA'"'-\-P{n)  is  divisible  by  D  for  every  integral 
value  of  7}  if  and  only  if  the  difference  A''f{0)  of  the  Ath  order  is  di\"isible 
by  D  for  k  =  0,  1,.  . .,  p-\-l.  Thus,  if  p  =  l,  the  conditions  are  that  /(O), 
/(l),/(2)  be  divisible  by  D. 

*H.  Verhagen^^  gave  theorems  on  the  g.  c.  d.  and  1.  c.  m. 

H.  H.  ]\Iitchell^  determined  the  number  of  pairs  of  residues  a,  b  modulo  X 
whose  g.  c.  d.  is  prime  to  X,  such  that  ka,  kb  is  regarded  as  the  same  pair  as 
a,  b  when  k  is  prime  to  X,  and  such  that  X  and  ax  +  by  have  a  given  g.  c.  d. 

W.  A.  Wijthoff^^  compared  the  values  of  the  sums 

S  (-l)'"-WF{(w,  a)},      "s    m'F{{m,a)},    s  =  l,  2, 

m=l  m=l 

where  {m,  a)  is  the  g.  c.  d.  of  m,  a,  while  F  is  any  arithmetical  function. 

F.  G.  W.  Brown  and  C.  M.  Ross^^  wTote  h,  U,  ...,ln  for  the  1.  c.  m 
the  pau-s  Ai,  A^;  A^,  Az;  . .  . ;  A„,  Ai,  and  gi,  g^,  ■  ■ .,  gn  for  the  g 
these  pairs,  respectively.     If  L,  G  are  the  1.  c,  m.  and  g.  c.  d.  of  Ai, 
A„,  then 

gm . .  .gn  =  G'', 


.  c 

^2, 


of 
d.  of 


9i92 


C.  de  Polignac^^  obtained  for  the  g.  c.  d 

(a\btJi)={a,by{\,fx).(--\ 

\{a,  b)    (X,  m) 

Sylvester^*  and  others  considered  the  g.  c.  d 


9n  G' 

(a,  6)  of  a,  6  results  like 
fi    \     /     b  X 


J     \(a,  6)'  (X,  mV 


6)'  (X,  m)> 

of  Z)„  and  Z)„+i  where  D^ 
is  the  n-rowed  determinant  whose  diagonal  elements  are  1,  3,  5,  7,  .  .  ., 
and  having  1,  2,  3,  4,  ...  in  the  line  parallel  to  that  diagonal  and  just  above 
it,  and  units  in  the  parallel  just  below  it,  and  zeros  elsewhere. 

On  the  g.  c.  d.,  see  papers  33-88,  215-6,  223  of  Ch.  V,  Cesaro"  of  Ch. 
X,  Cesaro^' '  of  Ch.  XI,  and  Kronecker^^  of  Ch.  XIX. 

"L'interm6diaire  des  math.,  20,  1913,  112,  183-4,  228;  21,  1914,  36-7. 

^''Ibid.,  21,  1914,  91-2. 

•'Sitzungs.  Ak.  Wiss.  Wien  (Math.),  101,  1892,  II  a,  425-494. 

""Annaes  Sc.  Acad.  Polyt.  do  Porto,  8,  1913,  248-253. 

^Mathesis,  (4),  4,  1914,  57. 

"Nieuw  Tijdschria  voor  Wiskunde,  2,  1915,  143-9. 

"Annals  of  Math.,  (2),  18,  1917,  121-5. 

"Wiskundige  Opgaven,  12,  1917,  249-251. 

"Math.  Quest,  and  Solutions,  5,  1918,  17-18. 

«'Xouv.  Corresp.  Math.,  4,  1878,  181-3. 

6»Math.  Que.st.  Educ.  Times,  36,  1881,  97-8;  correction,  117-8. 


CHAPTER  XII. 

CRITERIA  FOR  DIVISIBILITY  BY  A  GIVEN  NUMBER. 

In  the  Talmud^  lOOa+6  is  stated  to  be  divisible  by  7  if  2a+b  is  divis- 
ible by  7. 

Hippolytos^",  in  the  third  century,  examined  the  remainder  on  the 
division  of  certain  sums  of  digits  by  7  or  9,  but  made  no  appHcation  to 
checking  numerical  computation. 

Avicenna  or  Ibn  Sina  (980-1037)  is  said  to  have  been  the  discoverer 
of  the  familiar  rule  for  casting  out  of  nines  (cf .  Fontes^^) ;  but  it  seems  to 
have  been  of  Indian  origin.-^'' 

Alkarkhi^^  (about  1015)  tested  by  9  and  11. 

Ibn  Musa  Alchwarizmi^'*  (first  quarter  of  the  ninth  century)  tested  by  9. 

Leonardo  Pisano^^  gave  in  his  Liber  Abbaci,  1202,  a  proof  of  the'  test 
for  9,  and  indicated  tests  for  7,  11. 

Ibn  Albanna^-'^  (born  about  1252),  an  Arab,  gave  tests  for  7,  8,  9. 

In  the  fifteenth  century,  the  Arab  Sibt  el-Maridini^''  tested  addition  by 
casting  out  multiples  of  7  or  8. 

Nicolas  Chuquet^^  in  1484  checked  the  four  operations  by  casting  out  9's. 

J.  Widmann^''  tested  by  7  and  9. 

Luca  Paciuolo^  tested  by  7,  as  well  as  by  9,  the  fundamental  operations, 
but  gave  no  rule  to  calculate  rapidly  the  remainder  on  division  by  7. 

Petrus  Apianus^"  tested  by  6,  7,  8,  9. 

Robert  Recorde^''  tested  by  9. 

Pierre  ForcadeP  noted  that  to  test  by  7  =  10  —  3  we  multiply  the  first 
digit  by  3,  subtract  multiples  of  7,  add  the  residue  to  the  next  digit,  then 
multiply  the  sum  by  3,  etc. 

Blaise  Pascal^  stated  and  proved  a  criterion  for  the  divisibility  of  any 
number  N  by  any  number  A.  Let  ri,  r2, 7*3, .  . . ,  be  the  remainders  obtained 
when  10,  lOfi,  lOrg, ...  are  divided  by  A.  Then  iV  =  a+ 106  + 100c+  ...  is 
divisible  by  A  if  and  only  if  a-\-rib-\-r2C+ .  .  .  is  divisible  by  A. 

'Babylonian  Talmud,  Wilna  edition  by  Romm,  Book  Aboda  Sara,  p.  96. 

i«M.  Cantor,  Geschichte  der  Math.,  ed.  3,  I,  1907,  461. 

^^Ibid.,  511,  611,  756-7,  763-6. 

i^Cf.  Carra  de  Vaux,  Bibliotheca  Math.,  (2),  13,  1899,  33-4. 

i<^M.  Cantor,  Geschichte  der  Math.,  ed.  3,  I,  1907,  717. 

i^Scritti,  1,  1857,  8,  20,  39,  45;  Cantor,  Geschichte,  2,  1892,  8-10. 

'/Le  TaUfhys  d'Ibn  Albanna  public  et  traduit  par  A.  Marre,  Atti  Accad.  Pont.  Nuovi  Lincei, 

17,  1863-4,  297.     Cf.  M.  Cantor,  Geschichte  Math.,  I,  ed.  2,  757,  759;  ed.  3,  805-8. 
iffLe  Triparty  en  la  science  de  nombres.  Bull.  Bibl.  St.  Sc.  Math.,  13,  1880,  602-3. 
^''Behede  vnd  hubsche  Rechnung .  . . ,  Leipzig,  1489. 

^Summa  de  arithmetica  geometria  proportion!  et  proportionalita,  Venice,  1494,  f.  22,  r. 
2"Ein  newe.  .  .Kauffmans  Rechnung,  Ingolstadt,  1527,  etc. 
^^The  Grovnd  of  Artes,  London,  c.  1542,  etc. 
^L'Arithmeticqve  de  P.  Forcadel  de  Beziers,  Paris,  1556,  59-60. 
*De  numeris  multiphcibus,  presented  to  the  Acad^mie  Parisienne,  in  1654,  first  published  in 

1665;  Oeuvres  de  Pascal,  3,  Paris,  1908,  311-339;  5,  1779,  123-134. 

337 


338  History  of  the  Theory  of  Numbers.  [Chap,  xii 

D'Alembert^  noted  that  if  N  =  A-10'"+B-W-\-.  ..+E  is  divisible  by 
10-6,  then  Ab"'+Bb"-\- .  ..+E  is  divisible  by  10-6;  if  A'  is  divisible  by 
10+6,  then  A(-6)'"+B(-6)"+ .  .  . +^  is  divisible  by  10+6.  The  case 
6  =  1  gives  the  test  for  divisibiUty  by  9  or  11.  By  separating  A''  into  parts 
each  with  an  even  number  of  digits,  N  =  A-10"'+  .  .  .  +E,  where  m, .  .  .are 
even;  then  if  A^  is  di\'isible  by  100-6,  Ah"*^^ -\- .  .  .  +  E  is  divisible  by 
100-6. 

De  Fontenelle^  gave  a  test  for  divisibility  by  7  which  is  equivalent  to 
the  case  6  =  3  of  D'Alembert;  to  test  3976  multiply  the  first  digit  by  3  and 
add  to  the  second  digit;  it  remains  to  test  1876.  For  proof  see  F.  Sanvitali, 
Hist.  Literariae  Italiae,  vol.  6,  and  Castelvetri.^ 

G.  W.  Kraft^  gave  the  same  test  as  Pascal  for  the  factor  7. 

J.  A.  A.  Castelvetri^  gave  the  test  for  99:  Separate  the  digits  in  pairs, 
add  the  two-digit  components,  and  see  if  the  sum  is  a  multiple  of  99.  For 
999  use  triples  of  digits. 

Castelvetri^  tested  1375,  for  example,  for  the  factor  11  by  noting  that 
13+75  =  88  is  divisible  by  11.  If  the  resulting  sum  be  composed  of  more 
than  two  digits,  pair  them,  add  and  repeat.  To  test  for  the  factor  111, 
separate  the  digits  into  triples  and  add.  The  proof  follows  from  the  fact 
that  lO-*"  has  the  remainder  1  when  divided  by  11. 

J.  L.  Lagrange^°  modified  the  method  of  Pascal  by  using  the  least 
residue  modulo  A  (between  — .4/2  andyl/2)  in  place  of  the  positive  residue. 
He  noted  that  if  a  number  is  written  to  any  base  a  its  remainder  on  division 
by  a  —  1  is  the  same  as  for  the  sum  of  its  digits. 

J.  D.  Gergonne^^  noted  that  on  di\dding  iV  =  Ao+Ai6"*+A26^'"+ .  . ., 
written  to  base  6,  by  a  di\'isor  of  6'"  — 1,  the  remainder  is  the  same  as  on 
dividing  the  sum  A0+A1+A2+ ...  of  its  sets  of  m  digits.  Similarly  for 
6'"+l  and  A0-A1+A2-A3+ •  •  •• 

C.  J.  D.  HilP^  gave  rules  for  abbre\dating  the  testing  for  a  prime  factor 
p,  for  p<300  and  certain  larger  primes. 

C.  F.  Liljevalch^^a  ^^^^^  ^^^^^  jf  lO^a-/?  is  di\^sible  by  p  then  a- 10^6 
will  be  a  multiple  of  p  if  and  only  if  aa  — /36  is  a  multiple  of  p. 

J.  ]\I.  Argardh"  used  Hill's  symbols,  treating  divisors  7,  17,  27,  1429. 

F.  D.  Herter^^  noted  that  a  +  106+100c+ ...  is  divisible  by  10n±l  if 

'Manuscript  R.  240*  6  (8°),  Bibl.  Inst.  France,  21,  ff.  316-330,  Sur  une  propri^t^  des  nombres. 
•Histoire  Acad.  Paris,  ann^e  1728,  51-3.  'Comm.  Ac.  Sc.  Petrop,  7,  ad  annos  1734-5,  p.  41. 
»De  Bononiensi  Scientiarum  et  Artium  Institute  atque  Academia  Comm.,  4,  1757;  commen- 

tarii,  113-139;  opuscula,  242-260. 
•De  Bononiensi  Scientiarum  et  Artium  Institute  atque  Academia  Comm.,  vol.  5,  1767,  part  1, 

pp.  134-144;  part  2,  108-119. 
"Lemons  6\6m.  sur  les  math,  donn^es  k  I'^cole  normale  en  1795,  Jour,  de  I'^cole  polytechnique, 

vols.  7,  8,  1812,  194-9;  OemTes,  7,  pp.  203-8. 
"Annales  de  math,  (ed.,  Gergonne),  5,  1814-5,  170-2. 
"Jour,  fur  Math.,  11,  1834,  251-261;  12,  1834,  355.     Also,  De  factoribua  numerorum  com- 

positonim  dignoscendis,  Lund,  1838. 
"<»De  factoribus  numerorum  compositorura  dignoscendis,  Lund,  1838. 
"De  residuis  ex  divisione.  .  .,  Diss.  Lund,  1839. 
"Ueber  die  Kennzeichen  der  Theiler  einer  Zahl,  Progr.  Berlin,  1844. 


Chap.  XII]  CRITERIA   FOR  DIVISIBILITY.  339 

a=F&/nH-c/n^=F  .  .  .  is  divisible  by  10n±l,  with  a  like  test  for  10n±3 
(replacing  1/n  by  3/n),  and  deduced  the  usual  tests  for  9,  11,  7,  13,  etc. 

A.  L.  Crelle^^  noted  that  to  test  XmA""-^ .  .  .  +XiA-i-Xo  for  the  divisor  s 
we  may  select  any  integer  n  prime  to  s,  take  r=nA  (mod  s),  and  test 

for  the  divisor  s.  For  example,  if  A  =  10,  s  =  7,  10^=— 1  (mod  7),  so  that 
Xo  —  Xi-\-X2—.  .  .  ±0:^  is  to  be  tested  for  the  divisor  7,  where  Xq,  .  .  .are  the 
three-digit  components  of  the  proposed  number  from  right  to  left.  Simi- 
larly for  s=9,  11,  13,  17,  19. 

A.  Transon^^  gave  a  test  for  the  divisibility  of  a  number  by  any  divisor 
of  10"-n±l. 

A.  Niegemann^^  noted  that  354578385  is  divisible  by  7  since  35457 -f 
2X8385  is  divisible  by  7.  In  general  if  the  number  formed  by  the  last  m 
digits  of  A^  is  multiplied  by  k,  and  the  product  is  added  to  the  number  de- 
rived from  N  by  suppressing  those  digits,  then  N  is  divisible  by  d  if  the 
resulting  sum  is  divisible  by  d.  Here  k{0<k<d)  is  chosen  so  that  10'"/:  — 1 
is  divisible  by  d.     Thus  k  =  2  if  m  =  4,  d  =  7. 

Many  of  the  subsequent  papers  are  listed  at  the  end  of  the  chapter. 

H.  Wilbraham^^  considered  the  exponent  p  to  which  10  belongs  modulo 
m,  where  m  is  not  divisible  by  2  or  5.  Then  the  decimal  for  1/m  has 
a  period  of  p  digits.  If  any  number  N  be  marked  off  into  periods  of  p- 
digits  each,  beginning  with  units,  so  that  A^  =  ai  +  10^a2+10^^a34- •  •  •, 
then  ai-\-a2-]-  ■  ■  ■  =  N  (mod  m),  and  N  is  divisible  by  m  if  and  only  if 
«i+<J2+  •  •  ■  is  divisible  by  m. 

E.  B.  Elliott^^  let  10''  =  MD+r^,.  Thus  iV  =  10%-h . . . +10ni+no  is 
divisible  by  D  if  N ='ZfnjMD-{-'Znjrj  is  divisible  by  D.  The  values  of  the  r's 
are  tabulated  for  D  =  S,  7,  S,  9,  11,  13,  17. 

A.  Zbikowski^°  noted  that  N  =  a-\-10kis  divisible  by  7  if  k  —  2a  is  divis- 
ible by  7.  If  8  is  of  the  form  lOn+1,  N  =  a-\-10k  is  divisible  by  5  if  A;  — na  is 
divisible  by  d ;  this  holds  also  if  5  is  replaced  by  a  divisor  of  a  number  10n+ 1 . 

V.  ZeipeP^  tests  for  a  divisor  h  by  use  of  nh  =  10d-\-l.  Then  10a2+ai  is 
divisible  by  6  if  a2  —  aid  is  divisible  by  b. 

J.  C.  Dupain^^  noted,  for  use  when  division  by  p  —  1  is  easy,  that 
N={p  —  1)Q+R  is  divisible  by  p  if  R  —  Q  is  divisible  by  p. 

F.  Folie^^  proved  that  if  a,  c  are  such  that  ak'^ck  =  mp  then  AB-\-C  is 
divisible  by  the  prime  p  =  aB-{-c  if  Ak'=^Ck  =  m'p,  provided  a,  c,  k,  k'  are 

isjour.  fur  Math.,  27,  1844,  125-136. 

16N0UV.  Ann.  Math.,  4,  1845,  173-4  (cf.  81-82  by  O.  R.). 

i^Entwickelung  u.  Begrlindung  neuer  Gesetze  iiber  die  Theilbarkeit  der  Zahlen.     Jahresber. 

Kath.  Gym.  Koln,  1847-8. 
i^Cambridge  and  Dublin  Math.  Jour.,  6,  1851,  32. 
"The  Math.  Monthly  (ed.  Runkle),  1,  1859,  45-49. 
"oBull.  ac.  sc.  St.  Petersbourg,  (3),  3,  1861,  151-3;  Melanges  math.  astr.  ac.  St.  P^tersbourg, 

3,  1859-66,  312. 
2iOfversigt  finska  vetensk.  forhandl.,  Stockhohn,  18,  1861,  425-432. 
«Nouv.  Ann.  Math.,  (2),  6,  1867,  368-9. 
23M6m.  Soc.  Sc.  Liege,  (2),  3,  1873,  85-96. 


340  History  of  the  Theory  of  Numbers.  [Chap,  xii 

not  multiples  of  p.  Application  is  made  to  the  primes  p^37.  Again,  if 
p  is  a  prime  and 

aB-+cB+d  =  ak"-\-ck'-\-dk  =  Ak"-{-Ck'-\-Dk  =  mp, 

where  k,  k',  k"  are  prime  to  p,  then  AB~-\-CB-[-D  is  divisible  by  p  provided 
k'^  —  kk"  is  a  multiple  of  p. 

C.  F.  IMoller  and  C.  Holten-^  would  test  the  divisibility  of  n  by  a  given 
prime  p  by  seeking  a  such  that  ap=  =*=  1  (mod  10)  and  subtracting  from  n 
such  a  multiple  of  ap  that  the  difference  ends  with  zero. 

L.  L.  Hommel"-^  made  remarks  on  the  preceding  method. 

V.  SchlegeP^  noted  that  if  the  di\isor  to  be  tested  ends  with  1,  3,  7  or  9, 
its  product  by  1,  7,  3  or  9  is  of  the  form  (i  =  lOX+1.  Then  a,  with  the  final 
digit  u,  is  divisible  by  d\i  ai  =  {a  —  ud)/\0  is.     Then  treat  Oj  aswe  did  a,  etc. 

P.  Otto"^  would  test  Z  for  a  given  prime  factor  p  by  seeking  a  number  n 
such  that  if  the  product  by  n  of  the  number  formed  by  the  last  s  digits  of  Z 
be  subtracted  from  the  number  represented  by  the  remaining  digits,  the 
remainder  is  di\'isible  by  p  if  and  only  if  Z  is.  ^Material  is  tabulated  for  the 
application  of  the  method  when  p<100. 

N.  V.  Bougaief-^"  noted  that  a^. .  .Ci  to  base  B  is  divisible  by  D  if 
fli . .  .a„  to  base  d  is  divisible  by  D,  where  dB=  1  (mod  D).  For  jB  =  10  and 
Z)  =  10/1+9,  1,  3,  7,  we  may  take  c?  =  n  +  l,  9?i+l,  3nH-l,  7n+5,  respec- 
tively.    Again,  kB--\-aB-\'h  is  di\'isible  by  D  if  kB-\-a-\-hd  is  divisible. 

W.  Mantel  and  G.  A.  Oskamp'^  proved  that,  to  test  the  di\isibility  of  a 
number  to  any  base  by  a  prime,  the  value  of  the  coefficient  required  to 
eliminate  one,  two, .  .  .  digits  on  subtraction  is  periodic.  Also  the  number  of 
terms  of  the  period  equals  the  length  of  the  period  of  the  periodic  fraction 
arising  on  division  by  the  same  prime. 

G.  Dostor-^''  noted  that  \{)t-\-u  is  divisible  by  any  divisor  a  of  10A±  1  if 
t=^Au  is  di\dsible  by  a.    [A  case  of  Liljevalch^-''.] 

Hocevar^^  noted  that  if  N,  wTitten  to  base  a,  is  separated  into  groups 
Gi,  (x2, .  .  .  each  of  q  digits,  N  is  di\'isible  by  a  factor  of  a'+l  if  Gi  — G2+G3 
-  ...  is  divisible.  Thus,  for  a  =  2,  g  =  4,  A'' =  104533,  or  11001100001010101 
to  base  2  is  divisible  by  17  since  0101-0101  +  1000-1001  +  1  =  0. 

J.  Delboeuf^°  stated  that  if  p,  q  are  such  that  pa-\-qh  is  a  multiple  of  Z)  and 
if  N  =  Aa.-\-B^  is  a  multiple  of  Z)  =  aa  +  6/3,  then  pA+qB  is  a  multiple  of  Z). 

E.  Catalan  {ibid.,  p.  508)  stated  and  proved  the  preceding  test  in  the 
following  form:   If  a,  h  and  also  a',  h'  are  relatively  prime,  and 

iV  =  aa'+66',  Nx  =  Aa-\-Bh,  Nx' =  A'a'+B'h', 

then  AA'-\-BB'  is  a  multiple  of  A^  (and  a  sum  of  2  squares  if  N  is). 

"Tidsskrift  for  Math.,  (3^,  5,  1875,  177-180.  «*Tidsskrift  for  Math.,  (3),  6,  1876,  15-19. 

"Zeitschrift  Math.  Phys.,  21,  1876,  365-6.  »'Zeitschrift  Math.  Phys.,  21, 1876,  366-370. 

^"''Mat.  Sbornik  (Math.  Soc.  Moscow),  8,  1876,  I,  501-5. 

"Nieuw  Archief  voor  Wiskunde.  Amsterdam,  4,  1878,  57-9,  83-94. 

"-^.Ajchiv  Math.  Phys.,  63,  1879,  221-4. 

"Zur  Lehre  von  der  Teilbarkeit.  . .,  Prog.  Imisbruck,  1881. 

"La  Revue  Scientifique  de  France,  (3),  38,  1886,  377-8. 


Chap.  XII]  CRITERIA   FOR  DIVISIBILITY.  341 

Noel  (ibid.,  378-9)  gave  tests  for  divisors  11,  13,  17,.  .  .,  43. 

Bougon  {ibid.,  508)  gave  several  tests  for  the  divisor  7.  For  example, 
a  number  is  divisible  by  7  if  the  quadruple  of  the  number  of  its  tens  dimin- 
ished by  the  units  digit  is  divisible  by  7,  as  1883  since  188-4  —  3  =  749  is 
divisible  by  7.     J.  Heilmann  (ibid.,  187)  gave  a  test  for  the  divisor  7. 

P.  Breton  and  Schobbens  {ibid.,  444-5)  gave  tests  for  the  divisor  13. 

S.  Dickstein^^  gave  a  rule  to  reduce  the  question  of  the  divisibility  of  a 
number  to  any  base  by  another  to  that  for  a  smaller  number. 

A.  Loir^2  gave  a  rule  to  test  the  divisibihty  of  N,  having  the  units  digit  a, 
by  a  prime  P.  From  {N  —  a)/10,  subtract  the  product  of  a  by  the  number, 
say  (mP  — 1)/10,  of  tens  in  such  a  multiple  mP  of  P  that  the  units  digit  is  1. 
To  the  difference  obtained  apply  the  same  operation,  etc.,  until  we  exhaust 
N.     If  the  final  difference  be  P  or  0,  N  is  divisible  by  P. 

R.  Tucker^^  started  with  a  number  N,  say  5443,  cut  off  the  last  digit  3 
and  defined  ^2  =  544  — 2-3  =  538,  ^3  =  53  — 2-8,  etc.  If  any  one  of  the  ^^'s  is 
divisible  by  7,  N  is  divisible  by  7.  R.  W.  D.  Christie  (p.  247)  extended  the 
test  to  the  divisors  11, 13, 17,  37,  the  respective  multiphers  being  1,  9,  5,  11, 
provided  always  the  number  tested  ends  with  1,  3,  7  or  9. 

R.  Perrin^^  would  find  the  minimum  residue  of  N  modulo  p  as  follows. 
Decompose  N,  written  to  base  x,  into  any  series  of  digits,  each  with  any 
number  of  digits,  say  A,  Bi,  Cj,. .  .,  where  Bi  has  i  digits.  Let  p  be  any 
integer  prime  to  x  and  find  qi  so  that  qiX^^  1  (mod  p).  Let  a  be  any  one  of 
the  integers  prime  to  p  and  numerically  <p/2.  Let  j8  be  the  ith  integer 
following  a  in  that  one  of  the  series  containing  a  which  are  defined  thus: 
as  the  first  series  take  the  residues  modulo  p  of  1,  g,  g^, .  .  . ;  as  the  second 
series  take  the  products  of  the  preceding  residues  by  any  new  integer  prime 
to  p;  etc.  Let  y  be  the  jth  integer  following  /S  in  the  same  series,  etc. 
Then  N'  =  Aa-\-BS-{-Cjy-\-...  is  or  is  not  divisible  by  p  according  as 
A''  is  or  not.  By  repetitions  of  the  process,  we  get  the  minimum  residue 
of  N  modulo  p.     The  special  case  A-{-Biqi,  with  p  a  prime,  is  due  to  Loir.^^ 

Dietrichkeit^^  would  test  Z  =  \Ok-\-a  for  the  divisor  n  by  testing  k  —  xa, 
where  10a:+l  is  some  multiple  of  n.  To  test  Z  (pp.  316-7)  for  the  divisor 
7,  test  the  sum  of  the  products  of  the  units  digit,  tens  digit, ...  by  1,  3,  2,  6, 
4,  5,  taken  in  cyclic  order  beginning  with  any  term  (the  remainders  on  con- 
verting 1/7  into  a  decimal  fraction) .    Similarly  for  1/n,  when  n  is  prime  to  10. 

J.  Pontes^ ^  would  test  N  for  a  divisor  M  by  using  a  number <iV  and 
=  N  (mod  M),  found  as  follows.  For  the  base  B,  let  q  be  the  absolutely 
least  residue  of  B""  modulo  M.  Commencing  at  the  right,  decompose  N 
into  sets  of  m  digits,  as  X,„, .  .  . ,  a^,  and  set  f{x)=a^x"'+^jn^"'~^-\- .  .  .  +X;;,, 
whence  N=f{B'^).  By  expanding  N=f{q+M^),  we  see  that  f{q)  is  the 
desired  number  <  N  and  =  N  (mod  M) . 

S.  Levanen^^  gave  a  table  showing  the  exponent  to  which  10  belongs  for 

siLemberg  Museum  (Polish),  1886.      "Comptes  Rendus  Paris,  106,  1888,  1070-1;  errata,  1194. 
^'Nature,  40,  1889,  115-6.  34 Assoc,  franp.  avanc.  sc,  18,  1889,  II,  24-38. 

'^Zeitschr.  Math.  Phys.,  36, 1891,  64.   3«Comptes  Rendus  Paris,  115,  1892,  1259-61. 
"Ofversigt  af  finska  vetenskaps-soc.  forhandUngar,  34,  1892,  109-162.    Cf .  Jahrbuch  Fortschr. 
Math.,  24,  1892,  164-5. 


342  History  of  the  Theory  of  Numbers.  [Chap,  xii 

primes  6<200  and  certain  larger  primes,  from  which  are  easily  deduced 
tests  for  the  divisor  6. 

Several""  noted  that  if  10  belongs  to  the  exponent  n  modulo  d,  and  if 
Si,  Si,  ■  ■  .denote  the  sums  of  every  nth  digit  of  N  beginning  with  the  first, 
second, ...  at  the  right,  the  remainder  on  the  division  of  A^  by  d  is  that  of 
S1  +  IOS2+IO-S3+ .  .  . 

J.  Fontes^^  would  find  the  least  residue  of  A^  modulo  M.  If  10"  has  the 
residue  q  modulo  M,  we  do  not  change  the  least  residue  of  N  if  we  multiply 
a  set  of  n  digits  of  A^  by  the  same  power  of  q  as  of  10".  Thus  for  M  =  19, 
iV=10433  =  10'+4-10H33,  10"  has  the  residue  5  modulo  19  and  we  may 
replace  N  by  5- +4 ',5+ 33.     The  method  is  applied  to  each  prime  M^  149. 

Fontes^^  gave  a  history  of  the  tests  for  divisibility,  and  an  "extension 
of  the  method  of  Pascal,"  similar  to  that  in  his  preceding  paper. 

P.  Valerio*°  would  test  the  divisibility  of  N  by  39,  for  example,  by  sub- 
tracting from  N  a  multiple  of  39  with  the  same  ending  as  N. 

F.  Belohldvek^^  noted  that  10A-\-B  is  divisible  by  10p±l  if  A=FpB  is. 

C.  Borgen'^^  ^oted  that  Z  =  a„-10"+ .  .  .  +ai-10+ao  is  divisible  by  A^  if 

"T'  (a_„+rlO''-'+  .  .  .  +a,)(10"-A^)''/'' 
..=0 

is  divisible  by  N.  For  A''  =  7,  take  a  =  1 ;  then  10"— AT  =  3  and  Z  is  divisible 
by  7  if  ao+3ai+2a2  — ^3  — 3a4  — 2a5+  ...  is  divisible  by  7. 

J.  J.  Sylvester^^"  noted  that,  if  the  r  digits  of  A'^,  read  from  left  to  right, 
be  multiplied  by  the  first  r  terms  of  the  recurring  series  1,  4,  3,  — 1,-4,  — 3; 
1,  4, .  .  .  [the  residues,  in  reverse  order,  of  10,  10^, .  .  .,  modulo  13],  the  sum 
of  the  products  is  divisible  by  13  if  and  only  if  N  is  divisible  by  13. 

C.  L.  Dodgson^^**  discussed  the  quotient  and  remainder  on  division  by 
9  or  11. 

L.  T.  Riess^'  noted  that,  if  p  is  not  divisible  by  2  or  5,  106+a(a<10) 
is  divisible  by  p  if  b  —  xa  is  divisible  by  p,  where  mp  =  10x-\-a  (a<  10)  and 
m  =  l,  7,  3,  9  according  as  p=l,  3,  7,  9  (mod  10),  respectively. 

A.  Loir^^  gave  tests  for  prime  divisors  <  100  by  uniting  them  by  twos 
or  threes  so  that  the  product  P  ends  in  0 1 ,  as  7  -43  =  30 1 .  To  test  N,  multiply 
the  number  formed  of  the  last  two  digits  of  A'^  by  the  number  preceding  01  in 
P,  subtract  the  product  from  A^,  and  proceed  in  the  same  manner  with  the 
difference.  Then  P  is  a  factor  if  we  finally  get  a  difference  which  is  zero. 
If  a  difference  is  a  multiple  of  a  prime  factor  p  of  P,  then  N  is  divisible  by  p. 

Plakhowo"*^  gave  the  test  by  Bougaief,  but  without  using  congruences. 

'"'Math.  Quest.  Educ.  Times,  57,  1892,  111. 

"Assoc,  frang.  avanc.  sc,  22,  1893,  II,  240-254. 

»M4m.  ac.  sc.  Toulouse,  (9),  5,  1893,  459-475. 

"La  Revue  Scientifique  de  France,  (3),  52,  1893,  765 

"Casopis,  Prag,  23,  1894,  59.  "Mature,  57,  1897-8,  54. 

«MEducat.  Times,  March,  1897.     Proofs.  Math.  Quest.  Educ.  Times,  66,  1897,  108.     Cf.  W.  E. 

Heal,  Amer.  Math.  Monthly,  4,  1897,  171-2. 
"'^Nature,  56,  1897,  565-6. 

«Russ.  Nat.,  1898,  329.     Cf.  Jahrb.  Fortschritte  Math.,  29,  1898,  137. 
"Assoc,  frang.  avanc.  sc,  27,  1898,  II,  144-6. 
«»Bull.  des  sc.  math,  et  phys.  61(§mentaires,  4,  1898-9,  241-3, 


1 


Chap.  XII]  CRITERIA  FOR  DIVISIBILITY.  343 

To  testiV  =  ao+Oi-S+  •  •  •  +On5"for  the  divisor  D  prime  to  B,  determine  d  and 
X  so  that  Bd  =  Dx+l.  Multiply  this  equation  by  Oq  and  subtract  from  N. 
Thus 

N=-BN'-DaoX,  N'  =  aod-\-{ai+a2B-\- . . .  +a^B''-^)B. 

Hence  N  is  divisible  by  D  if  and  only  if  N'  is  divisible  by  D.  Now,  N'  is 
derived  from  N  by  supressing  the  units  digit  do  and  adding  to  the  result  the 
product  aod.     Next  operate  with  N'  as  we  did  with  A^. 

J.  Malengreau^^  would  test  N  for  a  factor  q  prime  to  10  by  seeking  a 
multiple  11 ...  1  (to  m  digits)  of  q,  then  an  exponent  t  such  that  the  number 
of  digits  of  lO'-A^  is  a  multiple  of  m.  From  each  set  of  m  digits  of  lO'-A^ 
subtract  the  nearest  multiple  of  1 ...  1  (to  m  digits) .  The  sum  of  the  resi- 
dues is  divisible  by  q  if  and  only  if  A''  is  divisible  by  q. 

G.  Loria^^  proved  that  N  =  aQ-\-gai-\- .  .  .-\-g''ak  is  divisible  by  a  if  and 
only  if  a  divides  the  sum  ao+  •  •  .  +a^  of  the  digits  of  N  written  to  a  base  g  of 
the  form  /ca+ 1 ;  or  if  a  divides  Oq  —  ai +02  —  •  •  •  when  the  base  g  is  of  the  form 
ka  —  1.  Taking  g  =  IC",  we  have  the  test,  in  Gelin's  Arithm^tique,  in  terms 
of  groups  of  m  digits.  We  may  select  m  to  be  |0(a)  or  a  number  such  that 
lO'^il  has  the  factor  a.  Inplace  of  00+^1+ •  • .  wheng'  =  10'",  we  may  employ 
pao+Xai  +  10Xa2+  .  .  .  +10"-'Xa^_i 

+  s\o^™-^(a,^+10a,^+i+  . . .  +10"'-'a,^+m-i), 

k  =  l 

where  X  =  l,  2  or  5,  and  p  is  determined  by  10p/X=l  (mod  o).  Taking 
a  =  7,  13,  17,  19,  23,  special  tests  for  divisors  are  obtained. 

G.  Loria^^  proved  that,  if  ao,  ai,.  .  .  are  successive  sets  of  t  digits  of  N, 
counted  from  the  right,  and  o-  =  ao='=cti+02=^«3+  •  •  •,  then 

N-(T  =  a,{10'=Fl)-\-a2{10^'-l)+as{10^'=pl)  + .  .  ., 

so  that  a  factor  of  10'=f1  divides  A^  if  and  only  if  it  divides  a. 
A.  Tagiuri^^  extended  the  last  result  to  any  base  g.    We  have 

N  =  ao+ga,+  . . .  =Nom+g"'Nr^+g''^N2^-\- . . . 

if  N,m  =  o,pm+apm+i9-\-  •  •  •  +«pm+m-i^"'"^     Heuce,  if  9"^=  ±  1  (mod  a), 

N=Nom^Nim+N2m=^...   (mod  a). 

L.  Ripert^"  noted  that  lOD-\-uis  divisible  by  lOS+i  if  Di—bu  is  divisible, 
and  gave  many  tests  for  small  divisors. 

G.  Biase^^  derived  tests  that  \Od-\-u  has  the  factor  7  or  19  from 

2{l{)d+u)^2u-d  (mod  7),  2{lQd+u)=2u+d  (mod  19). 

O.  Meissner^^  reported  on  certain  tests  cited  above. 

"Mathesis,  (3),  1,  1901,  197-8. 

*^Rendiconti  Accad.  Lincei  (Math.),  (5),  10,  1901,  sem.  2,  150-8.    Mathesis,  (3),  2,  1902,  33-39. 

"II  Boll.  Matematica  Gior.  Sc.-Didat.,  Bologna,  1,  1902.     Cf.  A.  Bindoni,  ibid.,  4,  1905,  87. 

"Periodico  di  Mat.,  18,  1903,  43-45.  ^oL'enseignement  math.,  6,  1904,  40-46. 

"II  Boll.  Matematica  Gior.  Sc.-Didat.,  Bologna,  4,  1905,  92-6. 

^''Math.  Naturw.  Blatter,  3,  1906,  97-99. 


344  History  of  the  Theory  of  Numbers.  [Chap,  xii 

E.  NanneF  employed  ri=Oi  — aoX,  r2  =  a2—riX,.  .  .  (a-<10).  Then,  if 
r„  =  0,  A''  =  10"a„4- . .  .  +10ai+Oo  is  divisible  by  lOx-fl  and  the  quotient  has 
the  digits  r„_i,  r„_2, .  .  ■ ,  7*i,  Qo-  The  cases  x  =  1,  2  are  discussed  and  several 
tests  for  7  deduced.  For  a:=  1/3,  we  conclude  that,  if  r„  =  0,  N  is  divisible 
by  13  and  the  digits  of  the  quotient  are  r„_i/3, .  .  . ,  r^/S,  ao/3. 

A.  Chiari^  employed  D'Alembert's^  method  for  10+6,  6  =  3,  7,  9. 

G.  Bruzzone^^  noted  that,  to  find  the  remainder  R  when  N  is  divided  by 
an  integer  x  of  r  digits,  we  may  choose  y  such  that  x-\-y  =  10'',  form  the 
groups  of  r  digits  counting  from  the  right  of  N,  and  multiply  the  successive 
groups  (from  the  right)  hy  l,y,y^,.  .  .  or  by  their  residues  modulo  x;  then  R 
equals  the  remainder  on  dividing  the  sum  of  the  products  by  x.  If  we  choose 
x  —  y  =  lO^,  we  must  change  alternate  signs  before  adding.  For  practical  use, 
take  y  =  l. 

Fr.  Schuh^^  gave  three  methods  to  determine  the  residue  of  large  numbers 
for  a  given  modulus. 

Stuyvaert^^  let  a,  6, ...  be  the  successive  sets  of  n  digits  of  A'' to  the  baseB, 
so  that  iV  =  a+6jB'*+c52''+  rj.^^^  ^  -^  (^^.^isibie  ^^y  ^  factor  D  of  B''=pR'' 

if  and  only  if  a=^bR''+cR~''^  ...  is  divisible  by  D.  For  R  =  l,  B  =  10, 
n  =  1,  2, .  .  .,  we  obtain  tests  for  divisors  of  9,  99,  11,  101,  etc.  A  divisor, 
prime  to  B,  of  niB+l  divides  N  =  a+bB  if  and  only  if  it  divides  h—ma. 

Further  Papers  Giving  Tests  for  a  Given  Divisor  d. 

J.  R.  Young  and  Mason  for  d  =  l,  13  [Pascal^],  Ladies'  Diary,  1831,  34-5,  Quest. 

1512. 
P.  Gorini  [Pascal^],  Annali  di  Fis.,  Chim.  Mat.,  (ed.,  Majocchi),  1,1841,  237. 
A.  Pinaud  for  d  =  l,  13,  Mem.  Acad.  Sc.  Toulouse,  1,  1844,  341,  347. 
*Dietz  and  Vincenot,  Mem.  Acad.  Metz,  33,  1851-2,  37. 
Anonymous  writer  for  d  =  9,  11,  Jour,  fiir  Math.,  50,  1855,  187-8. 
*H.  Wronski,  Principes  de  la  phil.  des  math.     Cf.  de  Montferrier,  Encyclop^die 

math.,  2,  1856,  p.  95. 
O.  Terquem  for  d^l9,  23,  37,  101,  Nouv.  Ann.  Math.,  14,  1855,  118-120. 
A.  P.  Reyer  for  d  =  l,  Archiv  Math.  Phys.,  25,  1855,  176-196. 
C.  F.  Lindman  for  d  =  l,  13,  ibid.,  26,  1856,  467-470. 
P.  Buttel  for  d  =  7,  9,  11,  17,  19,  ibid.,  241-266. 

De  Lapparent  [Herter^^],  Mem.  soc.  imp.  sc.  nat.  Cherbourg,  4,  1856,  235-258. 
Karwowski  [Pascal^],  Ueber  die  Theilbarkeit .  . .,  II,  Progr.,  Lissa,  1856. 
*D.  van  Langeraad,  Kenmerken  van  deelbarheid  der  geheele  getallen,  Schoonho- 

ven,  1857. 
Flohr,  Ueber  Theilbarkeit  und  Reste  der  Zahlen,  Progr.,  Berlin,  1858. 
V.  Bouniakowsky  for  d  =  37,  989,  Nouv.  Ann.  Math.,  18,  1859,  168. 
Elefanti  for  d  =  l-n,  Proc.  Roy.  Soc.  London,  10,  1859-60,  208. 
A.  Niegemann  for  d  =  10'"-n+a,  Archiv  Math.  Phys.,  38,  1862,  384-8. 
J.  A.  Grunert  for  d  =  7,  11,  13,  ibid.,  42,  1864,  478-482. 
V.  A.  Lebesgue,  Tables  diverses  pour  la  decomposition  des  nombres,  Paris,  1864, 

p.  13. 

"II  Pitagora,  Palermo,  13,  1906-7,  54-9. 

»/6r(f.,  14,  1907-8,  35-7. 

"/6td.,  15,  1908-9,  119-123. 

"Supplem.  De  Vriend  der  Wiskunde,  24,  1912,  89-103. 

"Les  Nombres  Positifs,  Gand,  1912,  59-62. 


Chap.  XII]  CrITEEIA    FOR   DIVISIBILITY.  345 

C.  M.  Ingleby  for  d  =  9,  11,  British  Assoc.  Report,  35,  1865,  7  (trans.). 

M.  Jenkins  for  any  prime  d,  Math.  Quest.  Educ.  Times,  8,  1868,  69,  111. 

F.  Unferdinger  [Gergonnei^],  Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  59,  1869,  II, 

465-6. 
H.  Anton  for  d  =  9,  11,  13,  101,  Archiv  Math.  Phys.,  49,  1869,  241-308. 
W.  H.  Walenn,  British  Assoc.  Report,  40,  1870,  16-17  (trans.);  Phil.  Mag.,  (4), 

36,  1868,  346-8;  (4),  46,  1873,  36-41;  (4),  49,  1875,  346-351;  (5),  2,  1876, 

345;  4,  1877,  378;  9,  1880,  56,  121,  271. 
M.  A.  X.  Stouff  for  d<  100,  Nouv.  Ann.  Math.,  (2),  10,  1871,  104. 
J.  Lubin,  ibid.,  (2),  12,  1874,  528-30  (trivial). 
Szenic  for  d  =  7,  9,  37,  Von  der  Kongruenz  der  Z.,  Progr.  Schrimm,  1873. 

E.  Brooks  for  d  =  7,  Des  Moines  Analyst,  2,  1875,  129. 

W.  J.  Greenfield  and  M.  Collins  for  d  =  47,  73,  Math.  Quest.  Educ.  Times,  22, 
1875,  87. 

F.  da  Ponte  Horta  for  d  =  7,  9,  11,  13,  Jornal  de  Sciencias  Mat.  Ast.,  1, 1877,  57-62. 
Mennesson  for  d  =  7,  Nouv.  Corresp.  Math.,  4, 1878,  151;  generahzation  by  Cesaro, 

p.  156. 
C.  Lange,  for  d  =  7, 13, 17, 19,  Ueber  die  Teilbarkeit  der  Zahlen,  Progr.,  Berlin,  1879. 

F.  Jorcke  for  d  =  7,  9,  11,  Ueber  Zahlenkongruenzen .  .  .,  Progr,  Fraustadt,  1878. 
K.  Broda  for  any  base,  Archiv  Math.  Phys.,  63,  1879,  413-428. 

A.  Badoureau  for  d  =  19,  Nouv.  Ann.  Math.,  (2),  18,  1879,  35-6. 
S.  M.  Drach  for  d  =  7,  Math.  Quest.  Educ.  Times,  35,  1881,  71-2. 
W.  A.  Pick  for  d  =  7,  ibid.,  38,  1883,  64. 

A.  Evans  for  d  =  7,  Des  Moines  Analyst,  10,  1883, 134. 
K.  Haas,  Theilbarkeitsregeln .  . .,  Progr.,  Wien,  1883. 

G.  Wertheim,  Elemente  der  Zahlentheorie,  1887,  31-33. 

B.  Adam  for  d<100,  Ueber  die  Teilbarkeit.  . .,  Progr.  Gym.  Clausthal,  1889. 
A.  Loir  for  d<138.  Jour,  de  math,  elem.,  1889,  66,  107-10,  121-3. 

A.  G.  Fazio  [SchlegeP^],  Sui  caratteri.  . .,  Palermo,  1889. 

E.  Gelin,  Mathesis,  (2),  2,  1892,  65,  93;  (2),  12,  1902,  65-74,  93-99  (extract  in 

Mathesis,  (3),  10,  1910,  Suppl.  I);  Ann.  Soc.  Sc.  Bruxelles,  34,  1909-10,  66; 

Recueil  de  problemes  d'arith.,  1896.     Extracts  by  M.  Nasso,  Revue  de  Math. 

(ed.,  Peano),  7,  1900-1,  42-52. 
Speckmann,  Dorsten,  Haas,  Dorr,  Zeitschrift  Math.  Phys.,    37,   1892,  58,  63, 

128,  192,  383. 
Lalbaletrier,  Jour,  de  Math,  (ed.,  de  Longchamps),  1894,  54. 
H.  T.  Burgess  [Pascal*],  Nature,  57,  1897-8,  8-9,  30,  55. 
A.  Conti  [Pascal*],  Periodico  di  Mat.,  13,  1898,  180-6,  207-9. 

F.  Mariantoni,  ibid.,  149-151,  191-2,  217-8. 

T.  Lange  for  d<30,  Archiv.  Math.  Phys.,  (2),  16,  1898,  220-3. 

W.  J.  Greenstreet,  Math.  Gazette,  1,  1900,  186-7. 

Christie  for  d  =  2^p,5'^p  (p  prime).  Math.  Quest.  Educ.  Times,  73,  1900,  119. 

A.  Cunningham  and  D.  Biddle  for  d  =  rp=i=l,  ibid.,  75,  1901,  49-50. 
M.  Zuccagni  for  d  =  7,  Suppl.  al  Periodico  di  Mat.,  6,  fasc.  V. 
Calvitti  for  d  =  7,  ibid.,  8,  fasc.  IV. 

S.  Dickstein,  Wiad.  Mat.,  Warsaw,  6,  1902,  253-7  (Pohsh). 

B.  Niewenglowski,  ibid.,  252-3. 

Pietzker  ford  =  7,  11,  13,  27,  37,  Unterrichtsblatter  Math.  Naturwiss.,  9,  1903, 

85-110. 
A.  Church  for  d  =  7,  13,  17,  Amer.  Math.  Monthly,  12,  1905,  102-3. 
E.  A.  Cazes,  Assoc,  frang.,  36,  1907,  55-63. 
A.  Gerardin  for  d  =  7, 13, 17,  37,  43,  Sphinx-Oedipe,  1907-8,  2. 
M.  Morale  for  d  =  7,  Suppl.  al  Periodico  di  Mat.,  11,  1908,  103. 
*T.  Ghezzi,  ibid.,  12,  1908-9,  129-130. 
Lenzi,  II  Boll.  Matematica  Gior.  Sc.-Didat.,  7,  1908. 


346  History  of  the  Theory  of  Numbers.  [Chap,  xiii 

R.  Polpi,  ibid.,  8,  1909,  281-5. 

M.  Morale  for  d  =  7,  13,  Suppl.  al  Periodico  di  Mat.,  13,  1909-10,  38-9. 

A.  L.  Csada,  ibid.,  56-8. 

*A.  La  Paglia,  ibid.,  14,  1910-11,  136-7,  extension  of  Morale  to  any  d. 

A.  V.  Filippov,  8  methods  for  d  =  9,  Kagans  Bote,  1910,  88-92,  No.  520. 

P.  Cattaneo  for  rf=  11,  II  Boll.  Matematica  Gior.  Sc.-Didat.,  9,  1910,  305-6. 

*L.  Miceli,  Condizioni  di  divisibility  di  un  numero  N  per  un  numero  a .  .  . ,  Matera, 

1911,  8  pp. 
R.  Ayza  for  d  =  a-10''±l,  Revista  sociedad  mat.  espanola,  Madrid,  1,  1911,  162-6. 
*Paoletti,  II  Pitagora,  Palermo,  18,  1911-12,  128-132. 
*R.  La  Marca,  Criteri  di  congruenza  e  criteri  di  divisibilita,  Torre  del  Greco,  1912, 

30  pp. 
K.  W.  Lichtenecker,  Zeitschr.  fur  Realschulwesen,  37,  1912,  338-49. 
R.  E.  Cicero,  Sociedad  Cientifica  Antonio  Alzate,  32,  1912-3,  317-331. 
J.  G.  Gal6  for  d  =  7,  Revista  sociedad  mat.  espanola,  3,  1913-4,  46-7.  .. 

C.  F.  lodi  for  d  =  7,  13,  17,  19,  Suppl.  al  Periodico  di  Mat.,  18,  1914,  20-3.  > 

E.  Kylla  for  d=ll,  Unterrichtsblatter  Math.  Naturwiss.,  20,  1914,  156. 
R.  Krahl  for  d  =  7,  Zeitschrift  Math.  Naturw.  Unterricht,  45,  1914,  562. 
P.  A.  Fontebasso,  II  Boll.  Matematica,  13,  1914-5. 
G.  M.  Persico,  Periodico  di  Mat.,  32,  1917,  105-124. 
Sammlung  der  Aufgaben  in  Zeitschrift  Math.  Naturw.  Unterricht,  1898:  ford=7, 

II,  337;   IV,  404,  407;  for  d  =  9,  11,  XXIV,  606;  XXV,  587-8;  for  d  =  37, 
etc.,  XXVI,  18,  25-27. 

Criteria  for  di\dsibility  in  connection  with  tables  were  given  by  Barlow ,^^ 
Tarry«6  ^nd  Lebon"  of  Ch.  XIII,  and  Harmuth^^  of  Ch.  XIV. 

Papers  on  Divisibility  not  Available  for  report. 

Joubin,  Jour.  Acad.  Soc.  Sc.  France  et  de  I'Etranger,  Paris,  2,  1834,  230. 

J.  Lenth^ric,  Th^orie  de  la  divisibility  des  nombres,  Paris,  1838. 

R.  Volterrani,  Saggio  sulla  divisione  ragionata  dei  n.  interi,  Pisa,  1871. 

F.  Tirelli,  Teoria  della  divisibilita  de'  numeri,  Napoli,  1875. 
E.  Tiberi,  Teoria  generale  sulle  condizioni  di  divisibility .  .  . ,  Arezzo,  1890. 
J.  Kroupa,  Casopis,  Prag,  43,  1914,  117-120. 

G.  Schroder,  Unterrichtsblatter  fiir  Math.  Naturwiss.,  21,  1915,  152-5. 


CHAPTER  XIII. 

FACTOR  TABLES.  LISTS  OF  PRIMES. 

Eratosthenes  (third  century  B.C.)  gave  a  method,  called  the  sieve  or 
crib  of  Eratosthenes,  of  determining  all  the  primes  under  a  given  limit  I, 
which  serves  also  to  construct  the  prime  factors  of  numbers  <l.  From 
the  series  of  odd  numbers  3,  5,  7, ... ,  strike  out  the  square  of  3  and  every 
third  number  after  9,  then  the  square  of  5  and  every  fifth  number  after  25, 
etc.  Proceed  until  the  first  remaining  number,  directly  following  that  one 
whose  multiples  were  last  cancelled,  has  its  square  >l.  The  remaining 
numbers  are  primes. 

Nicomachus  and  Boethius^  began  with  5  instead  of  with  5^,  7  instead  of 
with  7^,  etc.,  and  so  obtained  the  prime  factors  of  the  numbers  <l. 

A  table  containing  all  the  divisors  of  each  odd  number  ^113  was  printed 
at  the  end  of  an  edition  of  Aratus,  Oxford,  1672,  and  ascribed  to  Eratos- 
thenes by  the  editor,  who  incorrectly  considered  the  table  to  be  the  sieve  of 
Eratosthenes.  Samuel  Horsley^  believed  that  the  table  was  copied  by 
some  monk  in  a  barbarous  age  either  from  a  Greek  commentary  on  the 
Arithmetic  of  Nicomachus  or  else  from  a  Latin  translation  of  a  Greek 
manuscript,  published  by  Camerarius,  in  which  occurs  such  a  table  to  109. 

Leonardo  Pisano^  gave  a  table  of  the  21  primes  from  11  to  97  and  a 
table  giving  the  factors  of  composite  numbers  from  12  to  100;  to  determine 
whether  n  is  prime  or  not,  one  can  restrict  attention  to  divisors  ^  ^/n. 

Ibn  Albanna  in  his  Talkhys^  (end  of  13th  century)  noted  that  in  using 
the  crib  of  Eratosthenes  we  may  restrict  ourselves  to  numbers  ^  -y/l. 

Cataldi^  gave  a  table  of  all  the  factors  of  all  numbers  up  to  750,  with  a 
separate  list  of  primes  to  750,  and  a  supplement  extending  the  factor  table 
from  751  to  800. 

Frans  van  Schooten®  gave  a  table  of  primes  to  9979. 

J.  H.  Rahn^  (Rhonius)  gave  a  table  of  the  least  factors  of  numbers,  not 
divisible  by  2  or  5,  up  to  24000. 

T.  Brancker^  constructed  a  table  of  the  least  divisors  of  numbers,  not 
divisible  by  2  or  5,  up  to  100  000.     [Reprinted  by  Hinkley.^^] 

*Introd.  in  Arith.  Nicomachi;  Arith.  Boethii,  lib.  1,  cap.  17  (full  titles  in  the  chapter  on  perfect 
numbers).  Extracts  of  the  parts  on  the  crib,  with  numerous  annotations,  were  given  by 
Horsley.2    Cf.  G.  Bernhardy,  Eratosthenica,  Berlin,  1822,  173-4. 

2Phil.  Trans.  London,  62,  1772,  327-347. 

311  Liber  Abbaci  di  L.  Pisano  (1202,  revised  1228),  Roma,  1852,  ch.  5;  Scritti,  1,  1857,  38. 

*Transl.  by  A.  Marre,  Atti  Accad.  Pont.  Nuovi  Lincei,  17,  1863-4,  307. 

"Trattato  de'  numeri  perfetti,  Bologna,  1603.  Libri,  Histoire  des  Sciences  Math,  en  Italic, 
ed.  2,  vol.  4,  1865,  91,  stated  erroneously  that  the  table  extended  to  1000. 

«Exercitat.  Math.,  libri  5,  cap.  5,  p.  394,  Leiden,  1657. 

^Algebra,  Zurich,  1659.     WaUis,!"*  p.  214,  attributed  this  book  to  John  Pell. 

*An  Introduction  to  Algebra,  translated  out  of  the  High-Dutch  [of  Rahn's'  Algebra]  into 
EngHsh  by  Thomas  Brancker,  augmented  by  D.  P.  [=Dr.  Pell],  London,  1668.  It  is 
cited  in  Phil.  Trans.  London,  3,  1668,  688.  The  Algebra  and  the  translation  were  de- 
scribed by  G.  Wertheim,  BibUotheca  Math.,  (3),  3,  1902,  113-126. 

347 


348  History  of  the  Theory  of  Numbers.  (Chap,  xiii 

D.  Schwenter^  gave  all  the  factors  of  the  odd  numbers  <  1000. 

John  Wallis^°  gave  a  list  of  errata  in  Brancker's^  table. 

John  Harris,"  D.  D.,  F.  R.  S.,  reprinted  Brancker's^  table. 

De  Traytorens^^  emphasized  the  utility  of  a  factor  table.  To  form  a 
table  showing  all  prime  factors  of  numbers  to  1000,  begin  by  multiplying 
2,  3, ..  .  by  all  other  primes  <  1000,  then  multiply  2X3  by  all  the  primes, 
then  2X3X5,  etc. 

Joh.  Mich.  Poetius^^  gave  a  table  (anatomiae  numerorum)  of  all  the 
prime  factors  of  numbers,  not  divisible  by  2,  3,  5,  up  to  10200.  It  was 
reprinted  by  Christian  Wolf,"  Willigs,^^  and  Lambert. ^- 

Johann  Gottlob  Krliger^*  gave  a  table  of  primes  to  100  999  (not  to  1 
million,  as  in  the  title),  stating  that  the  table  was  computed  by  Peter 
Jager  of  Niirnberg. 

James  Dodson^®  gave  the  least  di\'isors  of  numbers  to  10000  not  divisible 
by  2  or  5  and  the  primes  from  10000  to  15000. 

Etienne  FranQois  du  Tour^^  described  the  construction  of  a  table  of  all 
composite  odd  numbers  to  10000  by  multiplying  3,  5, ... ,  3333  by  3, ... ,  99. 

Giuseppe  Pigri^^  gave  all  prime  factors  of  numbers  to  10000. 

Michel  Lorenz  Willigs^^  (Willich)  gave  all  di\dsors  of  numbers  to  10000. 

Henri  Anjema-°  gave  all  divisors  of  numbers  to  10000. 

Rallier  des  Ourmes-^  gave  as  if  new  the  sieve  of  Eratosthenes,  placing 
3  above  9  and  every  third  odd  number  after  it,  a  7  above  49,  etc.  He 
expressed  each  number  up  to  500  as  a  product  of  powers  of  primes. 

J.  H.  Lambert^^  described  a  method  of  making  a  factor  table  and  gave 
Poetius'^^  table  and  expressed  a  desire  for  a  table  to  102  000.  Lagrange 
called  his  attention  to  Brancker's^  table. 

Lambert-^  gave  [Ivriiger's^^]  table  showing  the  least  factor  of  numbers 
not  di\'isible  by  2, 3, 5  up  to  102000,  and  a  table  of  primes  to  102  000,  errata 
in  which  were  noted  by  KliigeP^. 

•Geometria  Practica,  Numb.,  1667,  I,  312. 

loTreatise  of  Algebra,  additional  treatise,  Ch.  Ill,  §22,  London,  1685. 
"Lexicon  Technicum,  or  an  Universal  English  Dictionary  of  Arts  and  Sciences,  London,  vol.  2, 

1710  (under  Incomposite  Numbers).     In  ed.  5,  London,  2,  1736,  the  table  was  omitted, 

but  the  text  describing  it  kept.     WaUis,  Opera,  2,  ,511,  listed  30  errors. 
"Histoire  de  I'Acad.  Roy.  Science,  ann6e  1717,  Paris,  1741,  Hist.,  42-47. 
"Anleitung  zu  der  Arith.  Wissenschaft  vermittelst  einer  parallel  Algebra,  Frkf .  u.  Leipzig,  1728. 
"VoUst.  Math.  Lexicon,  2,  Leipzig,  1742,  530. 
'*Gedancken  von  der  Algebra,  nebst  den  Primzahlen  von  1  bis  1  000  000,  Halle  im  Magd.,  1746, 

Cf.  Lambert. ^a 
"The  Calculator. .  .Tables  for  Computation,  London,  1747. 
•"Histoire  de  I'Acad.  Ro>.  Sc,  Paris,  ann6e  1754,  Hist.,  8&-90. 
"Nuove  tavole  degli  elementi  dei  numeri  dall'  1  al  10  000,  Pisa,  1758. 
"Griindhche  Vorstellung  der  Reesischen  allgemeinen   Regel . . .  Rechnungsarten,    Bremen  u. 

Gottingen,  2,  1760,  831-976. 
^Table  des  diviseurs  de  tous  les  nombres  naturels,  depuis  1  jusqu'4  10  000,  Leyden,  1767,  302  pp. 
"M^m.  de  math,  et  de  physique,  Paris,  5,  1768,  485-499. 

"Bej-trage  znm  Gebrauche  der  Math.  u.  deren  Anwendung,  Berlin,  1770,  II,  42. 
"Zusatze  zu  den  logarithm ischen  imd  trig.  Tabellen,  BerUn,  1770. 
"Math.  Worterbuch,  3,  1808,  892-900. 


Chap.  XIII]  FACTOR  TABLES,    LiSTS   OF   PrIMES.  349 

J.  Ozanam^^  gave  a  table  of  primes  to  10000. 

A.  F.  Marci^^  gave  in  1772  a  list  of  primes  to  400  000. 

Jean  Bernoulli^^"  tabulated  the  primes  16n+l  up  to  21601. 

L.  Euler"  discussed  the  construction  of  a  factor  table  to  one  miUion. 
Given  a  prime  p  =  30a±i  (^  =  1,  7,  11,  13),  he  determined  for  each  r  =  l,  7, 
11,  13,  17,  19,  23,  29,  the  least  q  for  which  SOq+r  is  divisible  by  p,  and 
arranged  the  results  in  a  single  table  with  p  ranging  over  the  primes  from 
7  to  1000.  He  showed  how  to  use  this  auxiliary  table  to  construct  a  factor 
table  between  given  limits. 

C.  F.  Hindenburg^^  employed  in  the  construction  of  factor  tables  a 
"patrone"  or  strip  of  thick  paper  with  holes  at  proper  intervals  to  show 
the  multiples  of  p,  for  the  successive  primes  p. 

A.  FelkeP^  gave  in  1776  a  table  of  all  the  prime  factors  (designated  by 
letters  or  pairs  of  letters)  of  numbers,  not  divisible  by  2,  3,  or  5,  up  to 
408  000,  requiring  for  entry  two  auxiliary  tables.  In  manuscript^",  the 
table  extended  to  2  million;  but  as  there  were  no  purchasers  of  the  part 
printed,  the  entire  edition,  except  for  a  few  copies,  was  used  for  cartridges 
in  the  Turkish  war.  The  imperial  treasury  at  Vienna,  at  the  cost  of  which 
the  table  was  printed,  retained  the  further  manuscript.     [See  Felkel.^^] 

L.  Bertrand^^  discussed  the  construction  of  factor  tables. 

The  Encyclopedie  of  d'Alembert,  ed.  1780,  end  of  vol.  2,  contains  a 
factor  table  to  100  000. 

Franz  Schaffgotsch^^  gave  a  method,  equivalent  to  that  of  a  stencil  for 
each  prime  p,  for  entering  the  factor  p  in  a  factor  table  with  eight  headings 
SOm+k,  /c  =  1,  7,  11,  13,  17, 19,  23,  29,  and  hence  of  numbers  not  divisible  by 
2,  3,  or  5.     Proofs  were  given  by  Beguelin  and  Tessanek,  ibid.,  362,  379. 

The  strong  appeals  by  Lambert^^  that  some  one  should  construct  a  fac- 
tor table  to  one  million  led  L.  Oberreit,  von  Stamford,  Rosenthal,  Felkel, 
and  Hindenburg  to  consider  methods  of  constructing  factor  tables  and  to 
prepare  such  tables  to  one  million,  with  plans  for  extension  to  5  or  10 

^^Recreations  Math.,  new  ed.,  Paris,  1723,  1724,  1735,  etc.,  I,  p.  47. 

^^Primes  "in  quater  centenis  millibus,"  Amstelodami,  1772. 

26aNouv.  M6m.  Ac.  Berlin,  ann^e  1771,  1773,  323. 

"Novi  Comm.  Acad.  Petrop.,  19,  1774,  132;  Comm.  Arith.,  2,  64. 

''^Beschreibung  einer  ganz  neuen  Art  nach  einem  bekannten  Gesetze  fortgehende  Zahlen  durch 

Abzahlen  oder  Abmessen  bequem  u.  sicher  zu  finden.     Nebst  Anwendung  der  Methode 

auf  verschiedene  Zahlen,  besonders  auf  eine  damach  zu  fertigende  Factorentafel .  .  . , 

Leipzig,  1776,  120  pp. 
2*Tabula  omnium  factorum  simphcium,  numerorum  per  2,  3,  5  non  divisibilium  ab  1  usque 

10  000  000  [!].     Elaborata  ab  Antonio  Felkel.     Pars  I.     Exhibens  factores  ab  1  usque 

144  000,  Vindobonae,  1776.     Then  there  is  a  table  to  408  000,  given  in  three  sections. 

There  is  a  copy  of  this  complete  table  in  the  Graves  Library,  University  College,  London. 

Tafel  aller  einfachen  Factoren  der  durch  2,  3,  5  nicht  theilbaren  Zahlen  von  1  bis  10  000  000. 

Entworfen  von  Anton  Felkel.     I.  Theil.     Enthaltend  die  Factoren  von  1  bis  144  000, 

Wien,  1776.     There  is  a  copy  of  this  incomplete  table  in  the  hbraries  of  the  Royal  Society 

of  London  and  Gottingen  University. 
"Cf.  Zach's  Monatliche  Correspondenz,  2, 1800,  223;  Allgemeine  deutsche  BibUothek,  33,  II,  495. 
'^Develop,  nouveau  de  la  partie  ^1.  math.,  Geneve,  1774. 
'''Gesetz,  welches  zur  Fortsetzung  der  bekannten  Pellischen  Tafehi  dient,  Abhand.  Privatgesell- 

schaft  in  Bohmen,  Prag,  5,  1782,  354-382. 


350  History  of  the  Theory  of  Numbers.  [Chap,  xiii  | '^ ' 

million.  Their  extended  correspondence  with  Lambert^^  was  published. 
Of  the  tables  constructed  by  these  computers,  the  only  one  published  is  that 
by  Felkel.-^  The  history  of  their  connection  with  factor  tables  has  been 
treated  by  J.  W.  L.  Glaisher.^ 

Johann  Neumann^^  gave  all  the  prime  factors  of  numbers  to  100  100. 

Desfaviaae  gave  a  like  table  in  the  same  year. 

F.  Maseres^^  reprinted  the  table  of  Brancker.^ 

G.  Vega^^  gave  all  the  prime  factors  of  numbers  not  divisible  by  2,  3,  or  5 
to  102  000  and  a  list  of  primes  from  102  000  to  400  031.  Chernac  hsted  errors 
in  both  tables.  In  Hiilsse's  edition,  1840,  of  Vega,  the  Ust  of  primes  extends 
to  400  313. 

A.  Felkel,^^  in  his  Latin  translation  of  Lambert's'^^  Zusatze,  gave  all  the 
prime  factors  except  the  greatest  of  numbers  not  divisible  by  2,  3,  5  up  to 
102  000,  large  primes  being  denoted  by  letters.  In  the  preface  he  stated 
that,  being  unable  to  obtain  his  extensive  manuscript^"  in  1785,  he  calculated 
again  a  factor  table  from  408  000  to  2  856  000. 

J.  P.  Griison^^  gave  all  prime  factors  of  numbers  not  divisible  by  2,  3,  5 
to  10500.     He^^''  gave  a  table  of  primes  to  10000. 

F.  W.  D.  Snell^°  gave  the  prime  factors  of  numbers  to  30000. 

A.  G.  Kastner^^  gave  a  report  on  factor  tables. 

K.  C.  F.  Krause'*-  gave  a  table  of  22  pages  showing  all  products  <  100  000 
of  two  primes,  a  table  of  primes  <  100  000  with  letters  for  01,  03, ... ,  99, 
and  (pp.  25-28)  a  factor  table  to  10000  by  use  of  letters  for  numbers  <  100. 

N.  J.  Lidonne^^  gave  all  prime  factors  of  numbers  to  102  000. 

Jacob  Struve"*^"  made  a  factor  table  to  100  by  de  Traytorens'^^  method. 

L.  Chernac^  gave  all  the  prime  factors  of  numbers,  not  divisible  by 
2,  3  or  5,  up  to  1  020  000. 

J.  C.  Burckhardt*^  gave  the  least  factor  of  numbers  to  3  million.  He  did 
not  compute  the  first  million,  but  compared  Chernac's  table  with  a  manu- 
script (mentioned  in  Briefwechsel,^^  p.  140)  by  Schenmarck  which  extended 
to  1  008  000.     Cf.  iVIeissel.''^ 

^'Joh.  Heinrich  Lamberts  deutscher  gelehrter  Briefwechsel,  herausgegeben  von  Joh.  Bernoulli, 

Berlin,  1785,  Leipzig,  1787,  vol.  5.  "Proc.  Cambridge  Phil.  Soc,  3,  1878,  99-138. 

'^Tabellen  der  Primzahlcn  und  der  Faktoren  der  Zahlen,  welche  unter  100  100,  und  durch  2,  3 

Oder  5  nicht  theilbar  sind,  Dessau,  1785,  200  pp. 
'*The  Doctrine  of  Permutations  and  Combinations.  .  .,  London,  1795. 
'^Tabulae  logarithmico-trigonometricae,  1797,  vol.  2. 
**J.  H.  Lambert,  Supplementa  tab.  log.  trig.,  Lisbon,  1798. 
"Pinaeoth6que,  ou  collection  de  Tables.  .  .,  Berlin,  1798. 
''"Enthiillte  Zaubereyen  u.  Geheimnisse  d.  Arith.,  Berlin,  1796,  I,  82-4. 
*°Ueber  eine  neue  und  bequeme  Art,  die  Factorentafeln  einzurichten,  nebst  einer  Kupfertafel 

der  einfachen  Factoren  von  1  bis  30000,  Gicssen  and  Darmstadt,  1800. 
"Fortsetzung  der  Rechenkunst,  ed.  2,  Gottingen,  1801,  566-582. 

^'Factoren-  und  Primzahlentafel  von  1  bis  100  000  neu  berechnet,  Jena  u.  Leipzig,  1804. 
"Tables  de  tous  les  diviseurs  des  nombres  <  102  000,  Paris,  1808. 
^'''Handbuch  der  Math.,  Altona,  II,  1809,  108. 
**Cribrum  Arithmeticum . . .  Daventriae,  Isil,  1020  pp.     Reviewed  by  Gauss,  Gottingische 

gelehrte  Anzeigen,  1812;  Werke  2,  181-2.     Errata,  Cunningham.*' 
"Tables  des  diviseurs. . .  1  ^  3  036  000,  Paris,  1817, 1814,  1816  (for  the  respective  three  milliona), 

and  1817  (in  one  volume). 


•'I;i     Chap.  XIII]  FACTOR  TABLES,   LiSTS   OF  PrIMES.  351 

P.  Barlow^®  gave  the  prime  and  power  of  prime  factors  of  numbers  to 
10000  and  a  list  of  primes  to  100  103. 

C.  Hutton^^  gave  the  least  factor  of  numbers  to  10000. 

Rees'  Cyclopaedia,  1819,  vol.  28,  Hsts  the  primes  to  217  219. 

Peter  Barlow^^  gave  a  two-page  table  for  finding  factors  of  a  number 
iV<  100  000.  The  primes  p  =  7  to  p  =  313  are  at  the  head  of  the  columns, 
while  the  18  numbers  1000, .  .  . ,  9000, 10000,  20000, .  .  . ,  90000  are  in  the  left- 
hand  column.  In  the  body  of  the  table  is  the  remainder  of  each  of  the 
latter  when  divided  by  the  primes  p.  To  test  if  p  is  a  factor  of  N,  add  its 
last  two  digits  to  the  remainders  in  the  line  of  hundreds  and  thousands  in 
the  column  headed  p  and  test  whether  the  sum  is  divisible  by  p. 

J.  P.  Kulik'^^  gave  a  factor  table  to  1  million. 

J.  HantschP°  gave  a  factor  table  to  18277;  J.  M.  Salomon,^^  to  102  Oil. 

A.  L.  Crelle^^  gave  the  number  of  primes  4n+ 1  and  the  number  of  primes 
4n+3  in  each  thousand  up  to  the  fiftieth. 

A.  Guyot^^  hsted  the  primes  to  100  000. 

A.  F.  Mobius,^^"  using  square  ruled  paper,  inserted  from  right  to  left 
0,  1,  2, ...  in  the  top  row  of  cells,  and  inserted  n  in  each  cell  of  the  nth  row 
below  the  top  row  whenever  the  corresponding  number  in  the  top  row  is 
divisible  by  n.  We  thus  have  a  factor  table.  Certain  numbers  of  the  table 
Ue  in  straight  lines,  others  in  parabolas,  etc. 

P.  A.  G.  Colombier^^^  discussed  the  determination  of  the  primes  <V, 
given  those  <  I. 

H.  G.  Kohler'^  gave  a  factor  table  to  21524. 

E.  Hinkley^^  gave  a  factor  table  to  100  000,  listing  all  factors  of  odd 
numbers  to  20000  and  of  even  numbers  to  12500. 

F.  Schallen^^"gave  the  prime  and  prime-power  factors  of  numbers  <  10000. 
F.  Landry^*^  gave  factor  and  prime  tables  to  10000. 
A.  L.  Crelle^^  discussed  the  expeditious  construction  of  a  factor  table,  and 

in  particular  a  method  of  extending  Chernac's^  table  to  7  million. 
J.  HoiieP^  gave  a  factor  table  to  10841. 
Jacob  Philip  Kulik  (1773-1863)  spent  20  years  constructing  a  factor 

*^New  Mathematical  Tables,  London,  1814.     Errata,  Cunningham.^ 

<Thil.  and  Math.  Dictionary,  1815,  vol.  2,  236-8. 

^8New  Series  of  Math.  Repository  (ed.,  Th.  Leybourn),  London,  4,  1819,  II,  30-39. 

^'Tafeln  der  einfachen  Faktoren  aUer  Zahlen  unter  1  million,  Graz,  1825. 

60Log.-trig.  Handbuch,  Wien,  1827.  "Log.  Tafeln,  Wien,  1827. 

^2Jour.  fur  Math.,  10,  1833,  208. 

'^Thdorie  g^nerale  de  la  divisibihte  des  nombres,  suivie  d'applications  varices  et  d'une  table  de 

nombres  premiers  compris  entre  0  et  100  000,  Paris,  1835. 
63a Jour,  fvir  Math.,  22,  1841,  276-284.  "^Nouv.  Ann.  Math.,  2,  1843,  408-410. 

"Log.-trig.  Handbuch,  Leipzig,  1848.     Errata,  Cunningham.** 
^^Tables  of  the  prime  numbers  and  prime  factors  of  the  composite  numbers  from  1  to  100  000, 

Baltimore,  1853.     Reproduction  of  Brancker's*  table. 
ssaprirjizahlen-Tafel  von  1  bis  10000.  . .,  Weimar,  1855.     For  99  errata,  see  Cunningham.*^ 
6*Tables  des  nombres  entiers  non  divisibles  par  2,  3,  5,  et  7,  jusqu'  k  10201,  avec  leurs  diviseurs 

simples  en  regard,  et  des  carres  des  1000  premiers  nombres,  Paris,  1855.     Tables  des 

nombres  premiers,  de  1  £l  10000,  Paris,  1855. 
"Jour,  ftir  Math.,  51,  1856,  61-99.  "Tables  de  log.,  Paris,  1858. 


352  History  of  the  Theory  of  Numbers.  [Ch.u'.  xiii 

table  to  100  million;  the  manuscripts^  has  been  in  the  library  of  the  Vienna 
Royal  Academy  since  1867.  Lehmer^-  gave  an  account  of  the  first  of  the 
eight  volumes  of  the  manuscript,  listed  226  errors  in  the  tenth  million,  and 
concluded  that  Kulik's  manuscript  is  certainly  not  accurate  enough  to 
warrant  publication,  though  of  inestimable  value  in  checking  a  newly 
constructed  table.  Lehmer^^  gave  a  further  account  of  this  manuscript 
which  he  examined  in  Vienna.  Volume  2,  running  from  12  642  600  to 
22  852  800  is  missing.     The  eight  volumes  contained  4,212  pages. 

B.  Goldberg*^"  gave  all  factors  of  numbers  prime  to  2,  3,  5,  to  251  647. 
Zacharias  Dase,^^  in  the  introduction  to  the  table  for  the  seventh  million, 

printed  a  letter  from  Gauss,  dated  1850,  giving  a  brief  history-  of  previous 
tables  and  referring  to  the  manuscript  factor  table  for  the  fourth,  fifth  and 
sixth  milUons  presented  to  the  Berlin  Academy  by  A.  L.  Crelle.  Although 
Gauss  was  confident  this  manuscript  would  be  pubhshed,  and  hence  urged 
Dase  to  undertake  the  seventh  million,  etc.,  the  Academy  found  the  manu- 
script to  be  so  inaccurate  that  its  publication  was  not  ad\'isable.  Dase  died 
in  1861  lea\'ing  the  seventh  million  complete  and  remarkably  accurate, 
the  eighth  nearly  complete,  and  a  large  part  of  the  factors  for  the  ninth  and 
tenth  millions.  The  work  was  completed  by  Rosenberg,  but  ^vith  numerous 
errors.  The  table  for  the  tenth  million  has  not  been  printed ;  the  manuscript 
was  presented  to  the  Berlin  Academy  in  1878,  but  no  trace  of  it  was  found 
when  Lehmer^-  desired  to  compare  it  with  his  table  of  1909. 

C.  F.  Gauss^-  gave  a  table  showing  the  number  of  primes  in  each  thousand 
up  to  one  million  and  in  each  ten  thousand  from  one  to  three  million,  with  a 
comparison  with  the  approximate  formula  jdx/log  x. 

V.  A.  Lebesgue^^  discussed  the  formation  of  factor  tables  and  gave  that 
to  115500  constructed  by  Hoiiel. 

W.  H.  Oakes^  used  a  complicated  apparatus  consisting  of  three  tables  on 
six  sheets  of  various  sizes  and  nine  perforated  cards  (cf.  Committee, ^^  p.  39). 

W.  B.  Da\as^s  considered  numbers  in  the  vicinity  of  10^,  and  of  10^^ 

E.  MeisseP^  computed  the  number  of  primes  in  the  successive  sets  of 
100  000  numbers  to  one  million  and  concluded  that  Burckhardt's*^  table 
gives  correctly  the  primes  to  one  million. 

••Cited  by  Kulik.  Abh.  Bohm.  Gesell.  Wiss.,  Prag,  (5),  11,  1860,  24,  footnote.  A  report  on  the 
manuscript  was  made  by  J.  Petzval,  Sitzungsberichte  Ak.  Wiss.  Wien  (Math.),  53,  1866, 
II,  460.     Cited  by  J.  Perott,  I'interm^iaire  des  math.,  2,  1895,  40;  11,  1904,  103. 

•"Primzahlen-  u.  Faktortafeln  von  1  bis  251  647,  Leipzig,  1862.     Errata,  Cunningham." 

•'Factoren-Tafeln  fur  alle  Zahlcn  der  siebenten  MilUon .  . . ,  Hamburg,  1862; .  .  .der  achten  Mil- 
Uon,  1863;. .  .der  neuhten  MilUon  (erganzt  von  H.  Rosenberg),  1865. 

•^Posthumous  manuscript,  Werke,  2,  1863,  435-447. 

••Tables  diverses  pour  la  decomposition  des  nombres  en  leurs  facteurs  premiers,  M6m.  soc.  sc. 
phys.  et  nat.  de  Bordeaux,  3,  cah.  1,  1864,  1-37. 

•♦Machine  table  for  determining  primes  and  the  least  factors  of  composite  numbers  up  to 
100  000,  London,  1865. 

••Jour.de  Math.,  (2),  11,  1866,  188-190;  Proc.  London  Math.  Soc,  4,  1873,  416-7.  Math. 
Quest.  Educ.  Times,  7,  1867,  77;  8,  1868,  30-1. 

••Math.  Annalen,  2,  1870,  63&-642.     Cf.  3,  p.  523;  21,  1883,  p.  304;  25,  1885,  p.  251. 


I 


Chap.  XIII]  FACTOR  TABLES,  LiSTS  OF   PRIMES.  353 

J.  W.  L.  Glaisher"  gave  for  the  second  and  ninth  millions  the  number  of 
primes  in  each  interval  of  50000  and  a  comparison  with  lix'  —  lix,  where 
lix  =  jdx/log  X  [more  precise  definition  at  the  end  of  Ch.  XVIII]. 

A  committee^^  consisting  of  Cayley,  Stokes,  Thompson,  Smith,  and 
Glaisher  prepared  the  Report  on  Mathematical  Tables,  which  includes 
(pp.  34-9)  a  list  of  factor  and  prime  tables. 

J.  W.  L.  Glaisher^^  described  in  detail  the  method  used  by  his  father^" 
and  gave  an  account  of  the  history  of  factor  tables. 

Glaisher^^"  enumerated  the  primes  in  the  tables  of  Burckhardt  and  Dase. 

Glaisher^^^  tabulated  long  sets  of  consecutive  composite  numbers.  He^^" 
enumerated  the  prime  pairs  (as  11,  13)  in  each  successive  thousand  to  3 
million  and  in  the  seventh,  eighth,  and  ninth  millions. 

E.  Lucas^^''  wrote  P(q)  for  the  product  of  all  the  primes  ^  q,  where  q 
is  the  largest  prime  <  n.  If  xP(g)±l  are  both  composite,  xP{q)—n,. .  ., 
xP{q),. . .,  xP{q)-\-n  give  2n+l  composite  numbers. 

Glaisher^^'  enumerated  the  primes  4n4-l  and  the  primes  4n+3  for  inter- 
vals of  10000  in  the  kth  milUon  for  k  =  l,2,  3,  7,  8,  9. 

James  Glaisher'^°  filled  the  gap  between  the  tables  by  Burckhardt^^  and 
Dase".  The  introduction  to  the  table  for  the  fourth  million  gives  a  history 
of  factor  tables  and  their  construction.  Lehmer^^  praised  the  accuracy  of 
Glaisher's  table,  finding  in  the  sixth  million  a  single  error  besides  two  mis- 
prints. 

Tuxen'^^  gave  a  process  to  construct  tables  of  primes. 

Groscurth  and  Gudila-Godlewksi,  Moscow,  1881,  gave  factor  tables. 

*V.  Bouniakowsky'^^"  gave  an  extension  of  the  sieve  of  Eratosthenes. 

W.  W.  Johnson'^^''  repeated  Glaisher's'^°  remarks  on  the  history  of  tables. 

P.  Seelhoff^^  gave  large  primes  /c-2"+l  {k<  100)  and  composite  cases. 

Simony'^^  gave  the  digits  to  base  2  of  primes  to  2^^  =  16384. 

L.  Saint-Loup^^  gave  a  graphical  exposition  of  Eratosthenes'  sieve. 

H.  Vollprecht'^^  discussed  the  construction  of  factor  tables. 

"Report  British  Association  for  1872,  1873,  trans.,  19-21.     Cf.  W.  W.  Johnson,  Des  Moines 

Analyst,  2,  1875,  9-11. 
68Report  British  Association  for  1873,  1874,  pp.  1-175.     Continued  in  1875,  305-336;  French 

transl.,  Sphinx-Oedipe,  8,  1913,  50-60,  72-79;  9,  1914,  8-14. 
"Proc.  Cambridge  Phil.  Soc,  3,  1878,  99-138,  228-9. 
69'»/6id.,  17-23,  47-56;  Report  British  Assoc,  1877,  20  (sect.).     Extracts  by  W.  W.  Johnson, 

Des  Moines  Analyst,  5,  1878,  7. 
s'^-Messenger  Math.,  7,  1877-8,  102-6,  171-6;  French  transl.,  Sphinx-Oedipe,  7,  1912,  161-8. 
^^'^Ibid.,  8,  1879,  28-33. 
«9«*/6id.,  p.  81.     C.  Gill,  Ladies'  Diary,  1825,  36-7,  had  noted  that  xP(q)+j  is  composite  for 

j  =  2,...,q-l. 
BseReport  British  Assoc,  1878,  470-1;  Proc.  Roy.  Soc.  London,  29,  1879,  192-7. 
^"Factor  tables  for  the  fourth,  fifth  and  sixth  millions,  London,  1879,  1880,  1883. 
"Tidsskrift  for  Mat.,  (4),  5,  1881,  16-25. 

'i«Memoirs  Imperial  Acad.  Science,  St.  Petersburg,  41,  1882,  Suppl,  No.  3,  32  pp. 
"^Annals  of  Math.,  1,  1884-5,  15-23. 

"Zeitschrift  Math.  Phys.,  31,  1886,  380.     Reprinted,  Sphinx-Oedipe,  4,  1909,  95-6. 
"Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  96,  II,  1887,  191-286. 
^^Comptes  Rendus  Paris,  107,  1888,  24;  Ann.  de  I'^cole  norm.,  (3),  7,  1890,  89-100. 
"Ueber  die  Herstellung  von  Faktorentafehi,  Diss.  Leipzig,  1891. 


354  History  of  the  Theory  of  Numbers.  [Chap,  xiii 

C.  A.  Laisant'^^"  would  exhibit  a  factor  table  by  use  of  shaded  and  un- 
shaded squares  on  square-ruled  paper  without  using  numbers  for  entries. 

G.  Speckmann'^''  made  tri\aal  remarks  on  the  construction  of  a  list  of 
primes. 

P.  Valerio'®  arranged  the  odd  numbers  prime  to  5  in  four  columns 
according  to  the  endings  1,  3,  7,  9.  From  the  first  column  cross  out  the  first 
multiple  21  of  3,  then  the  third  following  number  51,  etc.  Similarly  for  the 
other  columns.     Then  use  the  primes  7,  11,  etc.,  instead  of  3. 

J.  P.  Gram"  pubhshed  the  computation  by  N.  P.  Bertelsen  of  the 
number  of  primes  to  ten  million  in  intervals  of  50000  or  less,  which  led  to 
the  detection  of  numerous  errors  in  the  tables  of  Burckhardt^^  and  Dase." 

G.  L.  Bourgerel"^  gave  a  table  with  0,  1, .  .  . ,  9  in  the  first  row,  10, .  . . ,  19 
in  the  second  row  (with  10  under  0),  etc.  Then  all  multiples  of  a  chosen 
number  lie  in  straight  lines  forming  a  paralellogram  lattice,  with  one  branch 
through  0.  For  example,  the  multiples  of  3  appear  in  the  Une  through  0,  12, 
24,  36, ... ,  the  parallel  through  3,  15,  27, ... ,  the  parallel  21,  33,  45, ... ;  also 
in  a  second  set  of  parallels  3,  12,  21,  30;  6,  15,  24,  33,  42,  51,  60;  etc. 

E.  Suchanek"  continued  to  100  000  Simony's^^  table  of  primes  to  base  2. 

D.  von  Sterneck^"  counted  the  number  of  primes  100  n-|- 1  in  each  tenth  of 
a  million  up  to  9  million  and  noted  the  relatively  small  variation  from  one- 
fortieth  of  the  total  number  of  primes  in  the  interval. 

H.  Vollprecht^^  discussed  the  determination  of  the  number  of  primes  <N 
by  use  of  the  primes  <  v^- 

A.  Cunningham  and  H.  J.  WoodalP^  discussed  the  problem  to  find  all 
the  primes  in  a  given  range  and  gave  many  successive  primes  >9  million. 

They^^a  ^^^^^^  jj^y  primes  between  224±1020. 

H.  Schapira^^''  discussed  algebraic  operations  equivalent  to  the  sieve  of 
Eratosthenes. 

*V.  Di  Girio,  Alba,  1901,  applied  indeterminate  analysis  of  the  first 
degree  to  define  a  new  sieve  of  Eratosthenes  and  to  factoring. 

John  Tennant^  wrote  numbers  to  the  base  900  and  used  auxiliary  tables. 

A.  Cunningham^"  gave  long  lists  of  primes  between  9-10^  and  10^^ 

Ph.  Jolivald^  noted  that  a  table  of  all  factors  of  the  first  2n  numbers 
serves  to  tell  readily  whether  a  number  <4n+2  is  prime  or  not. 

^'"Assoc.  frang.,  1891,  II,  165-8.  7**Archiv  Math.  Phys.,  (2),  11,  1892,  439-441. 

'«La  revue  scientifique  de  France,  (3),  52,  1893,  764-5. 

"Acta  Math.,  17,  1893,  301-314.    List  of  errors  reproduced  in  Sphinx-Oedipe,  5,  1910,  49-51. 

^*La  revue  scientifique  de  France,  (4),  1,  1894,  411-2. 

"Sitzunpsber.  Ak.  Wiss.  Wien  (Math.),  103,  II  a,  1894,  443-610. 

'"Anzeiger  K.  Akad.  Wiss.  Wien  (Math.),  31,  1894,  2-4.     Cf.  Kronecker,  p.  416  below. 

"Zeitschrift  Math.  Phys.,  40,.  1895.  118-123. 

"Report  British  Assoc,  1901,  553;  1903,  561;  Messenger  Math.,  31,  1901-2,  165;  34,  1904-5, 
72,  184;  37,  1907-8,  6.5-83;  41,  1911,  1-16.  s^^Report  British  Assoc,  1900,  646. 

»«'Jahresber.  d.  Deutschen  Math.  Verein.,  5,  1901,  I,  69-72. 

"Quar.  Jour.  Math.,  32,  1901,  322-342. 

»*^Ibid.,  35,  1903,  10-21;  Mess.  Math.,  36,  1907,  145-174;  38,  1908,  81-104;  38,  1909,  145-175; 
39,  1909,  33-63,  97-128;  40,  1910,  1-36;  45,  1915,  49-75;  Proc  London  Math.  Soc,  27, 
1896,  327;  28,  1897,  377-9;  29,  1898,  381-438,  518;  34,  1902,  49. 

•♦L'intermfidiaire  des  math.,  11,  1904,  97-98. 


Chap.  XIII]  FaCTOK  TaBLES,    LiSTS   OF  PRIMES.  355 

A.  Cunningham^^  noted  errata  in  various  factor  tables. 

*J.  R.  Akerlund^^"  discussed  the  determination  of  primes  by  a  machine. 

Gaston  Tarry^^  would  use  an  auxiliary  table  (as  did  Barlow  in  1819) 
to  tell  by  the  addition  of  two  entries  (<  |p)  if  a  given  number  <  iV  is  divisible 
by  a  chosen  prime  p.  For  N  =  10000,  he  used  the  base  6  =  100,  and  gave  a 
table  showing  the  numerically  least  residues  of  the  numbers  r<h  and  the 
multiples  of  b  for  each  prime  p<b.  Then  nh-\-r  is  divisible  by  p  if  the 
residues  of  nh  and  r  are  equal  and  of  opposite  sign.  For  A^  =  100  000,  he 
used  6  =  60060  =  2-91-330  and  wrote  numbers  in  the  form  m6+330g+r, 
q<90,  r<330;  or,  again,  6  =  20580.  Ernest  Lebon"  used  such  tables  with 
the  base  30030  =  2-3-5-7-lM3,  or  its  product  by  17. 

Ernest  Lebon,^^  J.  Deschamps,^^  and  C.  A.  Laisant^''''  discussed  the  con- 
struction of  factor  tables. 

J.  C.  Morehead^°  extended  the  sieve  of  Eratosthenes  to  numbers 
ma^+6  (m  =  l,  2,  3,.  .  .)  in  any  arithmetical  progression.  The  case  a  =  2, 
6=  ±1,  is  discussed  in  detail,  with  remarks  on  the  construction  of  a  table 
to  serve  as  a  factor  table  for  numbers  m-2''=t  1. 

L.  L.  Dines^^  treated  the  case  a  =  6,  6  =  =fcl,  and  the  factorization  of 
numbers  m-Q'^^l. 

D.  N.  Lehmer^^  gave  a  factor  table  to  10  million  and  listed  the  errata  in 
the  tables  by  Burckhardt,  Glaisher,  Dase,  Dase  and  Rosenberg,  and 
Kulik's  tenth  million,  and  gave  references  to  other  (shorter)  lists  of  errata. 

E.  B.  Escott^^"  listed  94  pairs  of  consecutive  large  numbers  all  of  whose 
prime  factors  are  small. 

L.  Aubry^^°  proved  that  a  group  of  30  consecutive  odd  numbers  does  not 

contain  more  than  15  primes  or  numbers  all  of  whose  prime  factors  exceed  7. 

Cunningham^^"  listed  the  numbers  of  5  digits  with  prime  factors  ^  11 . 

85Messenger  Math.,  34,  1904-5,  24-31;  35,  1905-6,  24. 

ss^Nyt  Tidsskrift  for  Mat.,  Kjobenhavn,  16A,  1905,  97-103. 

8«Bull.  Soc.  Philomathique  de  Paris,  (9),  8,  1906,  174-6,  194-6;  9,  1907,  56-9.  Sphinx-Oedipe, 
Nancy,  1906-7,  39-41.  Tablettes  des  Cotes,  Gauthier-Villars,  Paris,  1906.  Assoc, 
frang.  avanc.  sc,  36,  1907,  II,  32-42;  41,  1912,  38-43. 

"Comptes  Rendus  Paris,  151, 1905,  78.  Bull.  Amer.  Math.  Soc,  13, 1906-7,  74.  L'enseignement 
math.,  9,  1907,  185.  Bull.  Soc.  PhUomathique  de  Paris,  (9),  8,  1906, 168,  270;  (9),  10, 
1908,  4-9,  66-83;  (10),  2,  1910,  171-7.  Assoc,  frang.  avanc.  sc,  36,  1907,  II,  11-20, 
49-55;  37,  1909,  33-6;  41,  1912,  44-53;  43,  1914,  29-35.  Rend.  Accad.  Lincei,  Rome,  (5), 
15,  1906,  I,  439;  26,  1917,  I,  401-5.  Sphinx-Oedipe,  1908-9,  81,  97.  BuU.  Sc.  Math. 
El6m.,  12,  1907,  292-3.  II  Pitagora,  Palermo,  13,  1906-7,  81-91  (table  serving  to  factor 
numbers  from  30030  to  510  510).  Table  de  caract6ristiques  relatives  a  le  base  2310  des 
facteurs  premiers  d'un  nombre  inf^rieur  k  30030,  Paris,  1906,  32  pp.  Comptes  Rendus 
Paris,  159,  1914,  597-9;  160,  1915,  758-760;  162,  1916,  346-8;  163,  1916,259-261;  164, 
1917,  482-4. 

*8Jomal  de  sciencias  math.,  phys.  e  nat.,  acad.  sc.  Lisbona,  (2),  7,  1906,  209-218. 

89Bull.  Soc.  Philomathique  de  Paris,  (9),  9,  1907,  112-128;  10,  1908,  10-41. 

s'^Assoc.  frang.,  41,  1912,  32-7. 

soAnnals  of  Math.,  (2),  10,  1908-9,  88-104.  ^Ubid.,  pp.  105-115. 

s^Factor  table  for  the  first  ten  milhons,  Carnegie  Inst.  Wash.  Pub.  No.  105,  1909. 

'^''Quar.  Jour.  Math.,  41,  1910,  160-7;  I'interm^diaire  des  math.,  11,  1904,  65;  Math.  Quest, 
Educ.  Times,  (2),  7,  1905,  81-5. 

"bSphinx-Oedipe,  6,  1911,  187-8;  Problem  of  Lionnet,  Nouv.  Ann.  Math.,  (3),  2,  1883,  310. 

'^'^Math.  Quest.  Educ.  Times,  (2),  21,  1912,  82-3. 


356  History  of  the  Theory  of  Numbers.  [Chap,  xni 

E.  Lebon''  stated  that  he  constructed  in  1911  a  table  of  residues  p,  p' 
permitting  the  rapid  factorization  of  numbers  to  100  million,  the  manuscript 
being  in  the  Bibliotheque  de  I'lnstitut. 

H.  W.  Stager^  gave  theorems  on  numbers  which  contain  no  factors  of 
the  form  p{kp-\-l),  where  k>0  and  p  is  a  prime,  and  listed  all  such  numbers 
<  12230. 

Lehmer'^  listed  the  primes  to  ten  million. 

A.  G^rardin^^  discussed  the  finding  of  all  primes  between  assigned  limits 
by  use  of  stencils  for  3,  5,  7,  11,.  . ..  He^^  described  his  manuscript  of 
an  auxiliary  table  permitting  the  factoring  of  numbers  to  200  million.  He^^" 
gave  a  five-page  table  serving  to  factor  numbers  of  the  second  million.  Cor- 
responding to  each  prime  M^  14867  is  an  entr\'  P  such  that  A^  =  1 000  000+P 
is  diWsible  by  M.  If  a  value  of  P  is  not  in  the  table,  A^  is  prime  (the  P's 
range  up  to  28719  and  are  not  in  their  natural  order).  By  a  simple  division 
one  obtains  the  least  odd  number  in  any  million  which  is  divisible  by  the 
given  prime  M^  14867. 

C.  Boulogne^^  made  use  of  lists  of  residues  modulis  30  and  300. 

H.  E.  Hansen^^  gave  an  impracticable  method  of  forming  a  table  of 
primes  based  on  the  fact  that  all  composite  numbers  prime  to  6  are  products 
of  two  numbers  6x±  1,  while  such  a  product  is  QN=^  1,  where  N  =  6xy='X+y 
or  Qxy—x  —  y.  A  table  of  values  of  these  A^'s  up  to  k  serves  to  find  the  com- 
posite numbers  up  to  6A-.  To  apply  this  method  to  factor  6N=^  1,  seek  an 
expression  for  A^  in  one  of  the  above  three  forms. 

N.  AUiston^°°  described  a  sieve  (a  modification  of  that  by  Eratosthenes) 
to  determine  the  primes  4n+l  and  the  primes  An  —  l. 

H.  W.  Stager^"^  expressed  each  number  <  12000  as  a  product  of  powers 
of  primes,  and  for  each  odd  prime  factor  gave  the  values  >0  of  A:  for  all 
divisors  of  the  form  p{kp-{-l).  The  table  thus  gives  a  list  of  numbers  which 
include  the  numbers  of  Sylow  subgroups  of  a  group  of  order  ^  12000. 

In  Ch.  XVI  are  cited  the  tables  of  factors  of  a^+1  by  Euler,^'  Escott,^^ 
Cunningham^^  and  WoodalP;  those  of  a--\-k-  (^*  =  1, .  . . ,  9)  of  Gauss";  those 
of  ?/"  +  l,  2/^±2,  y'=t.  1,  x^zti/^,  2«=tg,  etc.,  of  Cunningham.^^-  ^"^  Concern- 
ing the  sieve  of  Eratosthenes,  see  No\'iomagus-^  of  Ch..I,  Poretzkj^^  of  Ch. 
V,  :MerUn"^  and  de  Polignac^*^"-^  of  Ch.  XVIII.  Saint-Loup"  of  Ch.  XI, 
Re>Tnond^^^  and  Kempner^^^  of  Ch.  XIV,  represented  graphically  the  divi- 
sors of  numbers,  while  Kulik^^  gave  a  graphical  determination  of  primes. 

»»L'interin6diaire  des  math.,  19,  1912,  237. 

•♦University  of  California  Public,  in  Math.,  1,  1912,  No.  1,  1-26. 

•*LiHt  of  prime  numbers  from  1  to  10,006,721.    Carnegie  Inst.  Wash.  Pub.  No.  165, 1914.     The 
introduction  gives  data  on  the  distribution  of  primes. 

"Math.  Gazette,  7,  1913-4,  192-3. 

•'Assoc,  frang.  avanc.  sc,  42,  1913,  2-8;  43,  1914,  26-8. 

»«/bw/.,  43,  1914,  17-26. 

••"^Sphinx-Oedipe,  s^rie  sp4ciale.  No.  1,  Dec,  1913. 

••L'enseignement  math.,  17,  1915,  93-9.     Cf.  pp.  244-5  for  remarks  by  G^rardin. 
"""Math.  Quest.  Educat.  Times,  28,  1915,  53. 

'"A  Sylow  factor  table  of  the  first  twelve  thousand  numbers.     Carnegie  Inst.  Wash.  Pub.  No. 
151,  1916. 


CHAPTER  XIV. 

METHODS  OF  FACTORING. 
Factoring  by  Method  of  Difference  of  Two  Squares. 

Fermat^  described  his  method  as  follows:  "An  odd  number  not  a  square 
can  be  expressed  as  the  difference  of  two  squares  in  as  many  ways  as  it 
is  the  product  of  two  factors,  and  if  the  squares  are  relatively  prime  the 
factors  are.  But  if  the  squares  have  a  common  divisor  d,  the  given  number 
is  divisible  by  d  and  the  factors  by  Vrf-  Given  a  number  n,  for  example 
2027651281,  to  find  if  it  be  prime  or  composite  and  the  factors  in  the  latter 
case.  Extract  the  square  root  of  n.  I  get  r  =  45029,  with  the  remainder 
40440.  Subtracting  the  latter  from  2r+l,  I  get  49619,  which  is  not  a 
square  in  view  of  the  ending  19.  Hence  I  add  90061  =  2+2r+l  to  it. 
Since  the  sum  139680  is  not  a  square,  as  seen  by  the  final  digits,  I  again 
add  to  it  the  same  number  increased  by  2,  i.  e.,  90063,  and  I  continue  until 
the  sum  becomes  a  square.  This  does  not  happen  until  we  reach  1040400, 
the  square  of  1020.  For  by  an  inspection  of  the  sums  mentioned  it  is  easy 
to  see  that  the  final  one  is  the  only  square  (by  their  endings  except  for 
499944).  To  find  the  factors  of  n,  I  subtract  the  first  number  added, 
90061,  from  the  last,  90081.  To  half  the  difference  add  2.  There  results 
12.  The  sum  of  12  and  the  root  r  is  45041.  Adding  and  subtracting  the 
root  1020  of  the  final  sum  1040400,  we  get  46061  and  44021,  which  are  the 
two  numbers  nearest  to  r  whose  product  is  n.  They  are  the  only  factors 
since  they  are  primes.  Instead  of  11  additions,  the  ordinary  method  of 
factoring  would  require  the  division  by  all  the  numbers  from  7  to  44021." 

Under  Fermat,^^^  Ch.  I,  was  cited  Fermat's  factorization  of  the  number 
100895598169  proposed  to  him  by  Mersenne  in  1643. 

C.  F.  Kausler^  would  add  1^,  2^, .  .  .  to  iV  to  make  the  sum  a  square. 

C.  F.  Kausler^  proceeded  as  follows  to  express  4m+l  in  the  form  p^  —  q^. 
Then  q  is  even,  q  =  2Q.  Set  p-q  =  2^-\-l.  Then  w  =  Q(2i3+l)+/3(|8+l). 
Subtract  from  m  in  turn  the  pronic  numbers  i8(/3+l),  a  table  of  which  he 
gave  on  pp.  232-267,  until  we  reach  a  difference  divisible  by  2/3+1. 

Ed.  Collins,^  in  factoring  N  by  expressing  it  as  a  difference  of  two  squares, 
let  g^  be  the  least  odd  or  even  square  >  A^,  according  as  N=  1  or  3  (mod  4), 
and  set  N  =  g^  —  r.  If  r  is  not  a  square,  set  r  =  h^  —  c,  where  h^  is  the  even 
or  odd  square  just  >r,  according  as  r  is  even  or  odd,  whence  c  =  4d,  N  =  g^  — 
h^-\-4:d.  By  trial  find  integers  x,  y  such  that  both  g^-\-x  and  h^-\-y  are 
squares,  while  x  —  y  =  4id.     Then  N  will  be  a  difference  of  two  squares. 

^Fragment  of  a  letter  of  about  1643,  Bull.  Bibl.  Storia  Sc.  Mat.,  12,  1879,  715;  Oeuvres  de  Fer- 
mat,  2,  1894,  256.  At  the  time  of  his  letter  to  Mersenne,  Dec.  26, 1638,  Oeuvres,  2,  p.  177, 
he  had  no  such  method. 

"Euler's  Algebra,  Frankfort,  1796,  III,  2.  Anhang,  269-283.  Cf.  Kausler,  De  Cribro  Eratos- 
thenis.  1812. 

»Nova  Acta  Acad.  Petrop.,  14,  ad  annos  1797-8  (1805),  268-289. 

<BuU.  Ac.  Sc.  St.  Pdtersbourg,  6,  1840,  84-88. 

357 


358  History  of  the  Theory  of  Numbers.  Chap,  xivi 

F.  Landr>'*  used  the  method  of  Ferinat,  eliminating  certain  squares  by 
their  endings  and  others  by  the  use  of  moduU. 

C.  Henry^  stated  that  Landry's  method  is  merely  a  perfection  of  the 
method  given  in  the  article  "nombre  premier"  in  the  Dictionnaire  des 
Math^matiques  of  de  Montferrier.  It  is  improbable  that  the  latter  in- 
vented the  method  (based  on  the  fact  that  an  odd  prime  is  a  difference 
of   two  squares  in  a  single  way),  since  it  was  given  by  Fermat. 

F.  Thaarup^  gave  methods  to  limit  the  trials  for  x  in  x'^  —  y^  =  n.  We 
may  multiply  n  by  f  =  a^  —  b^  and  investigate  nf  =  X"  —  Y^,  X  =  ax  —  by, 
Y  =  bx  —  ay.  We  may  test  small  values  of  y,  or  apply  a  mechanical  test 
based  on  the  last  digit  of  n. 

C.  J.  Busk^  gave  a  method  essentially  that  by  Fermat.  It  was  put 
into  general  algebraic  form  by  W.  H.  H.  Hudson.^  Let  N  be  the  given 
number,  n"  the  next  higher  square.     Then 

N=^n'-ro={n+iy-r,=  ..., 

where  ri,  r^,...  are  formed  from  ro  by  successive  additions  of  2?i  +  l,  2nH-3, 
2n+5, .  . ..  Thus  r^  =  ro+27?27i+??r.  If  r^  is  a  square,  iV  is  a  difference 
of  two  squares.  A.  Cunningham  {ibid.,  p.  559)  discussed  the  conditions 
under  w^hich  the  method  is  practical,  noting  that  the  labor  is  prohibitive 
except  in  favorable  cases  such  as  the  examples  chosen  by  Busk. 

J.  D.  Warner^*"  would  make  N  =  A~—B'^  by  use  of  the  final  two  digits. 

A.  Cunningham^^  gave  the  22  sets  of  last  two  digits  of  perfect  squares,  as 
an  aid  to  expressing  a  number  as  a  difference  of  two  squares,  and  described 
the  method  of  Busk,  which  is  facilitated  by  a  table  of  squares. 

F.  W.  Lawrence^ ^  extended  the  method  of  Busk  (practical  only  when 
the  given  odd  number  iV  is  a  product  of  two  nearly  equal  factors)  to  the 
case  in  which  the  ratio  of  the  factors  is  approximately  l/m,  where  I  and  m 
are  small  integers.  If  I  and  ???  are  both  odd,  subtract  from  bnN  in  turn  the 
squares  of  a,  a+1, .  .  . ,  where  o^  just  exceeds  ImN,  and  see  if  any  remainder 
is  a  perfect  square  (6") .     If  so,  ImN  =  (a+  T)'^  —  6^. 

G.  Wertheim^^  expressed  in  general  form  Fermat's  method  to  factor 
an  odd  number  ?n.  Let  a^  be  the  largest  square  <m  and  set  m  =  a~+r. 
If  p=2a+l— r  is  a  square  (n^),  we  eliminate  r  and  get  m  =  {a-{-l-\-n) 
X  (a+1  —  n).  If  p  is  not  a  square,  add  to  p  enough  terms  of  the  arithmetic 
progression  2a +3,  2a+5, .  . .  to  give  a  square: 

p+(2a+3)-}-...H-(2a+2n-l)=s". 

'Aux  math^maticiens  de  toutes  les  parties  du  monde:  communication  sur  la  decomposition  des 
nombres  en  leurs  facteurs  simples,  Paris,  1867.  Letter  from  Landry  to  C.  Henry,  Bull. 
Bibl.  Storia  Sc.  Mat..  13,  1880,  469-70. 

•Assoc,  frang.  av.  sc,  1880,  201;  Oeuvres  de  Fermat,  4,  1912,  208;  Sphinx-Oedipe,  4,  1909,  3« 
Trimestre,  17-22.  ^Tidsskrift  for  Mat.,  (4),  5,  1881,  77-85. 

•Nature,  39,  1889,  413-5.  'Nature,  39,  1889,  p.  510. 

»'Proc.  Amer.  Assoc.  Adv.  Sc,  39,  1890,  54r-7. 

"Mess.  Math.,  20,  1890-1,  37-45.     Cf.  Meissner.i"  137-8. 

"/Wd.,  24,  1894-5,  100. 

»»Zeit8chrift  Math.  Naturw.  Unterricht,  27,  1896,  256-7. 


Chap.  XIV]  METHODS   OF  FACTORING.  359 

Then  2an+n^-r  =  s^  and  m  =  {a-\-ny-s^.  The  method  is  the  more  rapid 
the  smaller  the  difference  of  the  two  factors. 

M.  Neumann^^  proved  that  this  process  of  adding  terms  leads  finally 
to  a  square  and  hence  to  factors,  one  of  which  may  be  1. 

F.  W.  Lawrence^^  denoted  the  sum  of  the  two  factors  of  n  by  2a  and  the 
difference  by  2b,  whence  n  =  a^  — 6^.  Let  q  be  the  remainder  obtained  by 
dividing  n  by  a  chosen  prime  p,  and  write  down  the  pairs  of  numbers  <  p 
such  that  the  product  of  two  of  a  pair  is  congruent  to  q  modulo  p.  If 
p  =  7,  q  =  S,  the  pairs  are  1  and  3,  2  and  5,  4  and  6,  whence  2a=4,  0  or  3 
(mod  7).  Using  various  primes  p  and  their  powers,  we  get  limitations  on 
a  which  together  determine  a.  The  work  may  be  done  with  stencils.  The 
method  was  used  by  Lawrence^^  to  show  that  five  large  numbers  are  primes, 
including  10, 11  and  12  place  factors  of  3^^  —  1, 10^^  —  1, 10^^  —  1,  respectively. 
The  same  examples  were  treated  by  other  methods  by  D.  Biddle.^^ 

A.  Cunningham^'^  remarked  that  in  computing  by  Busk's  method  a 
k  for  which  {s+ky—N  is  a,  square,  we  may  use  the  method  of  Lawrence, 
just  described,  to  limit  greatly  the  number  of  possible  forms  of  k. 

F.  J.  Vaes^^  expressed  N  in  the  form  a^  —  b"^  by  use  of  the  square  a^  just 
>A^  and  then  increasing  a  by  1,  2, ... ,  and  gave  (pp.  501-8)  an  abbreviation 
of  the  method.  He  strongly  recommended  the  method  of  remainders 
(p .  425) :  If  p  is  a  factor  oiG  =  h^  —  g^,  and  iig={G  —  l)/2  has  the  remainder 
r  when  divided  by  p,  then  h={G+l)/2  must  have  the  remainder  r+1, 
so  that  p  is  a  factor  of  2r-\-l=G.     For  example,  let  G  =  80047,  whence 

^  =  200H23  =  20M99+24  =  202-198+27,.... 

For  r  =  24,  27,  32, .  .  .  we  see  that  2r+l  is  not  a  multiple  of  201,  202, .  . . 
until  we  reach  gr  =  209-191 +p,  p=104,  2p+ 1  =  209.     Thus  209  divides  G. 

P.  F.  Teilhet^^  wrote  N  =  a^-b  in  the  form  (a+kY-P,  where  P  =  k^ 
-{-2ak-{-b.  Give  to  k  successive  values  1,  2, .  .  .  (by  additions  to  P),  until 
P  becomes  a  square  v^.  To  abbreviate  consider  the  residues  of  P  for  small 
prime  moduli. 

E.  Lebon^°  proceeded  as  had  Teilhet^^  and  then  set  /=a+A;  — y.     Then 

2kf={a-fy-b, 

and  we  examine  primes  /<  a  to  see  if  k  is  an  integer. 

M.  Kraitchik^^  would  express  a  given  odd  number  A  in  the  form 
if—x^  by  use  of  various  moduli  p.     Let  A  =  r  (mod  p)  and  let  Oi, . .  . ,  a„  be  the 

"Zeitschrift  Math.  Naturw.  Unfcerricht,  27,  1896,  493-5;  28,  1897,  248-251. 

"Quar.  Jour.  Math.,  28,  1896,  285-311.     French  transl.,  Sphinx-Oedipe,  5,  1910,  98-121,  with 

an  addition  by  Lawrence  on  g^  +1. 
isProc.  Lond.  Math.  Soc,  28,  1897,  465-475.     French  transl.,  Sphinx-Oedipe,  5,  1910,  130-6. 
i^Math.  Quest.  Educat.  Times,  71,  1899,  113-4;  cf.  93-99. 
"/bid.,  69,  1898,  111. 
i«Proc.  Sect.  Sciences  Akad.  Wetenschappen  Amsterdam,  4,  1902,  326-336,  425-436,  501-8 

(EngUsh);  Verslagen  Ak.  Wet.,  10,  1901-2,  374-384,  474-486,  623-631  (Dutch). 
i^L'intermediaire  des  math.,  12,  1905,  201-2.     Cf.  Sphinx-Oedipe,  1906-7,  49-50,  55. 
"Assoc,  franc,  av.  sc,  40,  1911,  8-9. 
^^Sphinx-Oedipe,  Nancy,  Mai,  1911,  num^ro  special,  pp.  10-16. 


360  History  of  the  Theory  of  Numbers.  [Chap,  xiv 

quadratic  residues  of  p.  Then  r-\-x'^=ai  (mod  p).  Thus  a,  —  r  must  be 
a  quadratic  residue.  Reject  from  Oi, .  .  . ,  a^  the  terms  for  which  a^  —  r  is  not 
in  the  set.  We  get  the  possible  residues  of  x  modulo  p.  His  method  to  fac- 
tor 0"=*=  1  is  the  same  as  Dickson's^^^  and  is  applied  to  show  that  the  factor 
(273_|_237  4-i)/(5.239-9929)  of  2"^+l  is  a  prime  in  case  it  has  no  factor 
between  10500  and  108000. 

Kraitchik"  extended  the  method  of  Lawrence. 

F.  J.  Vaes^^  applied  his^^  method  to  factor  Mersenne's^  number.  The 
same  was  factored  by  various  methods  in  L'lnterm^diaire  des  Math6mat- 
iciens,  19, 1912,  32-5.  J.  Petersen,  ibid.,  5,  1898,  214,  noted  that  its  product 
by  8  equals  k^+k,  where  A:  =  898423. 

Method  of  Factoring  by  Sum  of  Two  Squares. 

Frenicle  de  Bessy^^  proposed  to  Fermat  that  he  factor  h  given  that 

h  =  a^+b''-=c'+(f,  as  221  =  100+121  =  196+25. 

In  1647,  Mersenne^^  (of  Ch.  I)  noted  that  a  number  is  composite  if  it  be 
a  sum  of  two  squares  in  two  ways. 

L.  Euler^^  noted  that  iV  is  a  prime  if  it  is  expressible  as  a  sum  of  two 
squares  in  a  single  way,  while  if  iV  =  a^+5^  =  c^+d^,  N  is  composite : 

{{a-cY+{h-d)'}  {{a+cy+ih-d)'] 
4(6 -d)2 

Euler^'  proved,  that,  if  a  number  A'"  =  4n+1  is  expressible  as  the  sum  of 
two  relatively  prime  squares  in  a  single  way,  it  is  a  prime.  For,  if  iV  were 
composite,  then  N={d^-{-b^){c^+d^)  is  the  sum  of  the  squares  of  ac^bd  and 
ad=f^bc,  contrary  to  hypothesis.  If  iV  =  a^+6^  =  c^+d^,  N  is  composite; 
for  if  w^e  set  a  =  c-\-x,  d  =  b-\-y,  and  assume*  that  the  common  value  of 
2cx-\-x^  and  2by-\-'if'  is  of  the  form  xyz,  we  get 

2c  =  yz-x,  2b  =  xz-y,  N  =  b'^+c^+xyz  =  \{x'^-]-y^){l+z^), 

whence  x^-\-y^  or  {x^+y^)/4:  is  a  factor  of  N.  To  express  iV  as  a  sum  of  two 
squares  in  all  possible  ways,  use  is  made  of  the  final  digit  of  N  to  limit  the 
squares  x^  to  be  subtracted  in  seeking  differences  N  —  x^  which  are  squares. 
Several  numerical  examples  of  factoring  are  treated  in  full. 

Euler-^  gave  abbreviations  of  the  work  of  applying  the  preceding  test. 
For  example,  if  4n+l=5m+2  =  x^+!/^,  then  x  and  y  are  of  the  form 

"Sphinx-Oedipe,  1912,  61-4. 

i^L'enseigncment  math.,  15,  1913,  333-4. 

"Oeuvres  de  Fermat,  2.  1894,  232,  Aug.  2,  1641. 

"Letters  to  Goldbach,  Feb.  16,  1745,  May  6,  1747;  Corresp.  Math.  Phys.  (ed.,  Fuss),  I,  1843, 
313,  416-9. 

"Novi  Comm.  Ac.  Petrop.,  4,  1752-3,  p.  3;  Comm.  Arith.,  1,  1849,  165-173. 

*Euler  gave  a  faultless  proof  in  the  margin  of  his  posthimious  paper,  Tractatus,  §570,  Comm. 
Arith.,  2,  573;  Opera  postuma,  I,  1862,  73.  We  have  {a+c)ia-c)  =  {b+d){d-b)  =pqrs, 
a+c  =  pq,a—c  =  rs,  b+d  =  pr,  d—b  =  qs  [since,  if  pbe  theg.  c.  d.  of  a+c,  b+d,  then  g(a—c) 
is  divisible  by  r,  whence  a—c  =  rs].     Hence  a2+6^  =  (p*+a^)(g'+r')/4. 

»»Novi  Comm.  Ac.  Petrop.,  13,  1768,  67;  Comm.  Arith.,  1,  379. 


Chap.  XIV]  METHODS   OF  FaCTOKING.  361 

5p±  1.  To  express  a  number  as  x^+y^,  subtract  squares  in  turn  and  seek  a 
remainder  which  is  a  square. 

N.  Beguehn^^  proposed  to  find  x  such  that  4pV+ 1  is  a  prime  by  exclud- 
ing the  values  x  making  the  sum  composite.     The  latter  is  the  case  if 

4pV+l=462+(2c+l)2,  a;2  =  ^!±^. 

Set  X  =  q+b/p.  Then  b  is  expressed  rationally  in  terms  of  c  and  the  known 
p.  Taking  p  =  l,  he  derived  a  tentative  process  for  finding  a  prime,  of  the 
form  4x^+1,  which  exceeds  a  given  number  a. 

L.  Euler^°  proved  that  1000^+3^  is  prime  since  not  expressible  as  a  sum  of 
two  squares  another  way. 

A.  M.  Legendre^""  factored  numbers  represented  as  a  sum  of  two  squares 
in  two  ways. 

J.  P.  Kulik's^"^  tables  VIII  and  IX,  relating  to  the  ending  of  squares, 
serve  to  test  if  4n+l  is  a  sum  of  two  squares  and  hence  to  test  if  it  be  prime 
or  composite. 

Th.  Harmuth^^  suggested  testing  a^-j-b^  for  factors,  where  a  and  b  are 
relatively  prime,  by  noting  that  it  is  divisible  by  5  if  a=  =t  1,  6=  ±  2  (mod  5) , 
and  similar  facts  for  p  =  13,  17,  29,  37,  there  being  p  —  1  sets  of  values  of  a,  b 
for  each  prime  p  =  4n+l. 

G.  Wertheim^^  explained  in  full  Euler's^''  method  of  factoring. 

R.  W.  D.  Christie  and  A.  Cunningham^^  granted  N  =  A^+B^  =  C^+D'^ 
and  showed  how  to  find  a,...,dso  that  N={a^+b^){c^-\-cP).  Similarly,  if 
N  =  x^-\-Py'^  in  two  ways. 

Factoring  by  Use  of  Binary  Quadratic  Forms. 

L.  Euler^^  noted  that  a  number  is  composite  if  it  be  expressible  in  two  v^  /*  .fi'*^ 
ways  in  the  form/  =  ax^+i(32/^.  The  product  of  two  numbers  of  the  form/ 
is  of  the  form  g  =  aPx^+y^;  the  product  of  a  number  of  the  form  /  by  one  of 
the  form  g  is  of  the  form /.  If  for  m>2  a  composite  number  mp  is  express- 
ible in  a  single  way  in  the  form  /,  there  exist  an  infinitude  of  composite 
numbers  mq  expressible  in  a  single  way  by  /.  He  called  (§34)  a  number 
A''  idoneal  (numerus  idoneus)  if,  for  a^  =  N,  every  number  representable 
hy  f  =  ax^+^y^  (with  ax  relatively  prime  to  ^y)  is  a  prime,  the  square  of  a 
prime,  the  double  of  a  prime  or  a  power  of  2,  so  that  a  number  representable 
by / in  a  single  way  is  a  prime.  It  suffices  to  test  N+y^<4N,  y  prime  to  N. 
He  gave  (§39,  p.  208)  the  65  idoneal  numbers  1,  2, . . .,  1848  less  than  10000. 

"Nouv.  M^m.  Acad.  Sc.  BerUn,  1777,  ann6e  1775,  300. 

soNova  Acta  Petrop.,  10,  1792  (1778),  63;  Comm.  Arith.,  2,  243-8. 

^'^Theorie  des  nombres,  ed.  3,  i830,  I,  310.     Simplification  by  Vuibert,  Jour,  de  math.  6\em., 

10,  p.  42.     Cf.  I'interm^diaire  des  math.,  1,  1894,  167,  245;  18,  1911,  256. 
3obTafebi  der  Quadrat  und  Kubik  Zahlen  ...  bis  hundert  Tausend,  Leipzig,  1848. 
"Archiv  Math.  Phys.,  67,  1882,  215-9. 
32Elemente  der  Zahlentheorie,  1887,  295-9. 
"Math.  Quest.  Educat.  Times,  (2),  11,  1907,  52-3,  65-7,  89-90. 
"Nova  Acta  Petrop.,  13,  1795-6  (1778),  14;  Comm.  Arith.,  2,  198-214. 


362  History  of  the  Theory  of  Numbers.  [Chap,  xiv 

Euler^*  used  the  idoneal  number  232  to  find  all  values  of  a  <  300  for 
which  232a- +  1  is  a  prime,  by  excluding  the  values  of  a  for  which  232a^-hl 
=  232x~-\-y-,y>l. 

Euler^^  noted  that  N  =  a~ -\-\b- =  x^ +\y^  imply 

N  =  l(\nr+n'^){\p^-\-q^),  a=^x  =  \mp,  nq;  y=^h  =  mq,  np, 

so  that  Xp^  +  g-,  or  its  half  or  quarter,  is  a  factor  of  A''.  He  gave  (p.  227) 
his^'  former  table  of  65  idoneal  numbers.  Given  one  representation  by 
cur 4-/3?/",  where  a/3  is  idoneal,  he  sought  a  second  representation.  If 
A^  =  4n  +  2  is  idoneal,  4iV  is  idoneal. 

Euler^*^  called  mx--\-ny-  a  congruent  form  if  everj'  number  representable 
by  it  in  a  single  way  (with  x,  y  relatively  prime)  is  a  prime,  the  square  of  a 
prime,  the  double  of  a  prime,  a  power  of  2,  or  the  product  of  a  prime  by  a 
factor  of  vin.  Then  also  vinx~-\-y~  is  a  congruent  form  and  conversely. 
The  product  mn  is  called  an  idoneal  or  congruent  number.  His  table  of  65 
idoneal  numbers  is  reproduced  (§18,  p.  253).  He  stated  rules  for  deducing 
idoneal  numbers  from  given  idoneal  numbers.  He  factored  numbers 
expressed  in  two  ways  by  ax^+/3i/-,  where  a/S  is  idoneal,  and  noted  that  a 
composite  number  may  be  expressible  in  a  single  way  in  that  form  if  a^  is 
not  idoneal. 

Euler^^  proved  that  the  first  five  squares  are  the  only  square  idoneal 
numbers. 

C.  F.  Kausler^-  proved  Euler's  theorem  that  a  prime  can  be  expressed  in 
a  single  way  in  the  form  mx'^-\-ny'^  ii  m,  n  are  relatively  prime.  To  find  a 
prime  v  exceeding  a  given  number,  see  whether  38a:- +5?/^  =  v  has  a  single  set 
of  positive  solutions  x,  y;  or  use  1848x"+?/^.  As  the  labor  is  smaller  the 
larger  the  idoneal  number  38-5  or  1848,  it  is  an  interesting  question  if  there 
be  idoneal  numbers  not  in  Euler's  list  of  65.     Cf.  Cunningham. ^^ 

Euler"*^  gave  the  65  idoneal  numbers  n  (with  44  a  misprint  for  45)  such 
that  a  number  representable  in  a  single  way  by  nx~-\-y'^  {x,  y  relatively 
prime)  is  a  prime.     By  using  n  =  1848,  he  found  primes  exceeding  10  million. 

N.  Fuss^  stated  the  principles  due  to  Euler.^^ 

E.  Waring^^  stated  that  a  number  is  a  prime  if  it  be  expressible  in  a  single 
way  in  the  form  ar-\-mh^  and  conversely. 

A.  M.  Legendre"*^  would  express  the  number  A  to  be  factored,  or  one  of 
its  multiples  kA ,  in  the  form  t~+air,  where  a  is  as  small  as  possible  and  within 
the  limits  of  his  Tables  III-YII  of  the  linear  forms  of  divisors  of  f^au^. 

»»Nova  Acta  Petrop.,  14,  1797-8  (1778).  3;  Ck)mm.  Arith.,  2,  215-9. 

»»/Wd.,  p.  11;  Comm.  Arith.,  2,  220-242.     For  X  =  2,  Opera  postuma,  I,  1862,  159. 

"/Wd..  12,  1794  (1778),  22;  Comm.  Arith.,  2,  249-260. 

"/6wf.,  15,  ad  annos  1799-1802  (1778),  29;  Comm.  Arith.,  2,  261-2. 

"Ibid.,  156-180. 

"Nouv.  M^m.  Berlin,  ann6e  1776,  1779,  337;  letter  to  Beguelin,  May,  1778;  Comm.  Arith.,  2, 

270-1. 
**Ibid.,  340-6 

«Medit.  Algebr.,  ed.  3,  1782,  352. 
«^h6orie  des  nombres,  1798,  pp.  313-320;  ed.  2,  1808,  pp.  287-292.     German  transl.  by  Maser, 

1,  329-336.     Cf.  Sphinx-Oedipe,  1906-7,  51. 


Chap.  XIV]  METHODS  OF  FACTORING.  363 

Then  the  divisors  of  A  are  included  among  these  linear  forms.  When 
VS  is  converted  into  a  continued  fraction,  let  (VM +/)/!)  be  a  complete 
quotient,  and  p/q  the  corresponding  convergent.  Then  ^D  =  p^  —  kAq^, 
so  that  the  divisors  of  A  are  divisors  oi  p^=f=D. 

C.  F.  Gauss'*^  stated  that  the  65  idoneal  numbers  n  of  Euler  and  no 
other  numbers  have  the  two  properties  that  all  classes  of  quadratic  forms 
of  determinant  —n  are  ambiguous  and  that  any  two  forms  in  the  same 
genus  (Geschlecht)  are  both  properly  and  improperly  equivalent. 

Gauss^^  gave  a  method  of  factoring  a  number  M  based  on  the  deter- 
mination of  various  small  quadratic  residues  of  M. 

Gauss^^  gave  a  second  method  of  factoring  M  based  on  the  finding  of 
representations  of  M  by  forms  x^+D,  where  D  is  idoneal. 

F.  Minding^°  gave  an  exposition  of  the  method  of  Legendre.^^ 

P.  L.  Tchebychef^^  gave  a  rapid  process  to  find  many  forms  x^^ay^ 
which  represent  a  given  number  A  or  a  multiple  of  A.  Then  a  table  of 
the  linear  forms  of  the  divisors  of  x^^ay^  serves  to  limit  the  possible  factors 
of  A. 

Tchebychef^^  gave  theorems  on  the  limits  between  which  lie  at  least 
one  set  of  integral  solutions  of  x^  —  Dy^  =  ±  iV.  If  there  are  two  sets  of  solu- 
tions within  the  limits,  N  is  composite.  There  are  given  various  tests  for 
primality  by  use  of  quadratic  forms. 

C.  F.  Gauss^^  left  posthumous  tables  to  facilitate  factoring  by  use  of 
his*^  second  method. 

F.  Grube^^  criticized  and  completed  certain  of  Euler's  proofs  relating  to 
idoneal  numbers,  here  called  Euler  numbers.  While  Gauss^^  said  it  is  easy 
to  prove  Euler's*^  criterion  for  idoneal  numbers,  Grube  could  prove  only 
the  following  modification:  Let  Q,  be  the  set  of  numbers  D+n^^4Z)  in 
which  n  is  prime  to  D.  According  as  all  or  not  all  numbers  of  12  are  of  the 
form  q,  2q,  q"^,  2^  {q  a  prime),  D  is  or  is  not  an  idoneal  number. 

E.  Lucas^^  proved  that  if  p  is  a  prime  and  A;  is  a  positive  integer,  and 
p  =  x^+ky^,  then  pT^Xi^+ky^^  for  values  Xi,  yi  distinct  from  ^x,  ^y. 

P.  Seelhoff^^  made  use  of  170  determinants  (including  the  65  idoneal 
numbers  of  Euler  and  certain  others  of  Legendre),  such  that  every  reduced 
form  in  the  principal  genus  is  of  the  type  ax^-\-by^.  To  factor  A^  seek 
among  the  numbers  m  of  which  iV  is  a  quadratic  residue  several  values 

<^Disq.  Arith.,  1801,  Art.  303. 

"7btd.,  Arts.  329-332. 

*mid.,  Arts.  333-4. 

s^Anfangsgriinde  der  Hoheren  Arith.,  1832,  185-7. 

"Theorie  der  Congruenzen  (in  Russian),  1849;  German  transl.  by  H.  Schapira,  1889,  Ch.  8, 

pp.  281-292. 
"Jour,  de  Math.,  16,  1851,  257-282;  Oeuvres,  1,  73. 
«Werke,  2,  1863,  508-9. 
"Zeitschrift  Math.  Phys.,  19,  1874,  492-519. 
"Nouv.  Corresp.  Math.,  4,  1878,  36.     [Euler."] 
"Archiv  Math  Phys.,  (2),  2,  1885,  329;  (2),  3,  1886,  325;  Zeitschrift  Math.  Phys.,  31,  1886, 166, 

174,  306;  Amer.  Jour.  Math.,  7,  1885,  264;  8,  1886,  26-44. 


364  History  of  the  Theory  of  Numbers.  [Chap,  xiv 

for  which  N  is  represent  able  by  x^-\-my-.     For  example,  if  iV  =  31-2^*-f  1, 

Eliminating  19-83  between  the  first  two,  we  get  nN  =  w'^  —  7f.  This  with 
the  third  leads  to  factors  of  N.  In  general,  when  elimination  of  common 
factors  of  the  77i's  has  led  to  representations  of  two  multiples  of  N  by  the 
same  form  x~+ny^,  we  may  factor  N  unless  it  be  prime. 

H.  Weber^^  computed  the  class  invariants  for  the  65  determinants  of 
Euler  and  remarked  that  there  is  no  known  proof  of  the  fact  found  by 
induction  by  Euler  and  Gauss  that  there  are  only  65  determinants  such  that 
all  classes  belonging  to  the  determinant  are  ambiguous  and  hence  each 
genus  has  only  one  class. 

T.  Pepin^^  developed  the  theory  of  Gauss'^^  posthumous  tables  and  the 
means  of  deducing  complete  tables  from  the  given  abridged  tables.  Pepin^* 
showed  how  to  abridge  the  calculations  in  using  the  auxiliary  tables  of  Gauss 
in  factoring  a"  —  1,  where  a  and  n  are  primes. 

D.  F.  Seliwanoff^°  noted  that  the  factoring  of  numbers  of  the  form 
t'  —  Du-  reduces  to  the  solution  of  {D/x)  =  \,  all  solutions  of  which  are 
easily  found  by  use  of  six  relations  by  Euler  on  these  Jacobi  symbols  {D/x) . 

E.  Lucas"  gave  a  clear  proof  of  Euler's  remark  that  a  prime  can  not  be 
expressed  in  two  ways  in  the  form  Ax~-\-By-,  if  A,  B  are  positive  integers. 

S.  Levanen^^  showed  and  illustrated  by  examples  and  tables  how  binary 
quadratic  forms  may  be  applied  to  factoring. 

G.  B.  ^Mathews*"^  gave  an  exposition  of  the  subject. 

T.  Pepin^  applied  determinants  —  8n  — 3  for  which  each  genus  has  three 
classes  of  quadratic  forms.  The  paper  is  devoted  mainly  to  the  solution 
of  X-+ (871+3)?/""  =  4A,  where  A  is  the  number  to  be  factored. 

T.  Pepin^^  assumed  that  the  given  number  N  had  been  tested  and  found 
to  have  no  prime  factor  ^p.  Let  Xx+1,  Xy+1  be  the  two  factors  of  N, 
each  between  p  and  N/p.  The  sum  of  the  factors  lies  between  2VW  and 
p+N/p.     Let  x—y  =  u,  x+y=pz.    Then  {N  —  l)/\  =  Xxy+x+y  gives 

in  which  special  values  are  assigned  to  p.     This  equation  yields  a  quadratic 
congruence  for  w"  with  respect  to  an  arbitrary  prime  modulus,  used  as  an 
excludant.     The  method  applies  mainly  to  numbers  a^=^l. 
E.  Cahen^^  used  the  linear  divisors  of  x^  +  Dy'^. 

'"Math.  Annalen,  33,  1889.  390-410. 

"Atti  Accad.  Pont.  Nuovi  Lincei,  48,  1889,  135-156. 

"Ibid.,  49,  1890.  163-191. 

"Moscow  Math.  Soc,  15,  1891,  789;  St.  Petersburg  Math.  See,  12,  1899. 

•'Th6orie  des  nombres,  1891,  356-7. 

"^vereigt  af  finska  Vetenskaps-Soc.  forhandlingar,  34,  1892,  334-376. 

"Theory  of  Numbers,  1892,  261-271.     French  transl.,  Sphinx-Oedipe,  1907-8,  155-8,  161-70. 

"Memorie  Accad.  Pontif.  Nuovi  Lincei,  9,  I,  1893,  46-76.     Cf.  Pepin,"  332. 

«;6iV/.,  17,  1900-1,  321-344;  Atti,  54,  1901,  89-93.     Cf.  Meissner"*,  121-2. 

"filaments  de  la  th^orie  des  nombres,  1900,  324-7.     Sphinx-Oedipe,  1907-8,  149-155. 


Chap.  XIV]  METHODS   OF  FACTORING.  365 

A.  Cunningham"  and  J.  Cullen  listed  the  188  prime  numbers  x^+18482/^ 
between  10"^  and  101-10^,  with  x  prime  to  1848?/. 

A.  Cunningham^ ^  noted  that  two  representations  of  N  by  ixx^-\-vy^ 
lead  to  factors  of  N  under  certain  conditions. 

A.  Cunningham^^  recalled  that  an  idoneal  number  /  has  the  property 
that,  if  an  odd  number  A''  is  expressible  in  only  one  way  in  the  form 
N  =  mx^-\-ny^,  where  mn  =  I,  and  mx^  is  prime  to  ny^,  then  A"  is  a  prime  or 
the  square  of  a  prime.  Euler's  largest  I  is  1848.  There  is  no  larger  I  under 
50000,  a  computation  checked  by  J.  Cullen.  Cunningham  noted  on  the 
proof-sheets  of  this  history  that  this  limit  has  been  extended  to  100  000. 

A.  Cunningham^"  noted  conditions  that  an  odd  prime  be  expressible  by 
f^qu"  when  q  or  —q'l^  idoneal. 

F.  N.  Cole'^^  discussed  Seelhoff's^^  method  of  factoring. 

Al.  Laparewicz^^  described  and  applied  Gauss'  ^^'^^  two  methods. 

P.  Meyer^^  discussed  Euler's  theorem  that,  if  n  is  idoneal,  a  number 
representable  only  once  by  x^-\-ny'^  is  a  prime. 

R.  BurgwedeF^  gave  an  exposition  and  completion  of  the  method  of 
Euler^^'^^  and  an  exposition  of  the  methods  of  Gauss.^^'*^ 

L.  Valroff  stated  and  A.  Cunningham'^^"  proved  that  (Dx'^  —  a^)  (Dy"^  —  a^) 
=  Dz^  —  a^  implies  that  one  factor  is  composite  unless  x^  =  y^  =  4:  when 
o  =  1,  D  =  2,  and  in  the  remaining  cases  if  the  two  factors  are  distinct  and  >  1. 

A.  Gerardin'^^  gave  a  method  illustrated  for  N  =  a^  —  5-29^,  where  a  =  6326. 
We  shall  have  a  second  such  representation  N  =  (a-{-5xy  —  5y^  if 

E=5x^+2ax+841=y^. 

Use  is  made  of  various  moduli  ?«  =  4,  3,  7,  25, ... .  On  square-ruled  paper, 
mark  x  =  0,  1,  2, .  .  .  at  the  head  of  the  columns.  On  the  line  for  modulus  m, 
shade  the  square  under  the  heading  x  when  x  makes  E  a  quadratic  non- 
residue  of  m.  Then  examine  the  column  in  which  occurs  no  shaded  square. 
Up  to  x^l5,  these  are  x  =  0  (excluded),  and  x  =  4,  which  gives  A  =  6346^ 
—  5-227^  and  the  factor  99^  —  5-2^.  The  same  diagram  serves  for  all  num- 
bers 1050  ff +671,  our  N  being  given  by  i7  =  38108.  To  apply  the  method 
to  A  =  (2a;)*+1  =  (4x^+1)^  — 2(2x)^,  seek  a  second  representation  N=(4x^ 
+2p  +  lY-2{2uy.  The  condition  is  {2p-\-l)x^+Mp-j-l)=u^,  solutions 
of  which  are  found  for  p  =  1,  8,  9, .  .  . ,  6^,  35^, ...  Or  we  may  choose  x,  say 
X  =  48,  and  find  p  =  S,  u  =  198. 

s^Brit.  Assoc.  Reports,  1901,  552.     The  entry  10098201  is  erroneous. 

"Proc.  London  Math.  Soc,  33,  1900-1,  361. 

«976id.,  34,  1901-2,  54. 

■"^Ibid.,  (2),  1,  1903,  134. 

7iBuU.  Amer.  Math.  Soc,  10,  1903-4,  134-7. 

"Prace  mat.  fiz.,  Warsaw,  16,  1905,  45-70  (PoUsh). 

^'Beweis  eines  von  Euler  entdeckten  Satzes,  betreffend  die  Bestimmung  von  Primzahlen,  Diss., 

Strassburg,  1906. 
'^Ueber  die  Eulerschen  und  Gausschen  Methoden  der  Primzahlbestimnaung,  Diss.,  Strassburg, 

1910,  101  pp. 
'*»Sphinx-Oedipe,  7,  1912,  60,  77-9. 
"Wiskundig  Tijdschrift,  10,  1913,  52-62. 


366  History  of  the  Theory  of  Numbers.  [Chap,  xiv 

G^rardin^®  gave  a  note  on  his  machine  to  factor  large  numbers,  espe- 
cially those  of  the  form  2x^  —  1. 

Factoring  by  Method  of  Final  Digits. 

Johann  Tessanek^°  gave  a  tedious  method  of  factoring  N,  not  divisible 
by  2,  3,  or  5,  when  A^/10  is  within  the  limits  of  a  factor  table.  For  example, 
let  N=10a+l;  its  factors  end  in  1,1  or  3,7  or  9,9.  To  treat  one  of  the 
four  cases,  consider  a  factor  lOx+3,  the  quotient  being  102+7.  Then  z  is 
the  quotient  of  a  — 2  — 7x  by  lOx+3.  Give  to  x  the  values  1,  2, . .  .,  and  test 
a  — 9  for  the  factor  13,  a  — 16  for  23,  etc.,  by  the  factor  table.  He  gave  a 
lengthy  extension^^  to  di\'isors  100j+10/+gr.  Again,  to  factor  iV  =  2a  +  l, 
given  a  table  extending  to  N/2,  note  that  if  2xH-l  is  a  divisor  of  N,  it 
di\'ides  a—x,  which  falls  in  the  table.  F.  J.  Studnicka^^  quoted  the  last 
result. 

N.  Beguelin^  would  factor  iV  =  4p+3  by  considering  the  final  digit  of 
TT  =  (A''  —  1 1)/4  and  hence  find  the  proper  line  in  an  auxihary  table  (pp.  291-2) , 
each  line  containing  four  fractional  expressions.  Proceed  with  each  until 
we  reach  a  fraction  whose  numerator  is  zero.  Then  its  denominator  is  a 
factor  of  A^. 

Georg  Simon  KliigeP  noted  that  a  number,  not  divisible  by  2,  3  or  5, 
is  of  the  form  30x+m  (m  =  l,  7,  11,  13,  17,  19,  23,  29).  Suppose  10007  = 
(30x+w)(30!/+n).  Then  {m,  n)  =  (l,  17),  (7,  11),  (13,  29)  or  (19,  23). 
For  m  =  1,  n  =  17,  we  get 

333-?/ 

^=3o^Ti7'   ^<^'  y<^' 

But  X  is  not  integral  for  y  =  0,  1,  2,  3. 

Johann  Andreas  von  Segner  {ibid.,  217-225)  took  two  pages  to  prove 
that  any  number  not  divisible  by  2  or  3  is  of  the  form  6n='=  1  and  noted  that, 
given  a  table  of  the  least  prime  factor  of  each  6n=tl,  he  could  factor  any 
number  within  the  limits  of  the  table! 

Sebastiano  Canterzani^^  would  factor  10^  +  1,  by  noting  the  last  digits 
1,  1  or  3,  7  or  9,  9  of  its  factors.  If  one  factor  ends  in  7,  there  are  10 
possibilities  for  the  digit  preceding  7;  if  one  ends  in  1  or  9,  there  are  five 
cases;  hence  20  cases  in  all.     A.  Niegemann^^"  used  the  same  method. 

Anton  Niegemann^®  gave  a  method  of  computing  a  table  of  squares 
arranged  according  to  the  last  two  digits.     Thus,  if  A76  =  {l0x  —  Qy,  then 

™\s30c.  fran?.  avanc.  sc,  43,  1914,  26-8.     Proc.  Fifth  Internat.  Congress,  II,  1913,  572-3; 

Brit.  Assoc.  Reports,  1912-3,  405. 
••Abhandl.  einer  Privatgesellschaft  in  Bohmen,  zur  Aufnahme  der  Math.,  Geschichte, . . . ,  Prag,  I, 

1775,  1-64. 
"M.  Cantor,  Geschichte  Math.,  4,  1908,  179. 
"Casopis,  14,  1885,  120  (Fortschr.  der  Math.,  17,  1885,  125). 
"Nouv.  Mdm.  Ac.  Berhn,  ann^e  1777  (1779),  265-310. 
•*Leipziger  Magazin  fiir  reine  u.  angewandte  Math,  (eds.,  J.  Bernoulli  und  Hindenburg),  1, 1787, 

199-216. 
"Memorie  dell'  Istituto  Xazionale  Italiano,  Classe  di  Fis.  e  Mat.,  Bologna,  2,  1810,  II,  445-476. 
""Entwickelung . . .  Theilbarkeit,  Jahresber.  Kath.  Gymn.  Kobi.,  1847-8,  23. 
"Archiv  Math.  Phys.,  45,  1866,  203-216  . 


^'11    Chap.  XIV]  METHODS   OF  FACTORING.  367 

A0  =  10x^  —  12x  —  4:,  whence  12a:+4  is  divisible  by  10,  so  that  a:  =  5(i  — 2. 
Then  A=25d'^  —  2Qd+Q.  Thus  if  we  delete  the  last  two  digits  7  ,  6  of  squares 
A7Q,  we  obtain  numbers  A  whose  values  for  d  =  l,  2, .  .  .  can  be  derived 
from  the  initial  one  5  by  successive  additions  of  49,  49+50,  49+2-50, .  .  . . 
He  gave  such  results  for  every  pair  of  possible  endings  of  squares. 

A  similar  method  is  applied  to  any  composite  number.  One  case  is 
when  the  last  two  digits  are  m,  1  and  Aml  =  (10a;  — 1)(10?/  — 1).     Then 

A0  =  10xy  —  y  —  x  —  m,        y+x+m  =  10a,        A  =  10ax—x'^—mx  —  a. 

The  discriminant  of  the  last  equation  must  be  a  square.  A  table  of  values 
of  A  for  each  a  may  be  formed  by  successive  additions. 

G.  Speckmann^^  noted  that  the  two  factors  of  AT"  =  2047  end  in  1  and  7 
or  3  and  9.  Treating  the  first  case,  we  see  that,  if  a  and  b  are  the  digits  in 
tens  place,  6+7a=4  (mod  10),  so  that  the  factors  end  in  01  and  47,  or  11  and 
77,  etc. 

G.  Speckmann^^  wrote  the  given  number  prime  to  3  in  the  form  9a +6 
(6<9),  so  that  the  sum  of  its  digits  is  =6  (mod  9).  By  use  of  a  small 
auxiliary  table  we  have  the  residues  modulo  9  of  the  sums  of  the  digits  of 
every  possible  pair  of  factors. 

R.  W.  D.  Christie^^  and  D.  Biddle^°  made  an  extensive  use  of  terminal 
digits. 

E.  Barbette^ ^  noted  that  lOd+u  has  a  divisor  lOw  — 1  if  and  only  if 
d+?nw  has  that  divisor.     Set  d-\-'mu  =  n(10m  —  l),  d  =  10d'+u'.     Then 
mn  =  d'-\-x,  lOx  =  'mu-\-n-\-u'. 

Eliminating  n,  we  get  a  quadratic  for  m.  Its  discriminant  is  a  quadratic 
function  of  x  which  is  to  be  made  a  square.     Similarly  for  lOm  +  1,  10m ±3. 

A.  Gerardin^^"  developed  Barbette's^^  method. 

R.  Rawson^^  found  Fermat's^  factors  of  a  number  proposed  by  Mersenne 
by  writing  it  to  the  base  100  and  expressing  it  as  (a- 10^ +23)  (6 -10^ +3). 

J.  Deschamps^^  would  use  the  final  digits  and  auxiliary  tables. 

A.  Gerardin®^  would  factor  N  (prime  to  2,  3,  5)  by  use  of 

iV=120n+i^=(120x+a)(120i/+6), 

and  a  table  showing,  for  each  of  the  32  values  of  K<  120,  the  16  pairs  o,  b 
(each<  120)  such  that  ab=K  (mod  120).     He  factored  Mersenne's  number.^ 

Factoring  by  Continued  Fractions  or  Fell  Equations. 
Franz  von  Schaffgotsch^""  would  factor  a  by  solving  az^-^l  =  x^  (having 

"Archiv  Math.  Phys.,  (2),  12,  1894,  435.  ssArchiv.  Math.  Phys.,  14,  1896,  441-3. 

"Math.  Quest.  Educat.  Times,  69,  1898,  99-104.     Cf.  Meissner,"^  138-9. 

'"/feid.,  87-88,  112-4;  71,  1899,  93-9;  Mess.  Math.,  28,  1898-9, 120-149, 192  (correction).     Cf. 

Meissner,"8 137-8.  siMathesis,  (2),  9,  1899,  241. 

"«Sphinx-Oedipe,  1906-7,  [1-2,  17,  33],  49-50,  54,  65-7,  77-8,  81-4;  1907-8,  33-5;  5,  1910, 

145-7;  6,  1911,  157-8.  s^Math.  Quest.  Educat.  Times,  71,  1899,  123-4. 

"^BuU.  See.  Philomathique  de  Paris,  (9),  10,  1908,  10-26. 
s'Assoc.  frang.,  38,  1909,  145-156;  Sphinx-Oedipe,  Nancy,  1908-9,  129-134,  145-9;  4,  1909, 

3«  Trimestre,  17-25. 
i»»Abh.  Bohmischen  Gesell.  Wiss.,  Prag,  2,  1786,  140-7. 


368  History  of  the  Theory  of  Numbers.  [Chap,  xiv 

solutions  if  a  is  not  a  square)  and  testing  x"  —  1  for  a  factor  in  common  with  a. 
Further,  U  ay-\-l=x'^  does  not  hold  for  l<x<a  — 1,  then  a  is  a  power  of  a 
prime  and  conversely  [false  if  a  =  10]. 

Marcker^*'^  noted  that  if  there  are  2n  terms  in  the  period  of 

and  Q  =  0,  Q'  =  a,  Q"  =  a'P' -Q',.  .  .,^ 

p_i  p/_^~Q''  ptf  _^~Q"^ 

r  —  1,  r    —       p      '  pf      )  •  ■  •  i 

then  the  nth  P  or  its  half  is  a  factor  of  A.  If  A  is  a  prime,  then  the  nth 
Pis  2. 

J.  G.  Birch^°-  derived  a  factor  of  N  from  a  solution  x  of  x'^  =  Ny+l.  The 
continued  fraction  for  x/{N —x)  is  of  the  form 

1        J_    J_  ill 

Go- 1  +01+02+  ■  •  •  +a2+ai+ao' 
and  N  is  the  continuant  defined  as  the  determinant  with  Oq,  Oi,.  .  .,  a„_i, 
On)  On-i>- •■>  Oi,  Oo  in  the  main  diagonal,  elements  +1  just  above  this 
diagonal,  elements  —1  just  below,  and  zeros  elsewhere.     Then  the  continu- 
ant with  the  diagonal  ao>  •  •  •>  o„_i  is  a  factor  of  N. 

W.  W.  R.  BalP°^  appHed  this  method  to  a  number  of  Mersenne.^ 

A.  Cunningham^*^  noted  that  a  set  of  solutions  of  if—Dx^  =  —  1  gives  at 
sight  factors  of  7/"  +  l. 

M.  V.  Thielmann^o^  illustrated  his  method  by  factoring  /c  =  36343817. 
The  partial  denominators  in  the  continued  fraction  for  \/^  are  1,  1,  2,  1,  1, 
12056.  Drop  the  last  term  and  pass  to  the  ordinary  fraction  7/12.  Hence 
set  (12x+7)^  =  12^i/+l.  The  least  solution  is  x  =  4,  y  =  2\.  Using  the  part 
of  the  period  preceding  the  middle  term  it»  =  2,  we  get 

Y^  =  -^,         P  =  l,        M  =  2,         Q  =  iyM+2P  =  6,  u  =  MQ  =  \2. 

Hence  ^"  — 21m^  =  1  has  the  solution  <  =  55.     For  a  suitably  chosen  n, 

ifc  =  wV+2^n+21=  (2g2n+^^  Uu^n+^-^X 

where  q  is  the  largest  integer  ^  Q/2.  Here  n  =  502  and  the  factors  of  k  are 
2-3"n  +  7  and  2-22n+3. 

D.  N.  Lehmer^''^  noted  that  if  jR  =  pg'  is  a  product  of  two  odd  factors 
whose  difference  is  <2y/R,  so  that  Kp— 9)^<V7^,  then 

x'-Rf  =  \{V-q? 
has  the  integral  solutions  x  =  (p+g')/2,  y  =  \.     Hence  i(p  — g')^  is  a  denomi- 
nator of  a  complete  quotient  in  the  expansion  of  y/R,  as  a  continued  fraction, 

""Jour,  fur  Math.,  20,  1840,  355-9.     Cf.  I'interm6diaire  dea  math.,  20,  1913,  27-8. 
J^Mcsa.  Math.,  22,  1892-3.  52-5. 

'o'/6id.,  p.  82-3.     French  transl.,  with  Birch"^,  Sphinx-Oedipe,  1913,  86-9. 
>"/6id.,  35,  1905-6,  166-185;  abat.  in  Proc.  London  Math.  Soc.  3,  1905,  xxii. 
'""Math.  Annalen,  62,  1906,  401. 

Bull.  Amer.  Math.  Soc,  13,  1906-7,  501-2.     French  transl.,  Sphinx-Oedipe,  6,  1911,  138-9. 


IM 


Chap.  XIV] 


Methods  of  Factoring. 


369 


in  view  of  the  theorem  of  Lagrange:  If  x^—Ry'^=^D  has  relatively  prime 
integral  solutions  x,  y,  where  D<  \/r,  then  D  is  a  denominator  of  a  com- 
plete quotient  in  the  expansion  of  \/r  as  a  continued  fraction. 

Factoring  by  use  of  Various  Moduli. 

C.  F.  Gauss^^°  gave  a  "method  of  exclusion,"  based  on  the  use  of  various 
small  moduli,  to  express  a  given  number  in  a  given  form  mx^+ny^. 

V.  Bouniakowsky^^^  noted  that  information  as  to  the  prime  factors  of  a 
number  N  may  be  obtained  by  comparing  the  solution  x  =  (l){N)  of  2''=1 
(mod  N)  with  the  least  positive  solution  x  =  a  found  by  a  direct  process  such 
as  the  following :  Since  2"  =  NK+l,  multiply  the  given  N  by  the  unknown 
K,  each  expressed  in  the  binary  scale  (base  2),  add  1  and  equate  the  result  to 
10.  .  .0.     The  digits  of  K  are  found  seriatim  and  very  simply. 

H.  J.  WoodalP^^  expressed  the  number  N  to  be  factored  in  the  form 
^a_|_^6_^  .  . .  +r,  where  r<  1000,  while  a,  /3, . . .  are  small,  but  not  necessarily 
distinct.  Hence  the  residues  of  N  with  respect  to  various  moduli  are 
readily  found  by  tables  of  residues. 

F.  Landry^^^  employed  the  method  of  exclusion  by  small  moduli. 

D.  Biddle^^^  investigated  factors  2Ap  +  l  by  using  moduli  A^,  4A^. 

C.  E.  Bickmore,  A.  Cunningham  and  J.  CuUen^^^  each  treated  the  large 
factor  of  10^^+1  by  use  of  various  moduli,  and  proved  it  is  prime. 

J.  Cullen"^"  gave  an  effective  graphical  process  to  factor  numbers  by 
the  use  of  various  moduli;  the  numbers  to  be  searched  for  in  a  diagram 
are  all  small. 

Alfred  Johnsen^^^  used  Rt(p)  to  denote  the  numerically  least  residue  of 
p  modulo  t.     Then,  for  every  p,  t,  k, 

[Rmf+Ri(p-k')^Rt(p)  (modO. 
If  Hs  a  factor  of  the  given  number  p,  the  left  member  will  be  divisible  by  t. 
In  practice  take  k^  to  be  the  nearest  square  to  p,  larger  or  smaller.     For 
example,  let  p  =  4699,  k^  =  4624  =  68^  p-k''  =  75.    Then 


t 

[^.(68)P 

Rt(75) 

Sum 

7 
13 

37 

4 
9 

36 

-2 
-3 

"i 

2 
6 

37 

Thus  37  is  the  least  factor  of  p. 


""DJsq.  Arith.,  1801,  Arts.  323-6. 

i"M6m.  Ac.  Sc.  St.  P^tersbourg,  Math.-Phys.,  (6),  2,  1841,  447-69.    Extract  in  Bull.  Ac.  Sc,  6, 

p.  97.     Cf.  Nordlund.i" 
"2Math.  Quest.  Educat.  Times,  70,  1899,  68-71;  71,  1899,  124. 
"sproc^d^s  nouveaux .  .  . ,  Paris,  1859.     Cf.  A.  Aubry,"^  pp.  214-7. 
i"Mess.  Math.,  30,  1900-1,  98,  190.     Math.  Quest.  Educat.  Times,  74,  1901,  147-152. 
"'Math.  Quest.  Educat.  Times,  72   1900,  99-103. 
"sa/bid.,  73,  1900,  133-5;  75,  1901,  10^4.     Proc.  London  Math.  Soc,  34,  1901-2,  323-334; 

(2),  2,  1905,  138-141.  "'Nyt  Tidsskrift  for  Mat.,  15  A,  1904,  109-110. 


370  History  of  the  Theory  of  Numbers.  [Chap,  xiv 

K.  P.  Nordlund^^^  would  use  the  exponent  e  to  which  2  belongs  modulo 
iV  [Bouniakowsky^^^].  For  A^  =  91,  e  =  12  is  not  a  divisor  of  iV  — 1,  so  that 
A^  is  composite,  and  we  expect  the  factor  13. 

L.  E.  Dickson"^  found  the  factors  of  56'±1,  26'Hl,  34^^-}-l,  52^Hl 
by  an  expeditious  method.     For  example,  each  factor  of 

is  =l(mod  14).     Let  6  =  (l  +  14A;)(l  +  14A'i).     Then 

A'+;i-i  +  14A'A-i=4Ar,  /:+A'i  =  4+14/1. 

Thus  h  and  A-j  are  the  roots  of  a  quadratic  whose  discriminant  Q  is  of  the 
second  degree  in  In.  By  use  of  various  moduli  which  are  powers  of  small 
primes,  the  form  of  h  is  limited  step  by  step,  until  finally  at  most  a  half 
dozen  values  of  h  remain  to  be  tested  directly. 

L.  E.  Dickson^ ^®  gave  further  illustrations  of  the  last  method. 

J.  Schatunovsky^^°  reduced  to  a  minimum  the  number  of  trials  in 
Gauss'^^°  method  of  exclusion,  taking  the  simplest  case  m  =  \.  He  gave 
theorems  on  the  linear  forms  of  the  factors  of  a"~H-D6^,  which  lead  easily  to 
all  its  odd  factors  when  D  is  an  odd  prime. 

H.  C.  Pocklington^^^  would  use  Fermat's  theorem  to  tell  whether  A"  is 
prime  or  composite.  Choose  an  integer  x  and  find  the  least  positive  residue 
of  x^"^  modulo  A'';  if  p^l,  A''  is  composite.  But  if  it  be  unity,  let  p  be  a 
prime  factor  (preferably  the  largest)  of  A^— 1  and  contained  a  times  in  it. 
Find  the  remainder  r  when  x"*  is  divided  by  A',  where  7n  =  (A'  — l)/p.  If 
Tt^I,  let  6  be  the  g.  c.  d.  of  r— 1  and  A^.  If  6>1,  we  have  a  factor  of  A^. 
If  6  =  1,  all  prime  factors  of  A^  are  of  the  form  Ap^  +  l.  But  if  r  =  1,  replace 
m  by  mlq^  where  q  is  any  prime  factor  of  m  and  proceed  as  before. 

D.  Biddle^^^"  made  use  of  various  small  moduli. 

A.  Gerardin^^^''  used  various  moduli  to  factor  77073877. 

See  papers  14,  15,  21,  22,  48,  65. 

Factoring  Into  Two  Numbers  6n±l. 
G.  W.  Kraft^"  ^^^^^  ^hat  6a+l  =  (6w+l)(6n+l)  implies 

_  a—m 
^~6m  +  l' 
Find  which  7??  =  1,  2,  3, .  .  .  makes  n  an  integer. 

Ed.  BartP^  tested  6-186+5  for  a  prime  factor  less  than  31,  just  less 
than  its  square  root,  by  noting  that  186,  185,  184,  183,  182  are  not  divisible 
by  5,  11,  17,  23,  29,  respectively;  while  the  last  of  7,  13,  19  is  a  factor. 

'"Gotcborps  Kunpcl.  Vetenskaps.  Hand!.,  (4),  1905,  VII-VIII,  pp.  21-4. 

»«Amer.  Math.  Monthly,  15,  1908,  217-222.  "'Quar.  Jour.  Math.,  40,  1909,  40-43. 

""Der  Grosste  GemeinschaftUche  Teller  von  Algebr.  Zahlen  zweiter  Ordnung,  Diss.  Strassburg, 
Leipzig,  1912.  >"Proc.  Cambridge  Phil.  Sec,  18,  1914-5,  29-30. 

"'"Math.  Quest.  Educat.  Times,  (2),  25,  1914,  43-6. 
"'"L'enseignement  math.,  17,  1915,  244-5. 
'"Novi  Comm.  Ac.  Petrop.,  3,  ad  annos  1750-1,  117-8. 
""Zur  Theorie  der  Primzahlen,  Progr.  Mies,  Pilsen,  1871. 


Chap.  XIV] 


Methods  of  Factoring. 


371 


F.  Landry ^^^  treated  the  possible  pairs  6n±  1  and  6n'±  1  of  factors  of  N. 
Taking  for  example  the  case  of  the  upper  signs,  we  have 


Set  n+n'  =  6/i+r. 


Qnn'+n-\-n'  =  — - —  =  Qq-\-r. 
6 

Then  nn'  =  q  —  h,  whence 

-     q—n'(r—n') 
h  =  - 


6n'  +  l 

Give  to  n'  values  such  that  6n'+l  is  a  prime  <  ViV. 

K.  P.  Nordlund^25  treated  Qp-l  =  {Qm+l){Qn-l)  solved  for  m. 
D.  Biddle^^^  applied  the  method  to  6n±l. 
Hansen,^^  of  Ch.  XIII,  used  this  method. 

Miscellaneous  Methods  of  Factoring. 

Matsunaga^^^  wrote  the  number  to  be  factored  in  the  form  r^+R.  For 
r  odd,  set  r  =  Bi,  Bi  —  2  =  B2,  Bg  — 2  =  ^3, ...  and  perform  the  following 
calculations : 

R  =  Q,B,+Au 

Ai+K2'  =  Q2B2+A2, 
A2-\-Ks'  =  Q3Bs+As, 

etc.,  until  we  reach  A„  =  0;  then  Bn  is  a  factor, 
and  replace  i?  by  jR  +  1  in  what  precedes. 

J.  H.  Lambert ^^°  used  periodic  decimals  [see  Lambert,^  Ch.  VI]. 

Jean  Bernoulli^^^  gave  a  method  based  on  that  of  Lambert  (Mem.  de 
Math.  Allemands,  vol.  2).  Let  A=a^+b  have  the  factors  a—x  and 
a+x+y.  Then  x^  =  ay— xy  —  b.  Solve  for  a:.  Thus  2/^+ 4a?/  — 46  must  be 
a  square.     Take  y  =  l,  2, .  .  .  and  use  a  table  of  squares. 

J.  Gough^^^  gave  a  method  to  find  the  factors  r,  s  of  each  number  f^  —  c 
between  (/— 1)^  and/^.  For  example,  let/=3  and  make  a  double  row  for 
each  r  =  l,.  .  ., /.  In  the  upper  row  for  r  =  l,  insert  2/— 1,. ..,  1,  0;  in  the 
lower,  (/— 1)^, .  . .,  /^.     In  the  upper  row  for  r  =  2,  insert  1  (the  remainder 


K2  =  2Q2-\-K2', 
K3  =  2Qs+Ks', 


K2'^^K,+4, 

K3' =  7^2+8, 

If  r  is  even,  set  r  —  1  =  Bi 


r  =  l 

c  =  5 

s  =  4 

4 
5 

3 

6 

2 

7 

1 
8 

0 
9 

r  =  2 

c  =  5 

s  =  2 

3 
3 

1 
4 

r  =  3 

c  = 

s  = 

0 
3 

i2'»Assoc.  fran?.  avanc.  sc,  9,  1880,  185-9. 

i25Nyt  Tidsskrift  for  Mat.,  Kjobenhavn,  15  A,  1904,  36-40. 

i^^Math.  Quest.  Educ.  Times,  69,  1898,  87-8;  (2),  22,  1912,  38-9,  84-6. 

i^'Japanese  manuscript,  first  half  eighteenth  century,  Abhandl.  Geschichte  Math.  Wiss.,  30, 

1912,  236-7.  isoNoya  Acta  Eruditorum,  1769,  107-128. 

"iNouv.  Mem.  Ac.  BerHn,  ann6e  1771,  1773,  323. 
"2Jour.  Nat.  Phil.  Chem.  Arts  (ed.,  Nicholson),  1,  1809,  1-4. 


372  History  of  the  Theory  of  Numbers.  [Chap,  xiv 

on  dmding/^  by  2),  1+2,  1+2+2  under  1,  3,  5  of  the  first  row  for  r  =  l;  in 
the  lower  row,  insert  4  (the  quotient),  4  —  1,  4  — 2.  To  factor/^  — c,  locate 
the  column  headed  by  the  given  c;  thus,  for  c  =  3,  the  factors  are  s  =  6, 
r  =  1  and  s  =  3,  r  =  2.     Since  c  =  2  occurs  only  in  the  first  row,  9  —  2  is  prime. 

Joubin,^^^  J.  P.  Kulik,^^^  0.  V.  Kielsen,i36  ^^d  G.  K.  Winter^^e  published 
papers  not  accessible  to  the  author. 

E.  Lucas  gave  methods  of  factoring  and  tests  for  primes  (Ch.  XVII). 

D.  Biddle^"  wrote  the  proposed  number  N  in  the  form  S^-\-A,  where 
S-  is  the  largest  square  <  A^.  Write  three  rows  of  numbers,  the  first  begin- 
ning with  A,  or  A  —S  if  A  >S;  the  second  beginning  with  S  (or  S+1)  and 
increasing  by  1 ;  the  third  beginning  with  S  and  decreasing  by  1.  Let  A^, 
B„,  C„  be  the  nth  elements  in  the  respective  rows.     Then 

except  that,  when  i4„>C„,  we  subtract  C„  from  An  as  often  (say  k  times) 
as  will  leave  a  positive  remainder,  and  then  B„  =  5„_i  +  1+A:.  WTien  we 
reach  a  value  of  n  for  which  i4„  =  0,  we  have  N  =  BJJn-  For  example,  if 
iV  =  589  =  24H 13,  the  rows  are 

13        14        17  1  9  0 

24        25        26        28        29        31         (factors  31,  19). 
24        23        22        21        20        19 
It  may  prove  best  to  start  with  2N  instead  of  with  N. 
O.  Meissner^^^  reviewed  many  methods  of  factoring. 
R.  W.  D.  Christie^^^  gave  an  obscure  method  by  use  of  "roots." 
Christie""  noted  that,  if  N  =  AB, 
A  =  {4hN+d''-d)/i2b),  B  =  (46Ar+dHd)/2,  d=a-hc, 

whence  d''  =  {B-h Ay. 

D.  Biddle"^  gave  a  method  of  finding  the  factors  of  N  given  those  of 
AT+l.  SetL  =  iV-l.  Try  to  choose  i^  and  M  so  that  i(:iV/  =  iV+ 1  and 
so  that  1 +Z  is  a  factor  of  N.  Since  2N={l-\-K)  M-\-L  —  M,we  will  have 
L-M=il+K)m,  whence  2N={l+K){M+m).  For  iV=1829,  A^+1 
=  2-3-5-61.  Take  K  =  30,  M  =  61.  Then  m  =  57,  M+w  =  2-59,  iV  =  31-59. 
He  gave  {ibid.,  p.  43)  the  theoretical  test  that  N  =  S^+A  is  composite  if 
the  sum  of  r  terms  of 

is  an  integer  for  some  value  of  r. 

'»*Sur  lea  facteura  num6riques,  Havre,  1831. 

>«Abh.  K.  Bohm.  Gesell.  Wiss.,  1,  1841  (2,  1842-3,  47,  graphical  determination  of  primes). 

"HDra  et  heel  tals  upplosning  i  factorer,  Kjobenhavn,  1841. 

"•Madras  Jour.  Lit.  Sc,  1886-7,  13. 

"'Mesa.  Math.,  28,  1898-9,  116-20;  Math.  Quest.  Educat.  Timea,  70,  1899,  100,  122;  75,  1901, 

48;  extension,  (2),  29,  1916,  43-6. 
"«Math.  Naturw.  Blatter,  3,  1906,  97,  117,  137. 
"'Math.  Quest.  Educat.  Times,  (2),  12,  1907,  90-1,  107-8. 
"o/bid.,  (2),  13,  1908,  42-3,  62-3. 
"i/btd.,  (2),  14,  1908,  34.     The  process  is  well  adapted  to  factoring  2P-1,  (2),  23, 1913,  27-8. 


I 


Chap.  XIV]  METHODS    OF   FACTORING.  373 

E.  Lebon^^^"  would  first  test  AT  for  prime  factors  P  just  <  VN.  Let 
Q  be  the  quotient  and  R  the  remainder  on  dividing  N  by  P.  If  Q  and  R 
have  a  common  factor,  it  divides  N)  if  not,  N  is  not  divisible  by  any  factor 
of  Q  or  of  R. 

D.  Biddle^^^  considered  N  =  S'^+A  =  {S+u){S-v),  wrote  w  =  iVi  and 
obtained  hke  equations  in  letters  with  subscripts  unity.  Then  treat 
UiVi  =  N2  similarly,  etc. 

A.  Cunningham^^^  noted  that  the  number  of  steps  in  Biddle's^^^  process 
is  approximately  the  value  of  A;  in  2'''  =  N,  and  developed  the  process. 

E.  Lebon^^^  treated  the  decomposition  of  forms 

a:"±a:^±a;T±  .  .  .±1  (a>|8>7...) 

of  degrees  ^  9  into  two  such  forms,  using  a  table  of  those  forms  of  degrees 
^  4  with  all  coefficients  positive  which  are  not  factorable.  The  base  most 
used  in  the  examples  is  x  =  10.    But  bases  2  and  3  are  considered. 

E.  Barbette^^^  quoted  from  his^^^  text  the  theorem  that  any  integer  N 
can  be  expressed  in  each  of  the  four  forms 

where  ^j:  =  x{x-\-l)/2.  The  resulting  new  methods  of  factoring  are  now 
simplified  by  use  of  triangular  and  quadratic  residues.  The  first  formula 
implies  N={x—y){x-\ry+\)/2.     In  his  text,  he  considered  the  sum 

iv=(2/+i)+(y+2)+ .  ..+{x-\)+x=^,-^y 

of  consecutive  integers.  Treating  four  types  of  numbers  N,  he  proved  that 
this  equation  has  1,  2  or  more  than  2  sets  of  integral  solutions  x,  y,  according 
as  iV  is  a  power  of  2,  an  odd  prime,  or  a  composite  number  not  a  power  of  2. 
He  proved  independently,  but  again  by  use  of  sums  of  consecutive  integers, 
that  every  composite  number  not  a  power  of  2  can  be  given  the  form*  N  =  u 
(2y  — w+l)/2,  where  u  and  v  are  integers  and  v^u'^S.  Solving  for  u,  and 
setting  x  =  2v-\-l,  we  get  2u  =  x+(x'^  —  8Ny^^.  Hence  x'^—SN  =  y^  is  solva- 
ble in  integers  [evidently  by  a:  =  2A'+l,  y  =  2N—l].  Finally,  Nz=Ax  is 
equivalent  to  (2a;+l)^  =  8A^2;+l.  For  four  types  of  numbers  N,  the  solu- 
tions of  y^  =  SNz-\-l  are  found  and  seen  to  involve  at  least  two  arbitrary 
constants. 

A.  Aubry^^^  reviewed  various  methods  of  factoring. 

"wil  Pitagora,  Palermo,  14,  1907-8,  96-7. 

i«Math.  Quest.  Educat.  Times,  (2),  19,  1911,  99-100;  22,  1912,  38-9;  Educat.  Times,  63,  1910, 
500;  Math.  Quest,  and  Solutions,  2,  1916,  36-42. 

"»/6id.,  (2),  20,  1911,  59-64;  Educat.  Times,  64,  1911,  135. 

i^BuU.  soc.  philomathique  de  Paris,  (10),  2,  1910,  45-53;  Sphinx-Oedipe,  1908-9,  81-3,  97-101 

"^L'enseignement  math.,  13,  1911,  261-277. 

"^Les  sommes  de  p-i&mes  puissances  distinctes  ^gales  k  une  p-ieme  puissance,  Paris,  1910,  20-76. 
*This  follows  from  the  former  result  N  =  {x —y)(x+y +1) /2  by  setting  x=v,  y  =  v—u.  To 
give  a  direct  proof,  take  u  to  be  the  least  odd  factor  >  1  of  the  composite  number  N  not 
a  power  of  2;  then  q  =  N  /u  can  be  given  the  form  y  — (u— 1)/2  by  choice  of  y.  If  t;<M, 
then  g<(«+l)/2<u,  so  that  q  has  no  odd  factor  and  2  =  2''.  But  N='2^U  is  of  the 
desired  form  if  we  take  v^u/2'^N. 
"'Sphinx-Oedipe,  num^ro  sp6c.,  June,  1911,  1-27.  Errata  and  addenda,  num6ro  3p6c.,  Jan., 
1912,  7-9,  14.     L'enseignement  math.,  15,  1913,  202-231. 


374  History  of  the  Theory  of  Numbers.  [Chap,  xiv 

S.  Bisman^"**  noted  that  N  is  composite  if  and  only  if  there  exist  two 
integers  A,  B  such  that  A +2 B  and  A-\-2BN  divide  2{N-1)  and  (A''-1)A, 
respectively.  But  there  is  no  convenient  maximum  for  the  smaller  integer 
B.     To  find  the  factor  641  of  2'^^  +  l  there  are  16  cases. 

A.  G^rardin^^^  gave  a  report  on  methods  of  factoring. 

J.  A.  Gmeiner,^^°  to  factor  a,  prime  to  6,  determined  b  and  e  so  that 
9a  =  166+e,  0^€<16.  Let  cu^  be  the  largest  square  <6  and  set  b  =  oi^-\-p, 
cr  =  p— CO.     Hence  9a  =  16(co  — a:)(w+a;H-l)+r(a:),  where 

t(x)  =  16(7+€+16x(x+1). 

Since  t{x)=t{x  —  1)+S2x,  we  may  rapidly  tabulate  the  values  of  r(x)  for 
x  =  0,  1,  2, .  . . .  If  we  reach  the  value  zero,  we  have  two  factors  of  a.  To 
prove  that  a  is  a  prime,  we  need  extend  the  table  until  co+x  +  l  is  the 
largest  square  <a.     To  modify  the  process,  use  4a  =  76  +  €. 

A.  Reymond^^^  used  the  graphs  of  y  =  x/n  (n  =  l,  2,  3,  5, . .  .)>  marking 
on  each  the  points  with  integral  coordinates.  He  omitted  y  =  x/4:  since 
its  integral  points  are  ony  =  x/2.  Since  17  is  not  the  abscissa  of  an  integral 
point  on  y  =  x/n  for  l<n<  17,  17  is  a  prime.     [Mobius^^"  of  Ch.  XIII.] 

A.  J.  Kempner^^^  found,  by  use  of  a  figure  perspective  to  Reymond's^^^ 
how  to  test  the  primality  of  numbers  by  means  of  the  straight  edge. 

D.  Biddle  and  A.  Cunningham^^^  factor  a  product  A^  of  two  primes  by 
finding  A^i<A^  and  N2>N  such  that  N2-N  =  N-Ni+2,  while  each  of  Ni 
and  A''2  is  a  product  of  two  even  factors,  the  two  smaller  factors  differing 
by  2  and  the  two  larger  factors  differing  by  2. 

'"Mathesis,  (4),  2,  1912,  58-60. 
"'Assoc.  fraiiQ.  avanc.  sc,  41,  1912,  54-7. 
"oMonatshefte  Math.  Phys.,  24,  1913,  3-26. 
»"L'enseignement  math.,  18,  1916,  332-5. 
'"Amer.  Math.  Monthly,  24,  1917,  317-321. 
'"Math.  Quest,  and  Solutions,  3,  1917,  21-23. 


I 


CHAPTER  XV. 

FERMAT  NUMBERS  F„  =  2'"+l. 

Fermat^  expressed  his  belief  that  every  F^  is  a  prime,  but  admitted 
that  he  had  no  proof.  Elsewhere^  he  said  that  he  regarded  the  theorem 
as  certain.  Later^  he  impHed  that  it  may  be  proved  by  "descent."  It 
appears  that  Frenicle  de  Bessy  confirmed  this  conjectured  theorem  of  Fer- 
mat's.  On  several  occasions  Fermat^  requested  Frenicle  to  divulge  his  proof, 
promising  important  applications.  In  the  last  letter  cited,  Fermat  raised 
the  question  if  (2/0)^""+ 1  is  always  a  prime  except  when  divisible  by  an  F„. 

C.  F.  Gauss^  stated  that  Fermat  affirmed  (incorrectly)  that  the  theorem 
is  true.     The  opposite  view  was  expressed  by  P.  Mansion^  and  R.  Baltzer.'^ 

F.  M.  Mersenne^  stated  that  every  F^  is  a  prime.  Chr.  Goldbach^ 
called  Euler's  attention  to  Fermat's  conjecture  that  F„  is  always  prime,  and 
remarked  that  no  F^  has  a  factor  <  100;  no  two  F^  have  a  common  factor. 

L.  Euler^o  found  that 

^5  =  2^2+1  =  641-6700417. 

Euler^^  proved  that  if  a  and  h  are  relatively  prime,  every  factor  of 
a^  +6^  is  2  or  of  the  form  2"+^ A:  +  land  noted  that  consequently  any 
factor  of  F5  has  the  form  64fc  +  l,  k  =  10  giving  the  factor  641. 

Euler^^"  and  N.  Beguelin^^  used  the  binary  scale  to  find  the  factor 
641  =  1+2^+2^  of  F5. 

C.  F.  Gauss^^  proved  that  a  regular  polygon  of  m  sides  can  be  constructed 
by  ruler  and  compasses  if  m  is  a  product  of  a  power  of  2  and  distinct  odd 
primes  each  of  the  form  F„,  and  stated  that  the  construction  is  impossible 
if  m  is  not  such  a  product.    This  subject  will  be  treated  under  Roots  of  Unity. 

Sebastiano  Canterzani^^  treated  twenty  cases,  each  with  subdivisions 
depending  on  the  final  digits  of  possible  factors,  to  find  the  factor  641  of  F^, 

iQeuvres,  2,  1894,  p.  206,  letter  to  Frenicle,  Aug.  (?)  1640;  2,  1894,  p.  309,  letter  to  Pascal, 
Aug.  29,  1654  (Fermat  asked  Pascal  to  undertake  a  proof  of  the  proposition,  Pascal, 
III,  232;  IV,  1819,  384);  proposed  to  Brouncker  and  Wallis,  June  1658,  Oeuvres,  2, 
p.  404  (French  transl.,  3,  p.  316).  Cf.  C.  Henry,  BuU.  Bibl.  Storia  So.  Mat.  e  Fis.,  12, 1879, 
500-1,  716-7;  on  p.  717,  42 ...  1  should  end  with  7,  ihid.,  13,  1880,  470;  A.  Genocchi,  Atti 
Ac.  Sc.  Torino,  15,  1879-80,  803. 

^Oeuvres,  1,  1891,  p.  131  (French  transl.,  3,  1896,  p.  120). 

^Oeuvres,  2,  433-4,  letter  to  Carcavi,  Aug.,  1659. 

^Oeuvres,  2,  208,  212,  letters  from  Fermat  to  Frenicle  and  Mersenne,  Oct.  18  and  Dec.  25,  1640. 

^Disq.  Arith.,  Art.  365.  Cf.  Werke,  2,  151,  159.  Same  view  by  Klugel,  Math.  Worterbuch, 
2,  1805,  211;  3,  1808,  896. 

«Nouv.  Corresp.  Math.,  5,  1879,  88,  122. 

'Jour,  fur  Math.,  87,  1879,  172. 

^Novarum  Physico-Mathematicarum,  Paris,  1647,  181. 

^Corresp.  Math.  Phys.  (ed.,  Fuss),  I,  1843,  p.  10,  letter  of  Dec.  1729;  p.  20,  May  22,  1730;  p.  32, 
July  1730. 

"Comm.  Ac.  Petrop.,  6,  ad  annos  1732-3  (1739),  103-7;  Comm.  Arith.  Coll.,  1,  p.  2. 

"Novi  Comm.  Petrop.,  1,  1747-8,  p.  20  [9,  1762,  p.  99);  Comm.  Arith.  CoU.,  1,  p.  55  [p.  357]. 
""Opera  postuma,  I,  1862,  169-171  (about  1770). 

"Nouv.  Mem.  Ac.  Berlin,  ann^e  1777,  1779,  239. 

"Disq.  Arith.,  1801,  Arts.  335-366;  German  transl.  by  Maser,  1889,  pp.  397-448,  630-652. 

"Mem.  1st.  Naz.  Italiano,  Bologna,  Mat.,  2,  II,  1810,  459-469. 

375 


376  History  of  the  Theory  of  Numbers.  [Chap,  xv 

and  proved  in  the  same  lengthy  dull  manner  that  the  quotient  is  a  prime. 
An  anonymous  writer^^  stated  that 

(1)  2+1,  2-+1,  2-'+l,  22''+!,... 

are  all  primes  and  are  the  only  primes  2*+l.     See  Malvy.'^ 

Joubin^^  suggested  that  these  numbers  (1)  are  possibly  the  ones  really 
meant  by  Fermat/  evidently  without  having  consulted  all  of  Fermat's 
statements. 

G.  Eisenstein^'  set  the  problem  to  prove  that  there  is  an  infinitude  of 
primes  F„. 

E.  Lucas^^  stated  that  one  could  test  the  primality  of  F^  in  30  hours  by 
means  of  the  series  3, 17,  577, . . . ,  each  term  being  one  less  than  the  double  of 
the  square  of  the  preceding.  Then  F„  is  a  prime  if  2""^  is  the  rank  of  the  first 
term  divisible  by  F„,  composite  if  no  term  is  di\'isible  by  F„.  Finally,  if  a  is 
the  rank  of  the  first  term  divisible  by  F„,  the  prime  divisors  of  F„  are  of  the 
form    2''g  +  l,  where  A;  =  a+1  [not  A:  =  2°+^].     See  Lucas." 

T.  Pepin^^  stated  that  the  method  of  Lucas^^  is  not  decisive  when  F„ 
divides  a  term  of  rank  a<2''~^;  for,  if  it  does,  we  can  conclude  only  that  the 
prime  di\'isors  of  F„  are  of  the  form  2''"''^g'+l,  so  that  we  can  not  say  whether 
or  not  F„  is  prime  if  a +2  ^2""^.  We  may  answer  the  question  unambigu- 
ously by  use  of  the  new  theorem:  For  n>  1,  F„  is  a  prime  if  and  only  if  it 
divides 

where  k  is  any  quadratic  non-residue  of  Fn,  as  5  or  10.    To  apply  this  test, 
take  the  minimum  residues  modulo  F„  of 

2"-l 

A/,         i^    )  "^    )  •    •    •  }  '^  ' 

Proof  was  indicated  by  Lucas-^  of  Ch.  XVII,  and  by  Morehead.^^ 
J.  Pervouchine^°  (or  Pervusin)  announced,  November  1877,  that 

Fi2=0  (mod  114689  =  7-2^^+1). 

E.  Lucas^^  announced  the  same  result  two  months  later  and  proved  that 
every  prime  factor  of  F„  is  =  1  (mod  2"'^^). 

Lucas^^  employed  the  series  6,  34,  1154, . .  . ,  each  term  of  which  is  2  less 
than  the  square  of  the  preceeding.  Then  F„  is  a  prime  if  the  rank  of  the  first 
term  divisible  by  F^  is  between  2""^  and  2"  —  !,  but  composite  if  no  term  is 
di\'isible  by  F„.     Finally,  if  a  is  the  rank  of  the  first  term  divisible  by  F„ 

"Annales  de  Math.  (ed.  Gergonne),  19,  1828-9,  256. 

"M^moire  sur  le3  facteurs  numdriques,  Havre,  1831,  note  at  end. 

»'Jour.  fiir  Math.,  27,  1844,  87,  Prob.  6. 

"€omptes  Rendus  Paris,  85,  1877,  136-9. 

'•Comptea  Rendus,  85,  1877,  329-331.     Reprinted,  with  Lucas"  and  Landry,"  Sphinx-Oedipe, 

5,  1910,  33-42. 
"Bull.  Ac.  St.  Pdtersbourg,  (3),  24,  1878,  559  (presented  by  V.  Bouniakowsky).      Melanges 

math.  ast.  sc  St.  P^tersbourg,  5,  1874-81,  505. 
"Atti  R.  Accad.  Sc.  Torino,  13,  1877-8,  271  (Jan.  27,  1878).     Cf.  Nouv.  Corresp.  Math.,  4, 

1878,  284;  5,  1879,  88.     See  Lucas"  of  Ch.  XVIL 
"Amer.  Jour.  Math.,  1,  1878,  313. 


Chap.  XV]  FerMAT   NUMBERS  F„=2^"  +  l.  377 

and  if  a<2''~\  the  prime  divisors  of  F„  are  of  the  form*  2*^+1,  where 
k  =  a-\-l  [cf .  Lucas^^].  He  noted  (p.  238)  that  a  necessary  condition  that  F^ 
be  a  prime  is  that  the  residue  modulo  F^  of  the  term  of  rank  2"  —  1  in  this 
series  is  zero.  He  verified  (p.  292)  that  F^  has  the  factor  641  and  again 
stated  that  30  hours  would  suffice  to  test  Fq. 

F.  Proth^^  stated  that,  if  A;  =  2",  2^+1  is  a  prime  if  and  only  if  it  divides 
m  =  3^  ~  + 1 .  He^^  indicated  a  proof  by  use  of  the  series  of  Lucas  defined  by 
Uo  =  0,  Ui  =  l,  .  .  .  ,  w„  =  3if„_i  +  l  and  the  facts  that  Up_i  is  divisible  by 
the  prime  p,  while  m=^U2^lu2^-^.     Cf.  Lucas. ^^ 

E.  Gelin^^  asked  if  the  numbers  (1)  are  all  primes.  Catalan^^  noted 
that  the  first  four  are. 

E.  Lucas^^  noted  that  Proth's^^  theorem  is  the  case  A;  =  3  of  Pepin's. ^^ 
Pervouchine"  announced,  February  1878,  that  7^23  lias  the  prime  factor 

5.225+1  =  167772161. 

W.  Simerka^^  gave  a  simple  verification  of  the  last  result  and  the  fact 
(Pervouchine^o)  that  7-2'Hl  divides  F^^. 

F.  Landry, ^^  when  of  age  82  and  after  several  months'  labor,  found  that 

7^6  =  274177-67280421310721, 

the  first  factor  being  a  prime.  He  and  Le  Lasseur  and  G^rardin^^"  each 
proved  that  the  last  factor  is  a  prime  (cf.  Lucas^^). 

K.  Broda^"  sought  a  prime  factor  p  of  a^^+1  by  considering 

n  =  (a^2-l)(a«Hl)(a'''+a^''+a2^«+a^''+l). 

Multiply  by  i^  =  (o^2_}_l)/p.  Thusnw  =  (a^^°-l)/p.  But  a^^°^l  (mod641). 
Since  each  factor  of  n  is  prime  to  p,  we  take  a  =  2  and  see  that  2^^+ 1  is  divis- 
ible by  641. 

E.  Lucas^^  stated  that  he  had  verified  that  F^  is  composite  by  his^^  test, 
before  Landry  found  the  factors. 

P.  Seelhoff^^  gave  the  factor  5-2^^+1  of  F^,^  and  commented  on  Beguelin.^^ 

*Lucas  wrote  fc  =  2"+^  in  error,  as  noted  by  R.  D.  Carmichael  on  the  proof-sheets  of  this 
History. 

23Comptes  Rendus  Paris,  87,  1878,  374. 

«Nouv.  Corresp.  Math.,  4,  1878,  210-1;  5,  1879,  31. 

MUd.,  4,  1878,  160. 

^HUd.,  5,  1879,  137. 

"Bull.  Ac.  St.  P6tersbourg,  (3),  25,  1879,  63  (presented  by  V.  Bouniakowsky) ;  Melanges  math, 
astr.  ac.  St.  P^tersbourg,  5,  1874-81,  519.  Cf.  Nouv.  Corresp.  Math.,  4,  1878,  284-5;  5, 
1879,  22. 

28Casopi3,  Prag,  8, 1879,  36,  187-8.     F.  J.  Studnicka,  iUd.,  11,  1881, 137, 

"Comptes  Rendus  Paris,  91,  1880,  138;  Bull.  Bibl.  Storia  Sc.  Mat.,  13,  1880,  470;  Nouv.  Cor- 
resp. Math.,  6,  1880,  417;  Lea  Mondes,  (2),  52,  1880.  Cf.  Seelhoflf,  Archiv  Math.  Phys., 
(2),  2,  1885,  329;  Lucas,  Amer.  Jour.  Math.  1,  1878,  292;  R6cr6at.  Math.,  2,  1883, 
235;  I'intermddiaire  des  math.,  16,  1909,  200. 

»»Sphinx-Oedipe,  5,  1910,  37-42. 

"Archiv  Math.  Phys.,  68,  1882,  97. 

"Recreations  Math.,  2,  1883,  233-5.     Lucas,"*  354-5. 

»2Zeitschr.  Math.  Phys.,  31, 1886,  172-4,  380.  For  Ft,  p.  329.  French  transl.,  Sphinx-Oedipe, 
1912,  84-90. 


378  History  of  the  Theory  of  Numbers.  [Chap,  xv 

J.  Hermes^^  indicated  a  test  for  composite  F„  by  Fermat's  theorem. 

R.  Lipschitz^  separated  all  integers  into  classes,  the  primes  of  one  class 
being  Fermat  numbers  F„,  and  placed  in  a  new  light  the  question  of  the 
infinitude  of  primes  F„. 

E.  I.ucas^^  stated  the  result  of  Proth,-^  but  with  a  misprint  [Cipolla^^]. 
H.  Scheffler^^  stated  that  Legendre  believed  that  every  F„  is  a  prime(!), 

and  obtained  artificially  the  factor  641  of  F5.     He  noted  (p.  167)  that 

F„F„+, .  .  .F„_,  =  l+22"+22-2"+23-2''+  .  .  .  +2^"-2". 
He  repeated  (pp.  173-8)  the  test  by  Pepin/^  with  k  =  3,  and  (p.  178)  expressed 
his  belief  that  the  numbers  (1)  are  all  primes,  but  had  no  proof  for  Fie- 

W.  W.  R.  Ball"  gave  references  and  quoted  known  results. 

T.  M.  Pervouchine^^  checked  his  verification  that  F12  and  F23  are  com- 
posite by  comparing  the  residues  on  di\'ision  by  10^  —  2. 

Mah'y^^  noted  that  the  prime  2^+1  is  not  in  the  series  (1). 

F.  Klein^°  stated  that  F7  is  composite. 

A.  Hurwitz"*^  gave  a  generalization  of  Proth's^^  theorem.  Let  F„(x) 
denote  an  irreducible  factor  of  degree  4>{n)  of  x"  — 1.  Then  if  there  exists 
an  integer  q  such  that  Fp_i(g)  is  divisible  by  p,  p  is  a  prime.  When 
p  =  2'-+l,Fp_i(x)^=x''"'  +  l. 

J.  Hadamard'*-  gave  a  very  simple  proof  of  the  second  remark  by  Lucas. ^^ 

A.  Cunningham^^  found  that  Fn  has  the  factor  319489-974849. 

A.  E.  \Yestern^  found  that  Fg  has  the  factor  2^^-37  +  1,  Fis  the  factor 
2'°-13  +  l,  the  quotient  of  Fn  by  the  known  factor  2^'*-74-l  has  the  factors 
2^^-397+1  and  2^^-7-139  +  l.  He  verified  the  primality  of  the  factor  2^^-3-f  1 
of  F38,  found  by  J.  Cullen  and  A.  Cunningham.  He  and  A.  Cunningham 
found  that  no  more  F„  have  factors  <  10^  and  similar  results. 

m-2 

M.  Cipolla^^  noted  that,  if  g  is  a  prime  >(9"^  -l)/2'"+^and  m>\, 
2'"g+ 1  is  a  prime  if  and  only  if  it  divides  3H 1  for  k  =  ^•2'"+^  He^^  pointed 
out  the  misprint  in  Lucas'^^  statement. 

Nazarevsky^^  proved  Proth's-^  result  by  using  the  fact  that  3  is  a  primi- 
tive root  of  a  prime  2''+l. 

"Archiv  Math.  Phys.,  (2),  4,  1886,  214-5,  footnote. 

"Jour,  fur  Math.,  105,  1889,  152-6;  106,  1890,  27-29. 

"Th^orie  des  nombres,  1891,  preface,  xii. 

»«Beitrage  zur  Zahlentheorie,  1891,  147,  151-2,  155  (bottom),  168. 

»^Math.  Recreations  and  Problems,  ed.  2,  1892,  26;  ed.  4,  1905,  36-7;  ed.  5,  1911,  39-40. 

"Math.  Papers  Chicago  Congress  of  1893,  I,  1896,  277. 

"L'interm6diaire  des  math.,  2,  1895,  41  (219). 

"Vortrage  iiber  aiisgewahlte  FragenderElementar  Geometric,  1895, 13;  French  transl.,  1896,  26; 

English  transl.,  "Famous  Problems  of  Elementary  Geometry,"  by  Beman  and  Smith, 

1897,  16. 
"L'interm^diaire  des  math.,  3,  1896,  214. 
«/6u/.,  p.  114. 
"Report  British  Assoc,  1899,  653-4.     The  misprint  in  the  second  factor  has  been  corrected 

to  agree  with  the  true  "  value  2".7.17  +  1. 
^Cunningham  and  Western,  Proc.  Lond.  Math.  Soc,  (2),  1,  1903, 175;  Educ.  Times,  1903,  270. 
«Periodico  di  Mat.,  18,  1903,  331. 
"Also  in  Annali  di  Mat.,  (3),  9,  1904,  141. 
*'L'interm6diaire  des  math.,  11,  1904.  215. 


Chap.  XV]  FeRMAT  NUMBERS  F„  =  2^"  +  l.  379 

A.  Cunningham^^"  noted  that  3,  5,  6,  7,  10,  12  are  primitive  roots  and 
13,  15,  18,  21,  30  are  quadratic  residues  of  every  prime  F„>5.  He  fac- 
tored F4'  +  S+iFoFiF2Fz)\ 

Thorold  Gosset^^  gave  the  two  complex  prime  factors  a±  6i  of  the  known 
real  factors  of  composite  F„,  n  =  5,  6,  9,  11,  12,  18,  23,  36,  38. 

J.  C.  Morehead'*^  verified  by  use  of  the  criterion  of  Pepin^^  with  k  =  S 
that  7^7  is  composite,  a  result  stated  by  Klein.^° 

A.  E.  Western^"  verified  in  the  same  way  that  Fy  is  composite.  The 
work  was  done  independently  and  found  to  agree  with  Morehead's. 

J.  C.  Morehead^^  found  that  Fy^  has  the  prime  factor  2^^-5+1. 

A.  Cunningham^^  considered  hyper-even  numbers 

Eo,  „  =  2",  E,,n  =  2^0.  n,  .  .  .  ,  E^+,,  „  =  2^r,  n. 

For  m  odd,  the  residues  modulo  m  of  Er,o,  -E'r,  i,-  •  •  have  a  non-recurrent 
part  and  then  a  recurring  cycle. 

A.  Cunningham^^  gave  tables  of  residues  of  Ei^  „,  E2,  n,  Er,  0,  3^"  and  5^" 
for  the  n's  forming  the  first  cycle  for  each  prime  modulus  <  100  and  for 
certain  larger  primes.  A  hyper-exponential  number  is  like  a  hyper-even 
number,  but  with  base  q  in  place  of  2.  He  discussed  the  quadratic,  quartic 
and  octic  residue  character  of  a  prime  modulo  F^,  and  of  F„  modulo  Fn+x- 

Cunningham  and  H.  J.  WoodalP^  gave  material  on  possible  factors  of  F^. 

A.  Cunningham^^  noted  that,  for  every  F„>5,  2Fn  =  t^  —  (Fn  —  2)u^ 
algebraically,  and  expressed  F5  and  Fq  in  two  ways  in  each  of  the  forms 
a^+6^  c^±2d2.  He^*^  noted  that  Fn^+E^^  is  the  algebraic  product  of 
n-\-2  factors,  where  ^n  =  2^",  and  that  M„  =  (F„H^„^)/(F„-fE„)  is  divisible 
by  Mn-r-     If  n  —  m^2,  F„i'^-\-FJ  is  composite. 

A.  Cunningham^^  has  considered  the  period  of  l/N  to  base  2,  where  N 
is  a  product  FmFm-i .  ■  -Fm-r  of  Fermat  numbers. 

J.  C.  Morehead  and  A.  E.  Western^^  verified  by  a  very  long  computation 
that  Fg  is  composite.  Use  was  made  of  the  test  by  Pepin^^  with  A;  =  3,  which 
was  proved  to  follow  from  the  converse  of  Fermat's  theorem. 

P.  Bachmann^^  proved  the  tests  by  Pepin^^  and  Lucas. ^^ 

A.  Cunningham*^^  noted  that  every  Fn>5  can  be  represented  by  4 
quadratic  forms  of  determinants  =^Gn,  =*=2(r„,  where  Gn  =  FoFi.  .  .Fn-i- 

Bisman^'^^  (of  Ch.  XIV)  separated  16  cases  in  finding  the  factor  641  of  F^. 

^^'^Math.  Quest.  Educ.  Times,  (2),  1,  1902,  108;  5,  1904,  71-2;  7,  1905,  72. 

"Mess.  Math.,  34,  1905,  153-4.  "BuU.  Amer.  Math.  Soc,  11,  1905,  543. 

"Proc.  Lond.  Math.  Soc,  (2),  3,  1905,  xxi. 

"Bull.  Amer.  Math.  Soc,  12,  1906,  449;  Annals  of  Math.,  (2),  10,  1908-9,  99.    French  transl.  in 

Sphinx-Oedipe,  Nancy,  1911,  49.  ^^Report  British  Assoc.  Adv.  Sc,  1906,  485-6. 

"Proc  London  Math.  Soc,  (2),  5,  1907,  237-274. 
"Messenger  of  Math.,  37,  1907-8,  65-83. 
55Math.  Quest.  Educat.  Times,  (2),  12,  1907,  21-22,  28-31. 
^HhU.,  (2),  14,  1908,  28;  (2),  8,  1905,  35-6. 
"Math.  Gazette,  4,  1908,  263. 

"Bull.  Amer.  Math.  Soc,  16,  1909,  1-6.     French  transl.,  Sphinx-Oedipe,  1911,  50-55. 
"Niedere  Zahlentheorie,  II,  1910,  93-95. 
'"Math.  Quest.  Educat.  Times,  (2),  20,  1911,  75,  97-98. 


380  History  of  the  Theory  of  Numbers.  [Chap,  xv 

A.  G^rardin"  noted  that  F„  =  (240j+97)(240!/+161)  for  all  the  F„ 
fully  factored  to  date,  and  specified  x  and  y  more  exactly  in  special  cases. 

C.  Henrv'^'-  gave  references  and  quoted  known  results. 

R.  D.  CannichaeP'  gave  a  test  for  the  primality  of  F„  equivalent  to 
Pepin's^^  and  a  further  generalization  (p.  65)  in  the  direction  of  Hurwitz's." 

R.  C.  Archibald"  cited  many  of  the  papers  hsted  above  and  collected  in  a 
table  the  known  factors  of  F„  with  the  exception  of  that  given  by  Morehead." 

For  a  remark  on  F„,  see  Cunningham^°^  of  Ch.  VII. 

"Sphinx-Oedipe,  7,  1912,  13. 
•HDeuvres  de  Fermat,  4,  1912,  202-4. 
"Annals  of  Math.,  (2),  15,  1913-4,  67. 
•'Amer.  Math.  Monthly,  21.  1914.  247-251. 


CHAPTER  XVI. 

FACTORS  OF  a^^b\ 

Fermat^  stated  that  (2''+l)/3  has  no  factors  other  than  2kp+l  if  p  is 
an  odd  prime. 

L.  Euler^  noted  that  o*+46*  has  the  factors  a^=i=2ah+2b^. 

Euler^  discussed  the  numbers  a  for  which  a^  +  1  is  divisible  by  a  prime 
4n+l=r^+s^.  Let  p/q  be  the  convergent  preceding  r/s  in  the  continued 
fraction  for  r/s ;  then  ps  —  qr==^l.  Thus  every  a  is  of  the  form  (4n + 1 )  m  ± 
k,  where  k  =  pr+qs. 

Euler^  gave  the  161  integers  a<  1500  for  which  a^  +  1  is  a  prime,  and  the 
cases  a  =  1,  2,  4,  6,  16,  20,  24,  34  for  which  a*+l  is  a  prime. 

Euler^  proved  that,  if  m  is  a  prime  and  a,  b  are  relatively  prime,  a  factor 
of  a"*  —  6"",  not  a  divisor  of  a  —  6,  is  of  the  form  kn-\-l.  If  p  =  A;n + 1  is  a  prime 
and  a  =p=^  pa,  then  a*'  —  1  is  divisible  by  p.  If  of"  —  bg""  is  divisible  by  a  prime 
p  =  'mn-{-l,  while/  and  g  are  not  both  divisible  by  p,  then  a""  — 6""  is  divisible 
by  p ;  the  converse  is  true  if  m  and  n  are  relatively  prime. 

Euler^  proved  the  related  theorems:  For  q  an  odd  prime,  any  prime 
divisor  of  a^—1,  not  a  divisor  of  a  —  1,  is  of  the  form  2nq-\-l.  If  a""  — 1  is 
divisible  by  the  prime  p  =  m7i+l,  we  can  find  integers  x,  y  not  divisible  by 
p  such  that  A  =  ax'*— ?/"  is  divisible  by  p  (since  the  quotient  of  a'"^""'— 2/"*" 
by  A  is  not  divisible  by  p\i  x,y  are  suitably  chosen) . 

Euler^  treated  the  problem  to  find  all  integers  a  for  which  a^  + 1  is  divisible 
by  a  given  prime  4n + 1  =  p^ + ?^-  If  a^ + b^  is  divisible  by  p'^-\-(f',  there  exist 
integers  r,  s  such  that  a  =  pr+g's,  6  =  ps—gr.  We  wish  6=  =±=1.  Hence  we 
take  the  convergent  r/s  preceding  p/q  in  the  continued  fraction  for  p/q. 
Thus  ps  —  qr=^\,  and  our  answer  is  a=  ± {pr-\-qs).  He  listed  all  primes 
P  =  4n+1<2000  expressed  as  p^+g^,  and  listed  all  the  a's  for  which  a^  +  1 
is  divisible  by  P.  The  table  may  be  used  to  find  all  the  divisors  <  a  of  a 
given  number  a^+1.  He  gave  his^  table  and  tabulated  the  values  a<  1500 
for  which  {a'^-\-l)/k  is  a  prime,  for  k=2,  5,  10.  He  tabulated  all  the 
divisors  of  a^+1  for  a^  1500. 

N.  Beguelin^  stated  that  2''+l  has  a  trinary  divisor  1+2^+2'  only  when 
n=10,  24,  32,  although  his  examples  (p.  249)  contradict  this  statement. 

Euler^  gave  a  factor  of  2"=^  1  for  various  composite  n's. 

iQeuvres,  2,  205,  letter  to  Frenicle,  Aug.  (?),  1640.     Bull.  Bibl.  St.  Sc.  Mat.  e  Fis.,  12, 1879,  716. 

''Corresp.  Math.  Phys.  (ed.,  Fuss),  I,  1843,  p.  145;  letter  to  Goldbach,  1742. 

^lUd.,  242-3;  letter  to  Goldbach,  July  9,  1743. 

mid.,  588-9,  Oct.  28,  1752.     Published,  Euler.^ 

^Novi  Comm.  Petrop.,  1,  1747-8,  20;  Coram.  Arith.  Coll.,  1,  57-61,  and  posthumous  paper, 

ibid.,  2,  530-5;  Opera  postuma,  I,  1862,  33-35.     Cf.  Euleri^z  of  Ch.  VII  and  the  topic 

Quadratic  Residues  in  Vol.  III. 
«Novi  Comm.  Petrop.,  7,  1758-9  (1755),  49;  Comm.  Arith.,  1,  269. 
^Novi  Comm.  Petrop.,  9,  1762-3,  99;  Comm.  Arith.,  1,  358-369.     French  transl.,  Sphinx- 

Oedipe,  8,  1913,  1-12,  21-26,  64. 
»M6m.  Ac.  Berlin,  annee  1777,  1779,  255.     Cf.  Ch.  XV  and  Henry.i^ 
'Posthumous  paper,  Comm.  Arith.,  2,  551;  Opera  postuma,  I,  1862,  51. 

381 


382  History  of  the  Theory  of  Numbers.  [Chap,  xvi 

Euler^"  discussed  the  divisors  of  numbers  of  the  iorm.fa^+gb'^. 

Anton  FelkeP"  gave  a  table,  incomplete  as  to  a  few  entries,  of  the  factors 
ofa"-l,n  =  l,...,  ll;a  =  2,3,...,  12. 

A.  M.  Legendre^^  proved  that  every  prime  divisor  of  a"+l  is  either  of  the 
form  2nx-{-l  or  divides  a"H-l  where  co  is  the  quotient  of  n  by  an  odd  factor; 
every  prime  divisor  of  a"  —  !  is  either  of  the  form  nx-\-l  or  divides  a'^  —  l 
where  co  is  a  factor  of  n.  For  n  odd,  the  divisors  must  occur  in  0(0"=^  1) 
=  ?/'-±  a  and  are  thus  further  limited  by  his  tables  III-XI  of  the  linear  forms 
of  the  divisors  of  <"±au". 

C.  F.  Gauss^^  obtained  by  use  of  the  quadratic  reciprocity  law  the  linear 
forms  of  the  divisors  of  x"— A. 

Gauss^'^  gave  a  table  of  2452  numbers  of  the  forms  a^+1,  a^+4, .  .  ., 
a^+81  and  their  odd  prime  factors  p,  for  certain  a's  for  which  the  p's  are 
all  <200. 

Sophie  Germain^^  noted  that  p'*+4g*  has  the  factors  p^=^2pq+2^ 
[Euler^].  Taking  p  =  l,  q  =  2\  we  see  that  2^'"''^+l  has  the  two  factors 
22.+i±2'+i  +  l. 

F.  Minding^^  gave  a  detailed  discussion  of  the  linear  forms  of  the  divisors 
of  x^  —  c,  using  the  reciprocity  law  for  the  case  of  primes.  He  reproduced 
(pp.  188-190)  the  discussion  by  Legendre.^^ 

P.  L.  Tchebychef^®  noted  that,  if  p  is  an  odd  prime,  every  odd  prime 
factor  of  a''— 1  is  either  of  the  form  2pz-\-l  or  is  a  factor  of  a  — 1,  and  more- 
over is  a  divisor  of  x^  —  ay'^.  Hence,  for  a  =  2,  it  is  of  the  form  2^2+1  and 
also  of  one  of  the  forms  8w=tl.  Every  odd  prime  factor  of  a^^+^  +  l  is 
either  of  the  form  2(2n+ 1)2+1  or  a  divisor  of  a+1  [cf.  Legendre"]. 

V.  A.  Lebesgue^^  noted  that  the  discussion  of  the  linear  forms  of  the 
divisors  of  z^—D,  where  D  is  composite,  is  simplified  by  use  of  Jacobi's 
generalization  (a/b)  of  Legendre's  symbol. 

C.  G.  Reuschle^'  denoted  (x"''-l)/(x''-l)  by  FM.  Set  a  =  ah+bi, 
6  =  0161+62,  61  =  0262+63,. ...     If  a,  6  are  relatively  prime, 

(T^  ^r)C    n  =  '^'^""nia(6-l-A)l+a;^  sVxjai(6i-l-A)| 

{X   —i){X  —L)       A=0  A=0 

+  ...  +a:^+^.+  -  ■  -+^-2  Vx^'"-'n„_i  {a._i(6„_i  -1-A)}  +x^+-  •  ■+^n-i. 

9^'Opera  postuma,  I,  1862,  161-7  (about  1773). 

loAbhandl.  d.  Bohmischen  Gesell.  Wiss.,  Prag,  1,  1785,  165-170. 

'•Th6orie  des  nombres,  1798,  pp.  207-213,  313-5;  ed.  2,  1808,  pp.  191-7,  286-8.     German 

transl.  by  Mascr,  p.  222. 
"Disq.  Arith.,  1801,  Arts.  147-150. 
"Werke,  2,  1863,  477-495.     Schering,  pp.  499-502,  described  the  table  and  its  formation  by 

the  compo.sition  of  binary  forms,  e.  g.,  (a^'  +  l)  { (a4-l)^4-l}  =  {a(o  +  l)+l}*  +  l. 
"Manuscript  9118  fonds  frangais  Bibl.  Nat.  Paris,  p.  84.     Cf.  C.  Henry,  Assoc,  frang.  avanc. 

sc,  1880,  205;  Oeuvres  de  Fermat,  4,  1912,  208. 
"Anfangsgrunde  dor  Hoheren  Arith.,  1832,  59-70. 
"Theorie  der  Congrucnzen,  in  Russian,  1849;  in  German,  1889;  §49. 
"Jour,  de  Math.,  15,  1850,  222-7. 
"Math.  Abhandlung,  Stuttgart,  1853,  II,  pp.  6-13. 


Chap.  XVI]  FACTORS  OF  a"  ±6".  383 

Reuschle's^^  table  A  gives  many  factors  of  a^=tl,  a*±l,  a^±l,  a^^  —  1 
for  a^lOO,  and  of  a"-l  for  n^42,  a  =  2,  3,  5,  6,  7,  10. 

Lebesgue^^"  proved  that  x*^^+  . .  .-\-x-\-l  has  no  prime  divisor  other 
than  the  prime  p  and  numbers  of  the  form  kp  +  1. 

Jean  Plana^^  gave  S^^  +  l  =4-6091g,  S^^-l=2-59r,  and  stated  that  q 
is  a  prime  and  that  r  has  no  factor  <  52259.     But  Lucas^^  noted  that 

q  =  523-5385997,  r  =  28537-20381027. 

E.  Kummer^^  proved  that  there  is  no  prime  factor,  other  than  t  and 
numbers  2m^±l,  of  the  cyclotomic  function 

obtained  from  (a'  — l)/(a  — 1)  by  settinga+a~^  =  a:;,  t  being  a  prime  2e  +  l. 

E.  Catalan^^  stated  that,  if  n  =  a=Fl  is  odd,  a"=Fl  is  divisible  by  n^,  but 
not  by  nK     Proof  by  Soons,  Mathesis,  (3),  2,  1902,  109. 

H.  LeLasseur  and  A.  Aurifeuille^^  noted  that  2^"'''^  +  l  has  the  factors 
22n+i±  2^^+1  +  1  [cf.  Euler,2  S.  Germain^^]. 

E.  Lucas^^  proved  that  (2^°+l)/(2^+l)  is  a  prime  and  gave  the  factors  of 
30^^±1,  2*^  +  1. 

Theorems  by  Lucas  on  the  factors  of  a"  ±6**,  given  in  various  papers  in 
1876-8,  are  cited  in  Ch.  XVII. 

Lucas^^  factored  (2m)"'=tl  for  m  =  7,  10,  11,  12,  14,  15,  and  corrected 
Plana.20 

Lucas^^  gave  tables  due  to  LeLasseur  and  Aurifeuille  of  functions 

5!il!    („odd),         ^^, 

x^y  x^-i-y^ 

expressed  in  the  form  V^^pxyZ^,  which  is  factorable  if  xy  =  ptP.  Factors 
of  x^°-\-y^^  are  given  for  various  x's,  y's.  He  gave  LeLasseur's  table  of  the 
proper  divisors  of  2"  — 1  for  all  odd  values  of  n<100  except  n  =  61,  67, 
71,  77,  79,  83,  85,  89,  93,  97;  the  proper  divisors  of  2"+l  for  n  odd  and  <71 
(except  n  =  61,  67)  and  for  n  =  73,  75,  81,  83,  99,  135;  the  proper  divisors 
of  22^1  for  2A;^74  (except  64,  68)  and  for  2k  =  7S,  82,  84,  86,  90,  94,  102, 
126,  etc.  Lucas  proved  (pp.  790-4)  that  the  proper  divisors  of  2^"+l  are  of 
the  form  16ng  +  l,  those  of  (j2a6n_|_^2a67i  ^^.^  q£  ^^^q  form  Sahnq-j-l;  for  n  odd, 
those  of  a"^"+6"^'^  are  of  the  form  4a6ng  +  l  if  a6  =  4/i  +  l,  those  of  a"''" -6"'''* 
are  of  the  form  4a6ng  +  l  if  a6  =  4/i+3. 

I'Math.  Abhandlung .  .  .Tabellen,  Stuttgart,  1856.     Full  title  in  Ch.  I. 

"aComptes  Rendus  Paris,  51,  1860,  11. 

20Mem.  Accad.  Sc.  Torino,  (2),  20,  1863,  139-141. 

"Cf.  Bachmann,  Kreistheilung,  Leipzig,  1872. 

22Revue  de  I'lnstruct.  publique  en  Belgique,  17,  1870,  137;  Melanges  Math.,  ed.  1,  p.  40. 

23Atti  R.  Ac.  Sc.  Torino,  8,  1871;  13,  1877-8,  279.     Nouv.  Corresp.  Math.,  4,  1878,  86,  98. 

Cf.  Lucas,25  p.  238;  Lucas,^^  784. 
2^Nouv.  Ann.  Math.,  (2),  14,  1875,  523-5. 
25Amer.  Jour.  Math.,  1,  1878,  293. 
26Bull.  Bibl.  Storia  Sc.  Mat.  e  Fis.,  11,  1878,  783-798. 


384  History  of  the  Theory  of  Numbers.  [Chap.  XVI 

Lucas"  gave  the  factors  of  2'"+!  for  w  =4n^60  and  for  72,  84;  also  for 
rM=4n+2^102  and  for  110,  114,  126,  130,  138,  150,  210. 

E.Catalan=«  noted  that  x^+2{q-r)x^-^q~  for  x^ ^  (2r)2*+i  has  the  rational 
factors  (2r)^*'*"^  =*=  (2r)*"'"^ +5.  The  case  r  =  q  =  \  gives  LeLasseur's^^  formula. 
Again,  3'*+'  +  l  has  the  factors  ^^'^'  +  \,  32'^+^±3*+^  +  l. 

S.  R^alis-^''  deduced  LeLasseur's'-^  formula  and  24"+22"+l=n(22''± 
2"+l). 

J.  J.  Sylvester'^^  considered  the  cyclotomic  function  ypt{x)  obtained  by 
setting  a  +  a"'  =x  in  the  quotient  by  a*""^  of 

(a'-l)n(a'^''''''-l) 
(1)  ^.(a)=^"    n(a'^'-l).  (^  =  P^'-  •  P"'"^' 

where  Pi,-  ■  ■,  Pn  are  distinct  primes.  He  stated  that  every  di\'isor  of  i/'t(x) 
is  of  the  form  kt^  1,  with  the  exception  that,  if  t  =  p\p=^l)/m,  p  is  a  divisor 
(but  not  p^).  Conversely,  every  product  of  powers  of  primes  of  the  form 
kt^l  is  a  divisor  of  i/',(x).  Proofs  were  given  by  T.  Pepin,  ibid.,  526;  E. 
Lucas,  p.  855;  Dedekind,  p.  1205  (by  use  of  ideals).  Lucas  added  that 
p  =  2^''+3  — 1  and  p  =  2^^''"''^  —  1  are  primes  if  and  only  if  they  divide  </'p+i(x) 
for  x  =  \/^  and  x  =  3\/^,  respectively. 

A.  Lefebure^°  determined  poljTiomipls  having  no  prime  factor  other  than 
those  of  the  form  HT+1,  where  H  is  given.  First,  let  T  =  n\  where  n  is  a 
prime.     For  A,  B  relatively  prime  integers, 

/ln_  Dn 

has,  besides  n,  no  prime  factor  except  those  of  the  form  Hn*+1,  when  A 
and  B  are  exact  n'~Hh  powers  of  integers.  Second,  let  T  =  n'm^,  where  n,  m 
are  distinct  primes.  The  integral  quotient  of  F^iu'",  v'")  by  F„(w,  t')  has  only 
prime  factors  of  the  form  Hn^m^  +  l  if  u,  v  are  powers  of  relatively  prime 
integers  with  the  exponent  w''~^n'~^  Similarly,  if  T  is  a  product  of  powers 
of  several  primes. 

Lef^bure^^  discussed  the  decomposition  into  primes  of  U^  —  V^,  where 
U,  V  are  powers  whose  exponents  involve  factors  of  R. 

E.  Lucas^-  stated  that  if  n  and  2/1  +  1  are  primes,  then  2n  +  l  is  a  factor 
of  2**  — 1  or  2"+l  according  as  n=3  or  n=l  (mod  4).  If  n  and  4n  +  l  are 
primes,  4n  +  l  is  a  factor  of  2^''+l.  If  n  and  Sn  +  l  =.4^  +  165^  are  primes, 
then  8n  +  l  is  a  factor  of  22" +  1  if  B  is  odd,  of  2-"±  1  if  5  is  even.  Also  ten 
theorems  stating  when  Qn  +  l=U.^+SM^,  12n  +  l=L^-\-12M^  or  24n  +  l 
=  L'^-\-4SM^  are  prime  factors  of  2'"'±1  for  certain  k's. 

*'Sur  la  s^rie  r^currente  de  Fermat,  Rome,  1879,  9-10.     Report  by  Cunningham.*' 

"Aesoc.  fran^.  avanc.  sc,  9,  1880,  228. 

»»^Nouv.  Ann.  Math.,  (2),  18,  1879,  500-9. 

"Comptes  Rendus  Paris,  90,  1880,  287,  345;  Coll.  Math.  Papers,  3,  428.     Incomplete  in  Math. 

Quest.  Educ.  Times,  40,  1884,  21. 
"Ann.  8C.  6cole  norm,  sup.,  (3),  1,  1884,  389-404;  Comptea  Rendus  Paris,  98,  1884,  293,  413, 

567,  613. 
".\nn.  BC.  6cole  norm,  sup.,  (3),  2,  1885,  113. 
"Assoc,  franc,  avanc.  sc,  15,  1886,  II,  101-2. 


Chap.  XVI]  FACTORS  OF  a" ±6".  385 


A.  S.  Bang^^  discussed  F<(a)  defined  by  (1).  If  p  is  a  prime,  F^i^{a)  has 
only  prime  factors  ap*+l  if  c?  =  a^  ~  —1  is  prime  to  p,  but  has  the  factor  p 
(and  not  p^)  if  d  is  divisible  by  p. 

Bang^^  proved  that,  if  a>l,  t>2,  Ft{a)  has  a  prime  factor  a^+1  except 
forF6(2). 

L.  Gianni^^  noted  that  if  p  is  an  odd  prime  dividing  a  — 1  and  p"  divides 
aF  —  \,  then  p""^  divides  a  — 1. 

L.  Kronecker^^  noted  that,  if  F^iz)  is  the  function  whose  roots  are  the 
<^(n)  primitive  nth  roots  of  unity, 


{x-yY^-^F^{^^=GAx,y') 


is  an  integral  function  involving  only  even  powers  of  y.  He  investigated  the 
prime  factors  q  of  Gn{x,  s)  for  s  given.  If  q  is  prime  to  n  and  s,  then  q  is 
congruent  modulo  n  to  Jacobi's  symbol  {s/q).  The  same  result  was  stated 
by  Bauer.^^ 

J.  J.  Sylvester^^  called  ^""  —  1  the  mth  Fermatian  function  of  6. 

Sylvester^^  stated  that,  for  6  an  integer  9^1  or  —1, 

ft  =?JZi 

contains  at  least  as  many  distinct  prime  divisors  as  m  contains  divisors  >  1 , 
except  when  0=  —  2,  m  even,  and  ^  =  2,  m  a  multiple  of  6,  in  which  two  cases 
the  number  of  prime  divisors  may  be  one  less  than  in  the  general  case. 

Sylvester^"  called  the  above  6^  a  reduced  Fermatian  of  injdex  m.  lim  =  np", 
n  not  divisible  by  the  odd  prime  p,  6^  is  divisible  by  p",  but  not  by  p"'^^,  if 
0  —  1  is  divisible  by  p.  If  m  is  odd  and  ^  —  1  is  divisible  by  each  prime  factor 
of  m,  then  dm  is  divisible  by  m  and  the  quotient  is  prime  to  m. 

Sylvester^^"  stated  that  if  P=l+p+ .  .  .+p''~^  is  divisible  by  q,  and 
p,  r  are  primes,  either  r  divides  q  —  1  or  r=q  divides  p  —  l.  li  P=q^  and 
p,  r,  j  are  primes,  j  is  a  divisor  of  q—r.  R.  W.  Genese  easily  proved  the 
first  statement  and  W.  S.  Foster  the  second. 

T.  Pepin^^  factored  various  a"  — 1,  including  a  =  79,  67,  43,  n  =  5;  a  =  7, 
n  =  ll',  a  =  S,  71  =  23;  a  =  5  or  7,  w  =  13  (certain  ones  not  in  the  tables  by 
Bickmore^^) . 

H.  Scheffler*^  discussed  the  factorization  of  2'"+!  by  writing  possible 
factors  to  the  base  2,  as  had  Beguelin.^  He  noted  (p.  151)  that,  if  m  =  2"~\ 
1^2(2-+!'"  =  (i+2")2{i_2m+(2m-l)2"-(2m-2)22" 

I  o  o(2m— 2)n_|_o(2m— l)n) 

His  formula  (top  p.  156),  in  which  2^*"^  is  a  misprint  for  2^''"^  is  equivalent 
to  that  of  LeLasseur.^^ 

"Tidsskrift  for  Mat.,  (5),  4,  1886,  70-80.     ^*Ibid.,  130-137.     sspgriodico  di  Mat.,  2,  1887,  114. 

36Berlin  Berichte,  1888,  417;  Werke,  3,  I,  281-292.     "jour,  fiir  Math.,  131,  1906,  265-7. 

"Nature,  37,  1888,  152.     ^Ubid.,  p.  418;  CoU.  Papers,  4,  1912,  628. 

"Comptes  Rendus  Paris,  106,  1888,  446;  CoU.  Papers,  4,  607. 

"«Math.  Quest.  Educ.  Times,  49,  1888,  54,  69. 

«Atti  Accad.  Pont.  Nuovi  Lincei,  49,  1890,  163.     Cf.  Escott,  Messenger  Math.,  33,  1903-4,  49. 

«Beitrage  zur  Zahlentheorie,  1891,  147-178. 


386  History  of  the  Theory  of  Numbers.  [Chap,  xvi 

E.  Lucas*^  gave  algebraic  factors  of 

K.  Zsigmondy^  proved  the  existence  of  a  prime  dividing  a'^  —  h'^,  but  no 
similar  binomial  with  a  lower  exponent,  exceptions  apart  (cf .  Bang,^^-  ^ 
Birkhoff^-). 

J.  W.  L.  Glaisher*^  gave  the  prime  factors  of  p^  — (  — l)^?-^)/^  ^^^  ^^^^i 

prime  p<100. 

T.  Pepin^^  proved  that  (31'-l)/30,  (83'-l)/82,  (2*^  +  l)/(3-83)  are 
primes. 

A.  A.  Markoff'^^  investigated  the  greatest  prime  factor  of  n^+1. 

W.  P.  Workman^^  noted  the  factors  of  3^*+Hl  [due  to  Catalan^^] 
and  2^"*  +  l,  and  stated  that  Lucas^^  (p.  326)  gave  erroneous  factors  of  2'^^+l. 

C.  E.  Bickmore*^  gave  factors  of  a"-l  for  n^50,  a  =  2,  3,  5,  6,  7,  10, 
11,  12. 

Several^^"  proved  that  n"  — 1  is  divisible  by  4n+l  if  47i+l  is  prime. 

A.  Cunningham^°  gave  43  primes  exceeding  9  milUon  which  are  factors  _ 
of  (x5±l)/(x±l),  and  factors  of  3'°+l,  3''-l,  S^^+l,  S'^^'+l,  5'^-l,  ] 
5^Hl,  5^'-l,  S^^+l,  53^-1. 

A.  Cunningham^^  considered  at  length  the  factorization  of  Aurifeuillians, 
i.  e.,  the  algebraically  irreducible  factors  of 

n+l 

{n,7?Y^+  {2n^fT,  (nix2)«+  ( -  \)~{n^'\?r  {n,n^  =  n), 

where  n^  and  x  are  relatively  prime  to  ^2  and  ?/,  while  n  has  no  square  factor, 
and  is  odd  in  the  second  case.  Aurifeuille  had  found  them  to  be  expressible 
algebraically  in  the  form  P^  —  Q'^.  There  are  given  factors  of  2"+!  for 
n  even  and  ^102,  and  for  n  =  110,  114,  126,  130,  138,  150,  210. 

A.  Cunningham^^  factored  numbers  a"=*=  1  by  use  of  tables,  complete  to 
p  =  101,  giving  the  lengths  I  of  the  periods  of  primes  p  and  their  powers 
<  10000  to  various  bases  q,  so  that  q^=  1  (mod  p  or  p^). 

A.  Cunningham  and  H.  J.  WoodalP^  gave  factors  of  A^  =  2''10"±l  for 
x^30,  a^  10,  and  for  further  sets;  also,  for  each  prime  p^3001,  the  least 
a  and  the  least  corresponding  x  for  which  p  is  a  divisor  of  N.  Bickmore 
(p.  95)  gave  the  linear  and  quadratic  forms  of  factors  of  A^. 

T.  Pepin^  factored  a^-1  for  a  =  37,  41,  79;  also^^  151^-1. 

"Th^orie  des  nombres,  1891,  132,  exs.  2-4. 

"Monatshefte  Math.  Phys.,  3,  1892,  283.  Details  in  Ch.  VII,  Zsigmondy." 

«Quar.  Jour.  Math.,  26,  1893,  47. 

*«Memorie  Accad.  Pont.  Nuovi  Lincei,  9,  I,  1893,  47-76. 

♦^Comptea  Rendus  Paris,  120,  1895,  1032.      "Messenger  Math.,  24,  1895,  67. 

"/6id.,  25,  1896,  1-44;  26,  1897,  1-38;  French  transl.,  Sphinx-Oedipe,  7,  1912,  129-44,  155-9. 

"^Math.  Quest.  Educ.  Times,  65,  1896,  78;  (2),  8,  1905,  97. 

»»Proc.  London  Math.  Soc.,  28,  1897,  377,  379.      "/bid.,  29,  1898,  381-438. 

"Messenger  Math.,  29,  1899-1900,  145-179.     The  line  of  iV'  =  532(p.  17)  is  incorrect. 

"Math.  Quest.  Educat.  Times,  73,  1900,  83-94.     [Some  errors.] 

"Mem.  Pont.  Ac.  Nuovi  Lincei,  17,  1900,  321-344;  errata,  18,  1901.     Cf.  Sphinx-Oedipe,  5, 

1910,  num6ro  special,  1-9.     Cf.  Jahrbuch  Fortschritte  Math.,  on  a  =  37. 
*»Atti  Accad.  Pont.  Nuovi  Lincei,  44,  1900-1,  89. 


Chap.  XVI]  FACTORS  OF  a"  =±=5".  387 

A.  Cunningham^^  factored  5'*  —  !  for  n  =  75,  105. 

L.  Kronecker^^"  proved  that  every  divisor,  prime  to  <,  of  (1)  is  =  1  (mod  t). 
H.  S.  Vandiver^®''  noted  that  the  proof  applies  to  the  homogeneous  form 
Ft{a,  b)  of  (1)  if  a,  b  are  relatively  prime. 

D.  Biddle^^  gave  a  defective  proof  that  3-2^^+1  is  a  prime. 

The  Math.  Quest.  Educational  Times  contains  the  factorizations  of: 

Vol.  66  (1897),  p.  97,  2i55_i  factor  3P.  Vol.  68  (1898),  p.  27,  p.  112,  272o_i. 
p.  114,  1012+4. 

Vol.  69  (1898),  p.  61,  3824+1;  p.  73,  x^-1,  x  =  500,  2000;  p.  117,  x'+y^;  p.  118, 
10^^+33,  33.IOI8+I. 

Vol.  70  (1899),  p.  32,  p.  69,  242i''+l;  p.  47,  3201^-1;  p.  64,  2^2+1,  8i*+l, 
20018+1;  p.  72,  2014-1;  p.  107,  9721^+1.  Vol.  71  (1899),  p.  63,  x^^+^-l;p.  72, 
x'-\-y\ 

Vol.  72  (1900),  p.  61,  (Sny^-l  factor  24n+l  if  prime;  p.  86,  72210+1;  p.  117, 
144010+1. 

Vol.  73  (1900),  p.  51,  3520+I;  p.  96,  711-I;  p.  104,  p.  114,  x'-\-y*. 

Vol.  74  (1901),  p.  27,  a  prime  2iq+l  divides  g^-1  if  k  =  2^-^;  p.  86,  rcio-5y. 

Vol.  75  (1901),  p.  37,  3^+y';  p.  90,  1792^+1;  p.  Ill,  7^5+1.  [Educ.  Times, 
(2),  54,  1901,  223,  260]. 

Ser.  2,  Vol.  1  (1902),  p.  46,  10082«+1;  p.  84,  x'-\-fxy^.  Vol.  2  (1902),  p.  33,  p. 
53,  iV4+l;  p.    118,  IP^+l. 

Vol.    3  (1903),  p.  49,  a'+b*  (cf.  74,  1901,  44);  p.  114,  a«+l,  a  =  60000. 

Vol.    6  (1904),  p.  62.  9618+1. 

Vol.    7  (1905),  p.  62,  20813-1;  pp.  106-7,  2126+I. 

Vol.    8  (1905),  p.  50,  9618+1;  p.  64,  212^+1. 

Vol.  10  (1906),  p.  36,  5418+I,  6^4+1. 

Vol.  12  (1907),  p.  54,  6*2+1,  24^0+1. 

Vol.  13  (1908),  p.  63,  106-7,  S'*-\-2'\ 

Vol.  14  (1908),  p.  17,  15018+1;  p.  71,  sextics;  p.  96,  7^5+1. 

Vol.  15  (1909),  p.  57,  S''-\-2'*;  p.  33,  3111+I,  12*5+1;  p.  103,  282i+l,  44ii+l, 
630+1. 

Vol.  16  (1909),  p.  21,  1924+1. 

Vol.  18  (1910),  pp.  53-5,  102-3,  x^-\-^y';  pp.  69-71,  a:«+27?/«;  p.  93,  y^^-l. 

Vol.  19  (1911),  p.  103,  c(^-\-y'  =  z^-\-w\  Vol.  23  (1913),  p.  92,  (x'^-Nx-^NY 
-{-Nix^-Ny. 

Vol.  24  (1913),  pp.  61-2,  x'^^^y^  y  =  5,  7,  11,  13;  pp.  71-2,  a;i2+2«,  x'^-\-S\ 
x3''+3i5. 

Vol.  26  (1914),  p.  23,  x^^-{-l  for  fc  =  6n+35^3^  p.  39,  a;i2+6«;  p.  42,  xl0-5^ 
^14+7^  a:22+llii,  a;26-13i3.  Vol.  27  (1915),  pp.  65-6,  451^-1,  20''-l,  fc^o+l  for  A; 
=  6,  8,  10;  p.  83,  x4+4y4  (when  four  factors).  Vol.  28  (1915),  p.  72,  503o+l.  Vol. 
29(1916),  p.  95,  9618+1. 

New  series,  vol.  1  (1916),  p.  86,  rc2o+10i<',  x28+14i4;  pp.  94-5,  x^'^-5^^,  x^'^+WK 

Vol.  2  (1916),  p.  19,  ajso-sis. 

Vol.  3  (1917),  p.  16,  x''-y'';  p.  52,  xH-l. 

E.  B.  Escott^^  gave  many  cases  when  1+x^  is  a  product  of  two  powers 
of  primes  or  the  double  of  such  a  product. 

«Proc.  London  Math.  Soc,  34,  1901,  49. 

^^''Vorlesungeu  iiber  Zahlentheorie,  1,  1901,  440-1. 

s6*Amer.  Math.  Monthly,  10,  1903,  171. 

"Messenger  Math.,  31,  1901-2,  116  (error);  33,  1903-4,  126. 

"L'mterm^diaire  des  math.,  7,  1900,  170. 


388  History  of  the  Theory  of  Numbers.  [Chap,  xvi 

P.  F.  Teilhet*^  gave  formulas  factoring  cases  of  l+ar^,  as 

(6H6+l)Hl  =  [(6+l)2-hl](6Hl), 

4(c+ir+l       =[(c+2)H(c+l)^[(c+l)Hca 
the  last  being  (10,  1903,  170)  a  case  of  the  known  formula  for  the  product 
of  two  sums  of  two  squares  (cf.  11,  1904,  50). 

Escott^°  repeated  Euler's^  remarks  on  the  integers  x  for  which  \-\-:i^ 
is  di\'isible  by  a  given  prime.  He  and  Teilhet  (11,  1904,  10,  203)  noted 
that  any  common  di\'isor  of  h  and  a±l  di\'ides  (a''='=l)/(a=fcl). 

G.  Wertheim"  collected  the  theorems  on  the  divisors  of  a"'=«=l. 

G.  D.  Birkhoff  and  H.  S.  Vandiver^-  employed  relatively  prime  integers 
a,  6  (a>6)  and  defined  a  primitive  divisor  of  F„  =  a"  — 6"  to  be  one  relatively 
prime  to  V^,  for  all  di\'isors  m  of  n.  They  proved  that,  if  n?^2,  F„  has  a 
primitive  divisor  7^  1  except  f orn  =  6,  a  =  2,  6  =  1. 

L.  E.  Dickson^^''  noted  that  {p^  —  l){p^  —  l)  has  no  factor=l  (mod  p^) 
if  p  is  prime. 

A.  Cunningham^^  gave  high  primes  ?/"  +  l,  (t/^+1)/2,  ?/^4-y+l. 

H.  J.  Woodall^  gave  factors  of  ?/"  +  l. 

J.  W.  L.  Glaisher^^  factored  2^''=t2^+l  for  r^ll,  in  connection  with 
the  question  of  the  similarity  of  the  nth  pedal  triangle  to  a  given  triangle. 

L.  E.  Dickson^^  gave  a  new  derivation  of  (1),  found  when  F,(a)  is  divisible 
by  pi  or  pi^,  where  pi  is  a  prime  factor  of  t,  and  proved  that,  if  a  is  an  integer 
>1,  F,(a)  has  a  prime  factor  not  di\'iding  0*^  —  1  (m<t)  except  in  the  cases 
f  =  2,  a  =  2*  —  1,  and  t  =  Q,  a  =  2;  whence  a'  —  1  has  a  prime  factor  not  dividing 
a"*—l{m<t)  except  in  those  cases  [cf.  Birkhoff,^-  CarmichaeP]. 

Dickson®^  applied  the  last  theorem  to  the  theory  of  finite  algebras  and 
gave  material  on  the  factors  of  p"  — 1. 

A.  Cunningham^^  treated  at  length  the  factorization  of  i/"+l  for  71  =  2, 
4,  8,  16,  and  (?/^''+l)/(2/"+l)  for  n  =  1,  2,  4,  8,  by  means  of  extensive  tables 
of  solutions  of  the  corresponding  congruences  modulo  p.  He  discussed  also 
x^+y",  n  =  4,  6,  8,  12. 

Cunningham^^''  factored  \{x^  —  i^)/{x  —  y)-\-iJL{x^+y^)/{x'^+y'^)  by  ex- 
pressing the  fractions  in  the  form  P^  —  kxyQ"^,  k=  o,  6. 

"L'intermediaire  des  math.,  9,  1902,  31&-8. 

^oibid.,  12,  1905,  38;  cf.  11,  1904,  195-6. 

"Anfangsgriinde  der  Zahlenlehre,  1902,  297-303,  314. 

"Annals  of  Math.,  5,  1903-4,  173.     Cf.  Zsigmondy,"  Dickson." 

•«»Amer.  Math.  Monthly,  11,  1904,  197,  238;  15,  1908,  90-1. 

•»Quar.  Jour.  Math.,  35,  1904,  10-21. 

•*Ibid.,  p.  95. 

**Ibid.,  36,  1905,  156. 

"Amer.  Math.  Monthly,  12,  1905,  86-89. 

•'Gottingen  Nachrichten,  1905,  17-23. 

"Messenger  Math.,  35,  1905-6,  16&-185;  36,  1907,  145-174;  38,  1908-9,  81-104,  145-175; 

39,  1909,  33-63,  97-128;  40,  1910-11,  1-36.     Educat.  Times,  60,  1907,  544;  Math.  Quest. 

Educat.  Times,  (2),  13,  1908,  95-98;  (2),  14,  1908,  37-8,  52-3,  73^;  (2),  15,  1909,  33-4, 

103-4;  (2),  17,  1910,  88,  99.     Proc.  London  Math.  Soc,  27,  1896,  98-111;  (2),  9,  1910, 

1-14. 
•»<»Math.  Quest.  Educ.  Times,  10,  1906,  58-9. 


I 


Chap.  XVI]  FACTORS  OF  a**=*=6''.  389 

L.  E.  Dickson  and  E.  B.  Escott^^  discussed  the  divisibility  of  p^^*  — 1 
by  d(p"^'^  — 1),  where  d  is  a  divisor  of  n,  and  d  of  d. 

R.  D.  CarmichaeF°  proved  that  if  P^^—R^"  is  divisible  by  8a  and  we 
set  Q  =  {P''—R'')/{a{P  —  R) } ,  then  Q/8  is  an  integer  if  and  only  if  a  is  divisible 
by  the  least  integer  e  for  which  P^  —  R^  is  divisible  by  each  prime  factor  of 
a  not  dividing  P  —  R,  and  5  is  a  divisor  of  Q.  Proof  for  the  case  R  =  l  had 
been  given  by  E.  B.  Escott'^^ 

A.  Cunningham^^  tabulated  the  factors  of  y^^^^l  for  ?/  =  2,  3,  5,  7,  12. 

K.  J.  Sanjana'^^  considered  the  factors  of 

Sanjana'^^"  applied  his  method  to  prove  the  statement  of  M.  Kannan  that 
20^5-1=  11. 19-31-61-251421-3001-261451-64008001-3994611390415801 

•4199436993616201. 

L.  E.  Dickson''^  factored  w"— 1  for  various  values  of  n. 

R.  D.  CarmichaeF^  employed  the  methods  of  Dickson^ ^  to  obtain  general- 
izations. Let  Q„(a,  /3)  be  the  homogeneous  form  of  Fn{a),  Let  n  =  lip/*, 
where  the  p's  are  distinct  primes,  and  let  c  be  a  divisor  of  n  and  a  multiple 
of  pi°'.  If  a,  j3  are  relatively  prime,  the  g.  c.  d.  of  5  =  a"^^'— iS'*''^'  and 
Qc{a,  /3)  is  1  or  pi  and  at  most  one  Qda,  /3)  contains  the  factor  pi  when 
d  contains  pi^;  if  pi>2  divides  5,  at  most  one  Qcio-,  jS)  contains  pi,  and  no 
one  of  them  contains  pi^.  If  a,  /3  are  relatively  prime  and  c  =  mpi"',  where 
m>l  and  m  is  prime  to  pi,  then  QXcl,  jS)  is  divisible  by  pi  if  and  only  if 
fjx—j^x  ^jj^Q^  p^)  holds  for  x  =  m,  but  not  for  0<a;<m;  in  all  other  cases 
Q=  1  (mod  m).  If  a,  jS  are  relatively  prime,  QXo-,  jS),  and  hence  also  a"—^", 
has  a  prime  factor  not  dividing  a*— j3'(s<c),  except  in  the  cases  (i)  c  =  2, 
^  =  1^  a  =  2^-l;  (ii)  Q,(a,  /3)  =p  =  greatest  prime  factor  of  c,  and  a"^^=/3"/^ 
(modp);  (iii)a(a,/3)  =  l. 

E.  Miot^^  noted  that  LeLasseur's^^  formula  is  the  case  m  =  n  =  l  of 


/02fc+1^2\  2  /  02t+1^2  \ 

(f_JL)  +^2^n(m+^-^±2*+in 
\    m    /  \  m  I 


Welsch  (p.  213)  stated  that  the  latter  is  no  more  general  than  the  case  A:  =  0, 
which  follows  from  the  known  formula  for  the  product  of  two  sums  of  two 
squares. 

A.  Cunningham^^  noted  the  decomposition  into  primes : 

2"+l  =  3-43-617-683-78233-35532364099. 

"L'interm^diaire  des  math.,  1906,  87;  1908,  135;  18,  1911,  200.     Cf.  Dickson." 

'"Amer.  Math.  Monthly,  14,  1907,  8-9. 

'i76id.,  13,  1906,  155-6. 

"Report  British  Assoc,  78,  1908,  615-6. 

"Proc.  Edinburgh  Math.  Soc,  26,  1908,  67-86;  corrections,  28,  1909-10,  viii. 

"aJour.  Indian  Math.  Club,  1,  1909,  212. 

'^Messenger  Math.,  38,  1908,  14-32,  and  Dickson"*"'  of  Ch.  XIV. 

'^Amer.  Math.  Monthly,  16,  1909,  153-9. 

'«L'interm6diaire  des  math.,  17,  1910,  102. 

"Report  British  Assoc,  for  1910,  529;  Proc.  London  Math.  Soc,  (2),  8,  1910,  xiii. 


390  History  of  the  Theory  of  Numbers.  [Chap,  xvi 

A.  Cunningham"^  discussed  quasi-Mersenne  numbers  N'g  =  x^—y^,  with 
x  —  y=l,  q  a  prime,  tabulating  every  prime  factor  <  1000  for  q<50,  a:<20 
if  q>o,  a:<50  if  q  =  5,  and  treated  Aurifeuillians 

{X''=^Y')/{X=^Y),  X  =  ^^,  Y  =  qy]\ 

H.  C.  Pocklington"^  proved  that,  if  n  is  prime,  (x"  — ?/'*)/(x  — ?/)  is  divisible 
only  by  numbers  of  the  form  77m +  1  unless  x  —  y\s,  divisible  by  n  [Euler], 
and  then  is  divisible  only  by  n  and  numbers  of  the  forms  mn-\-\,  n{inn-\-\). 

G.  Fonten^*°  stated  that,  if  p  is  a  prime  and  x,  y  are  relatively  prime, 
each  prime  factor  of  {x^  —  y^)/{x  —  y)  is  of  the  form  A'p  +  1,  except  for  a 
factor  p,  occurring  if  x=y  (mod  p)  and  then  only  to  the  first  power  if  p>2. 

G.  Fonten^^^  considered  the  homogeneous  form/t(x,  y)  derived  from  (1) 
by  setting  a  =  x/y.     If  p"  is  the  highest  power  of  a  prime  p  dividing  n. 

The  main  theorem  proved  is  the  following:  If  x,  y  are  relatively  prime 
every  prime  divisor  of /„(x,  y)  is  of  the  form  kn+l,  unless  it  is  divisible  by 
the  greatest  prime  factor  (say  p)  of  n.  It  has  this  factor  p  if  p  —  1  is  divisible 
by  n/p°-  and  if  x,  y  satisfy /„/pa=0  (mod  p),  the  latter  having  for  each  y  prime 
to  p  a  number  of  roots  x  equal  to  the  degree  of  the  congruence.  In  par- 
ticular, if  n  is  a  power  of  a  prime  p,  every  prime  factor  of  /„  is  of  the  form 
kn-\-l,  with  the  exception  of  a  divisor  p  occurring  if  x=y  (mod  p),  and  then 
to  the  first  power  if  n5^2. 

J.  G.  van  der  Corput^^  considered  the  properties  of  the  factors  of  the 
expression  derived  from  a' +6'  as  (1)  is  derived  from  a'  — 1. 

A.  G^rardin^  factored  a^+6^  in  four  numerical  cases  and  gave 
(a2+3iS2)H(4ai3)*=n{(3a2=fc2ai3+3/32)2-2(2a2±2a/3)2). 

A.  Cunningham^  tabulated  factors  of  y'^^2,  2?/*±  1. 

R.  D.  CarmichaeP^  treated  at  length  the  numerical  factors  of  a"=tj8" 
and  the  homogeneous  form  Qnio-,  /3)  of  (1),  when  a+^S  and  a/5  are  relatively 
prime  integers,  while  a,  j8  may  be  irrational. 

A.  G^rardin^S'^  factored  xHl  for  x  =  373,  404,  447,  508,  804,  929;  inves- 
tigated x'* -2  for  x^  50,  y^-Sfory^ 75,  Sv^ - 1  for  y^ 25,  2w^ -Iforw^ 37, 
and  gave  ten  methods  of  factoring  numbers  Xa^  — 1. 

L.  Valroff^^''  factored  2x^-1  for  101^x^180,  8x^-1  for  x<128. 

A.  Gerardin^^*^  expressed  622833161  (a  factor  of  20^°+ 1)  as  a  sum  of  two 
squares  in  two  ways  to  get  its  prime  factors  2801  and  222361. 

"Messenger  Math.,  41,  1911-12,  119-145. 

"Proc.  Cambr.  Phil.  Soc,  16,  1911,  8. 

'"Nouv.  Ann.  Math.,  (4),  9,  1909,  384;  proof,  (4),  10,  1910,  475;  13,  1913,  383-4. 

"Ubid.,  (4),  12,  1912,  241-260. 

8«Nieuw  Archicf  voor  Wiskunde,  (2),  10,  1913,  357-361. 

"Wiskundig  Tijdschrift,  10,  1913,  59. 

"Messenger  Math.,  43,  1913-4,  34-57. 

«Annals  of  Math.,  (2),  15,  1913-4,  30-70. 

w^Sphinx-Oedipe,  1912,  188-9;  1913,  34-44;  1914,  20,  23-8,  34-7,  48. 

o^^lbid.,  1914,  5-6,  18-9,  28-30,  33,  37,  73. 

*^Ibid.,  39.     Stated  by  E.  Fauquembergue,  I'interm^diaire  des  math.,  21,  1914,  45. 


I 


Chap.  XVI]  FACTORS  OF  a**  ±6".  391 

A.  Cunningham^Habulated  factors  of  y^=^l,x'^=^y''",  and  gave  an  account 
of  printed  and  manuscript  tables  of  solutions  of  2/"*='=l  =  0  (mod  p^). 

Cunningham^^  tabulated  factors  of  x^^^y""  for  x^lQ  and  certain  y's  as 
high  as  31  when  x  =  2  or  4,  where  x,  y  are  relatively  prime  and  x>  1,  y>  1. 

Cunningham^^  noted  that  x-2''+l  is  composite  for  l<a;<233,  x 9^  14:1. 

A.  Cunningham  and  H.  J.  WoodalP^  tabulated  factors  of  2'^^q  and 
g.2*±l  for  5-^66,  and  tabulated  values  of  q  for  which  one  of  these  four 
functions  is  divisible  by  a  given  prime  p  or  power  of  p.  They  confirmed 
that  X.2'' + 1  is  composite  when  1<  x<  233  except  perhaps  when  x  =  141 .  In- 
cidentally (p.  15),  the  factors  of  2*"^^;  — 1  for  A; ^  17  are  given. 

For  factors  of  2^  —  1  and  10"  — 1,  see  Chapters  I  and  VI.  For  factor 
tables  of  numbers  m.2^±l,  see  Seelhoff^^  ^^^  Morehead^o  of  Ch.  XIII;  for 
m. 6^=1=1,  Dines^^  For  factors  of  several  numbers  d^  —  1,  see  Lawrence^^, 
Biddle^^  and  Kraitchik^i  of  Ch.  XIV.  For  the  form  of  factors  of  a'^+b" 
when  k  =  2^,  see  Euler"  of  Ch.  XV.  Various  results  in  Ch.  XVII  relate  to 
factors  of  a'*=t5'». 

Factors  of  Trinomials. 

Seven^^  primes  p  such  that  (p^  — 1)^  has  4  or  more  factors  px+l,x<p. 
List^^  of  algebraically  factorable  trinomials  x^+xy'^+y^,  etc. 
Factors"  of  14^14^+1,  7^+2-7'+l,  etc. 

Conditions  that  x^-\-Px'^-\-c^  be  a  product  of  4  rational  quadratic  factors.^^ 
Two^^  factors  of  x^-{-i4:m'^+Sm^+2)xY+y\ 
Factors^"*^  of  various  trinomial  expressions. 

For  factors  of  x'^+Qbx^+b^  see  Dirichlet^  of  Ch.  XVII.  See  papers  28, 
28a,  65,  89  above. 

8«Messenger  Math.,  45,  1915,  49-75. 

^Ubid.,  185-192. 

ssProc.  London  Math.  Soc,  (2),  4,  1907,  xviii;  (2),  15,  1916-7,  xxix. 

s^Messenger  Math.,  47,  1917,  1-38.     Math.  Quest.  Educ.  Times,  (2),  10,  1906,  44. 

»5Math.  Quest.  Educat.  Times,  (2),  15,  1909,  82-3.     Amer.  Math.  Monthly,  15,  1908,  67,  138. 

L'intermediaire  des  math.,  15,  1908,  121. 
9«Math.  Quest.  Educ.  Times,  (2),  16,  1909,  39-41. 
»Ubid.,  65-6. 

«8/6id.,  (2),  18,  1910,  64-5;  (2),  22,  1912,  20-1. 
s^Sphinx-Oedipe,  6,  1911,  8-9. 
""Math.  Quest.  Educ.  Times,  72,  1900,  26-8;  74,  1901,  130-1;  (2),  6,  1904,  97;  19,  1911,  85; 

20,  1911,  25-6,  76-8;  22,  1912,  54-61.     Math.  Quest,  and  Solutions,  3,  1917,  66;  4,  1917, 

13,39;  5,  1918,  38,  50-1. 


CHAPTER  XVII. 

RECURRING  SERIES;  LUCAS*  Un,  Vn. 

Leonardo  Pisano\  or  Fibonacci,  employed,  in  1202  (revised  manuscript, 
1228),  the  recurring  series  1,  2,  3,  5,  8, 13,  ...  in  a  problem  on  the  number  of 
offspring  of  a  pair  of  rabbits.  We  shall  write  Un  for  the  nth  term,  and  Un 
for  the  {n-\-l)th  term  of  0,  1,  1,  2,  3,  5, . . .  derived  by  prefixing  0,  1  to  the 
former  series. 

Albert  Girard^  noted  the  law  w„+2  =  ^<n+i+'Wn  for  these  series. 

Robert  Simson^  noted  that  this  series  is  given  by  the  successive  conver- 
gents  to  the  continued  fraction  for  (\/5  +  l)/2.  The  square  of  any  term  is 
proved  to  differ  from  the  product  of  the  two  adjacent  terms  by  =*=  1. 

L.  Euler^  noted  that  {a-\-\/b)''  =  Ak+Bk^  implies 

A,  =  h{{a-\-Vbr+ia-Vbr},        B,  =  -~^{ia+Vbr-{a-Vby]. 

J.  L.  Lagrange^  noted  that  the  residues  of  Ak  and  B^  with  respect  to  any 
modulus  are  periodic. 

Lagrange^  proved  that  if  the  prime  p  divides  no  number  of  the  form 
f—au^,  then  p  divides  a  number  of  the  form 

{{t-huVar+'-it-uV^)^']/Va. 

A.  M.  Legendre'  proved  that,  if  (f)^—A^^  =  l,  then  ((l>+\l/VAy  —  l  is 
of  the  form  r+sV^?  where  r  and  s  are  divisible  by  a  prime  w,  not  dividing 

8=<o-lit(i)  =  +l,  «=«+litg)  =  -l. 

C.  F.  Gauss*  proved  [Lagrange's^  result]  that,  if  6  is  a  quadratic  non- 
residue  of  the  prime  p,  then  Bp+i  is  divisible  by  p  for  every  integral  value 
of  a.  If  e  is  a  divisor  of  p+ 1,  then  Be  is  divisible  by  p  for  e  —  1  values  of  a, 
being  a  factor  of  B^+i. 

G.  L.  Dirichlet^  proved  that,  if  b  is  an  integer  not  a  square  and  x  is  any 
integer  prime  to  b,  and  if  U,  V  are  polynomials  in  x,  b  such  that 

{x+Vbr=^u-\-vvb, 

then  U  and  V  have  no  common  odd  divisors.  If  n  is  an  odd  prime,  no  prime 
of  which  6  is  a  quadratic  residue  is  a  factor  of  V  unless  it  be  of  the  form 
2mn+l.  No  prime  of  which  6  is  a  quadratic  non-residue  is  a  factor  of  V 
unless  it  be  of  the  form  2mn  —  l.     Lagrange^  had  proved  conversely  that  a 

iScritti,  I,  1857  (Liber  Abbaci),  283-4. 

^L'Arithm^tique  de  Simon  Stevin  de  Bruges,  par  Albert  Girard,  Leyde,  1634,  p.  677.     Lea 

Oeuvres  Math,  de  Simon  Stevin,  1634,  p.  169. 
3Phil.  Trans.  Roy.  Soc.  London,  48,  I,  1753,  368-376;  abridged  edition,  10,  1809,  430-4. 
*Novi  Comm.  Acad.  Petrop.,  18,  1773,  185;  Comm.  Arith.,  1,  554. 

^Additions  to  Euler's  Algebra,  2, 1774,  §§  78-9,  pp.  599-607.    Euler,  Opera  Omnia,  (1),  1, 619. 
•Nouv.  M^m.  Ac.  Berlin,  ann^e  1775  (1777),  343;  Oeuvres,  3,  782-3. 

'TWorie  des  nombres,  1798,  p.  457;  ed.  2,  1808,  p.  429;  ed.  3,  1830,  vol.  2,  Art.  443,  pp.  111-2. 
«Disq.  Arith.,  1801,  Art.  123.  »Deformishnearibus,Breslau,1827;  Werke,  1,51.  Cf .  Kronecker." 

393 


394  History  of  the  Theory  of  Numbers.  [Chap,  xvn 

prime  of  which  6  is  a  non-residue,  and  having  the  form  2mn  —  1,  vdW  di\'ide  V. 
li  b=  —n,  where  n  is  a  prime  4m-\-3,  no  prime  divides  V  unless  it  is  of  the 
form  A-n=fc  1,  and  conversely.  The  divisors  of  U  are  discussed  for  the  case 
n  a  power  of  2;  in  particular,  of  U  =  x^+Qbx^-\-b^  when  n  =  4. 

J.  P.  M.  Binet^°  noted  that  the  number  of  terms  of  a  solution  t'„,  expressed 
as  a  function  of  n,  ro,  . . .,  of  the  equation  t'„+2  =  i'„+i+''«i'n  in  finite  differ- 
ences is 


^(i^-'-(^)-*')- 


This  equals  f/„  as  shown  by  taking  each  r„  to  be  unity. 

G.  Lam6^^  used  the  series  of  Pisano^  to  prove  that  the  number  of  divi- 
sions necessary  to  find  the  g.  c.  d.  of  two  integers  by  the  usual  process  of 
division  does  not  exceed  five  times  the  number  of  digits  in  the  smaller 
integer.  Lionnet^-  added  that  the  number  of  divisions  does  not  exceed  three 
times  it  when  no  remainder  exceeds  half  the  corresponding  di\'isor.  See 
also  Serret,  Traits  d'Arithmetique;  C.  J.  D.  Hill,  Acta  Univ.  Lundensis, 
2,  1865,  No.  1;  E.  Lucas,  Nouv.  Corresp.  Math.,  2,  1876,  202,  214;  4, 
1878,  65,  and  Th^orie  des  Nombres,  1891,  335,  Ex.  3;  P.  Bachmann,  Niedere 
Zahlentheorie,  1902,  116-8;  L.  Grosschmid,  Math.-Naturwiss.  Blatter,  8, 
1911,  125-7,  for  an  elementary  proof  by  induction;  Math,  es  Phys.  Lapok, 
23,  1914,  5-9;  R.  D.  Carmichael,  Theory  of  Numbers,  p.  24,  Ex.  2. 

H.  Siebeck^^  considered  the  recurring  series  defined  by 

for  a,  c  relatively  prime.     By  induction, 

where  /3  =  0  or  1,  7  =  (r  — 1)/2  or  (r  — 2)/2,  according  as  r  is  odd  or  even; 

whence  A^^  is  divisible  by  N^-  If  P  and  q  are  relatively  prime,  Np  and 
A"g  are  relatively  prime  and  conversely.  If  p  is  a  prime,  6  =  a"+4c,  and 
s  =  (b/p)  is  Legendre's  symbol,  then 

Np=s,  Np.,=  0  (mod  p), 

so  that  either  Np+i  or  A"p_i  is  divisible  by  p. 

J.  Dienger^^  considered  the  question  of  the  number  of  terms  of  the  series 
of  Pisano  with  the  same  number  of  digits  and  the  problem  to  find  the  rank 
of  a  given  term. 

A.  Genocchi^^  took  a  and  b  to  be  relatively  prime  integers  and  proved 
that  B„„  is  divisible  by  B„  and  that  the  quotient  Q  has  no  odd  divisor  in 

"Comptea  Rendus  Paris,  17,  1843,  563. 

"Ibid.,  19,  1S44,  867-9.     Cf.  Binet,  pp.  937-9. 

'*Compl^ment  des  616ments  d'arithm^tique,  1857,  39-42. 

"Jour,  fur  Math.,  33,  1846,  71-6.  "Archiv  Math.  Phys.,  16,  1851,  120-4. 

"AnnaU  di  Mat.,  (2),  2,  1868-9,  256-267.     Cf.  Genocchi". ". 


Chap.  XVII]  Recueeing  Seeies;  Lucas'  Un,  v^.  395 

common  with  B^  other  than  a  divisor  of  n.  If  p  is  an  odd  divisor  of  B^^  and 
if  h  is  the  least  k  for  which  B^  is  divisible  by  p,  then  /i  is  a  divisor  of  m.  If  p 
is  an  odd  prime,  Bp_i  or  Bp^i  is  divisible  by  p  according  as  6  is  a  quadratic 
residue  or  non-residue  of  p,  whatever  be  the  value  of  a.  This  is  used  to 
prove  the  existence  of  primes  of  the  two  forms  n'2;±l(n  a  prime  >2)  and 
the  existence  of  an  infinitude  of  primes  of  each  of  the  forms  mz^l  [Ch .  XVIII] . 
E.  Lucas^^  stated  without  proof  theorems  on  the  series  of  Pisano.^  The 
sum  of  the  first  n  terms  equals  C/„+2  — 2;  the  sum  of  those  terms  taken  with 
alternate  signs  equals  (  — l)"C/„_i.     Also 

U  „_i  -\-U  n—  U2n,        UnUn+l  ~  U  n—\  U  n—2  =  U  2n}         U  n'^U  n+1  ~  U  „_i  =  C/ 3„+2' 

We  have  the  symbolic  formulas 

where,  after  expansion,  exponents  are  replaced  by  subscripts.  From 
E.  Catalan's  Manuel  des  Candidats  a  I'Ecole  Polytechnique,  I,  1857,  86,  he 
quoted 

Lucas^'^  employed  the  roots  a,boix^  =  x-\-l  and  set 

^  ^  n    \    1.71        ^2re  I 

a  —  0  Un 

The  u's  form  the  series  of  Pisano  with  the  terms  0,  1  prefixed,  so  that 
Uo=0,  Ui  =  U2=l,  U3  =  2.  Since  5w„^  —  z^„^  =  ± 4,  u^  and  Vn  have  no  common 
factor  other  than  2.  If  p  is  a  prime  ^2,  5,  we  have  Up=±l,  Vp  =  l  (mod 
p).    We  have  the  symbolic  formulas 

Given  a  law  Un+k  =  -^oUn+p+  ■  ■  ■  -i-ApUn  of  recurrence,  we  can  replace  the 
symbol  f/*  by  0  ( C/) ,  where 


(f){u)=Aou''+Aiu''-^+. .  .  +Ap_^u+A 


p) 


since  U^+kp^  U''{({){U)}^,  symbolically. 

E.  Lucas^^  stated  theorems  on  the  series  of  Pisano.    We  have 

2"\/5i/,  =  (l  +  V5r-(l-V5r,  ^n+i  =  l  +  (i)  +  (''2^)  +  --- 

and  his^^  symbolic  formulas  with  u's  in  place  of  U^s.  Up^  is  divisible  by  Up 
and  Uq,  and  by  their  product  if  p,  q  are  relatively  prime.  Set  Vn  =  U2n/un. 
Then 

Vn+2  =  Vn+l+Vn,  Vin  =  An-2,  y4„+2  =  «^Wl +2- 

«Nouv.  Corresp.  Math.,  2,  1876,  74r-5. 

^Ubid.,  201-6. 

"Comptes  Rendus  Paris,  82,  1876,  165-7. 


396  History  of  the  Theory  of  Numbers.  [Chap,  xvii 

If  the  term  of  rank  ^  + 1  in  Pisano's  series  is  divisible  by  the  odd  number  A 
of  the  form  10^=*=  3  and  if  no  term  whose  rank  is  a  divisor  of  A  + 1  is  divisible 
by  .4 ,  then  A  is  a  prime.  If  the  term  of  rank  A  —  1  is  divisible  by  A  =  10p=t  1 
and  if  no  term  of  rank  a  divisor  of  A  —  1  is  divisible  by  A,  then  A  is  a  prime. 
It  is  stated  that  A=2^^^  — 1  is  a  prime  since  A  =  10p  — 3  and  u^  is  never 
divisible  by  A  for  k  =  2",  except  for  n  =  127. 

Lucas^^  employed  the  roots  a,  6  of  a  quadratic  equation  x^— Px+Q  =  0, 
where  P,  Q  are- relatively  prime  integers.     Set 

a"  — 6" 

Un  = r-'  Vn  =  a"+h'*,  5  =  o  — 6. 

a  —  o 

The  quotients  of  5if„ V  —  1  and  y„  by  2^"^^  are  functions  analogous  to  the 
sine  and  cosine.     It  is  stated  that 

(1)  U2n  =  UnVn,  V^-bW  =  '^Q', 

(2)  2u^+ri  =  UmVn  +  U„Vm,  "J^n^ "  Itn-l^^n+l  =  Q'*"^ 

Not  counting  divisors  of  Q  or  5^,  we  have  the  theorems: 

(I)  Wpg  is  divisible  by  Up,  u^,  and  by  their  product  if  p,  q  are  relatively 
prime. 

(II)  Un,  Vn  are  relatively  prime. 

(III)  If  d  is  the  g.  c.  d.  of  m,  n,  then  Ud  is  the  g.  c.  d  of  u^,  Un. 

(IV)  For  n  odd,  u^,  is  a  divisor  of  x^  —  Qif'. 

By  developing  w„p  and  f„p  in  powers  of  Un  and  v„,  we  get  formulas  analo- 
gous to  those  for  sin  nx  and  cos  nx  in  terms  of  sin  n  and  cos  n,  and  thus  get 
the  law  of  apparition  of  primes  in  the  recurring  series  of  the  it„  [stated 
explicitly  in  Lucas"*^],  given  by  Fermat  when  5  is  rational  and  by  Lagrange 
when  5  is  irrational.  The  developments  of  uj'  and  v^^  as  linear  functions  of 
""nj  '^2n,  ■  ■  ■  are  like  the  formulas  of  de  Moivre  and  Bernoulli  for  sin^x  and 
cos^x  in  terms  of  sin  kx,  cos  kx.    Thus — 

(V)  If  n  is  the  rank  of  the  first  term  u^  containing  the  prime  factor  p 
to  the  power  X,  then  Uj^  is  the  first  term  divisible  by  p^"*"^  and  not  by  p^"*"^; 
this  is  called  the  law  of  repetition  of  primes  in  the  recurring  series  of  w„. 

(VI)  If  p  is  a  prime  4g+l  or  4g+3,  the  divisors  of  u^Ju^  are  divisors 
of  x^—py^  or  S^x^+pi/^,  respectively. 

(VII)  If  Up^i  is  divisible  by  p,  but  no  term  of  rank  a  divisor  of  p=*=l  is 
divisible  by  p,  then  p  is  a  prime. 

Lucas^*^  proved  the  theorems  stated  in  the  preceding  paper.  Theorems 
II  and  IV  follow  from  (I2)  and  (22),  while  (20  shows  that  every  factor 
common  to  w^^.„  and  u^  divides  Un  and  conversely. 

(VIII)  If  a  and  h  are  irrational,  but  real,  t^p+i  or  Wp_i  is  divisible  by  the 
prime  p,  according  as  6^  is  a  quadratic  non-residue  or  residue  of  p  (law  of 
apparition  of  primes  in  the  it's).  If  a  and  h  are  integers,  Up^i  is  divisible 
by  p.  Hence  the  proper  divisors  of  u^  are  of  the  form  /en +1  if  6  is  rational, 
/cn=*=l  if  5  is  irrational. 

"Comptes  Rendus  Paris,  82,  1876,  pp.  1303-5. 

"Sur  la  thdorie  des  nombres  premiers,  Atti  R.  Accad.  Sc.  Torino  (Math.),  11,  1875-6,  928-937. 


Chap.  XVII]  RECURRING   SERIES ;   LuCAS'   Un,   V^.  397 

The  law  V  of  repetition  of  primes  follows  from 

where  t  =  (p  —  l)/2.  Special  cases  of  the  law  are  due  to  Arndt,^^  p.  260, 
and  Sancery,^^  each  quoted  in  Ch.  VII.  Theorem  VII,  which  follows  from 
VIII,  gives  a  test  for  the  primality  of  2"±  1  which  rests  on  the  success  of 
the  operation,  whereas  Euler's  test  for  2^^  — 1  was  based  on  the  failure  of 
the  operation.  The  work  to  prove  that  2^^  —  1  is  prime  is  given,  and  it  is 
stated  that  2^''  — 1  was  tested  and  found  composite, ^^  contrary  to  Mersenne. 
Finally,  a^+Qy^  is  shown  to  have  an  infinitude  of  prime  divisors. 

A.  Genocchi^^  noted  that  Lucas'  w„,  y„  are  analogous  to  his^^  5„,  A^. 
[If  we  set  a  =  a+\/b,  ^  =  a  —  \/h,  we  have 

a — p 

Lucas^^  stated  that,  if  4m+3  is  prime,  p  =  2*'"+^  — 1  is  prime  if  the  first 
term  of  the  series  3,  7,  47,. .  .,  defined  by  r„_,.i  =  r„^  — 2,  which  is  divisible 
by  p  is  of  rank  4m+2;  but  p  is  composite  if  no  one  of  the  first  4m +2  terms 
is  divisible  by  p.  Finally,  if  a  is  the  rank  of  the  first  term  divisible  by  p, 
the  divisors  of  p  are  of  the  form  2"A;=t:l,  together  with  the  divisors  of 
x^  —  2y^.     There  are  analogous  tests  by  recurring  series  for  the  primality  of 

3.24m+3_i^  2-3*™+2±i^  2-3*'"+^-l,  2-52'"+'  +  l. 

Lucas^^  proposed  as  an  exercise  the  determination  of  the  last  digit  in  the 
general  term  of  the  series  of  Pisano  and  for  the  series  defined  by  w„+2 
=  aUn+i-\-bUn',  also  the  proof  of  VIII:  If  p  is  a  prime, 

(a-\-Vby-^-(a-Vby-'^ 


Up^l—- 


Vb 


is  divisible  by  p  if  6  is  a  quadratic  residue  of  p,  excepting  values  of  a  for 
which  a^  — 6  is  divisible  by  p;  and  the  corresponding  result  [of  Lagrange^  and 
Gauss^]  for  Up+i.  Moret-Blanc^^  gave  a  proof  by  use  of  the  binomial  theorem 
and  omission  of  multiples  of  p. 

Lucas^^  wrote  s„  for  the  sum  of  the  nth  powers  of  the  roots  of  an 
equation  whose  coefficients  are  integers,  the  leading  one  being  unity. 
Then  Snp  —  sJ'  is  an  integral  multiple  of  p.  Take  n  =  1.  Then  Si  =  0  impHes 
Sp=  0  (mod  p) .  It  is  stated  that  if  Si  =  0  and  if  Sj,  is  divisible  by  p  for  k  =  p, 
but  not  for  k<p,  then  p  is  a  prime. 

*iA.  Cunningham,  Proc.  Lond.  Math.  Soc,  27,  1895-6,  54,  remarked  that,  while  primality  is 
proved  by  Lucas'  process  by  the  success  of  the  procedure,  his  verification  that  a  number 
is  composite  is  indirect  and  proved  by  the  failure  of  the  process  and  hence  is  liable  to  error. 

22Atti.  R.  Accad.  Sc.  Torino,  11,  1875-6,  924. 

^Comptes  Rendus  Paris,  83,  1876,  1286-8. 

2*Nouv.  Ann.  Math.,  (2),  15,  1876,  82. 

^mid.,  (2),  20,  1881,  258  [p.  263,  for  primality  of  2"-l]. 

28Assoc.  frang.  avanc.  sc,  5,  1876,  61-67.     Cf.  Lucas^*. 


398  History  of  the  Theory  of  Numbers.  (Chap,  xvii 

By  use  of  (1)  and  (2),  theorems  I-IV  are  proved.  Theorem  VIII  is 
stated,  and  VII  is  proved.  Employing  two  diagrams  and  working  to  base  2, 
he  showed  that  2"^^  —  1  is  a  prime. 

Lucas"  considered  a  product  m  =  p^i* ...  of  powers  of  primes,  no  one 
dividing  Q.     Set  A  =  (a-6)^  (A/p)=0,  ^\=^^r>-l)/2  ^^^^  ^^^ 


[51? 


,W=."-V-...[p-g)][.-(f)] 


Then  w<=0  (mod  m)  for  f =^(w).  The  ranks  n  of  terms  u^  divisible  by  m 
are  multiples  of  a  certain  divisor /z  o{\p{m).  This  ii  is  the  exponent  to  which 
a  or  6  belongs  modulo  vi.  The  case  6  =  1  gives  Euler's  generalization  of 
Fermat's  theorem.  The  primality  test^^  is  reproduced  and  applied  to  show 
that  2^^  — 1  is  a  prime. 

Lucas^^  considered  the  series  of  Pisano.  Taking  a,  6  =  (l=tv5)/2,  we 
have  Ml  =1*2  =  1,  1*3  =  2,  etc.  According  as  n  is  odd  or  even  the  divisors  of 
u-iJun  are  divisors  of  5x^  —  3?/^  or  5a:^+32/";  those  of  u^Ju2n  are  divisors  of 
5x^  — 2zf  or  5x^'-f-2?/^;  those  of  v^Jvn  are  divisors  of  x^+Sy^  or  x^  —  3y^;  those 
of  V2n  are  divisors  of  x^+22/^  or  x^  —  2y^;  those  of  u^JUn  are  divisors  of  x^+Sy^ 
or  T'  —  hy^.  The  law  V  of  repetition  of  primes  and  theorem  III  are  stated. 
The  law  VIII  of  apparition  of  primes  now  takes  the  following  form:  If  p  is  a 
'  prime  10g=t  1,  Wp_iis  divisible  by  p ;  if  p  is  a  prime  lOg^  3,  i/p+i  is  divisible  by  p. 
The  test^^  for  the  primality  of  A  is  given  and  applied  to  show  that  2^^^  —  1 
and  2^^  — 1  are  primes.     There  is  a  table  of  prime  factors  of  u^  for  n^60. 

L Finally,  ^u^ju^  is  expressible  in  the  form  x^  —  'py^  or  bx^+py"^  according  as 
the  prime  p  is  of  the  form  45-+!  or  40^+3. 

Lucas^^  considered  the  series  defined  by  r„+i  =  r„^  — 2, 

Let  A  =  3or9  (mod  10),  g=0  (mod  4) ;  or  A  =  7,  9  (mod  10),  g=l(  mod  4); 
or  A=l,  7  (mod  10),  g=2  (mod  4);  or  A=l,  3  (mod  10),  q=S  (mod  4). 
Then  p  =  2'A  —  1  is  a  prime  if  the  rank  of  the  first  term  divisible  by  p  is  5 ; 
if  a  {a<q)  is  the  rank  of  the  first  term  divisible  by  p,  the  divisors  of  p  are 
either  of  the  form*  2aAk-{-l,  or  of  the  forms  of  the  divisors  of  x^  —  2y^ 
and  x^  —  2Ay'^.  Corresponding  tests  are  given  for  2^ A  -\- 1  and  S'^A  —  1.  The 
first  part  of  the  theorem  of  Pepin^'^  for  testing  the  primaUty  of  a„  =  2^"+l 
follows  from  theorem  VII  with  a  =  5,  5  =  1,  p  =  a„;  the  second  part  follows 
from  the  reciprocity  theorem  and  the  form  of  a„  —  1 . 

For  A=p,  let  the  above  rj  become  r.  When  p=7  or  9  (mod  10)  and 
p  is  a  prime,  then  2p  —  1  is  a  prime  if  and  only  if  r=0  (mod  2p  —  1).  When 
p  =  4g+3  is  a  prime,  2pH-l  is  a  prime  if  and  only  if  2''=1  (mod  2p  +  l). 
When  p  =  4^-4- 3  is  a  prime,  2p  — 1  is  a  prime  if  and  only  if 

*'Comptes  Rendus  Paris,  84,  1877,  439-442.     Corrected  by  Carmichael.*' 

"Bull.  Bibl.  Storia  Sc.  Mat.  e  Fis.,  10,  1877,  129-170.     Reprinted  as  "  Recherches  sur  plusieura 

ouvrages  de  Leonard  de  Pise."     Cf .  von  Sterneck"  of  Ch.  XIX. 
"Assoc,  frang.  avanc.  sc,  6,  1877,  1.59-166.     *Corrected  to  2MA'=tl  in  Lucas";  see  Lucas." 
'"KUomptes  Rendus  Paris,  85,  1877,  329-331.    See  Ch.  XV,  Pepin",  Lucas,"'  "  Proth.^^ 


Chap.  XVII]  RECURRING   SERIES ;   LuCAS'   W„,    V^.  399 

:^{a+V2y-a-V2r}^o  (mod2p-i). 

To_test  the  primality  of  p  =  2^^^^  —  l,  use  x^  —  4x-{-l=0  with  the  roots 
2±  Vs.  Then  if  p  is  a  prime,  Wp+i  is  divisible  by  p.  We  use  the  residues 
of  the  series  2,  7,  97, .  .  .  defined  by  r„+i  =  2r„^  — 1. 

Lucas^^  stated  that  p  =  2'^'""*"^  —  1  is  a  prime  if  the  rank  of  the  first  term^^ 
of  3,  7, 47, .  .  .  divisible  by  p  is  between  2m  and  4m +2.  To  test  P  =  2^«+^  - 1, 
form  the  series 

ri  =  l,         r2=-l,        rs=-7,        r,  =  17,...,        r„+i  =  2r„2-32"-'; 

if  ^  is  the  least  integer  for  which  r^  is  divisible  by  P,  then  P  is  a  prime  when  I 
is  comprised  between  2g  and  4g+l,  composite  when  ?>4g'  +  l. 

Lucas^^  expressed  Un,  v^  as  polynomials  in  P  and  A  =  P^  —  4Q  =  5^,  obtained 
various  relations  between  them  corresponding  to  relations  between  sine  and 
cosine;  in  particular, 

Wn+2  —  P'^n+l  ~  Q'^ni  '^n+2r  —  ^r'^n+r  ~  Q  ^n> 

and  formulas  derived  from  them  by  replacing  uhy  v;  also  symbolic  formulas 
generalizing  those^^  for  the  series  of  Pisano. 

In  the  second  paper,  Un+i,  v^  are  expressed  as  determinants  of  order  n 
whose  elements  are  Q,  —  P,  2,  1,  0.  There  is  given  a  continued  fraction  for 
U(n+i)r/unr,  hoTCi  whlch  Is  derfved  (I2)  and  generalizations.  The  same 
fraction  is  developed  into  a  series  of  fractions. 

Lucas^^  noted  that  u^r  is  divisible  by  Ur  since 

where  ^  =  Jn  —  1  if  n  is  even,  f  =  J(n  —  1)  if  n  is  odd,  the  final  factor  being  then 
absent.  Proof  is  given  for  (2i)  and  2y^+„  =  ?;^y„+AM„w^.  From  these  are 
derived  new  formulas  by  changing  the  sign  of  n  and  applying 

To  show  that 

[m,  n\  = — 

is  integral,  apply  (2i)  repeatedly  to  get 

2[m,  n\  =  [m  —  l,  n]Vn+{m,  n  —  \]v^. 

Finally,  sums  of  squares  of  functions  Un,  v^  are  found. 

Lucas^^  gave  a  table  of  the  linear  forms  4A+?'  of  the  odd  divisors  of 
x^-\-Llf  and  x^—I\'if'  for  A  =  l,.  .  .,  30.  By  use  of  (I2),  it  is  shown  that  the 
terms  of  odd  rank  in  the  series  u^  are  divisors  of  x^  —  Qif' ;  the  terms  of  even  or 
odd  rank  in  the  series  v^  are  divisors  of  x^+A?/^  or  x^-\-QAy^,  respectively. 

3iMessenger  Math.,  7,  1877-8,  186. 

'^Sur  la  theorie  des  fonctions  numeriques  simplement  p^riodiques,  Nouv.  Corresp.  Math.,  3, 
1877,  369-376,  401-7.     These  and  the  following  five  papers  were  reproduced  by  Lucas.'* 
^Hhid.,  4,  1878,  1-8,  continuation  of  preceding. 
^'lUd.,  pp.  33-40. 


400  History  of  the  Theory  of  Numbers.  [Chap,  xvii 

Lucas^^  proved  III  by  use  of  (2i)  and  gave 

Lucas^^  determined  the  quadratic  forms  of  divisors  of  i'2n  from 
t;2n=Aw„2+2Q",  V2n  =  vJ'-2Q\ 

In  the  last,  take  Q  =  2f,  n  =  2/1+1;  thus  1^4^+2  factors  if  Q  is  the  double  of  a 
square.    As  a  special  case  we  have  the  result  by  H.  LeLasseur  (p.  86) : 

In  the  first  expression  for  t'2n,   take  n=juH-l,  A  =  ±2/i^,  Q==r^ff^;  thus 
Vi^+2  factors  when  QA  is  of  the  form  —  2f.     Similarly,  v^^  factors  if  A  =  —2f. 
Lucas"  gave  the  formulas 

developments  of  w„^,  t'/  as  linear  functions  of  Vkn,  k  =  p,  p  —  2,  p  —  i,.  . .,  and 
complicated  developments  of  w„r,  ^nr- 

Lucas^^  reproduced  the  preceding  series  of  seven  papers,  added  (p.  228) 
a  theorem  on  the  expression  of  4Upr/ur  as  a  quadratic  form,  a  proof  (p.  231) 
of  his^^  test  for  primality  by  use  of  the  s^,  and  results  on  primes  and  perfect 
numbers  cited  elsewhere. 

Lucas^^  considered  series  w„  of  the  first  kind  (in  which  the  roots  a,  h  are 
relatively  prime  integers)  and  deduced  Fermat's  theorem  and  the  analogue 
it<=0  (mod  m),  t  =  4>(m),  of  Euler's  generalization.  Proof  is  given  of  the 
earlier  theorems  VII,  VIII  and  (p.  300)  of  his"  generalization  of  the  Euler- 
Fermat  theorem.  The  primality  test^^  is  stated  (p.  305)  and  applied  to 
show  that  2^^  — 1  and  2^^  — 1  are  primes.  It  is  stated  (page  309)  that 
p  =  2'**+^  —  1  is  prime  if  and  only  if 

3=2coS7r/22«+i  (mod  p), 

after  rationalizing  with  respect  to  the  radicals  in  the  value  of  the  cosine. 
The  primality  tests^^  are  given  (page  310),  with  similar  ones  for  3'yl  +  l, 
2-5'A  +  l.  The  tests^^  for  the  primaUty  of  2p  +  l  are  given  (p.  314).  The 
primality  test^^  for  2'^'''^^  — 1  is  proved  (pp.  315-6). 

Lucas"*"  reproduced  his^^  earlier  results,  and  for  p  =  3,  5,  7,  11,  13,  17, 
expressed  ypr/i'2r  in  the  form  x^— 2pQV>  and,  for  p  a  prime  ^31,  expressed 

»»Nouv.  Corresp.  Math.,  4,  1878,  65-71. 

"Ihid.,  pp.  97-102. 

*Ubid.,  pp.  129-134,  225-8. 

»«Amer.  Jour.  Math.,  1,  1878,  184-220.     Errors  noted  by  Carmichael." 

"Ibid.,  pp.  289-321. 

"Atti  R.  Accad.  Sc.  Torino,  13,  1877-8,  271-284. 


Chap.  XVII]  RECURRING   SeRIES;  LucAS'  U„,   y„.  401 

Upr/Ur  in  the  form  Ax^^pQ''y'^.  The  prime  factors  of  3^^±1  are  given  on 
p.  280.  The  proper  divisors  of  2^"+l  are  known  to  be  of  the  form  Snq-\-l; 
it  is  shown  that  q  is  even.  Thus  for  2^^+l  the  first  divisor  to  be  tried  is 
641,  for  2^^^+!  the  first  one  is  114689;  in  each  case  the  division  is  exact 
(cf.  Ch.  XV).  The  following  is  a  generalization:  If  the  product  of  two  I 
relatively  prime  integers  a  and  h  is  of  the  form  4/i+l,  the  proper  divisors 
of  a^'^^^+b^"^'^  are  of  the  form  Sahnq+1.  A  primality  test  for  2'^^+^-!  isj 
given.     Finally,  p  =  2^"^^+^"+^  —  1   is  a  prime  if  and  only  if 

(2"+V2^^)^+(2"-V2^H4)^  =  0  (mod  p). 
T.  Pepin*  ^  gave  a  test  for  the  primality  of  g  =  2"  — 1.     Let 

^1-    ^2-1-52     (modg') 
and  form  the  series  Ui,  U2y. . .,  it„_i  by  use  of 

u^+i=u^  —  2  (mod  q). 

Then  g  is  a  prime  if  and  only  if  u^-i  is  divisible  by  q.  This  test  differs  from 
that  by  Lucas^^  in  the  choice  of  Ui. 

E.  Lucas"*^  reproduced  his^^  test  for  the  primality  of  2^ A  —  1,  etc.,  and  the 
test  at  the  end  of  another  paper,*"  with  similar  tests  for  2*'^+^  —  1  and  2^^*'+^  —  1 . 

G.  de  Longchamps*^  noted  that,  if  dk  =  Uk  —  aUk-i, 

d,  =  h''-\  d^d,  =  ¥+^-\ 

with  the  generalization 

X 

Jldj,.=d„    s  =  pi+. .  .+p^-x+l. 
Take  pi  =  . . .  =  p^;  =  p.     Hence 

[Up      CLUp^i)    =Upx—x+l       f^Upx—x' 

There  is  a  corresponding  theorem  for  the  v's. 

J.  J.  Sylvester**  considered  the  g.  c.  d.  of  u^,  u^+i  if 

w^  =  {2x  -  l)Ux-i  -  {x-l)u^_2' 

E.  Gelin*^ stated  and  E.  Cesaro*^  proved  by  use  of  ?7„+p=  UpUn+  C/p_iC/„_i 
that,  in  the  series  of  Pisano,  the  product  of  the  means  of  four  consecutive 
terms  differs  from  the  product  of  the  extremes  by  ±  1 ;  the  fourth  power  of 
the  middle  term  of  five  consecutive  terms  differs  from  the  product  of  the 
other  four  terms  by  unity. 

"Comptes  Rendus  Paris,  86,  1878,  307-310. 

«Bull.  Bibl.  Storia  Sc.  Mat.  e  Fis.,  11, 1878,  783-798.     The  further  results  are  cited  in  Ch.  XVI. 

Comptes  Rendus,  90,  1880,  855-6,  reprinted  in  Sphinx-Oedipe,  5,  1910,  60-1. 
«Nouv.  Corresp.  Math.,  4,  1878,  85;  errata,  p.  128. 
«Comptes  Rendus  Paris,  88,  1879,  1297;  Coll.  Papers,  3,  252. 
«Nouv.  Corresp.  Math.,  6,  1880,  384. 
"Ibid.,  423-4. 


402  History  of  the  Theory  of  Numbers.  [Chap,  xvii 

Magnon,'*^  in  reply  to  Lucas,  proved  that 

)  Oi     Oj     Og      ■  ■  ■      1  —  V5 

I    if  a„  —  1  is  the  sum  of  the  squares  of  the  first  n  —  1  terms  of  Pisano's  series. 

H.  Brocard^^  studied  the  arithmetical  properties  of  the  U's  defined  by 
Un+i  =  Un-\-2U„_i,  Uo  =  l,  Ui=3,  in  connection  with  the  nth  pedal  triangle. 

E.  Ces^ro^^  noted  that  if  Un  is  the  nth  term  of  Pisano's  series,  then 
(2C/+l)"-C/^'*  =  0,  symbolically. 

E.  Lucas^"  gave  his^^  test  for  the  primahty  of  2'*^+^  —  !. 

A.  Genocchi"  reproduced  his^^  results. 

M.  d'Ocagne^^  proved  for  Pisano's  series  that  [Lucas^®] 

iu,  =  u^2-h        2)(-iyii,=  (-l)X-i-l,  lim-^=(i±^', 

»=0  P=«  (*p-i        \        Z         / 

The  main  problem  treated  is  that  to  insert  p  terms  ai, .  .  .,  ap  between  two 
given  numbers  ao  =  a,  ap+i  =  h,  such  that  aj  =  a;_i+a,_2.     The  solution  is 

hUi-{-{  —  iyaUp+i.i 

Up+i 

Most  of  the  paper  is  devoted  to  the  question  of  the  maximum  number  of 
negative  terms  in  the  series  of  a's. 

E.  Catalan^2apj.Q^3^  ^^^^  uji^Ur,-pU,+p=  (-iy-'^^U\_i  for  Pisano's 
series. 

E.  Lucas^^  stated,  apropos  of  sums  of  squares,  that 

L.  Kronecker^^  obtained  Dirichlet's^  theorems  by  use  of  modular  systems. 
Lucas^"  proved  that,  if  w„=(a''— 5")/(a  — 6), 

Wp_l  —U(p_i)n/Un 

is  divisible  by  Up  when  p  is  a  prime  and  n  is  odd  and  not  divisible  by  p,  and 
by  Up  when  n  =  2p+l. 

L.  Liebetruth^^  considered  the  series  Pi  =  1 ,  P2  =  x, .  .  . ,  P^  =  ^Pn-i  —Pn-2) 
and  proved  any  two  consecutive  terms  are  relatively  prime,  and 

Pn  =  PxPn-X+l-Px-lPn-X  (X<n). 

Taking  n  =  2X,  3X, . .  . ,  we  see  that  Px  is  a  common  factor  of  P2X,  Psx,  •  •  •  • 
The  g.  c.  d.  of  P,„,  P„  is  Pj,  where  d  is  the  g.  c.  d.  of  ?n,  n.     Next, 

«'Nouv.  Corresp.  Math.,  6,  1880,  418-420.        '"Nouv.  Corresp.  Math.,  6,  1880,  145-151. 
"/bid.,  528;  Nouv.  Ann.  Math.,  (3),  2,  1883,  192;    (3),  3,  1884,  533.     Jornal  de  Sc.  Math. 

Astr.,  6,  1885,  17. 
»0R6cr6ation8  mathdmatiques,  2,  1883,  230.       "Coraptes  Rendus  Paris,  98,  1884,  411-3. 
"'Bull.  Soc.  Math.  France,  14,  1885-6,  20-41. 

^^M6m.  soc.  roy.  sc.  Li^ge,  (2),  13,  1886,  319-21  (  =  M61anges  Math.,  II). 
"Mathesis,  7,  1887,  207;  proofs,  9,  1889,  234-5. 

"Berlin  Berichte,  1888,  417-423;  Werke,  3,  I,  281-292.     Cf.  Kronecker'"  of  Ch.  XVI. 
""Assoc.  franQ.  avanc.  sc,  1888,  II,  30.  "Beitrag  zur  Zahlentheorie,  Progr.,  Zerbst,  1888. 


Chap.  XVII]  Recuering  Series;  Lucas'  u^,  y„.  403 

P1+P3+  ■  .  •  +P2n-X=Pn\  P2  +  -P4+  •  •  •  +P2n  =  PnPn+l. 

If  P^=P^  (mod  Px)  then  n=m  (mod  2X).    Also, 

P„=x"-+  S  (_i).(n-fe-l)...(n-2fc)^„.,,_,_ 
fc=i  1-2.  .  .A; 

If  a„/6„  is  the  nth  convergent  to  i_j+^_...,   then     a„+2  =  ^^n+i~cf„,  6„ 
=  o„+i.     Hence  a^^Pn  if  cti  =  1 ,  02  =  a;. 

Sylvester  stated  and  W.  S.  Foster^^"  proved  that  if  f{d)  is  a  polynomial 
with  integral  coefficients  and  Ux+i=f{Uj^,  Ui=f{0),  and  5  is  the  g.  c.  d.  of 
r,  s,  then  Us  is  the  g.  c,  d.  of  Ur,  Ug. 

A.  Schonfiies^®  considered  the  numbers  no  =  l,  Ui,...,  n^  defined  by 
n^  =  n^_n^-i+ri^-2_  .  .  .  +(-1)^         (X  =  0,  1,. . .) 

and  proved  geometrically  that  if  n^-i  is  the  least  of  these  numbers  which 
has  a  common  factor  with  n^,  then  r  is  a  divisor  of  g+1,  while  a  relation 

'mni=mnr+i  (mod  n^) 
holds  for  every  index  i. 

L.  Gegenbauer^^  gave  a  purely  arithmetical  proof  of  this  theorem. 

E.  Lucas^^  gave  an  exposition  of  his  theory,  with  an  introduction  to 
recurring  series. 

M.  Frolov^^  used  a  table  of  quadratic  residues  of  composite  numbers  to 
factor  Lucas'  numbers  v„. 

D.  F,  Seliwanov^°  proved  Lucas'  results  on  the  factors  of  w„,  ?;„. 

E.  Catalan^^  gave  the  first  43  terms  of  the  series  of  Pisano,  noted  that 
Un  divides  U2y,+i,  that  Uzn  is  a  sum  of  two  squares,  and  treated  the  series 

Un  =  aun-i + w„-2,  Ux  =  a,  W2  =  a^ + L 

Fontes^^"  proved  theorems  stated  by  Lucas^^  (p.  127),  and  found  in  an 
elementary  way  the  general  term  of  Pisano's  series,  as  given  by  Binet^^. 
E.  Maillet^^''  proved  that  a  necessary  condition  that  every  positive 
integer,  exceeding  a  certain  limit,  shall  equal  (up  to  a  limited  number  of 
units)  the  sum  of  the  absolute  values  of  a  finite  number  of  terms  of  a  recur- 
ring series,  satisfying  an  irreducible  law  of  recurrence  with  integral  coeffi- 
cients, is  that  all  the  roots  of  the  corresponding  generating  equation  be  roots 
of  unity. 

W.  ManteP^  noted  that,  if  the  denominator  F{x)  of  the  generating 
fraction  of  a  recurring  series  is  irreducible  modulo  p,  a  prime,  the  residues 
modulo  p  of  the  terms  of  the  recurring  series  repeat  periodically,  and  the 
length  of  a  period  is  at  most  p"  —  1 ;  the  proof  is  by  use  of  Galois'  general- 
ization of  Fermat's  theorem.     The  case  of  a  reducible  F{x)  is  also  treated. 

65<iMath.  Quest.  Educ.  Times,  50,  1889,  54-5.  ^«Math.  Annalen,  35,  1890,  537. 

"Denkschriften  Ak.  Wiss.  Wien  (Math.),  57,  1890,  528. 

68Theorie  des  nombres,  1891,  299-336;  30;  127,  ex.  1.     A   pamphlet,   pubhshed   privately   by 

Lucas  in  1891,  is  cited  in  I'intermediaire  des  math.,  5,  1898,  58. 
"Assoc,  frang.  avanc.  sc,  21,  1892,  149. 
soMath.  Soc.  Moscow,  16,  1892,  469-482  (in  Russian). 
"Mem.  Acad.  R.  Belgique,  45,  1883;  52,  1893-4,  11-14. 

""Assoc,  frang.  avanc.  sc,  1894,  II,  217-221.  "''Assoc,  frang.  avanc.  sc,  1896,  II,  78-89 

•^Nieuw  Archief  voor  Wiskunde,  Amsterdam,  1,  1895,  172-184. 


404  History  of  the  Theory  of  Numbers.  [Chap,  xvii 

R.  W.  D.  Christie^  stated  that,  for  the  recurring  series  defined  by 
a„+i  =  3a„  — a„_i,  27n  —  l  is  a  prime  if  and  only  if  a^  —  1  is  di\'isible  by 
2m  — 1.    The  error  of  this  test  was  pointed  out  by  E.  B.  Escott.^ 

S.  R^alis""  noted  that  two  of  A''  consecutive  terms  of  7,  13,  25, . . ., 
3(n^H-n)+7, .  .  .  are  di\'isible  by  iV  if  iV  is  a  prime  6/^2  + 1. 

C.  E.  Bickmore^**  discussed  factors  of  w„  in  the  final  series  of  Catalan^\ 
He^"  and  others  gave  known  formulas  and  properties  of  Pisano's  series. 

R.  Perrin®^  employed  r„  =  r„_2  +  r„_3,  ro  =  3,  ^'l  =  0,  ?'2  =  2.  Then  v^  is 
di\'isible  by  n  if  n  is  a  prime.  This  was  verified  to  be  not  true  when  n  is 
composite  for  a  wide  range  of  values  of  n.  The  same  subject  was  considered 
by  E.  jMalo^®  and  E.  B.  Escott®"  who  noted  that  Perrin's  test  is  incomplete. 

SeveraP^"  discussed  the  computation  of  Pisano's  w„  for  large  n's. 

E.  B.  Escott"^  computed  Sl/ti^.  E.  Landau^"'  had  evaluated  Sl/wgA 
in  terms  of  the  sum  of  Lambert's^  series  of  Ch.  X,  and  "Zl/uoh+i  in  relation 
to  theta  series. 

A.  Tagiuri^^  employed  the  series  Wi  =  l,  U2  =  l,  Us  =  2,...  of  Leonardo 
and  the  generalization  Ui,  U2,...,  where  t/„=  t/„_i  +  i[7„_2,  with  Ui  =  a, 
U2  =  b  both  arbitrary.     Writing  e  for  a^+ab  —  h^,  it  is  proved  that 

UJJ-Ur^.^U,^,  =  ( - 1)"- V^,+,_„e. 

{C/„4.a+(  — l)*?7„_j)/?7„  is  an  integer  independent  of  a,  h,  n;  it  equals 
Wj+i+Wj-i.  It  is  shown  that  u^  is  a  multiple  of  u^  if  and  only  if  r  is  a 
multiple  of  s. 

Tagiuri^^  obtained  analogous  results  for  the  series  defined  by  Vn  =  hVn-i 
-\-lVn-2,  and  the  particular  series  r„  obtained  by  taking  ri  =  l,  V2  =  h.  If 
h  and  I  are  relatively  prime,  v^  is  a  multiple  of  v^  if  and  only  if  r  is  a  multiple 
of  s.  Let  ^{i\)  be  the  number  of  terms  of  the  series  of  f's  which  are  ^  r,  and 
prime  to  it;  if  h>l,  <i>(t',)  is  Euler's  (f>{i);  but,  if  h  =  l,  <l>(r,)  =0(^)+0(^y2), 
the  last  term  being  zero  if  i  is  odd.  If  i  and  j  are  relatively  prime,  $(y,y) 
=*(j'.)$(r;). 

Tagiuri^"  proved  that,  for  his  series  of  v's,  the  terms  between  v^p  and 
v^p+i)  are  incongruent  modulo  t'^.  if  h>l,  and  for  /i  =  1  except  for  Vkp+i=Vkp+2- 
If  /x  is  not  divisible  by  k  and  e  is  the  least  solution  of  /-**=!  (mod  f*),  then 
Vj.=  v^  (mod  t';t)  if  a:=Ai  (mod  4A:e). 

If  /x  is  not  divisible  by  k,  and  k  is  odd,  and  €1  is  the  least  positive  solution  of 
P=l  (mod  Vk),  then  Vx=v^  (mod  ft)  if  x=ij,  (mod  2kei). 
A.  Emmerich^^  proved  that,  in  the  series  of  Pisano, 

"Nature,  56,  1897,  10.  «Math.  Quest.  Educat.  Times,  3,  1903,  46;  4,  1903,  52 

""Math.  Quest.  Educat.  Times,  66,  1897,  82-3;  cf.  72,  1900,  40,  71. 

"*/6m/.,  71,    1899,49-50.  "«/6ui.,  Ill;  4,  1903,  107-8;  9,  1906,  55-7. 

«L'interm6diaire  des  math.,  6,  1899,  76-7. 

*»Ibid.,  7,  1900,  281,  312.  "L'interm^diaire  des  math.,  8,  1901,  63-64. 

•'"/Wd.,  7,  1900,  172-7.  ^'>>Ibid.,  9,  1902,  43-4. 

•■<^BuU.  Soc.  Math.  France,  27, 1899, 198-300.  «'Periodico  di  Mat.,  16,  1901,  1-12. 

"Peridico  di  Mat.,  97-114.  "/bid.,  17,  1902,  77-88,  119-127. 

"Mathesis,  (3),  1,  1901,  98-9. 


Chap.  XVII]  Recuering  Series;  Lucas'  t/„,  v^.  405 

Un+5=Un  (mod  2),  w„+5=3w„  (mod  5),  ^n+6o=Wn  (mod  10), 

so  that  Uq,  Us,  Uq,  Ug,. . .  alone  are  even,  Uq,  %,  u^q, . . .  are  multiples  of  5. 

J.  Wasteels'^^  proved  that  two  positive  integers  x,  y,  for  which  y^—xy—x^ 
equals  +1  or  — 1,  are  consecutive  terms  of  the  series  of  Pisano.  If  5x^±4 
is  a  square,  a;  is  a  term  of  the  series  of  Pisano.  These  are  converses  of 
theorems  by  Lucas.  ^^ 

G.  Candido^^  treated  Un,  v^,  by  algebra  and  function- theory. 

E.  B.  Escott^^  proved  the  last  result  in  Lucas'  paper. ^° 

A.  Arista'^^  expressed  S"if it~^  in  finite  form. 

M.  Cipolla'^^  gave  extensive  references  and  a  collection  of  known  formulas 
and  theorems  on  w„,  v„.  His  apphcation  to  binomial  congruences  is  given 
under  that  topic. 

G.  Candido^^  gave  the  necessary  and  sufficient  conditions,  involving 
the  i/j,  that  a  polynomial  x  has  the  factor  x^  —  Px-\-Q,  whose  roots  are  a,  b. 

A.  Laparewicz'^^  treated  the  factoring  of  2'"='=  1  by  Lucas'  method.^^ 

E.  B.  Escott^^"  showed  the  connection  between  Pisano's  series  and  the 
puzzle  to  convert  a  square  into  a  rectangle  with  one  more  (or  fewer)  units    • 
of  area  than  the  square. 

E.  B.  Escott^^  applied  Lucas'  theory  to  the  case  it„  =  2tt„_i-f-w„_2. 

L.  E.  Dickson'^^"  proved  that  if  Zj,  is  the  sum  of  the  kth  powers  of  the  roots 
of  a"'-]-pid^~^+  . . .  +Pm  =  0,  where  the  p's  are  integers  and  pi  =  0,  then,  in 
the  series  defined  by  Zx+m-\-piZx+m-i+  •  •  •  +PmZx  =  0,  Zt  is  divisible  hy  i\it  is 
a  prime. 

E.  Landau^"  proved  theorems  on  the  divisors  of  V^,  Y^,  where 

{x^iT=  VJx)^iYSx),  i=  \/^. 
P.  Bachmann^^  treated  at  length  recurring  series. 
C.  Ruggieri^^  used  Pisano's  series  for  w_„  to  solve  for  ^  and  r\ 

E.  Zeuthen^^  proposed  a  problem  on  the  series  of  Pisano.  ^ 

H.  Mathieu^^  noted  that  in  1,  3,  8, . . .,  x,i+i  =  3a:„— a:„_i,  the  expressions     | 
a^n^n+i  +  1,  Xn-iXn+i-\-l  are  squares.  — 

Valroff^^  stated  in  imperfect  form  theorems  of  Lucas. 
A.  Aubry^^  gave  a  summary  of  results  by  Genocchi^^  and  Lucas. 

"Mathesis,  (3),  2,  1902,  60-62. 

"Periodico  di  Mat.,  17,  1902,  320-5;  I'interm^diaire  des  math.,  23,  1916,  175-6. 
T*L'mterm6diaire  des  math.,  10,  1903,  288.  "Giornale  di  Mat.,  42,  1904,  186-196. 

"Rendiconto  Ac.  Sc.  Fis.  e  Mat.  Napoh,  (3),  10,  1904,  135-150. 
"Periodico  di  Mat.,  20,  1905,  281-285. 

"Wiadomosci  Matematyczne,  Warsaw,  11,  1907,  247-256  (Polish). 

78oThe  Open  Court,  August,  1907.     Reproduced  by  W.  F.  White,  A  Scrap-Book  of  Elementary 
Mathematics,  Notes,  Recreations,  Essays,  The  Open  Court  Co.,  Chicago,  1908,  109-113. 
"L'interm6diaire  des  math.,  15,  1908,  248-9.  "OAmer.  Math.  Monthly,  15,  1908,  209. 

soHandbuch.  .  .Verteilung  der  Primzahlen,  I,  1909,  442-5. 

"Niedere  Zahlentheorie,  II,  1910,  55-96,  124.  s^pgrjodico  di  Mat.,  25,  1910,  266-276. 

"Njrt  Tidsskr.  for  Math.,  Kjobenhavn,  A  22,  1911,  1-9.     Solution  by  Fransen  and  Damm. 
»*L'interm6diaire  des  math.,  18,  1911,  222;  19,  1912,  87-90;  23,  1916,  14  (generalizations). 
^Ihid.,  19,  1912,  145,  212,  285.  "L'enseignement  math.,  15,  1913,  217-224 


406  History  of  the  Theory  of  Numbers.  [Chap,  xvii 

R.  Niewiadomski^^  noted  that,  for  a  series  of  Pisano, 
Uff^a=Ula-hi     or     -Uia-1  (rnodN), 

according  as  the  prime  A^  =  10m±lorlOw±3.  He  showed  how  to  compute 
rapidly  distant  terms  of  the  series  of  Pisano  and  similar  series,  and  factored 
numerous  terms. 

L.  Bastien^^  employed  a  prime  p  and  integer  a^Kp  and  determined 
02,03,  .  .  .,each  <p,bymeansof  0102=  Q,  02+ 03—-^, 0304— Q,a4+a5=-P,  • . . 
(mod  p).    Then 

a2.H-i^f^±^^^^-^''  (modp),  K,+,  =  PK,-QK,_,. 

The  types  of  series  are  found  and  enumerated.  Every  divisor  of  Kp  is  of 
the  form  Xp±l.     Some  of  Lucas'  results  are  given. 

R.  D.  CarmichaeP^  generalized  many  of  Lucas'^^'^^  theorems  and 
corrected  several.  The  following  is  a  generalization  (p.  46)  of  Fermat's 
theorem:  If  a-\-^  and  aj3  are  integers  and  aj3  is  prime  to  n  =  pC\  .  .pk'', 
where  p\,  .  .  .,Pk  are  distinct  primes,  Uy,  =  {a^—0^)/{a—^)  is  divisible  by  n 
when  X  is  the  1.  c.  m.  of 

(3)  p;rMPi-(a,/3)pj         {i=i,...,k). 

Here,  if  p  is  an  odd  prime,  the  symbol  (a,  ^)p  denotes  0,  +1  or  —  1,  according 
as  {a—^Y  is  divisible  by  p,  is  a  quadratic  residue  of  p,  or  is  a  quadratic  non- 
residue  of  p;  while  (a,  /3)2  denotes  +1  if  a/3  is  even,  0  if  a^  is  odd  and  a-\-^  is 
even,  and  —1  if  aj8(a+j3)  is  odd.  In  particular,  if  </>  is  the  product  of  the 
numbers  (3),  •w^=0(mod  n),  which  is  the  corrected  form  of  the  theorem  of 
Lucas'". 

Relations  have  been  noted®°  between  terms  of  recurring  series  defined  by 
one  of  the  equations 

^n  +  W„  +  l='W„+2,       'W„  +  Wn+2  =  W„+3,       y„+i +  t'„_i  =  4y„,       ^1=1,^2  =  3. 

E.  Malo^^  and  Prompt^  ^  considered  the  residues  with  respect  to  a  prime 
modulus  10m='=l  of  the  series  Uq,  Ui,  U2  =  Uo-\-Ui,. .  .,  ii„  =  w„_i+w„_2. 

A.  Boutin^^  noted  relations  between  terms  of  Pisano's  series. 

A.  Agronomof^^  treated  it„  =  it„_i+it„_2+w„_3. 

Boutin^^  and  Malo^^  treated  sums  of  terms  of  Pisano's  series. 

A.  Pellet^''  generalized  Lucas' ^^  law  of  apparition  of  primes. 

A.  G^rardin^^  proved  theorems  on  the  divisors  of  terms  of  Pisano's 
series. 

»'L'interm6diaire  des  math.,  20,  1913,  51,  53-6. 

"Sphinx-Oedipe,  7,  1912,  33-38,  145-155. 

"Annals  of  Math.,  (2),  15,  1913,  30-70. 

•"Math.  Quest.  Educat.  Times,  23,  1913,  55;  25,  1914,  89-91. 

"L'intermddiaire  des  math.,  21,  1914,  86-8. 

"Ibid.,  22,  1915,  31-6.  "Mathesis,  (4),  4,  1914,  125. 

"Mathesis,  (4),  4,  1914,  126.  "L'interm^diaire  des  math.,  23,  1916,  42-3. 

••L'intermediaire  des  math.,  23,  1916,  64-7  "Nouv.  Ann.  Math.,  (4),  16,  1916,  361-7. 


Chap.  XVII]  ALGEBRAIC    THEORY   OF   RECURRING    SERIES.  407 

E.  Piccioli^^  noted  that  in  Pisano's  series  1,  1,  2,  3, . . ., 

according  as  A;  is  odd  or  even. 

T.  A.  Pierce®^  proved  for  the  two  functions  HlZiil^aD  of  the  roots  ai 
of  an  equation  with  integral  coefficients  properties  analogous  to  those  of 
Lucas'  Un,  Vn. 

Algebraic  Theory  of  Recurring  Series. 

J.  D.  Cassini^°°  and  A.  de  Moivre^"^  treated  series  whose  general  term  is 
a  sum  of  a  given  number  of  preceding  terms  each  multiplied  by  a  constant. 
D.  Bernoulli^°^  used  such  recurring  series  to  solve  algebraic  equations.  J. 
Stirling^°^  permitted  variable  multipUers. 

L.  Euler^°^  studied  ordinary  recurring  series  and  their  application  to 
solving  equations. 

J.  L.  Lagrange^°^  made  the  subject  depend  on  the  integration  of  linear 
equations  in  finite  differences,  treating  also  recurring  series  with  an  additive 
term.     The  general  term  of  such  a  series  was  found  by  V.  Riccati.^°® 

P.  S.  Laplace^"^  made  systematic  use  of  generating  functions  and  applied 
recurring  series  to  questions  on  probability. 

J.  L.  Lagrange^°^  noted  that  if  Ay^-\-Byi.^i-}- .  .  .-\-Nyt+n  =  0  is  the 
recurring  relation  and  if  A+Bt-{- . .  .+iVT  =  0  has  distinct  roots  a,  jS, .  .  ., 
the  general  term  of  the  series  is  y^  =  aa^+&/3'^+  •  ■  •  •  For  the  case  of  multiple 
roots  he  stated  a  formula  which  G.  F.  Malfatti^°^  proved  to  be  erroneous; 
the  latter  gave  a  new  process  explained  for  2,  3  or  4  equal  roots. 

Lagrange^^°  had  noticed  independently  his  error  and  now  gave  the 
general  term  of  a  recurring  series  in  the  case  of  multiple  roots  by  a  more 
direct  process  than  that  of  Malfatti. 

Pietro  Paoli^^^  investigated  the  sum  of  a  recurring  series. 

98Periodico  di  Mat.,  31,  1916,  284-7. 
s'Annals  of  Math.,  (2),  18,  1916,  53-64. 
""Histoire  acad.  roy.  sc.  Paris,  annee  1680,  309. 
"iPhil.  Trans.  London,  32,  1722,  176;  Miscellanea  analytica,  1730,  27,  107-8;  Doctrine  of 

chances,  ed.  2,  1738,  220-9. 
i»2Comm.  Acad.  Petrop.,  3,  ad  annum  1728,  85-100. 
lO'Methodus  differentialis,  London,  1730,  1764. 
"^Introductio  in  analysin  infinitorum,  1748,  I,  Chs.  4,  13,  17.      Cf .  C.  F.  Degen,  Det  K.  Danske 

Vidensk.  Selskabs  Afhand.,  1,  1824,  135;  Oversigt. .  .Forhand.,  1818-9,  4. 
ii^Miscellanea  Taurinensia,  1,  1759,  Math.,  33-42;  Oeuvres,  I,  23-36. 
"^Mem.  present^s  div.  sav.  Paris,  5,  1768,  153-174;  Comm.  Bonon.,  5,  1767.      Cf.  M.  Cantor, 

Geschichte  Math.,  lY,  1908,  261. 
"^Mem.  sav.  etr.  ac.  sc.  Paris,  6,  annee  1771,  1774,  p.  353;  7,  annee  1773, 1776;  Oeuvres,  VIII, 

5-24,  69-197.     M6m.  ac.  roy.  sc.  Paris,  ann^e,  1779,  1782,  207;  Oeuvres,  X,  1-89  (ann^e 

1777,  99). 
"«Nouv.  Mem.  Ac.  Berlin,  annle  1775,  1777,  183-272;  Oeuvres,  IV,  151. 
losMem.  mat.  fis.  soc.  Ital.,  3,  1786-7,  571. 

""Nouv.  M6m.  Ac.  Sc.  Berlin,  ann^es  1792-3,  247;  Oeuvres,  V,  625-641  (p.  639  on  the  error). 
">Mem.  Acad.  Mantova,  1,  1795,  121.     See  Partitions  in  Vol.  Ill  of  this  History. 


408  History  of  the  Theory  of  Numbers.  [Chap,  xvii 

J.  B.  Fourier's"^"  error  in  appl^dng  recurring  series  to  the  solution  of 
numerical  equations  was  pointed  out  by  R.  Murphy.^^^'' 

P.  Frisiani"^"  applied  recurring  series  to  the  solution  of  equations. 

E.  Betti^^^'^  emploj'ed  doubly  recurring  series  to  solve  equations  in  two 
unknowns,  by  extending  the  method  of  Bernoulli. ^°' 

W.  Scheibner^^-  considered  a  series  with  a  three-term  recursion  formula, 
deduced  the  linear  relation  between  any  three  terms,  not  necessarily  con- 
secutive, and  applied  his  results  to  continued  fractions  and  Gauss'  h>TDer- 
geometric  series. 

D.  Andr^^^^  deduced  the  generating  equation  of  a  recurring  series  F» 
from  that  of  a  recurring  series  L'„  given  a  Unear  homogeneous  relation 
between  the  terms  F,  multiplied  by  constants  and  the  terms  C/„,  C/„_i, . . ., 
multipUed  by  polynomials  in  n. 

D.  Andr^^^^  considered  a  series  Ui,  U2, . . .,  with 

where  w„,  X„  are  given  functions  of  n,  X„  being  an  integer  ^n  — 1,  while 
Ai"^  is  a  given  function  of  k,  n.     It  is  proved  that 

C/„  =  2  ^(n,  p)u„  ^(n,  p)  =i:AiyA[y ...., 

p=i 

where  the  second  summation  extends  over  all  sets  of  integral  solutions  of 

k\-\-k2+  ■  ■  ■=n-p,  ni  =  ki+p,  n<  =  A\+n<_i         (0</:<<Xn,). 

Application  is  made  to  eight  special  types  of  series. 

D.  Andr^"^  discussed  the  sums  of  the  series  whose  general  terms  are 


n(/i+l) .  .  .  {n+p - 1)'  {an+j3) ! 

where  w„  is  the  general  term  of  any  recurring  series. 

G.  de  Longchamps"^"  proved  the  first  result  by  Lagrange^^^  and 
expressed  y^  as  a  sj-mmetric  function  of  the  distinct  roots  a,  /3, .  .  . .  He"^* 
reduced  Un  =  AiU„_i+ .  . +A^L''„_„+/(n),  where  /  is  a  polj^nomial  of  de- 
gree p,  to  the  case  f(n)  =  0  by  making  a  substitution  ?7„=  F„-f-Xon^+  .  .  -\-\p. 

C.  A.  Laisant^^^*"  studied  the  ratios  of  consecutive  terms  of  recurring 
series,  in  particular  for  Pisano's  series. 

'""Analyse  des  Equations,  Paris,  1831. 

'»»Phil.  Mag.,  (3),  11,  1837,  38-40. 

"'•^Effemeridi  .\stronomiche  di  Milano,  1850,  3. 

"I'^Annali  di  Sc.  Mat.  Fis.,  8,  1857,  48-61. 

"«Berichte  Gesell.  Wiss.  Leipzig  (Math.),  16,  1864,  44-68. 

i"Bull.  Soc.  Math.  France,  6,  1877-8,  166-170. 

"«Aim.  8C.  r^cole  norm,  sup.,  (2),  7,  1878,  375-408;  9,  1880,  209-226.     Summary  in  Bull,  dea 

Sc.  Math.,  (2),  1,  I,  1877,  350-5. 
"HIbmptes  Rendus  Paris,  86,  1878,  1017-9;  87,  1878,  973-5. 
'"^Assoc.  frang.,  9,  1880,  91-6. 
"»*7&id.,  1885,  II,  94-100. 
»"«Bull.  dea  Sc.  Math.,  (2),  5,  I,  1881,  218-249. 


Chap.  XVII]  AlGEBEAIC    ThEORY   OF   RECURRING  SeRIES.  409 

M.  d'Ocagne^^^  considered  the  recurring  series  Ui  with 

and  with  Uq,  . .  .,  Up_i  arbitrary;  and  the  series  u  with  the  same  law,  but 
with   Ui  =  0  {i  =  0,.  .  .,  p  —  2),  Up_i  =  l.     Then 

Un=UoUn+p-i-\-{Ui-aiUo)Un+p-2+  ■  ■  •  +  (  C/p_i  "  ^i  f7p_2  -  .  .  .  -ap_iC/o)^n- 

For  each  series  he  found  the  sum  of  any  fixed  number  of  consecutive  terms 
and  the  Hmit  of  that  sum. 

M.  d'Ocagne^^^  treated  Up+^  =  Up^n-i+  ■  ■  ■  +Un'  He^^^  discussed  the  con- 
vergents  to  a  periodic  continued  fraction  by  use  of  t«„  =  atW„_i  +  (  — l)*w„_2, 

L.  Gegenbauer^^^"  found  the  solution  Pm  of  gnPn  =  2^UnPn—i+i/nPn-2, 
where 

S.  Pincherle^^^^  applied  p„+i(x)  =  (a;  — aj(x— i8„)p„(a;)  to  developments  in 
series. 

E.  Study "^^  showed  how  to  express  the  general  term  of  a  recurring  series 
as  a  sum  of  the  general  terms  of  simpler  recurring  series,  exhibited  explicitly 
the  general  term  when  n  =  3,  and  applied  the  theory  to  bilinear  forms. 

M.  d'Ocagne^^^  considered  a  recurring  series  with  the  law  of  recurrence 

(^1, .  .  . ,  Ap) :  F„+AiF„_i+  .  . .  -{-ApYn—p^O 

of  order  p  and  generating  equation 

^{x)=x''+Aix''-'+ . . .  +Ap-=0. 
Set 
Qi{x)=x'+A,x'-'+ . . .  +A„         ^{x)  =  Yp_,+Q,{x)Yp-2+  ■  ■  •  +Qp-iix)Yo. 

The  existence  of  a  conomon  root  a  of  $(x)  =0,  '^  (a;)  =0  is  a  necessary  and 
sufficient  condition  that  the  Y's  satisfy  also  a  law  of  recurrence  of  order 
p  —  1,  viz.,  (Qi(a),.  .  .,  Qp_i(a)),  and  then  the  initial  law  of  recurrence  is 
said  to  be  reducible  to  one  of  order  p  —  1. 

M.  d'Ocagne^^''  considered  the  series  with  the  law  of  recurrence 

U^n  —  ^*0  '^  n—l  +  a  itt  n—2  "T  •  •  •  +  Ct  Pi—lU  n—Pi 

and  generating  equation 

(l>iix)  =x^'  —  a\  of *~^  —  ...  —  a*p.._i, 

"«Nouv.  Ann.  Math.,  (3),  2,  1883,  220-6;  3,  1884,  65-90;  9,  1890,  93-7;  11,  1892,  526-532  (5, 
1886,  257-272).  BuU.  Soc.  Math.  France,  12,  1883-4,  78-90  (case  p  =  2);  15,  1886-7, 
143-4;  19,  1890-1,  37-9  (minor  applications).  Nieuw  Archief  voor  Wiskunde,  17,  1890, 
229-232  (applications  to  sin  ma  as  function  of  sin  a  and  cos  a). 

"'Comptes  Rendus  Paris,  104,  1887,  419-420;  errata,  534. 

"8/bid.,  108,  1889,  499-501. 

"s^Sitzungsber  Ak.  Wiss.  Wien  (Math.),  97,  Ila,  1888,  82-89. 

"s^Atti  R.  Accad.  Lincei,  Rendiconti,  5,  1889,  I,  8-12,  323-7. 

"8cMonatshefte  Math.  Phys.,  2,  1891,  22-54. 

"'Bull.  Soc.  Math.  France,  20,  1892,  121-2. 

"oComptes  Rendus  Paris,  115,  1892,  790-2;  errata,  904. 


410 


History  of  the  Theory  of  Numbers. 


[Chap.  XVII 


such  that,  for  I  =  0,  Uq=  . . .  =  Wp_2  =  0,  Up_i  =  1 .     If  4>o{x)  =  </)i(x)-  •  •  </>„(x) , 

U  n+p-l  =2W  r^+p^-l'  '  "^"""w+P^-l' 

summed  for  all  combinations  of  n's  for  which  tzj  +  .  .  .  -\-n^  =  n.  Application 
is  made  to  the  sum  of  a  recurring  series  with  a  variable  law  of  recurrence. 

M.  d'Ocagne^-^  reproduced  the  last  result,  and  gave  a  connected  expo- 
sition of  his  earlier  results  and  new  ones. 

R.  Perrin^"  considered  a  recurring  series  U  of  order  p  ^ith  the  terms 
Uq,Ui,.  . ..    The  general  term  of  the  A-th  derived  series  of  U  is  defined  to  be 


u 


(*)_ 


Un+1 
Wn+2 


Un+k 


•  •     'Un+2k 

ies  is  zero,  the  law  of  recurrence  of 


If  any  term  of  the  (p  — l)th  derived  ser 
the  given  series  U  is  reducible  (to  one  of  lower  order) .  If  also  any  term  of 
the  (p  — 2)th  derived  series  is  zero,  continue  until  we  get  a  non- vanishing 
determinant;  then  its  order  is  the  minimum  order  of  U.  This  criterion  is 
only  a  more  convenient  form  of  that  of  d'Ocagne.^^^'^^^ 

E.  ]Maillet^"-^  noted  that  a  necessarj^  condition  that  a  law  of  recurrence 
of  order  p  be  reducible  to  one  of  order  p—q  is  that  ^(x)  and  ^{x)  of 
d'Ocagne^^^  have  q  roots  in  common,  the  condition  being  also  sufficient  if 
$(x)=0  has  only  distinct  roots.  He  found  independently  a  criterion  anal- 
ogous to  that  of  Perrin^--  and  studied  series  with  two  laws  of  recurrence. 

J.  Neuberg^^  considered  w„  =  aw„_i+6u„_2  and  found  the  general  term 
of  the  series  of  Pisano. 

C.  A.  Laisant^^^  treated  the  case  F  a  constant  of  d'Ocagne's^^^ 
u,{f{u)]=Fik). 

S.  Lattes^^^  treated  Wn+p=/(Wn+p-i>- •  •>  ^J»  where  /  is  an  analytic 
function. 

M.  .Amsler^^^  discussed  recurring  series  by  partial  fractions. 

E.  Netto,^-"''  L.  E.  Dickson,^-"'  A.  Ranum,^-^  and  T.  Hayashi'-^  gave 
the  general  term  of  a  recurring  series.  N.  Traverso^^"  gave  the  general 
term  for  Q„=  (n  — l)(Q„_i+Q„_2)  and  u„=  aUn-i+hu^_2. 

Traverso^^^  applied  the  theory  of  combinations  with  repetitions  to  express, 
as  a  function  of  p,  the  solution  of  Q„,  =  p(Qm-i+Qm-2+  •  •  •  +Qm-n)- 

'"Jour,  de  l'6cole  polyt.,  64,  1894,  151-224. 

i=»Comptes  Rendus  Paris,  119,  1894,  990-3. 

i»M6m.  Acad.  Sc.  Toulouse,  (9),  7,  1895,  179-180,  182-190;  Assoc,  fran?.,  1895,  III,  233  [report 

with  miscellaneous  Dioph.  equations  of  order  n,  Vol.  II);  Nouv.  Ann.  Math.,  (3),  14, 

1895,  152-7,  197-206. 
"*Mathesis,  (2),  6,  1896,  88-92;  Archive  de  mat.,  1,  1896,  230. 


"*Bull.  Soc.  Math.  France,  29,  1901, 145-9 

""Nouv.  Ann.  Math.,  (4),  10,  1910,  90-5. 

^"'-Amer.  Math.  Monthly,  10,  1903,  223-6. 

"'Bull.  Amef.  Math.  Soc,  17,  1911,  457-461. 

"•/Wd.,  18,  1912,  191-2. 

"oPeriodico  di  Mat.,  29,  1913-4,  101-4;  145-160. 

"'Ibid.,  31,  1915-6,  1-23,  49-70,  97-120,  145-163,  193-207 


i»>Comptes  Rendus  Paris,  150,  1910,  1106-9. 
"'"Monatshefte  Math.  Phys.,  6, 1895,  285-290. 


Chap.  XVII]  ALGEBRAIC    THEORY   OF   RECURRING    SERIES.  411 

F.  Nicita^^^  found  many  relations  like  2aJ^  — hn^=  —  {  —  !)''  between 
the   two   series  ai  =  l,   02  =  2,..,   an  =  2(an+i  — «n-i),  •  •  • ;   &i  =  l,  62  =  3,..., 

&n  =  l(^n+l~-^n-l)j  •  •  •• 

Reference  may  be  made  to  the  text  by  A.  Vogt^^^  and  to  texts  and  papers 
on  difference  equations  cited  in  Encyklopadie  der  Math.  Wiss.,  I,  2,  pp.  918, 
935;  Encyclopedie  des  Sc.  Math.,  I,  4,  47-85. 

A.  Weiss^^^  expressed  the  general  term  4  of  a  recurring  series  of  order 
T  linearly  in  terms  of  tq,  tg_i , .  . . ,  ^q-r+i,  where  q  is  an  integer. 

W.  A.  Whitworth^^^  proved  that,  if  Co+CiX+C2X^+ ...  is  a  convergent 
recurring  series  of  order  r  whose  first  2r  terms  are  given,  its  scale  of  relation 
and  sum  to  infinity  are  the  quotients  of  certain  determinants. 

H.  F.  Scherk^^^  Started  with  any  triangle  ABC  and  on  its  sides  con- 
structed outwards  squares  BCED,  ACFG,  ABJH.  Join  the  end  points  to 
form  the  hexagon  DEFGHJ.  Then  construct  squares  on  the  three  joining 
lines  EF,  GH,  JD  and  again  join  the  end  points  to  form  a  new  hexagon,  etc. 
If  tti,  hi,  Ci  are  the  lengths  of  the  joining  lines  in  the  iih.  set,  a„+i  =  5a„_i  —  a„_3. 
The  nth  term  is  found  as  usual. 

Sylvester^"  solved  Uj,=  u^_i-\-{x  —  \){x  —  2)Ux_2'  A._  Tarn^^^  treated 
recurring  series  connected  with  the  approximations  to  \^2,  Vs,  Vs. 

V.  SchlegeP^^  called  the  development  of  {1  —  x—x^— . . .  —x'')~'^  the 
(n  — l)th  series  of  Lame;  each  coefficient  is  the  sum  of  the  n  preceding. 
For  n=2,  the  series  is  that  of  Pisano. 

References  on  the  connection  between  Pisano's  series  and  leaf  arrange- 
ment and  golden  section  (Kepler,  Braun,  etc.)  have  been  collected  by  R.  C. 
Archibald."^ 

Papers  by  C.  F.  Degen,^^^  A.  F.  Svanberg,^^^  and  J.  A.  Vesz"^  were  not 
available  for  report. 

"2Periodico  di  Mat.,  32,  1917,  200-210,  226-36. 

^'^Theorie  der  Zahlenreihen  u.  der  Reihengleichung,  Leipzig,  1911,  133  pp. 

"4Jour.  fur  Math.,  38,  1849,  148-157. 

"^Oxford,  Cambridge  and  Dublin  Mess.  Math.,  3,  1866,  117-121;  Math.  Quest.  Educ.  Times, 

3,  1865,  100-1. 
i36Abh.  Naturw.  Vereine  zu  Bremen,  1,  1868,  225-236. 
•   i"Math.  Quest.  Educ.  Times,  13,  1870,  50. 
•38Math.  Quest,  and  Solutions,  1,  1916,  8-12. 
139E1  Progreso  Mat.,  4,  1894,  171-4. 
""Amer.  Math.  Monthly,  25,  1918,  232-8. 
i"M6m.  Acad.  Sc.  St.  Petersbourg,  1821-2,  71. 
i«2Nova  Acta  R.  Soc.  Sc.  UpsaUensis,  11,  1839,  1. 
i«EIrtekez.  a  Math.,  Magyar  Tudom.  Ak.  (Math.  Memoirs  Hungarian  Ac.  Sc),  3,  1875,  No.  1. 


CHAPTER  XVIII. 

THEORY  OF  PRIME  NUMBERS. 
Existence  of  an  Infinitude  of  Primes. 

Euclid^  noted  that,  if  p  were  the  greatest  prime,  and  M  =  2-S-5.  .  .p  is 
the  product  of  all  the  primes  ^p,  then  M+l  is  not  divisible  by  one  of 
those  primes  and  hence  has  a  prime  factor  >p,  thus  involving  a  contra- 
diction. 

L.  Euler^  deduced  the  theorem  from  the  [invahd]  equation 

s -=n(i--)   , 

„=in        V      p/ 

the  left  member  being  infinite  and  the  right  finite  if  there  be  only  a  finite 
number  of  primes.  Euler^  concluded  from  the  same  equation  that  "the 
number  of  primes  exceeds  the  number  of  squares." 

Euler^  modified  Euclid's^  argument  slightly.  The  number  of  integers 
<M  and  prime  to  M  is  <t>(M)  =2-4.  .  . (p  — 1),  so  that  they  include  integers 
which  are  either  primes  >p  or  have  prime  factors  >p. 

The  theorem  follows  from  Tchebychef's^^^  proof  of  Bertrand's  postulate. 

L.  Kronecker^  noted  that  we  may  rectify  Euler's^  proof  by  using 


2  -^=nfl-i)    '  (s>l). 


where  p  ranges  over  all  primes  >  1 .  If  there  were  only  a  finite  number  of 
p's,  the  product  would  remain  finite  when  s  approaches  unity,  while  the 
sum  increases  indefinitely.  He  also  gave  the  proof  a  form  leading  to  an 
interval  from  m  to  n  within  which  there  exists  a  new  prime  however  great 
m  is  taken. 

R.  Jaensch®  repeated  Euler's-  argument,  also  ignoring  convergency. 

E.  Kummer'^  gave  essentially  Euler's'*  argument. 

J.  Perott^  noted  that,  if  pi,.  .  .,  p„  are  the  primes  ^N,  there  are  2" 
integers  ^N  which  are  not  divisible  by  a  square,  and 


2">iV- 


"-mx'-^i^'^'-iyi- 


Hence  there  exist  infinitely  many  primes. 

L.  Gegenbauer^"  proved  the  theorem  by  means  of  2"ii*n~*. 

lElementa,  IX,  20;  Opera  (ed.,  Heiberg),  2,  1884,  388-91. 

^Introductio  in  analysin  infinitorum,  1,  Ch.  15,  Lausanne,  1748,  p.  235;  French  transl.  by 

J.  B.  Labey,  1,  218. 
'Comm.  Acad.  Petrop.,  9,  1737,  172-4. 

^Posthumous  paper.  Coram.  Arith.  Coll.,  2,  518,  Nos.  134-6;  Opera  Postuma,  I,  1862,  18. 
'Vorlesungen  uber  Zahlentheorie,  I,  1901,  269-273,  Lectures  of  1875-6. 
*Die  Schwierigeren  Probl.  Zahlentheorie,  Progr.  Rastenburg,  1876,  2. 
'Monatsber.  Ak.  Wiss.  Berlin  fiir  1878,  1879,  777-8. 
sBull.  sc.  math,  et  astr.,  (2),  5,  1881,  I,  183-4. 
8«Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  95,  II,  1887,  94-6;  97,  Ila,  1888,  374-7. 

413 


414  History  of  the  Theory  of  Numbers.  [Chap,  xviii 

J.  Perott'  applied  the  theory  of  commutative  groups  to  show  that, 
if  Q\,-  ■  ■)  Qn  are  primes,  there  exist  at  least  n  — 1  primes  between  g„  and 
M  =  qi-  ■  Qn' 

T.  J.  Stieltjes^°  expressed  the  product  P  of  the  primes  2,  3, . . .,  p  as  a 
product  AB  of  two  factors  in  any  way.  Since  A-{-B  is  not  divisible  by 
2, .  . . ,  p,  there  exists  a  prime  >p. 

J.  Hacks"  proved  the  existence  of  an  infinitude  of  primes  by  use  of  his 
formula  (Ch.  XI,  Hacks^^)  for  the  number  of  integers  ^m  not  divisible 
by  a  square. 

C.  0.  Boije  af  Gennas^^  showed  how  to  find  a  prime  exceeding  the  nth 
primep„>2.  Take  P  =  2'''3'''.  .  Pn*^,  each  ^^,^1.  Express  P  as  a  product 
of  relatively  prime  factors  5,  P/8,  where  Q  =  P/8—8>l.  Since  Q  is  divisible 
by  no  prime  ^  p„,  it  is  a  product  of  powers  of  primes  qi'^pn+2.  Take  5  so 
that  Q<  (p„+2)l     Then  Q  is  a  prime. 

Axel  Thue^^  proved  that,  if  (l+n)*<2",  there  exist  at  least  k-{-l 
primes  <2''. 

J.  Braun^'"  noted  that  the  sum  of  the  inverses  of  the  primes  ^p  is,  for 
p^  5,  an  irreducible  fraction  >  1 ;  hence  the  numerator  contains  at  least  one 
prime  >p.  He  attributed  to  Hacks  a  proof  by  means  of  11(1  — l/p-)~^  = 
2s~^  =  TT^/G ;  the  product  would  be  rational  if  there  were  only  a  finite  number 
of  primes,  whereas  tt  is  irrational. 

E.  Cahen^^  proved  the  ''identity  of  Euler"  used  by  Kjonecker.^ 

Stormer-^^  gave  a  proof. 

A.  Le\'y^^  took  a  product  P  of  k  of  the  first  n  primes  Pi,. . .,  p„  and 
the  product  Q  of  the  remaining  n  —  k.  Then  P+Q  is  either  prime  or  has 
a  prime  factor  >p„;  like\\'ise  for  P  —  Q.  If  p„  is  a  prime  such  that  p„+2  is 
composite,  there  exist  at  least  n  primes  >p„,  but  ^l+PiP2-  •  Pn-     When 

1  1 

± ±  db 

Pi        '"        Pn 

is  reduced  to  a  simple  fraction,  the  numerator  has  no  factor  in  common  with 
Pi .  .  .p„;  hence  there  is  a  prime  >Pn-  He  considered  (pp.  242-4)  the  primes 
defiiied  by  a;(x  — 1)  —  1  for  consecutive  integers  x. 

A.  Auric^^  assumed  that  pi, . .  . ,  pk  give  all  the  primes.  Then  the  number 
of  integers  <  71 =npi°'  is 

which  is  small  in  comparison  with  n,  whence  k  increases  indefinitely  with  n. 

»Amer.  Jour.  Math.,  11,  1888,  9&-138;  13,  1891,  235-308,  especially  303-5. 
"Annales  fac.  sc.  de  Toulouse,  4,  1890,  14,  final  paper. 
"Acta  Math.,  14,  1890-1,  335. 

'Hifversigt  K.  Sv.  Vetenskaps-Akad.  Forhand.,  Stockholm,  .50,  1893,  469-471. 
"Archiv  for  Math,  og  Xatur.,  Kristiania,  19,  1897,  No.  4,  1-5. 
'*^Das  Fortschreitungsgesetz   der   Primzahlen   durch   eine  transcendente   Gleichung  exakt 

dargestellt,  WLss.  Beilage  Jahresbericht,  Gymn.,  Trier,  1899,  96  pp. 
"filaments  de  la  th^orie  des  nombres,  1900,  319-322. 
"Bull,  de  Math.  £l6mentaires,  15,  1909-10,  33-34,  80-82. 
"L'interm^diaire  des  math.,  22,  1915,  252. 


I 


Chap.  XVIII]  INFINITUDE   OF  PrIMES.  415 

G.  M^trod^'^  noted  that  the  sum  of  the  products  n  —  1  at  a  time  of  the 
first  n  prunes  >  1  is  either  a  prime  or  is  divisible  by  a  prime  greater  than  the 
nth.    He  also  repeated  Euler's^  proof. 

Infinitude  of  Primes  in  a  General  Arithmetical  Progression. 

L.  Euler^''  stated  that  an  arithmetical  progression  with  the  first  term 
unity  contains  an  infinitude  of  primes. 

A.  M.  Legendre^^  claimed  a  proof  that  there  is  an  infinitude  of  primes 
2mx+ii  if  2m  and  /x  are  relatively  prime. 

Legendre^^  noted  that  the  theorem  would  follow  from  the  following 
lemma:  Given  any  two  relatively  prime  integers  A,  C,  and  any  set  of  k  odd 
primes  d,\,.  .  .,  co  [not  divisors  of  A],  and  denoting  the  zth.  odd  prime  by 
7r^^\  then  among  tt'*"^'  consecutive  terms  of  the  progression  A  —  C,  2A  —  C, 
3A  —  C,.  .  .  there  occurs  at  least  one  divisible  by  no  one  of  the  primes 
6,. . .,  CO.  Although  Legendre  supposed  he  had  proved  this  lemma,  it  is 
false  [Dupr^28j^ 

G.  L.  Dirichlet^^  gave  the  first  proof  that  mz-\-n  represents  infinitely 
many  primes  if  m  and  n  are  relatively  prime.  The  difficult  point  in  the 
proof  is  the  fact  that 

n=i    n 

where  x(^)  =0  if  ^j  ^  have  a  common  factor  >  1,  while,  in  the  contrary  case, 
xin)  is  a  real  character  different  from  the  chief  character  of  the  group  of 
the  classes  of  residues  prime  to  k  modulo  k.  This  point  Dirichlet  proved 
by  use  of  the  classes  of  binary  quadratic  forms. 

Dirichlet^'*  extended  the  theorem  to  complex  integers. 

E.  Heine^^  proved  "without  higher  calculus"  Dirichlet's  result 

VJ?  p[jHW^''^{b+2ay+'>~^  ■■■}'=  a 

A.  Desboves^^  discussed  the  error  in  Legendre's^^  proof. 
L.  Durand^^  gave  a  false  proof. 

A.  Dupre^^  showed  that  the  lemma  of  Legendre^"  is  false  and  gave 
(p.  61)  the  following  theorem  to  replace  it:     The  mean  number  of  terms, 

"L'intermediaire  des  math.,  24,  1917,  39-40. 

^oOpusc.  analytica,  2,  1785  (1775),  241;  Comm.  Arith.,  2,  116-126. 

2iMem.  ac.  sc.  Paris,  ann^e  1785,  1788,  552. 

22Theorie  des  nombres,  ed.  2,  1808,  p.  404;  ed.  3,  1830,  II,  p.  76;  Maser,  2,  p.  77. 

23Bericht  Ak.  Wiss.  Berlin,  1837,  108-110;  Abhand.  Ak.  Wiss.  Berlin,  Jahrgang  1837,  1839, 

Math.,  45-71;  Werke,  1,  1889,  307-12,  313-42.     French  transl.,  Jour,  de  Math.,  4,  1839, 

393-422.     Jour,  fiir  Math.,  19,  1839,  368-9;  Werke,  1,  460-1.     Zahlentheorie,  §132,  1863; 

ed.  2,  1871;  3,  1879;  4,  1894  (p.  625,  for  a  simplification  by  Dedekind). 
z^Abhand.  Ak.  Wiss.  BerUn,  Jahrgang  1841,  1843,  Math.,  141-161;  Werke,  1,  509-532.     French 

transl.,  Jour,  de  Math.,  9,  1844,  245-269. 
2«Jour.  fiir  Math.,  31,  1846,  133-5. 
26Nouv.  Ann.  Math.,  14,  1855,  281. 
^Ubid.,  1856,  296. 
^^Examen  d'une  proposition  de  Legendre,  Paris,  1859.     Comptes  Rendus  Paris,  48,  1859,  487. 


416  History  of  the  Theory  of  Numbers.  iChap.  xviii 

prime  to ^,X,. .  .,  co,  contained  in  tt**"^^  consecutive  terms  of  the  progression 
is  ^p-'Q7r"'-''-2,  where  P  =  3-5-7-ll .  .  .,  (?=  (3-l)(5-l) .  .  .. 

J.  J.  Sylvester-^  gave  a  proof. 

V.  I.  Berton^^"  found  h  such  that  between  x  and  xh  occur  at  least  2g 
primes  each  of  one  of  the  2g  linear  forms  2py+ri,  where  ri, .  .  .,  r2g  are  the 
integers  <  2p  and  prime  to  2p. 

C.  ]Moreau^°  noted  the  error  in  Legendre's--  proof. 

L.  Kronecker^  (pp.  442-92)  gave  in  lectures,  1886-7,  the  following 
extension*  of  Dirichlet's  theorem  (in  lectures,  1875-6,  for  the  case  m  a 
prime):  If  jjl  is  any  given  integer,  we  can  find  a  greater  integer  v  such  that, 
if  7u,  r  are  any  two  relatively  prime  integers,  there  exists  at  least  one  prime 
of  the  form  h7?i+r  in  the  interval  from  /x  to  ^'  (p.  11,  pp.  465-6).  Moreover 
(pp.  478-9),  there  is  the  same  mean  density  of  primes  in  each  of  the  <l){m) 
progressions  mh-\-ri,  where  the  r,  are  the  integers  <?n  and  prime  to  m. 

I.  Zignago^^  gave  an  elementary  proof. 

H.  Scheffler^^  devoted  31  pages  to  a  re\nsion  of  Legendre's  insufficient 
proof  and  gave  a  process  to  determine  all  primes  under  a  given  limit. 

G.  Speckmann^^  failed  in  an  attempt  to  prove  the  theorem. 

P.  Bachmann^^  gave  an  exposition  of  Dirichlet's^^  proof. 

Ch.  de  la  Vallee-Poussin^^  obtained  without  computations,  by  use  of 
the  theorj'  of  functions  of  a  complex  variable,  a  proof  of  the  difficult  point 
in  Dirichlet's^^  proof.  He^®  proved  that  the  sum  of  the  logarithms  of  the 
primes  hk-\-l^x  equals  x/4>{k)  asymptotically  and  concluded  readily  that 
the  number  of  primes  hk+l'^x  equals,  asjonptotically, 


4>  {x)   log  X 

F.  Mertens^^  proved  the  existence  of  an  infinitude  of  primes  in  an  arith- 
metical progression  by  elementarj^  methods  not  using  the  quadratic  reci- 
procity theorem  or  the  number  of  classes  of  primitive  binary  quadratic  forms. 
He  supplemented  the  theorem  by  showing  how  to  find  a  constant  c  such 
that  between  x  and  ex  there  lies  at  least  one  prime  of  the  progression  for 
every  x^l  [cf.  Kronecker,^  pp.  480-96]. 

"Proc.  London  Math.  Soc,  4, 1871,  7;  Messenger  Math.,  (2),  1, 1872,  143-4;  Coll.  Math.  Papers, 

2,  1908,  712-3. 
""Comptes  Rendus  Paris,  74,  1872,  1390. 

">Nouv.  Ann.  Math.,  (2),  12,  1873,  323-4.     Also,  A.  Piltz,  Diss.,  Jena,  1884. 
•Improvements  in  the  exposition  were  made  by  the  editor,  Hensel  (cf.  p.  508). 
«Annan  di  Mat.,  (2),  21,  1893,  47-55. 

**Beleuchtung  u.  Beweis  eines  Satzes  aus  Legendre's  Zahlentheorie  [H,  1830,  76],  Leipzig,  1893. 
"Archiv  Math.  Phys.,  (2),  12,  1894,  439-441.     Cf.  (2),  15,  1897,  326-8. 
"Die  analytische  Zahlentheorie,  1894,  51,  74-88. 
**M6ra.  couronnes.  .  .acad.  roy.  sc.  Belgique,  53,  1895-6,  No.  6,  24-9. 
*Annales  de  la  soc.  sc.  de  Bru.xeUes,  20,  1896,  II,  281-361.     Cf.  183-256,  361-397;  21,  1897,  I, 

1-13,  60-72; II,  251-368. 
"Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  106,  1897,  II a,  254-286.     Parts  published  earlier, 

ibid.,  104,  1895,  II a,  1093-1121,  1158-1166;  Jour,  fiir  Math.,  78,  1874,  46-62;  117,  1897, 

169-184. 


Chap.  XVIII]  INFINITUDE   OF  PRIMES.  417 

F.  Mertens^^  gave  a  proof,  still  simpler  than  his"  earlier  one,  of  the 
difficult  point  in  Dirichlet's-^  proof.  The  proof  is  very  elementary,  involv- 
ing computations  of  finite  sums. 

F.  Mertens^^  gave  a  simplification  of  Dirichlet's^  proof  of  his  general- 
ization to  complex  primes. 

H.  Teege^°  proved  the  difficult  point  in  Dirichlet's^^  proof. 

E.  Landau'*^  proved  that  the  number  of  prime  ideals  of  norm  ^  a:  of  an 
algebraic  field  equals  the  integral-logarithm  Li(x)  asymptotically.  By 
specialization  to  the  fields  defined  by  V  — 1  or  V  — 3?  we  derive  theorems'*^ 
on  the  number  of  primes  4:k^l  or  6A;=fcl  ^x. 

L.  E.  Dickson^^  asked  if  aifi+bi  {i  =  l,.  .  .,  m)  represent  an  infinitude  of 
sets  of  m  primes,  noting  necessary  conditions. 

H.  Weber^^  proved  Dirichlet's^^  theorem  on  complex  primes. 

E.  Landau^^  simplified  the  proofs  by  de  la  Vallee-Poussin^^  and  Mertens.^* 

E.  Landau*^'^^  simplified  Dirichlet's^^  proof.  Landau^^  proved  that, 
if  k,  I  are  relatively  prime,  the  number  of  primes  ky-\-l^x  is 

where  7  is  a  constant  depending  on  k.     For  0  see  Pfeiffer^°  of  Ch.  X. 

A.  Cunningham*^  noted  that,  of  the  N  primes  ^R,  approximately 
N/c{)(n)  occur  in  the  progressions  nx-\-a,  a<n  and  prime  to  n,  and  gave  a 
table  showing  the  degree  of  approximation  when  R  =  10^  or  5-10^,  with 
n  even  and  <  1928.  Within  these  limits  there  are  fewer  primes  nx-\-l  than 
primes  nx+a,  a>l. 

Infinitude  of  Primes  Represented  by  a  Quadratic  Form. 

G.  L.  Dirichlet^^  gave  in  sketch  a  proof  that  every  properly  primitive 
quadratic  form  {a,  h,  c),  a,  26,  c  with  no  common  factor,  represents  an  infini- 
tude of  primes. 

Dirichlet^^  announced  the  extension  that  among  the  primes  represented 
by  {a,  b,  c),  an  infinitude  are  representable  by  any  given  linear  form  Mx-\-N, 
with  M,  N  relatively  prime,  provided  a,  h,  c,  M,  N  are  such  that  the  linear 
and  quadratic  forms  can  represent  the  same  number. 

»8Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  108,  1899,  II a,  32-37. 

"/bid.,  517-556.     Polish  transl.  in  Prace  mat.  fiz.,  11,  1900,  194-222. 

"Mitt.  Math.  GeseU.  Hamburg,  4,  1901,  1-11. 

*iMath.  Amialen,  56,  1903,  665-670. 

*?Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  112,  1903,  II a,  502-6. 

"Messenger  Math.,  33,  1904,  155. 

"Jour,  fur  Math.,  129,  1905,  35-62.     Cf.  p.  48. 

«Sitzungsber.  Akad.  Berlin,  1906,  314-320. 

«Rend.  Circ.  Mat.  Palermo,  26,  1908,  297. 

"Handbuch  . .  .Verteilung  der  Primzahlen,  I,  1909,  422-35. 

"Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  117,  1908,  Ila,  1095-1107. 

"Proc.  London  Math.  Soc,  (2),  10,  1911,  249-253. 

"Bericht  Ak.  Wiss.  Berhn,  1840,  49-52;  Werke,  1,  497-502.     Extract  in  Jour,  fiir  Math.,  21, 

1840,  98-100. 
"^Comptes  Rendus  Paris,  10,  1840,  285-8;  Jour,  de  Math.,  5,  1840,  72-4;  Werke,  1,  619-623. 


418  History  of  the  Theory  of  Numbers.  [Chap,  xviii 

H.  Weber^^  and  E.  Schering^^  completed  Dirichlet's^^  proof  of  his  first 
theorem.     A.  Meyer^^  completed  Dirichlet's^^  proof  of  his  extended  theorem. 

F.  Mertens^"  gave  an  elementary  proof  of  Dirichlet's^^  extended  theorem. 

Ch.  de  la  Vall^e-Poussin^^  proved  that  the  number  of  primes  ^  x  repre- 
sentable  by  a  properly  primitive  definite  positive  or  indefinite^^  irreducible 
binary'  quadratic  form  is  asjTnptotic  to  gx/\ogx,  where  ^  is  a  constant;  and 
the  same  for  primes  belonging  also  to  a  linear  form  compatible  with  the 
character  of  the  quadratic  form. 

L.  Kronecker^  (pp.  494-5)  stated  a  theorem  on  factorable  forms  in 
several  variables  which  represent  an  infinitude  of  primes. 

Elementary  Proofs  of  the  Existence  of  an  Infinitude  of  Primes  mz+l, 

FOR  Any  Given  m. 

V.  A.  Lebesgue^^  gave  a  proof  for  the  case  m  a  prime,  using  the  fact 
that  x^~^  —  x"'~~y+  . . .  +y"'~^  has  besides  the  possible  factor  m  only  prime 
factors  2km +  1.    A  like  method  apphes^^  to  2mz  —  l. 

J.  A.  Serret^^  gave  an  incomplete  proof  for  any  m. 

F.  Landry^^  gave  a  proof  like  Lebesgue's.®^  If  ^  is  the  largest  prime 
2km-\-l  and  if  x  is  the  product  of  all  of  them,  a:'"+l  is  divisible  by  no  one 
of  them.  Since  (x'"+l)/(x+l)  has  no  prime  divisor  not  of  the  form 
2km+l,  there  exists  at  least  one  >d. 

A.  Genocchi^^  proved  the  existence  of  an  infinitude  of  primes  mz^  1 
and  n'2±l  for  n  a  prime  by  use  of  the  rational  and  irrational  parts  of 

(a+Vb)'- 

L.  Kronecker^  (pp.  440-2)  gave  in  lectures,  1875-6,  a  proof  for  the  case 
m  a  prime;  the  simple  extension  in  the  text  to  any  m  was  added  by  Hensel. 

E.  Lucas  gave  a  proof  by  use  of  his  w„  (Lucas,^^  p.  291,  of  Ch.  XYII). 

A.  Lefebure^°  of  Ch.  XVI  stated  that  the  theorem  follows  from  his 
results. 

L.  Kraus^^  gave  a  proof. 

A.  S.  Bang"''  and  Sylvester^  proved  it  by  use  of  cyclotomic  functions. 

K.  Zsigmondy"^  of  Ch.  VII  gave  a  proof.  Also,  E.  Wendt,'^  and 
Birkhoff  and  Vandiver^^  ^f  q^   ^VI. 

»^Math.  Annalen,  20,  1882,  301-329.     Elliptische  Functionen  (=  Algebra,  III),  ed.  2,  1908, 

613-6. 
»8Werke,  2,  1909,  357-365,  431-2. 

"Jour,  fur  Math.,  103,  1888,  98-117.     Exposition  by  Bachmaim,»<  pp.  272-307. 
•oSitzungsber.  Ak.  Wiss.  Wien   (Math.),   104,   1895,  Ila,   1093-1153,   1158.     Simplification, 

ibid.,  109,  1900,  Ila,  415-480. 
"Cf.  E.  Landau,  Jahresber.  D.  Math.  Verein.,  24,  1915,  250-278. 
"Jour,  de  Math.,  8,  1843,  51,  note.     Exercices  d'analyse  num^rique,  1859,  91. 
«''.Jour.  de  Math.,  (2),  7,  1862,  417. 
•«Jour.  de  Math.,  17,  1852,  186-9. 

•'Deuxidme  mdmoire  but  la  thdorie  des  nombres,  Paris,  1853,  3. 
"AnnaU  di  mat.,  (2),  2,  1868-9,  256-7.     Cf.  .Genocchi==-  "  of  Ch.  XVII. 
••Casopis  Math,  a  Fys.,  15,  1886,  61-2.     Cf.  Fortschritte,  1886,  134-5. 
^Tidsskrift  for  Math.,  (5),  4,  1886,  70-80,  130-7.     See  Bang".  ",  Ch.  XVI. 
"Jour,  fiir  Math.,  115,  1895,  85. 


i 


Chap.  XVIII]  INFINITUDE   OF  PRIMES.  419 

N.  V.  Bervi^^  proved  that  the  ratio  of  the  number  of  integers  cm+1  not 
>n  and  not  a  product  of  two  integers  of  that  form  to  the  number  of  all 
primes  not  >n  has  the  limit  unity  for  n=  oo . 

H.  C.  Pocklington^^  proved  that,  if  n  is  any  integer,  there  is  an  infinitude 
of  primes  mn+1,  an  infinitude  not  of  this  form  if  n>2,  and  an  infinitude 
not  of  the  forms  mn±l  if  n  =  5  or  n>6. 

E.  Cahen^^  proved  the  theorem  for  m  an  odd  prime. 

J.  G.  van  der  Corput'^^  proved  the  theorem. 

Elementary  Proofs  op  the  Existence  of  an  Infinitude  of  Primes  in 
Special  Arithmetical  Progressions. 

J.  A.  Serret^^  for  the  common  difference  8  or  12,  and  for  lOx+9. 

V.  A.  Lebesgue»°  for  4n±l,  Sn+k  (/c  =  l,  3,  5,  7).  Lebesgue^^  for  the 
same  and  2'"n+l,  Qn  —  1.  Also,  by  use  of  infinite  series,  for  the  common 
difference  8  or  12. 

E.  Lucas«2  f<3j.  5n-\-2,  8n+7. 

J.  J.  Sylvester^^  for  the  difference  8  or  12  and^  for  p^x  —  1,  p  a  prime. 

A.  S.  Bang^^  for  the  differences  4,  6,  8,  10,  12,  14,  18,  20,  24,  30,  42,  60. 

E.  Lucas^^  for  4n=i=l,  6n  — 1,  Sn+5. 

R.  D.  von  Sterneck"  for  an  — 1. 

K.  Th.  Vahlen^^  for  mz-\-l  by  use  of  Gauss'  periods  of  roots  of  unity. 
Also,  if  m  is  any  integer  and  p  a  prime  such  that  p  —  1  is  divisible  by  a  higher 
power  of  2  than  <^(m)  is,  while  A;  is  a  root  of  km  +  1^  —1  (mod  p),  the  linear 
form  mpx+km-^-l  represents  an  infinitude  of  primes;  known  special  cases 
are  mx+1  and  2px  —  l. 

J.  J.  Iwanow^^  for  the  difference  8  or  12. 

E.  Cahen^^  (pp.  318-9)  for  4a;±  1,  6x±  1,  8a:+5.  K.  HenseP  (pp.  438-9, 
508)  for  the  same  forms.     M.  Bauer^°  for  an  —  1. 

E.  Landau^^  (pp.  436-46)  for  /cn±l. 

I.  Schur^^  proved  that  if  f=  1  (mod  k)  and  if  one  knows  a  prime  '>(f>(k)/2 
of  the  form  kz+l,  there  exists  an  infinitude  of  primes  kz+l;  for  example, 

2»z-\-2''-^=i=l,     Smz-\-2m+l,     Smz+4m-\-l,     8m2+6m+l, 

where  m  is  any  odd  number  not  divisible  by  a  square. 

K.  Hensep2  f^j.  4^±i^  6n±l,  8n-l,  8n=t3,  12n-l,  lOn-1. 

"Mat.  Sbornik  (Math.  Soc.  Moscow),  18,  1896,  519. 

"Proc.  Cambr.  Phil.  Soc,  16,  1911,  9-10.  '^Nouv.  Ann.  Math.,  (4),  11,  1911,  70-2. 

^^Nieuw  Archief  voor  Wiskunde,  (2),  10,  1913,  357-361  (Dutch). 

8«Nouv.  Ann.  Math.,  15,  1856,  130,  236. 

"Exercices  d'analyse  num^rique,  1859, 91-5, 103-4, 145-6. 

s^Amer.  Jour.  Math.,  1, 1878,  309.  saQomptes  Rendus  Paris,  106,  1888,  1278-81,  1385-6. 

s^Assoc.  frang.  av.  sc,  17,  1888,  II,  118-120. 

s^Nyt  Tidsskrift  for  Math.,  Kjobenhavn,  1891,  2B,  73-82. 

s^Theorie  des  nombres,  1891,  353-4.  "Monatshefte  Math.  Phys.,  7,  1896,  46. 

8'Schriften  phys.-okon.  Gesell.  Konigsberg,  38,  1897,  47. 

"Math.  Soc.  St.  Petersburg,  1899,  53-8  (Russian). 

'"Jour,  fur  Math.,  131,  1906,  265-7;  transl.  of  Math.  6s  Phys.  Lapok,  14,  1905,  313. 

9iSitzungsber.  BerUn  Math.  Gesell.,  11,  1912,  40-50,  with  Archiv  M.  P. 

92Zahlentheorie,  1913,  304-5. 


420  History  of  the  Theory  of  Numbers.  [Chap,  xvin 

R.  D.  CannichaeP  for  p*n  — 1  (p  an  odd  prime)  and  2*-3m  — 1. 
M.  Bauer's^  paper  was  not  available  for  report. 

Polynomials  Representing  Numerous  Primes. 

Chr.  Goldbach^^"  noted  that  a  polynomial  fix)  cannot  represent  primes 
exclusively,  since  the  constant  term  would  be  unit}'-,  whereas  it  is  f{p) 
in  /(x+p). 

L.  Euler^"^  proved  this  by  noting  that,  if /(«)  =A,f(nA-\-a)  is  divisible 
by  A. 

Euler^°-  noted  that  x^  — t+41  is  a  prime  for  a:  =  1, .  . . ,  40. 

Euler^"^  noted  that  a:- +2:+ 17  is  a  prime  for  x  =  0,  1, .  .  .,  15  and  [error] 
16;  x'^+x+41  is  a  prime  for  x  =  0,  1,  . .  .,  15. 

A.  M.  Legendre^^  noted  that  x^+a:+41  is  a  prime  for  x  =  0,  1,. . .,  39, 
that  2x^+29  is  a  prime  for  x  =  0,  1, .  .  .,  28,  and  gave  a  method  of  finding 
such  functions.  [Replacing  x  by  x+1  in  Euler's^°-  function,  we  get 
X-+X+41.]  If  j8^-|-2(a+i3)x  — 13x^  is  a  square  only  when  x=0,  and  a  and 
j3  are  relatively  prime,  then  a^+2aj3+14/3-  is  a  prime  or  double  a  prime. 
He  gave  many  such  results. 

Chabert^"^"  stated  that  37i^-f  3n+l  represents  many  primes  for  n  small. 

G.  01tramare^°^  noted  that  x^+ax+6  has  no  prime  divisor  ^/z  and 
hence  is  a  prime  when  <fjr,  if  a^—4b  is  a  quadratic  non-residue  of  each  of 
the  primes  2,  3, .  .  .,  ix.  The  function  x^+ax+(a^+163)/4  is  suitable  to 
represent  a  series  of  primes.  Taking  x  =  0,  a=ii/v,  he  stated  that  u^+lQSv^ 
or  its  quotient  by  4  gives  more  than  100  primes  between  40  and  1763. 

H.  LeLasseur^"^  verified  that,  for  a  prime  A  between  41  and  54000, 
x^+x+A  does  not  represent  primes  exclusively  for  x  =  0,  1,.  .  .,  A— 2. 

E.  B.  Escott^""  noted  that  x"+x+41  gives  primes  not  only  forx  =  0,  1, 
. .  . ,  39,  but  also^*^^  for  x=  —  1,  —  2, .  .  . ,  —40.  Hence,  replacing  x  by  x— 40, 
we  get  x^  — 79x  +  1601,  a  prime  for  x  =  0,  1, .  . .,  79.  Several  such  functions 
are  given. 

Escott^"^  examined  values  of  A  much  exceeding  54000  in  x^+x+A 
without  finding  a  suitable  A>41.  Legendre's^*^  first  seven  formulas  for 
primes  give  composite  numbers  for  a  =  2,  the  eighth  for  a  =  3,  etc.  Escott 
foundthatx^+x^+17isaprimeforx=  — 14,  — 13, . .  ., +10.  Inx^— x^  — 17 
replace  x  by  x  — 10;  we  get  a  cubic  which  is  a  prime  for  x  =  0,  1, .  .  . ,  24. 

"Annals  of  Math.,  (2),  15,  1913,  63-5.  ^Archiv  Math.  Phvs.,  (3),  25,  1916,  131^. 

""Corresp.  Math.  Phys.  (ed.,  Fuss),  I,  1843,  595,  letter  to  Euler,  Nov.  18,  1752. 

'oiNovi  Comm.  Acad.  Petrop.,  9,  1762-3,  99;  Comm.  Arith.,  1,  357. 

i^M^m.  de  Berlin,  ann^e  1772,  36;  Comm.  Arith.,  1,  584. 

"'K)pera  postuma,  I,  1862,  185.     In  Pascal's  Repertorium  Hoheren  Math.,  German  transl.  by 

Schepp,  1900, 1,  518,  it  is  stated  incorrectly  to  be  a  prime  for  the  first  17  values  of  x;  Uke- 

wise  by  Legendre,  Th^orie  des  nombres,  1798,  10;  i808,  11. 
iMTh^oric  des  nombres,  1798,  10,  304-312;  ed.  2,  1808,  11,  279-285;  ed.  3,  1830,  I,  248-255; 

German  transl.  by  Maser,  I,  322-9.  '<»<'Nouv.  Ann.  Math.,  3,  1844,  250. 

i^M^m.  rinat.  Nat.  Genevois,  5,  1857,  No.  2,  7  pp. 

•"Nouv.  Corresp.  Math.,  5,  1879,  371;  quoted  in  rinterm^diaire  des  math.,  5,  1898,  114-5. 
'"L'intermddiaire  des  math.,  6,  1899,  10-11. 

"•The  same  40  primes  as  for  x=0, . . .,  39,  as  noted  by  G.  Lemaire,  ibid.,  16,  1909,  p.  197. 
'"/Wd.,  17,  1910,  271. 


Chap.  XVIII]  GoLDBACH's    THEOREM.  421 

E.  Miot^i^  stated  that  x^- 2999a; +2248541  is  a  prime  for  1460^0;^  1539. 

G.  Frobenius"^  proved  that  the  value  of  x^+xy+py^  is  a  prime  if  <p^, 
that  of  2x^+py^  (y  odd)  if  <p(2p+l),  that  of  x'^+2py^  (x  odd)  if  <p(p-\-2), 
and  noted  cases  in  which  an  indefinite  form  x^-\-xy—qy^  is  a  prime. 

Levy^^  examined  x^—x  —  1.  He"^  considered  f{x)  =  ax^-{-ahx-{-c,  where 
a,  h,  c  are  integers,  0^a<4.  Giving  to  x  the  values  0,  1,  2,  .  .  .,  we  get  a 
set  of  integers  such  that,  for  every  n  exceeding  a  certain  value,  f{n)  is 
either  prime  or  admits  a  prime  factor  which  divides  a  number  f(p),  where 
p<n.  For  example,  if  for  f{x)  =a;^  — a:+41  we  grant  that  /(O),  /(I),  /(2), 
/(3)  and  /(4)  are  primes,  we  can  conclude  that  f{x)  is  prime  for  x^^O. 
Likewise  when  41  is  replaced  by  11  or  17.  Again,  2a;^  — 2a:+19  and  3x^  — 3a; 
+23  give  successions  of  18  and  22  primes  respectively.  Bouniakowsky^^ 
of  Ch.  XI  considered  polynomials  which  represent  an  infinitude  of  primes. 

Braun^^"  proved  that  there  exists  no  quotient  of  two  polynomials  such 
that  the  greatest  integer  contained  in  its  numerical  value  is  a  prime  for  all 
integral  values  >  A;  of  the  variable. 

Goldbach's  Empirical  Theorem:  Every  Even  Integer  is  a  Sum  of 

Two  Primes. 

Chr.  Goldbach^^°  conjectured  that  every  number  N  which  is  a  sum  of  two 
primes  is  a  sum  of  as  many  primes  including  unity  as  one  wishes  (up  to  N), 
and  that  every  number  >2  is  a  sum  of  three  primes. 

L.  Euler^^^  remarked  that  the  first  conjecture  can  be  confirmed  from  an 
observation  previously  communicated  to  him  by  Goldbach  that  every  even 
number  is  a  sum  of  two  primes.  Euler  expressed  his  belief  in  the  last  state- 
ment, though  he  could  not  prove  it.  From  it  would  follow  that,  if  n  is 
even,  n,  n  —  2,  n  —  4:,.  .  .  are  the  sums  of  two  primes  and  hence  n  a  sum  of 
3,  4,  5, .  .  .  primes. 

R.  Descartes^^^  stated  that  every  even  number  is  a  sum  of  1,  2  or  3 
primes. 

E.  Waring^^^  stated  Goldbach's  theorem  and  added  that  every  odd 
number  is  either  a  prime  or  is  a  suiri  of  three  primes. 

L.  Euler^^^  stated  without  profbf  that  every  number  of  the  form  4n+2 
is  a  sum  of  two  primes  each  of  the  form  4A;+1,  and  verified  this  for  4n+2 
^110. 

"OL'intermediaire  des  math.,  19,  1912,  36.     [From  X2+X+41  by  setting  X=x-1500.] 

"iSitz.  Ak.  Wiss.  Berlin,  1912,  966-980. 

"^Bull.  Soc.  Math.  France,  1911,  Comptes  Rendus  des  Seances.     Extract  in  Sphinx-Oedipe, 

9,  1914,  6-7. 
i^oCorresp.  Math.  Phys.  (ed.,  P.  H.  Fuss),  1,  1843,  p.  127  and  footnote;  letter  to  Euler,  June  7, 

1742. 
i"/&id.,  p.  135;  letter  to  Goldbach,  June  30,  1742.     Cited  by  G.  Enestrom,  Bull.  Bibl.  Storia  Sc. 

Mat.  e  Fis.,  18,  1885,  468. 
^^'Posthumous  manuscript,  Oeuvres,  10,  298. 
i23Meditationes  Algebraicae,  1770,  217;  ed.  3,  1782,  379.     The  theorem  was  ascribed  to  Waring 

by  O.  Terquem,  Nouv.  Ann.  Math.,  18,  1859,  Bull.  Bibl.  Hist.,  p.  2;  by  E.  Catalan,  Bull. 

Bibl.  Storia  Sc.  Mat.  e  Fis.,  18,  1885,  467;  and  by  Lucas,  Th^orie  des  Nombres,  1891,  353. 
i2^Acta  Acad.  Petrop.,  4,  II,  1780  (1775),  38;  Comm.  Arith.  Coll.,  2,  1849,  135. 


422  History  of  the  Theory  of  Numbers.  [Chap,  xviii 

A.  Desboves^^^  verified  that  every  even  number  between  2  and  10000  is  a 
sum  of  two  primes  in  at  least  two  ways;  while,  if  the  even  number  is  the 
double  of  an  odd  number,  it  is  simultaneously  a  sum  of  two  piimes  of  the 
form  4n+l  and  also  a  sum  of  two  primes  of  the  form  4n  — 1. 

J.  J.  Sylvester^ ^^  stated  that  the  number  of  ways  of  expressing  a  very 
large  even  number  n  as  a  sum  of  two  primes  is  approximately  the  ratio  of  the 
square  of  the  number  of  primes  <  n  to  n,  and  hence  bears  a  finite  ratio  to  the 
quotient  of  n  by  the  square  of  the  natural  logarithm  of  n.     [Cf.  Stackel"^]. 

F.  J.  E.  Lionnet^-'  designated  by  x  the  number  of  ways  2a  can  be 
expressed  as  a  sum  of  two  odd  primes,  by  y  the  number  of  ways  2a  can  be 
expressed  as  a  sum  of  two  distinct  odd  composite  numbers,  by  z  the  number 
of  odd  primes  <2a,  and  by  q  the  largest  integer  ^a/2.  He  proved  that 
q^x  =  y-\-z  and  argued  that  it  is  very  probable  that  there  are  values  of  n 
for  which  q  =  y-\-z,  whence  a:  =  0. 

N.  V.  Bougaief  ^^^"  noted  that,  if  M{n)  denotes  the  number  of  ways  n  can 
be  expressed  as  a  simi  of  two  primes,  and  if  Oi  denotes  the  ith.  prime  >1, 

S(n-3^.)ilf(n-^.)=0. 

G.  Cantor^^^  verified  Goldbach's  theorem  up  to  1000.  His  table  gives 
the  number  of  decompositions  of  each  even  number  <  1000  as  a  sum  of  two 
primes  and  lists  the  smaller  prime. 

V.  Aubry^29  verified  the  theorem  from  1002  to  2000. 

R.  Haussner^^°  verified  the  law  up  to  10000  and  announced  results 
observed  by  a  study  of  his^"  tables  up  to  5000.  His  table  I  (pp.  25-178) 
gives  the  number  v  of  decompositions  of  every  even  n  up  to  3000  as  a  sum 
x-\-y  oi  two  primes  and  the  values  oi  x  (x^y),  as  in  the  table  by  Cantor. 
His  table  II  (pp.  181-191)  gives  v  for  2<n<5000;  this  table  and  further 
computations  enable  him  to  state  that  Goldbach's  theorem  is  true  for 
n<  10000.  Let  P(2p+1)  be  the  number  of  all  odd  primes  1,  3,  5, . . .  which 
are  ^2p+l,  and  set 

^(2p+l)=P(2p+l)-2P(2p-l)+P(2p-3),  P(-l)=P(-3)=0. 

Then  the  number  of  decompositions  of  2n  into  a  sum  of  two  primes  x,  y 

{x^y)  is  „_i 

S  P(2n-2p-l)^(2p+l). 

If  e  =  1  or  —  1  according  as  n  is  a  prime  or  not, 

»/  =  iSP(2n-2p-l)^(2p+l)+^- 


■1 


2 


i»Nouv.  Ann.  Math.,  14,  1855,  293. 

i"Proc.  London  Math.  Soc,  4,  1871-3,  4-6;  CoU.  M.  Papers,  2,  709-711. 

>"Nouv.  Ann.  Math.,  (2),  18,  1879,  356.     Cf.  Assoc,  frang.  av.  ec,  1894,  I,  p.  96. 

""KDomptes  Rendus  Paris,  100,  1885,  1124. 

"'Assoc,  fran?.  av.  sc,  1894,  117-134;  rinterm^diaire  des  math.,  2,  1895,  179. 

"»L'interm6diaire  des  math.,  3,  1896,  75;  4,  1897,  60;  10,  1903,  61  (errata,  p.  166,  p.  283). 

"•Jahresbericht  Deutschen  Math.-Verein.,  5,  1896,  62-66.     Verhandlungen  Gesell.  Deutscher 

Naturforscher  u.  Aerzte,  1896,  II,  8. 
»"Nova  Acta  Acad.  Caes.  Leop.-Carolinae,  72,  1899,  1-214. 


Chap.  XVIII]  GoLDBACh's  THEOREM.  423 

Table  III  gives  the  values  of  P  and  ^  for  each  odd  number  2p+l<5000. 
P.  StackeP^^  noted  that  Lionnet's^"  argument  is  not  conclusive,  and 
designated  by  G2r,  the  number  of  all  decompositions  of  2n  as  a  sum  of  two 
primes  (counting  p+g  and  g+p  as  two  different  decompositions).  If 
Pk  is  the  number  of  all  odd  primes  from  1  to  k, 

I  G2y'^  =  {l^x^f  =  {l-x^Y(%P,^^,:^''A\ 

where  p  ranges  over  all  the  odd  primes.    Approximations  to  G^2n  for  n  large 
in  terms  of  Euler's  0-function  are 

[P(2n- V2n)  -P{-V2^)f     n 


2n 


0(2n)  n-y/2n  <t>{2n) 

where  P{k)  is  written  for  P^  for  convenience  in  printing.  Lack  of  agree- 
ment with  Sylvester ^2^  is  noted;  cf.  Landau. ^^^  It  is  stated  that  the 
truth  of  Goldbach's  theorem  is  made  very  probable  [but  not  proved^^^]. 

Sylvester^^^"  stated  that  any  even  integer  2n  is  a  sum  of  two  primes,  one 
>  n/2  and  the  other  <  3n/2,  whence  it  is  possible  to  find  two  primes  whose 
difference  is  less  than  any  given  number  and  whose  sum  is  twice  that  number. 

F.  J.  Studnicka^^*  discussed  Sylvester's  statement. 

Sylvester^^**"  stated  that,  if  N  is  even  and  X, .  .  . ,  co  are  the  6  primes  >  \N 
and  <fA^  (excluding  ^A^"  if  it  be  prime),  the  number  of  ways  of  composing 
N  [by  addition]  with  two  of  these  primes  is  the  coefficient  of  x^  in 

(l^+-+T^)Vr(.-l)r-  (rfe2). 

E.  Landau^^^  noted  that  Stackel's  approximation  to  G„  is 


mn= 


n^ 


log^  n0(n) 

and  showed  that  S^=i(t„  has  the  true  approximation  ^x^/\o^x.  By  a  longer 
analysis,  he  proved  that  if  we  use  Stackel's  (SJ„  to  form  the  sum,  we  do  not 
obtain  a  result  of  the  correct  order  of  magnitude. 

L.  Ripert^^^  examined  certain  large  even  numbers. 

E.  Maillet^"  proved  that  every  even  number  ^350000  (or  10^  or  9-10®) 
is,  in  default  by  at  most  6  (or  8  or  14),  the  sum  of  two  primes. 

A.  Cunningham^^^  verified  Goldbach's  theorem  for  all  numbers  up  to 
200  million  which  are  of  the  forms 

(4-3)",     (4-5)",     2•10^     2'»(2"=f1),     a-2",     2a",     (2a)^     2(2^=Fa), 

for  a  =  1,  3,  5,  7,  9,  11.  He  reduced  the  formula  of  Haussner  for  i'  to  a  form 
more  convenient  for  computation. 

"2G6ttingen  Nachrichten,  1896,  292-9.  "'Encyclop6die  des  sc.  math.,  I,  17,  p.  339,  top. 

i33aNature,  55,  1896-7,  196,  269.  »"Casopis,  Prag,  26,  1897,  207-8. 

"40Educ.  Times,  Jan.  1897.     Proof  by  J.  Hammond,  Math.  Quest.  Educ.  Times,  26,  1914,  100. 

"^Gottingen  Nachrichten,  1900,  177-186. 

"«L'interm6diaire  des  math.,  10,  1903,  67,  74,  166  (errors,  p.  168). 

»"/6id.,  12,  1905,  107-9.  »"Messenger  Math.,  36,  1906,  17-30. 


^ 


f 


424  History  of  the  Theory  of  Numbers.  [Chap,  xviii 

J.  Merlin^^'  considered  the  operation  A (6,  a)  of  effacing  from  the  natural 
series  of  integers  all  the  numbers  ax-\-b.  The  effect  of  carrying  out  one  of 
the  two  sets  of  operations  A{ri,  pi),  ^(r,-,  p,),  A{r'i,  pi),  i  =  2,.  .  .,  n,  where 
p„  is  the  nth  prime  >  1,  is  equivalent  to  constructing  a  crib  of  Eratosthenes 
up  to  p„.  It  is  stated  that  in  every  interval  of  length  vp^  log  p„  there  is  at 
least  one  number  not  effaced,  if  v  is  independent  of  n.  It  is  said  to  follow 
that,  for  a  sufficiently  large,  there  exist  two  primes  having  the  sum  2a. 
Under  specified  assumptions,  there  exist  an  infinitude  of  n's  for  which 
Pn+i-Pn  =  2. 

M.  Vecchi^*°  wrote  Pn  for  the  nth  odd  prime  and  called  p^  and  p/,+„  of 
the  same  order  if  p^h>Ph+a-  Then  2n>  132  is  a  sum  of  two  primes  of  the 
same  order  in  [K0+1)]  ways  if  and  only  if  there  exist  </>  numbers  not 
>  /I  — p^+i  +  1  and  not  representable  in  any  of  the  forms 

Gi+Sx,  bi+5x,...,  li+PmX  (i=l,  2), 

where  p^+i  is  the  least  prime  p  for  which  p'^-\-p>  2n,  and  the  known  terms  a., 
. . .  are  the  residues  with  respect  to  the  odd  prime  occurring  as  coefficient  of  x. 
*G.  Giovannelli,  Sul  teorema  di  Goldbach,  Atri,  1913. 

Theorems  Analogous  to  Goldbach's. 

Chr.  Goldbach ^^^  stated  empirically  that  every  odd  number  is  of  the 
form  p-\-2a^,  where  p  is  a  prime  and  a  is  an  integer  ^  0.  L.  Euler^^^  verified 
this  up  to  2500.  Euler^24  verified  for  m  =  8iV+3^  187  that  m  is  the  sum 
of  an  odd  square  and  the  double  of  a  prime  4n+l. 

J.  L.  Lagrange ^"^^  announced  the  empirical  theorem  that  every  prime 
4n  — 1  is  a  sum  of  a  prime  4?n  +  l  and  the  double  of  a  prime  4/i  +  l. 

A.  de  Polignac^^^  conjectured  that  every  even  number  is  the  difference 
of  two  consecutive  primes  in  an  infinitude  of  ways.  His  verification  up 
to  3  million  that  every  odd  number  is  the  sum  of  a  prime  and  a  power  of 
2  was  later  "^^  admitted  to  be  in  error  for  959. 

M.  A.  Stern^^^  and  his  students  found  that  53-109  =  5777  and  13-641 
=  5993  are  neither  of  the  form  p-\-2a^  and  verified  that  up  to  9000  there  are 
no  further  exceptions  to  Goldbach's^'*^  assertion.  Also,  17,  137,  227,  977, 
1187  and  1493  are  the  only  primes  <9000  not  of  the  form  p+26^  6>0. 
Thus  all  odd  numbers  <9000,  which  are  not  of  the  form  6n+5,  are  of  the 
form  p+26^. 

E.  Lemoine^^''  stated  empirically  that  every  odd  number  >3  is  a  sum 
of  a  prime  p  and  the  double  of  a  prime  tt,  and  is  also  of  the  forms  p  —  2Tr 
and  27r'  — p'. 

'"Comptes  Rendus  Paris,  153,  1911,  516-8.     Bull.  des.  sc.  math.,  (2),  39,  I,  1915,  121-136.     In 

a  prefatory  note,  J.  Hadamard  noted  that,  while  the  proof  has  a  lacuna,  it  is  suggestive. 
""Atti  Reale  Accad.  Lincei,  Rendiconti,  (5),  22,  II,  1913,  654-9. 
'"Corresp.  Math.  Phys.  (ed..  Fuss),  1,  1843,  595;  letter  to  Euler,  Nov.  18,  1752. 
»«/W<i.,  p.  596,  606;  Dec.  16,  1752. 

"'Nouv.  M6ni.  Ac.  Berlin,  ann6e  1775,  1777,  356;  Oeuvres,  3,  795. 
"8Nouv.  Ann.  Math.,  8,  1849,  428  (14,  1855,  118). 
»«»''Comptes  Rendus  Paris,  29,  1849,  400,  738-9. 
"»Nouv.  Ann.  Math.,  15, 1856,  23.  ""L'interm^diaire  des  math.,  1,  1894,  179;  3, 1896,  151 


Chap.  XVIII]  PRIMES   IN   ARITHMETICAL   PROGRESSION.  425 

H.  Brocard^^^  gave  an  incorrect  argument  by  use  of  Bertrand's  postulate 
that  there  exists  a  prime  between  any  two  consecutive  triangular  numbers. 

G.  de  Rocquigny^^^  remarked  that  it  seems  true  that  every  multiple  of 
6  is  the  difference  of  two  primes  of  the  form  6n+l. 

Brocard^^^  verified  this  property  for  a  wide  range  of  values. 

L.  Kronecker^^^  remarked  that  an  unnamed  writer'^^  had  stated  empiri- 
cally that  every  even  number  can  be  expressed  in  an  infinitude  of  ways  as 
the  difference  of  two  primes.  Taking  2  as  the  number,  we  conclude  that 
there  exist  an  infinitude  of  pairs  of  primes  differing  by  2. 

L.  Ripert^^^  verified  that  every  even  number  <  10000  is  a  sum  of  a  prime 
and  a  power,  every  odd  one  except  1549  is  such  a  sum. 

E.  Maillet^^^  commented  on  de  Polignac's  conjecture  that  every  even 
number  is  the  difference  of  two  primes. 

E.  Maillet^"  proved  that  every  odd  number  <  60000  (or  9-10^)  is,  in 
default  by  at  most  8  (or  14),  the  sum  of  a  prime  and  the  double  of  a  prime. 

Primes  in  Arithmetical  Progression. 

E.  Waring^^^  stated  that  if  three  primes  (the  first  of  which  is  not  3)  are 
in  arithmetical  progression,  the  common  difference  d  is  divisible  by  6, 
except  for  the  series  1,  2,  3  and  1,  3,  5.  For  5  primes,  the  first  of  which  is 
not  5,  d  is  divisible  by  30;  for  7  primes,  the  first  not  7,  d  is  divisible  by 
2-3-5-7;  for  11  primes,  the  first  not  11,  d  is  divisible  by  2-3-5-7-11;  and 
similarly  for  any  prime  number  of  primes  in  arithmetical  progression,  a 
property  easily  proved.  Hence  by  continually  adding  d  to  a  prime,  we 
reach  a  number  divisible  by  3,  5, ... ,  unless  d  is  divisible  by  3,  5, .  .  . . 

J.  L.  Lagrange ^^"^  proved  that  if  3  primes,  no  one  being  3,  are  in  arith- 
metical progression,  the  difference  d  is  divisible  by  6;  for  5  primes,  no  one 
being  5,  d  is  divisible  by  30.  He  stated  that  for  7  primes,  d  is  divisible  by 
2-3-5'7,  unless  the  first  one  is  7,  and  then  there  are  not  more  than  7  consecu- 
tive prime  terms  in  a  progression  whose  difference  is  not  divisible  by  2-3 -5 -7. 

E.  Mathieu^^^  proved  Waring's  statement. 

M.  Cantor^ ®^  proved  that  if  P  =  2-3.  .  .p  is  the  product  of  all  the  primes 
up  to  the  prime  p,  there  is  no  arithmetical  progression  of  p  primes,  no  one 
of  which  is  p,  unless  the  common  difference  is  divisible  by  P.  He  conjec- 
tured that  three  successive  primes  are  not  in  arithmetical  progression  unless 
one  of  them  is  3. 

A.  Guibert^®^  gave  a  short  proof  of  the  theorem  stated  thus:  Let 
Pi,...,Pn  be  primes  ^  1  in  arithmetical  progression,  where  n  is  odd  and  >3. 
Then  no  prime  >1  and  ^n  is  a  Pi.     If  n  is  a  prime  and  is  a  pi,  then  i  =  l. 

i"L'intermediaire  des  math.,  4,  1897,  159.     Criticism  by  E.  Landau,  20,  1913,  153. 

is^/feid.,  5,  1898,  268.  i^L'intermediaire  des  math.,  6,  1899,  144. 

i64Vorlesungen  iiber  Zahlentheorie,  1,  1901,  68. 

i«L'interm6diaire  des  math.,  10,  1903,  217-8.  i"/6td.,  12,  1905,  108. 

^"Ibid.,  13,  1906,  9.  i«*Meditationes  Algebraicae,  1770;  ed.  3,  1782,  379. 

i««Nouv.  M6m.  Ac.  Berlin,  ann^e  1771,  1773,  134-7.  »"Nouv.  Ann.  Math.,  19,  1860,  384-5. 

"sZeitschrift  Math.  Phys.,  6,  1861,  340-3. 

"»Jour.  de  Math.,  (2),  7,  1862,  414-6. 


426  History  of  the  Theory  of  Numbers.  [Chap,  xviii 

The  common  difference  is  divisible  by  each  prime  ^n,  and  by  n  itself  if  n  is 
a  prime  not  in  the  series. 

H.  Brocard^^^"  gave  several  sets  of  five  consecutive  odd  integers,  four  of 
which  are  primes.  Lionnet^^^**  had  asked  if  the  number  of  such  sets  is  un- 
limited. 

G.  Lemaire^^°  noted  that  7+30n  and  107+30/1  (n  =  0,  1,. .  .,  5)  are  all 
primes;  also  7  +  150/1  and  47+210m  (n  =  0, .  .  .,  6). 

E.  B.  Escott^^^  found  conditions  that  a+210M  (n  =  0,  1,. .  .,  9)  be  all 
primes  and  noted  that  the  conditions  are  satisfied  if  a  =  199. 

De\'ignot^"  noted  the  primes  47+210«,  71+2310?i  (n  =  0,  1,.  . .,  6). 

A.  Martin^^^  gave  numerous  sets  of  primes  in  arithmetical  progression. 

Tests  for  Primality. 

The  fact  that  n  is  a  prime  if  and  only  if  it  divides  1  +  (n  —  1) !  was  noted 
by  Leibniz/  Lagrange, ^^  Genty,-^  Lebesgue,^^  and  Catalan/^^  cited  in  Chap- 
ter III,  where  was  discussed  the  converse  of  Fermat's  theorem  in  furnishing 
a  primality  test.  Tests  by  Lucas,  etc.,  were  noted  in  Ch.  XVII.  Further 
tests  have  been  noted  under  Cipolla^"  and^'^  Cole^"  of  Ch.  I,  Sardi"^  of 
Ch.  Ill,  Lambert^  of  Ch.  VI,  Zsigmondv"^  of  Ch.  VII,  Gegenbauer^o- »2  ^f 
Ch.  X,  Jolivald^  of  Ch.  XIII,  Euler,^^^  Tchebychef,"^  Schaffgotsch^""^  and 
Biddle^^^  of  Ch.  XIV,  Hurwitz"  and  CipoUa^^  of  Ch.  XV.  See  also  the 
papers  by  von  Koch,^^^  Hayashi,^^'  ^°  Andreoli,^"^  and  Petrovitch^  of  the 
next  section. 

L.  Euler^^^  gave  a  test  for  the  primality  of  a  number  N  =  4:m-\-l  which 
ends  with  3  or  7.  Let  R  be  the  remainder  on  subtracting  from  2N  the  next 
smaller  square  (5n)-  which  ends  with  5.  To  R  add  100(n  — 1),  100(n  — 3), 
100(n  — 5),  ....  If  among  R  and  these  sums  there  occurs  a  single  square, 
iV  is  a  prime  or  is  divisible  by  this  square.  But  if  no  square  occurs  or  if 
two  or  more  squares  occur,  A^  is  composite.  For  example,  if  A  =  637, 
(5n)2  =  1225,  R  =  49;  among  49,  649,  1049,  1249  occurs  only  the  square  49; 
hence  A^  is  a  prime  or  is  di\asible  by  49  [A  =  49- 13]. 

W.  L.  Kraft^"^  noted  that  Qin  + 1  is  a  prime  if  m  is  of  neither  of  the  forms 
Qxy=^{x+y);  6?m  — 1  is  a  prime  if  m9^6xy+x  —  y. 

A.  S.  de  Montferrier^'^  noted  that  an  odd  number  A  is  a  prime  if  and 
only  if  A-\-k'  is  not  a  square  for  A:  =  1,  2,  .  .  .,  (A  —  3)/2. 

AI.  A.  Stern^^°  noted  that  n  is  a  prime  if  and  only  if  it  occurs  n  — 1  times 
in  the  (n  — l)th  set,  where  the  first  set  is  1,  2,  1;  the  second  set,  formed  by 
inserting  between  any  two  terms  of  the  first  set  their  sum,  is  1,  3,  2,  3,  1 ;  etc. 

"'"Xouv.  Ann.  Math.,  (3),  15,  1896,  389-90.  i696Nouv.  Ann.  Math.,  (3),  1,  1882,  336. 

""L'intcrraddiaire  des  math.,  16,  1909,  194-5. 

"'Ibid.,  17,  1910,  285-6. 

"*Jbid.,  4.5-6.  ^"School  Science  and  Mathematics,  13,  1913,  793-7, 

'"*Doubt  as  to  the  suflBciency  of  Cole's  test  has  been  expressed,  Proc.  London  Math.  Soc,  (2). 

16,  1917-  8.  i"Opera  postuma,  I,  188-9  (about  1778). 

"»\ova  Acta  Acad.  Petrop.,  12,  1801,  hist.,  p.  76,  mem.,  p.  217. 
"•Corresp.  Math.  Phys.  (ed.,  Quetelet),  5,  1829,  94-6. 
""Jour,  fiir  Math.,  55,  1858,  202. 


1 


Chap.  XVIII]  Tests  FOR  Primality.  -         427 

L.  Gegenbauer^^^  noted  that  4n+l  is  a  prime  if 

L       4i/       J"L       4?/        J 

for  every  odd  y,  1<?/^  V4n+1,  and  gave  two  similar  tests  for  4n+3. 

D.  Gambioli^^^  and  O.  Meissner^^^  discussed  the  impracticabiUty  of  the 
test  by  the  converse  of  Wilson's  theorem. 

J.  Hacks^^^  gave  the  characteristic  relations  for  primes  p: 

K.  Zsigmondy^^^  noted  that  a  number  is  a  prime  if  and  only  if  not 
expressible  in  the  form  aia2+i3i/32,  where  the  a's  and  j3's  are  positive  integers 
such  that  ai+a2=7/3i— 1S2.  An  odd  number  C  is  a  prime  if  and  only  if 
C-\-k^  is  not  a  square  for  k  =  0,  1,.  .  .,  [(C-9)/6]. 

R.  D.  von  Sterneck^^^"  gave  several  criteria  for  the  (s  +  l)th  prime  by  use 
of  partitions  into  elements  formed  from  the  first  s  primes. 

H.  Laurent^^^*  noted  that 

equals  0  or  1  according  as  z  is  composite  or  prime. 

Fontebasso^^®  noted  that  A^  is  a  prime  if  not  divisible  by  one  of  the 
primes  2,  3, . . .,  p,  where  iV/p<p+4. 

H.  Laurent  ^^^  proved  that  if  we  divide 

Fnix)  =  n\l  -a:0(l  -x^^) .  . .  (1  -x^""^'-'') 

3  =  1 

by  (x"  — l)/(a:  — 1),  the  remainder  is  0  or  n"~^  according  as  n  is  composite  or 
prime.  If  we  take  a:  to  be  an  imaginary  root  of  a;"  =  1,  Fn{x)  becomes  0  or 
n"~^  in  the  respective  cases. 

Helge  von  Koch^^^  used  infinite  series  to  test  whether  or  not  a  number  is  a 
power  of  a  prime. 

Ph.  Jolivald^^^  noted  that,  since  every  odd  composite  number  is  the 
difference  of  two  triangular  numbers,  an  odd  number  iV  is  a  prime  if  and 
only  if  there  is  no  odd  square,  with  a  root  ^  (2iV  — 9)/3,  which  increased  by 
8A^  gives  a  square. 

S.  Minetola^^"  noted  that,  if  k—n  is  divisible  by  2n  +  l,  then  2A;+1  is 
composite.  We  may  terminate  the  examination  when  we  reach  a  prime 
2n+l  for  which  {k-n)/{2n+l)^n. 

A.  Bindoni^^^  added  that  we  may  stop  with  a  prime  giving  (k—n) 

isiSitzungsber.  Ak.  Wiss.  Wien  (Math.),  99,  Ila,  1890,  389. 

is^Periodico  di  Mat.,  13,  1898,  208-212.  '"Math.  Naturw.  Blatter,  3,  1906,  100 

>8*Acta  Mathematica,  17,  1893,  205.  ""Monatsh.  Math.  Phys.,  5,  1894,  123-8. 

"S'^Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  105,  Ila,  1896,  877-882. 

iss^Comptes  Rendus  Paris,  126,  1898,  809-810.  "^guppj^  Periodico  di  Mat.,  1899,  53. 

i"Nouv.  Ann.  Math.,  (3),  18,  1899,  234-241. 

"sOfversigt  Veten.-Akad.  Forhand.,  57,  1900,  789-794  (French). 

"9L'interm4diaire  des  math.,  9,  1902,  96;  10,  1903,  20. 

""11  Boll.  Matematica  Giorn.  Sc.-Didat.,  Bologna,  6,  1907,  100-4.  i"/6td.,  165-6. 


428  History  of  the  Theory  of  Numbers.  [Ch.u'.  xviii 

-^  (2n+l)^n+2a  —  1,  where  a  is  the  difiference  between  2/1+1  and  the  next 
greater  prime. 

F.  Stasi^^-  noted  that  iV  is  a  prime  if  not  divisible  by  one  of  the  primes 
2,  3, . . .,  p,  where  N/p<p-\-2a  and  a  is  the  difference  between  p  and  the 
prime  just  >p. 

E.  Zondadari^^^  noted  that 

Sin-TTX  *  TTX 


(7rx)^(l— x^)^n=2W  sin  irx/n 

is  zero  when  j  =  ±  p  (pa  prime)  and  not  otherwise. 

A.  Chiari'^"*  cited  known  tests  for  primes,  as  the  converse  of  Wilson's 
theorem. 

H.  C.  PockHngton^^^  employed  single  valued  functions  (}>(x),  rp(x), 
vanishing  for  all  positive  integers  a:  (as  0  =  i/'  =  sin  ttx),  and  real,  finite  and  not 
zero  for  all  other  positive  values  of  x.     Then,  for  the  gamma  function  F, 


^.(,)+^.(i±IM) 


is  zero  if  and  only  if  x  is  a  prime  [Wigert^^^"]. 

E.  B.  Escott^^^  stated  that  if  we  choose  ai, . .  . ,  a„,  b  so  that  the  coeflBicients 
of  x^",  x^"~^, .  . . ,  a:^  in  the  expansion  of 

(x"+aix'*-^+...+aj2(x+5) 

are  all  zero,  then  all  the  remaining  coefficients,  other  than  the  first  and  last, 
are  divisible  by  2?i  +  l  if  and  only  if  2/i  +  l  is  a  prime. 

J.  de  Barinaga^^^  concluded  from  Wilson's  theorem  that  if  (P  — 1)!  is 
divided  by  1+2+  .  .  .  +(P-1)  =P(P-l)/2,  the  remainder  is  P-1  when  P 
is  a  prime,  but  is  zero  when  P  is  composite  (not  excluding  P  =  4  as  in  the  con- 
verse of  Wilson's  theorem).  Hence  on  increasing  by  unity  the  least  positive 
residues  ?^0  obtained  on  di\'iding  1-2.  .  .x  by  1+2+ .  .  .  +x,  for  x=  1,  2, 
3, .  .  . ,  we  obtain  the  successive  odd  primes  3,  5, ... . 

M.  Vecchi^^°  noted  that,  if  x^  1,  A^>2  is  a  prime  if  and  only  if  it  be  of 
the  form  2^'— tt,  where  tt  is  the  product  of  all  odd  primes  ^p,  p  being 
the  largest  odd  primed  [\/iV],  and  where  tt'  is  a  product  of  powers  of 
primes  >p  with  exponents  ^0.  Again,  A^>  121  is  a  prime  if  and  only  if  of 
the  form  tt  — 2V  where  y^l. 

Vecchi^^^  gave  the  simpler  test:  A^>5  is  a  prime  if  and  only  ii  a—^=N, 
a+j8  =  7r,  for  a,  /3  relatively  prime,  where  tt  is  the  product  of  all  the  odd 
primes  ^[VA'']- 

G.  Rados^^^  noted  that  p  is  a  prime  if  and  only  if  {2!3!  .  . .  (p  — 2)! 
(p-l)!j4=i(modp). 

CarmichaeP^  gave  several  tests  analogous  to  those  by  Lucas. 

i«Il  Boll.  Matiraatica  Giorn.  Sc.-Didat.,  Bologna,  6,  1907.  120-1. 

"'Rend.  Accad.  Lined,  (5),  19, 1910, 1,  319-324.       i»^Il  Pitagora,  Palermo,  17,  1910-11,  31-33. 

"*Proc.  Cambr.  Phil.  Soc.,  16,  1911,  12.  »»«L'interm6diaire  des  math.,  19,  1912,  241-2. 

"'Revista  de  la  Sociedad  Mat.  Espanola,  2,  1912,  17-21. 

"»Periodico  di  Mat.,  29,  1913,  126-8.  "»Math.  6s  Term^s  firtesito,  34,  1916,  62-70" 


i 


Chap.  XVIII]  NUMBER   OF  PRIMES.  429 

Number  of  Primes  Between  Assigned  Limits. 

Formula  (5)  of  Legendre  in  Ch.  V  implies  that  if  0,  X, .  .  .  are  the  primes 
=  Vn,  the  mimber  of  primes  ^  n  and  >  Vn  is  one  less  (if  unity  be  counted  a 
prime)  than 

statements  or  proofs  of  this  result  have  been  given  by  C.  J.  Hargreave,^^* 
E.  de  Jonquieres,206  R.  Lipschitz,207  j.  j.  Sylvester,^^^  E.  Catalan,^^^  F.  Ro- 

gel,^^°  J.  Hammond^^^  with  a  modification,  H.  W.  Curjel,^^^"  S.  Johnsen,^^^ 
and  L.  Kronecker.^^^ 

E.  MeisseP^'*  proved  that  if  d{m)  is  the  number  of  primes  (including 
unity)  ^m  and  if 

*(P.- . . . p."")  =  ( - 1)-+   •+"" ("■+"^+- •■+»»)!, 


rill. .  .n-f 


1 


.$(l)^[f]+<i.(2)e[f]  +  ...+*Wflg]. 


E.  MeisseP^^  wrote  $(m,  n)  for  Legendre's  formula  for  the  number  of 
integers  ^  m  which  are  divisible  by  no  one  of  the  first  n  primes  pi  =  2, . .  . ,  p„. 
Then  .p^-,  x 

<l>(m,  n)  =^{m,  n  —  1)  — «l>  (    —   ,  n  —  1  j  • 

Let^(m)be  the  number  of  primes  ^m.   Setn+fx  =  d{\/m),n==d{^^/m).   Then 

e{m)=^{m,  n)+n(M+l)+^^^~'^^-l-  i  d(^^), 

which  is  used  to  compute  Birri)  for  m  =  ^-10^,  A;  =  1/2,  1,  10. 

MeisseP^^  applied  his  last  formula  to  find  ^(10^). 

Lionnet^^^''  stated  that  the  number  of  primes  between  A  and  2 A  is 
<B{A). 

N.  V.  Bougaief2i7  obtained  from  Q{n)  +^(n/2)  +^(V3)  +  •  •  •  =S[n/p],  by 
inversion  (Ch.  XIX), 

^(^) =49 -^4^] +«4J^]  - -45] +4Je] -<^c] + •  ■  • . 

where  a,  h,.  .  .  range  over  all  primes. 

2«6Lond.  Ed.  Dub.  Phil.  Mag.,  (4),  8,  1854,  118-122. 

^o^Comptes  Rendus  Paris,  95,  1882,  1144,  1343;  96,  1883,  231. 

^oUbid.,  95,  1882,  1344-6;  96,  1883,  58-61,  114-5,  327-9. 

2<>876id.,  96,  1883,  463-5;  Coll.  Math.  Papers,  4,  p.  88. 

2°9Mem.  Soc.  Roy.  Sc.  de  Liege,  (2),  12,  1885,  119;  Melanges  Math.,  1868,  133-5. 

JioArchiv  Math.  Phys.,  (2),  7,  1889,  381-8.  ^iiMessenger  Math.,  20,  1890-1,  182. 

2"«Math.  Quest.  Educ.  Times,  67,  1897,  27. 

"2Nyt  Tidsskrift  for  Mat.,  Kjobenhavn,  15  A,  1904,  41^. 

2"Vorlesungen  uber  Zahlentheorie,  I,  1901,  301-4.  "ujour.  fur  Math.,  48,  1854,  310-4. 

2i5Math.  Ann.,  2,  1870,  636-642.     Outline  in  Mathews'  Theory  of  Numbers,  273-8,  and  in  G. 

Wertheim's  Elemente  der  Zahlentheorie,  1887,  20-25. 
2i676id.,  3,  1871,  523-5.     Corrections,  21,  1883,  304. 
2i6aNouv.  Ann.  Math.,  1872,  190.     Cf.  Landau,  (4),  1,  1901,  281-2. 
21'Bull.  sc.  math,  astr.,  10, 1,  1876,  16.     Mat.  Sbomik  (Math.  Soc.  Moscow),  6,  1872-3,  I,  180. 


430 


History  of  the  Theory  of  Numbers. 


[Chap.  XVIII 


N, 


P.  de  Mond^sir-^^  wrote  Np  for  the  number  of  multiples  of  the  prime  p 
which  are  <  2.V  and  divisible  by  no  prime  <  p.  Then  the  number  of  primes 
<2.V  is  N—1iNp-\-n-\-l,  where  n  is  the  number  of  primes  <y/2N.    Also, 

.p\        Lapj        LahpA  ' 

where  a,  h,.  .  .  are  the  primes  <p.  By  this  modification  of  Legendre's 
formula,  he  computed  the  number  78490  of  primes  under  one  million. 

*L.  Lorenz-^^  discussed  the  number  of  primes  under  a  given  limit. 

Paolo  Paci"''  proved  that  the  number  of  integers  ^n  divisible  by  a 
prime  <\/ri  is 

where  r,  s,.  .  .  range  over  all  the  H  primes  2,  3, .  .  . ,  p  less  than  \/n.  Thus 
there  are  n  —  \  —N-\-H  primes  from  1  to  n.     The  approximate  value  of  N  is 

</>(2-3...p)" 


y  r       rs  }        y         2-3... p    J 


Vs"^'J     '\S       2-3... p 

K.  E.  Hoffmann--^  denoted  by  N  the  number  of  primes  <m,  by  X  the 
number  of  distinct  prime  factors  of  tw,  by  ^i  the  number  of  composite  integers 
<m  and  prime  to  m.  Evidently  N  =  4>{M)  —  ii-\-\.  To  find  A''  it  suffices 
to  determine  ix.  To  that  end  he  would  count  the  products  <mhy  twos, 
by  threes,  etc.  (with  repetitions)  of  the  primes  not  dividing  m. 

J.  P.  Gram-"  proved  that  the  number  of  powers  of  primes  ^n  is 

[Cf.  Bougaief.^^^]     Of  the  two  proofs,  one  is  by  inversion  from 

E.  Cesaro^^  considered  the  number  x  of  primes  ^qn  and  >n,  where 
g  is  a  fixed  prime.  Let  coi, . .  . ,  co,  be  the  primes  ^  n  other  than  1  and  q. 
Let5*^n<g*+\    Then 

Let  Ir,,  be  the  number  of  the  [qn/{o3i .  .  .co,)]  which  give  the  remainder  r  when 
divided  by  q.     Set  t,  =  'Ljlj,,.     Then 

x=(k+l)q-{k  +  2)-U  +  t2-h+ .  .  .. 

"'Assoc,  frang.  av.  sc,  6,  1877,  77-92.     Nouv.  Corresp.  Math.,  6,  1880,  256. 

«»Tidsskr.  for  Math.,  Kjobenhavn,  (4),  2,  1878,  1-3. 

""Sul  numero  de  numeri  primi  inferiori  ad  un  dato  numero,  Parma,  1879,  10  pp. 

»"Archiv  Math.  Phys.,  64,  1879,  333-6. 

«K.  Danake  Vidensk.  Selskabs.  Skrifter,  (6),  2,  1881-6,  183-288;  r6sum6  in  French,  289-308. 

See  pp.  220-8,  296-8. 
»M6m.  Soc.  Sc.  Li^e,  (2),  10,«1883,  287-8. 


Chap.  XVIII]  NuMBER   OF   PRIMES.  431 

E.  Catalan^^'*  obtained  the  preceding  results  for  the  case  g  =  2;  then  ti  is 
the  number  of  odd  quotients  [2n/(8],  (2  the  number  of  odd  quotients 
[2n/{^y)], .  .  . ,  where  I3,y,.  .  .  are  the  primes  >2  and  ^n. 

L.  Gegenbauer225  gave  eight  formulas,  (29)-(36),  of  the  type  of 
Legendre's,  a  special  case  of  one  being 


([^])/^(^)  =  1+L,(n),        SM^^^  t\ 


where  x  ranges  over  the  integers  divisible  by  no  prime  >  Vn,  while  fi(x)  is 
Merten's  function  (Ch.  XIX)  and  Lh{n)  is  the  sum  of  the  kth.  powers  of  all 
primes  >\/n  but  Sn.  The  case  A;  =  0  is  Legendre's  formula.  The  case 
A;  =  l  is  Sylvester's^"^ 

E.  MeisseP^^  computed  the  number  of  primes  <  10^ 
Gegenbauer-^*^"  gave  compUcated  expressions  for  d{n),  one  a  generaliza- 
tion of  Bougaief's.^^'^ 

A.  Lugli^"  wrote  4>{n,  i)  for  the  number  of  integers  ^n  which  are  divis- 
ible by  no  one  of  the  first  i  primes  Pi  =  2,  p2  =  3, .  .  . .  If  2;  is  the  number  of 
primes  S  -y/n  and  if  s  is  the  least  integer  such  that 


the  number  \l/{n)  of  primes  ^n,  excluding  1,  is  proved  to  satisfy 

This  method  of  computing  \l/{n)  is  claimed  to  be  simpler  than  that  by 
Legendre  or  Meissel. 

J.  J.  van  Laar^""  found  the  number  of  primes  <  30030  by  use  of  the 
primes  <1760. 

C.  Hossfeld^^^  gave  a  direct  proof  of 

^{gVi-  ■  ■Pn=^r,n)==g{pi-l).  .  .(p„-l)±$(r,  n), 

the  case  of  the  upper  signs  being  due  to  Meissel. ^^^ 

F.  Rogel"^^  gave  a  modification  and  extension  of  Meissel's^^^  formula. 
H.  Scheffler^^''  discussed  the  number  of  primes  between  p  and  q. 
J.  J.  Sylvester^^^  stated  that  the  number  of  primes  >n  and  <2n  is 

a  ab  abc 

ii  a,h,.  .  .  are  the  primes  ^  \/2n  and  Hx  denotes  x  when  its  fractional  part 

224Mem.  Soc.  Sc.  Lidge,  M6m.  No.  1. 

225Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  89,  II,  1884,  841-850;  95,  II,  1887,  291-6. 
22«Math.  Annalen,  25,  1885,  251-7. 

""ositzungsber.  Ak.  Wiss.  Wien  (Math.),  94,  II,  1886,  903-10. 
'"Giornale  di  Mat.,  26,  1888,  86-95. 
227aNieuw  Archief  voor  Wisk.,  16,  1889,  209-214. 
=28Zeitschrift  Math.  Phys.,  35,  1890,  382-i. 

22»Math.  Annalen,  36,  1890,  304-315.  ^'OBeitrage  zur  Zahlentheorie,  1891,  187. 

"iLucas,  Th^orie  des   nombres,  1891,  411-2.      Proof  by  H.  W.  Curjel,  Math.  Quest.  Educ. 
Times,  57,  1892,  113. 


432  History  of  the  Theory  of  Numbers.  [Chap,  xviii 

is  1/2,  but  the  nearest  integer  to  x  in  the  contrary  case.    L.  Gegenbauer^^^* 
gave  a  proof  and  generalization. 

Sylvester-^ ^''  noted  that,  if  din)  is  the  number  of  primes  ^u,  and  if 
pi  , . . . ,  Pi  be  the  primes  ^  \/x,  and  gi, .  ,gy  those  between  \/x  and  x,  then 

xd(x/p)-m^'/q)=  {e{Vx)\\ 

H.  W.  Curjel'-^^"  noted  that  the  number  of  primes  >p  and  <p-  is  ^p 
if  p  is  a  prime  ^5.  We  have  only  to  delete  from  1,  2, . .  .,  p^  multiples  of 
2,  3,  5, . . . ,  or  p. 

L.  Gegenbauer-^-  considered  the  integers  x  divisible  by  no  square  and 
formed  of  the  odd  primes  ^m,  when  n^w^\/2n.  Of  the  numbers  [2«/x] 
which  are  of  one  of  the  forms  4s +1  and  4s +2,  count  those  in  which  x  is 
formed  of  an  even  number  of  primes  and  those  in  which  x  is  formed  of  an 
odd  number;  denote  the  difference  of  the  counts  by  a.  He  stated  that  the 
interv^al  from  m  + 1  to  n  (limits  included)  contains  a  —  1  more  primes  than  the 
inter\'al  from  tz+I  to  2n. 

He  gave  (pp.  89-93)  an  expression  for  the  sum  of  the  values  taken  by  an 
arbitrary-  function  g(x)  when  x  ranges  over  the  primes  among  the  first  n 
terms  of  an  arithmetical  i)rogression;  in  particular,  he  enumerated  the 
primes  ^n  of  the  form  4s +1  or  4s  — 1. 

F.  Graefe-^^  would  find  the  number  of  primes  <m  =  10000  by  use  of 
tables  showing  for  each  prime  p,  5'^p^\/m,  the  values  of  n  for  which 
6n+l  or  6nH-5  is  divisible  by  p. 

P.  Bachmann-^  quoted  de  Jonquieres,^°^  Lipschitz,-°^  Sylvester,2°^  and 
Ces^ro."^ 

H.  von  Koch235  wrote  f{x)  =  (a:-l)(a:-2) .  . .  (x-n), 

fix) 

X=2L  A    J  t..^=2{x-IJLv)f'{t^v) 


0(a:)=njl-^j,  p{x)=  2    ,,    •;.:;;,..,.,  (fxvSn), 


and  proved  that,  for  positive  integers  x^n,  d(x)  =  1  or  0  according  as  x  is 
prime  or  composite.     The  number  of  primes  ^m^n  is  ^(1)+.  .  .+^(m). 
A.  Baranowski-^^  noted  the  formula,  simpler  than  Meissel's,^^^ 

xP{n)=<l>[n,  ^(v^)]H-^(v^)  -1 

for  computing  the  number  \f/(n)  of  primes  ^n. 

S.  Wigert^^^^  noted  that  the  number  of  primes  <  n  is 

1  Cf'{x)dx     ,      .,  .     .  2     ,  .  o   /i+r(x)\ 

TT^  I       ./  X    ,  where /(x)  =smVx+sm-7r  I ), 

2TnJ      f{x)  \      X      / 

"'"Denkschr.  Akad.  Wiss.  Wien  (Math.),  60,  1893,  47. 

"i^Math.  Quest.  Educ.  Times,  56,  1892,  67-8. 

"'olbid.,  58,  1893,  127. 

"^Monatshefte  Math.  Phys.,  4,  1893,  98. 

'"Zeitschrift  Math.  Phys.,  39,  1894,  38-50. 

»»<Die  Analytische  Zahlentheorie,  1894,  322-5. 

"*Comptes  Rendus  Paris,  118,  1894,  850-3. 

"•Bull.  Int.  Ac.  Sc.  Cracovie,  1894,  280-1  (German).     Cf.  ♦Rozpraw'y  Akad.  Umiej.,  Cracovie, 

(2),  8,  1895,  192-219. 
'"''Ofversigt  K.  Vetensk.  Ak.  Forhand.,  Stockhohn,  52,  1895,  341-7. 


I 


Chap.  XVIII]  NuMBER   OF   PRIMES.  433 

since  the  only  real  zeros  of  f{x)  are  the  primes.  The  integration  extends 
over  a  closed  contour  enclosing  the  segment  of  the  a;-axis  from  1  to  n  and 
narrow  enough  to  cofitain  no  complex  zero  of /(x). 

T.  Levi-Civita^^^^  gave  an  analytic  formula,  involving  definite  integrals 
and  infinite  series,  for  the  number  of  primes  between  a  and  /3. 

L.  Gegenbauer^"  gave  formulas,  similar  to  that  by  von  Koch,^"^  for  the 
number  of  primes  4s  ±1  or  6s  ±1  which  are  ^n. 

A.  P.  Minin^^'^''  wrote  ^{y)=0  or  1  according  as  y  is  composite  or  prime; 

*^®°         e{n-l)  =  [n-2]  +  [n-b]  +  [n-7]+  .  .  .-^^P{x-\)[n-x], 
summed  for  all  composite  integers  x. 

Gegenbauer^"''  proved  that  Sylvester's^^^  expression  for  the  number  of 
primes  >n  and  <2n  equals  S)u(x)[m/a:+l/2],  where  x  takes  those  integral 
values  S  2n  which  are  products  of  primes  S  y/2n. 

F.  RogeP^^  gave  a  recursion  formula  for  the  number  of  primes  ^m. 

T.  Hayashi^^^  wrote  Rf/q  for  the  remainder  obtained  on  dividing  /  by  q. 
By  Laurent's^^'^  result,  —RFn{x)/{x''  —  l)rf'~^  =  0  or  1  according  as  n  is 
composite  or  prime.     Hence  the  sum  of  the  jth  powers  of  the  primes  between 

.andHs  _j,;__FM__ 

nt.(a:"-l)n"-^-2' 
which  becomes  the  number  of  primes  for  j  =  0.     If  a  is  a  primitive  nth  root 
of  unity,  Wilson's  theorem  shows  that 

n-l 

Sa^"^  =  norO         (m  =  (n-l)!  +  l), 

;  =  0 

according  as  n  is  prime  or  composite.  Hence  ^x^"~^^7(a:"— 1)  =  1  or  0 
according  as  n  is  prime  or  composite.     Thus 

^2a;("-i)y(a;'»-l) 

n  =  s 

is  the  number  of  primes  between  s  and  t. 

Hayashi^^"  reproduced  the  second  of  his  two  preceding  results  and  gave 
it  the  form 

J"^^"^  „  ^  \ cos  (m—n)d  —  r''cos  md\dd    ^ 
„    '•  l-2r''cosng+r^'  =^"  "  «' 

according  as  n  is  prime  or  not,  and  gave  a  direct  proof. 

J.  V.  Pexider^^^  investigated  the  number  \l/{x)  of  primes  ^  x.     Write 


4g=G]-[^]'  ^"-[f]- 


"«&Atti  R.  Accad.  Lincei,  Rendiconti,  (5),  4,  1895,  I,  303-9. 

"'Monatshefte  Math.  Phys.,  7,  1896,  73. 

2"aBull.  Math.  Soc.  Moscow,  9,  1898,  No.  2;  Fortschritte,  1898,  165. 

2376Monatshefte  Math.  Phys.,  10,  1899,  370-3. 

238Archiv  Math.  Phys.,  (2),  17,  1900,  225-237. 

«9Jour.  of  the  Phys.  School  in  Tokio,  9,  1900;  reprinted  in  Abhand.  Gesch.  Math.  Wiss.,  28, 

1900,  72-5. 
»^oArchiv  Math.  Phys.,  (3),  1,  1901,  246-251. 
2«Mitt.  Naturforsch.  Gesell.  Bern,  1906,  82-91. 


434 


History  of  the  Theory  of  Numbers. 


[Chap.  XVIII 


Hence  the  number  of  integers  ^x  which  are  divisible  by  a,  but  not  by 
a-1,  a-2,.  .  .,  2,  is 

[x/a]  a-1 

(7„=s  n(i-3,). 

k=ln=2 

The  number  ^(a:),  of  primes  ^x  and  >v  =  [\/^]  is  [x]-l-2°Z20-,.     Let 
Pi,. . .,  Pa  be  the  primes  ^ \/a-     Let  p^  be  the  greatest  prime  ^ v.     Then 

<,(,)+i=M_[|]_jTn{i-A[^]}, 

from  which  follows  Legendre's  formula. 

S.  Minetola-^-  obtained  a  formula  to  compute  the  number  of  primes 
^K  =  2k  +  l,  not  presupposing  a  knowledge  of  any  primes  >2,  by  consid- 
ering the  sets  of  positive  integers  n,  n',. . .  for  which 

(2n  +  l)(2n'  +  l)^K,  (2n  +  l)(2n'  +  l)(2n"  +  l)^i^,.  .  .. 

F.  RogeP^^  started  with  Legendre's  formula  for  the  number  A{z)  of 
primes  ^z,  introduced  the  remainders  t—\t\,  and  wrote  i?„(z)  for  the  sum 
of  these  partial  remainders.  He  obtained  relations  between  values  of  the 
^'s  and  ^'s  for  various  arguments  z,  and  treated  sums  of  such  values.  For 
arbitrary  x's  (p.  1815), 

Pn+l-l 

p=i 
summed  for  the  primes  p  between  1  and  the  nth  prime  p„.    By  special 
choice  of  the  x^s,  we  get  formulas  involving  Euler's  ^-function  (p.  1818), 
and  the  number  or  sum  of  the  divisors  of  an  integer.     See  RogeP^  of  Ch.  XL 

G.  Andreoli^^  noted  that,  if  x  is  real,  and  F  is  the  gamma  function, 


^(x)  =  sin 


2(r(a:)  +  l)7r 


■sinVx 


is  zero  if  and  only  if  a:  is  a  prime.     Hence  the  number  of  primes  <  n  is 

1     r''^'{x)dx 
2TnX  .   ^ix) 
The  sum  of  the  A:th  powers  of  the  primes  <n  is  given  asymptotically. 
M.  Petrovitch^"*^  used  a  real  function  d{x,  u),  like 

a  cos  27ra;+6  cos  2Tru  —  a  —  h, 

which  is  zero  for  every  pair  of  integers  x,  u,  and  not  if  a:  or  w  is  fractional. 
Let$(x)  be  the  function  obtained  from  d{x,  u)  by  taking 

u=\\+nx)]/x. 
Thus  y=^{x)  cuts  the  a:-axis  in  points  whose  abscissas  are  the  primes. 

"=Giornale  di  Mat.,  47,  1909,  305-320. 

»«Sitzung8ber.  Ak.  Wiss.  Wien  (Math.),  121,  1912,  Ila,  1785-1824;  122,  1913,  II a,  669-700. 

"^Rendiconti  Accad.  Lincei,  (5),  21,  II,  1912,  404-7.     Wigert.'''*'' 

*«Nouv.  Ann.  Math.,  (4),  13,  1913,  406-10. 


Chap.  XVIII]  BeRTRAND's   POSTULATE.  435 

E.  Landau^''^  indicated  errors  in  rintermediaire  des  math^maticiens  on 
the  approximate  number  of  primes  ax+h<N. 

*M.  Kossler-^^  discussed  the  relation  between  Wilson's  theorem  and  the 
number  of  primes  between  two  limits. 

See  Cesaro^^  of  Ch.  V,  Gegenbauer^^  ^f  qj^  ^I,  and  papers  62-81  of 
Ch.  XIII. 

Bertrand's  Postulate. 

J.  Bertrand^®"  verified  for  numbers  <  6  000  000  that  for  any  integer 
n>6  there  exists  at  least  one  prime  between  n  —  2  and  n/2. 

P.  L.  Tchebychef^"  obtained  Hmits  for  the  sum  d{z)  of  the  natural 
logarithms  of  all  primes  ^z  and  deduced  Bertrand's  postulate  that,  for 
x>3,  there  exists  a  prime  between  x  and  2x  — 2.  His  investigation  shows 
that  for  every  e>  1/5  there  exists  a  number  ^  such  that  for  every  x^^  there 
exists  at  least  one  prime  between  x  and  (l  +  e)x. 

A.  Desboves,^^^  assuming  an  unproved  theorem  of  Legendre's,^^  con- 
cluded the  existence  of  at  least  two  primes  between  any  number  >6  and 
its  double,  also  between  the  squares  of  two  consecutive  primes;  also  at 
least  p  primes  between  2n  and  2n  —  k  for  p  and  k  given  and  n  sufficiently 
large,  and  hence  between  a  sufficiently  large  number  and  its  square. 

F.  Proth^^^  claimed  to  prove  Bertrand's  postulate. 
J.  J.  Sylvester264  reduced  Tchebychef's  e  to  0.16688. 

L.  Oppermann-^^  stated  the  unproved  theorem  that  if  n>l  there  exists 
at  least  one  prime  between  n{n  —  l)  and  n^,  and  also  between  n^  and  n(n+l), 
giving  a  report  on  the  distribution  of  primes. 

E.  C.  Catalan^^^  proved  that  Bertrand's  postulate  is  equivalent  to 


n\  n\ 


>a''/5^..7^^ 


where  a, .  .  . ,  tt  denote  the  primes  S  n,  while  a  is  the  number  of  odd  integers 
among  [2n/a],  [2n/a^],...,  h  the  number  among  [2n/^],  [2n//3^],.  . ..  He 
noted  (p.  31)  that  if  the  postulate  is  applied  to  6  —  1  and  6+1,  we  see  the 
existence  between  26  and  46  of  at  least  one  even  number  equal  to  the  sum 
of  two  primes. 

J.  J.  Sylvester^"  reduced  Tchebychef's  e  to  0.092;  D.  von  Sterneck^^^ 
to  0.142. 

24«L'intermediaire  des  math.,  20,  1913,  179;  15,  1908,  148;  16,  1909,  20-1. 

2"Casopis,  Prag,  44,  1915,  38-42. 

^^ojour.  de  I'ecole  roy.  polyt.,  cah.  30,  tome  17,  1845,  129. 

2"M^m.  Ac.  Sc.  St.  Petersbourg,  7,  1854  (1850),  17-33,  27;  Oeuvres,  1,  49-70,  63.     Jour,  de 

Math.,  17,  1852,  366-390,  381.     Cf.  Serret,  Cours  d'algebre  sup^rieure,  ed.  2,  2,  1854, 

587;  ed.  6,  2,  1910,  226. 
262NOUV.  Ami.  Math.,  14,  1855,  281-295. 
2"Nouv.  Corresp.  Math.,  4,  1878,  236-240. 
2«^Amer.  Jour.  Math.,  4,  1881,  230. 
2660versigt  Videnskabs  Selsk.  Forh.,  1882,  169. 

»««M6m.  Soc.  R.  Sc.  Liege,  (2),  15,  1888  (  =  M61anges  Math.,  Ill),  108-110. 
26'Messenger  Math.,  (2),  21,  1891-2,  120. 
268Sitzungsb.  Akad.  Wiss.  Wien,  109,  1900,  II  a,  1137-58. 


436  History  of  the  Theory  of  Numbers.  [Chap,  xvill 

T.  J.  Stieltjes  stated  and  E.  Cahen'^^^  proved  that  we  may  take  e  to  be 
any  positive  number  however  small,  since  d{z)  is  asjTnptotic^^^"®  to  z. 

H.  Brocard""  stated  that  at  least  four  primes  lie  between  the  squares 
of  two  consecutive  primes,  the  first  being  >3.  He  remarked  that  this  and 
the  similar  theorem  by  Desboves-^-  can  apparently  be  deduced  from  Ber- 
trand's  postulate;  but  this  was  denied  by  E.  Landau. ^^^ 

E.  ^laillet^^-  proved  there  is  at  least  one  prime  between  two  consecutive 
squares  <9-10^  or  two  consecutive  triangular  numbers  ^9-10^. 

E.  Landau"*^  (pp.  89-92)  proved  Bertrand's  postulate  and  hence  the 
existence  of  a  prime  between  x  (excl.)  and  2x  (incl.)  for  every  x^l. 

A.  Bonolis-"^  proved  that,  if  x>13  is  a  number  of  p  digits  and  a  is  the 
least  integer  >x/{lO{p-{-l)],  there  exist  at  least  a  primes  between  x  and 
[-|x  — 2],  which  implies  Bertrand's  postulate.  If  x>l3  is  a  number  of 
p  digits  and  /3  is  the  greatest  integer  <x/(3p— 3),  there  are  fewer  than 
^  primes  from  a:  to  [-|x  — 2]. 

Miscellaneous  Results  on  Primes. 

H.  F.  Scherk^^*^  stated  the  empirical  theorems:  Every  prime  of  odd 
rank  (the  nth  prime  1,  2,  3,  5, . .  .  being  of  rank  n)  can  be  composed  by 
addition  and  subtraction  of  all  the  smaller  primes,  each  taken  once;  thus 

13  =  1+2-3-5+7+11  = -1+2+3+5-7+11. 

Every  prime  of  even  rank  can  be  composed  similarly,  except  that  the  next 
earlier  prime  is  doubled;  thus 

17  =  1+2-3-5+7-11+2-13= -1-2+3-5+7-11+2-13. 

Marcker^^^  noted  that,  if  a,  6, .  .  . ,  m  are  the  primes  between  1  and  A 
and  if  p  is  their  product,  all  the  primes  from  A  to  A^  are  given  by 


K^?+  ■+^>+»)' 


and  each  but  once  if  each  numerator  is  positive  and  less  than  its  denominator. 

0.  Terquem"^-  noted  that  the  primes  <rr  are  the  odd  numbers  not 
included  in  the  arithmetical  progressions  q^,  q^-\-2q,  5^+4g, .  .  .  up  to  n^, 
for  g  =  3,  5, .  .  .,  n  — 1. 

H.  J.  S.  Smith-^^  gave  a  theoretical  method  of  finding  the  primes  between 
the  xth  prime  P^  and  P'^x+i,  given  the  first  x  primes. 

C.  de  Polignac^^^"  considered  the  primes  ^x  in  a  progression  Km-\-h. 

""Comptes  Rendus  Paris,  116,  1893,  490;  These,  1894,  45;  Ann.  ficole  Normale,  (3),  11,  1894. 

""L'intermddiaire  des  math.,  11,  1904,  149. 

'■'/bMf.,  20,  1913,  177. 

"'/6trf.,  12,  1905,  110-3. 

*"Atti  Ac.  Sc.  Torino,  47,  1911-12,  576-585. 

"ojour.  fur  Math.,  10,  1833,  201. 

"'/twi.,  20,  1840,  350. 

"'Nouv.  Ann.  Math.,  5,  1846,  609. 

»8»Proc.  Ashmolean  Soc,  3,  1857,  128-131;  Coll.  Math.  Papers,  1,  37. 

""K>)mptes  Rendus  Paris,  54,  1862,  158-9. 


Chap.  XVIII]  MISCELLANEOUS  RESULTS   ON   PRIMES.  437 

E.  Dormoy^^^  noted  that,  if  2,  3, . . .,  r,  s,  t,  u  are  the  primes  in  natural 
order,  all  primes  (and  no  others)  <u^  are  given  by 

2-3.  .  .stm-\-Dtat-\-tCtD,a,+tsCtC,Drar-\- . .  . 

-\-tsr. .  .7-5C,C,Cr. .  .C,D,as+ts. .  .5'3C,Cs. .  .C3, 

where  Ct  is  found  from  the  quotients  obtained  in  finding  the  g.  c.  d.  of  t 
and  2-3 ..  .rs  by  a  rule  which  if  applied  to  four  quotients  a,  h,  c,  d  consists  in 
forming  in  turn  1,  p  =  cZc+l,  p&-|-^,  {ph+d)a+p  =  Ct.  Further,  A  =  ^C^=i=l, 
the  sign  being  +  or  —  according  as  there  is  an  odd  or  an  even  number  of 
operations  in  the  g.  c.  d.  process. 

C.  de  Polignac^^^"  wrote  p„  for  the  nth  prime  and  discussed  the  express- 
ibility  of  all  numbers,  under  a  specified  limit  and  divisible  by  no  one  of 
pi, .  .  .,  pn-i,  in  the  form 

{P2,P3,--->Pn-hPn)  +  {p3,P4,-  ■  ■,PnPl)+.  .  .  +  (Pl,  .  .  . ,  Pn-l) , 

where  (a,  6, ... )  denotes  ±  a^b^ ....  For  example,  every  number  <  53  and 
divisible  by  neither  2  nor  3  is  of  the  form  ±3  "±2^. 

J.  J.  Sylvester^^^  proved  that  if  m  is  prime  to  i  and  not  less  than  n,  the 
product  (m+i)(m+2i) .  .  .  (w+m)  is  divisible  by  some  prime  >n. 

A.  A.  Markow^^*^  found  a  fragment  in  a  manuscript  by  Tchebychef 
aiding  him  to  prove  the  latter's  result  that  if  //  is  the  greatest  prime  divi- 
sor of  (1+2^)  (1+4^) .  .  .(1+4A^^),  then  fx/N  increases  without  limit  with 
N  (cf.  Hermite,  Cours,  ed.  4,  1891,  197). 

J.  Iwanow-^^  generalized  the  preceding  theorem  as  follows:  If  fx  is  the 
greatest  prime  divisor  of  (A  +  1^).  .  .(A+L^),  then  fi/L  increases  without 
limit  with  L. 

C.  Stormer^^^  concluded  the  existence  of  an  infinitude  of  primes  from 
Tchebychef's^^^resultandusedthelattertoprovethat2'(i  — l)(t  — 2) .  .  .(i—n) 
is  neither  real  nor  purely  imaginary  if  n  is  any  integer  5^  3,  and  i  =  \/  —  1. 

E.  Landau^'  (pp.  559-564)  discussed  the  topics  in  the  last  three  papers. 

Braun^^"  proved  that  the  (n+l)th  prime  is  the  only  root  Xt^I  oi 

where    ai  =  2,  02,     . . ,  «„  are  the  first  n  primes. 

C.  Isenkrahe^^^  expressed  a  prime  in  terms  of  the  preceding  primes. 

R.  Le  Vavasseur^^°  noted  that  all  primes  between  p„  and  p\+i,  where  p„ 
is  the  nth  prime,  are  given  by  Sjii  qiWiPJpi  (mod  P,J,  where  Pn  =  PiP2 
•  •  •  Pn  and  WiPJpi=l  (mod  p,). 

"^Comptes  Rendus  Paris,  63,  1866,  178-181. 

284<»Comptes  Rendus  Paris,  104,  1887,  1688-90. 

285Messenger  Math.,  21,  1891-2,  1-19,  192.     Math.  Quest.  Educ.  Times,  56,  1892,  25. 

286Bull.  Acad.  Sc.  St.  Petersbourg,  3,  1895,  55-8. 

^^Ubid.,  361-6. 

"sArchiv  Math,  og  Natur.,  Kristiania,  24,  1901-2,  No.  5. 

289Math.  Annalen,  53,  1900,  42. 

290M6m.  Ac.  Sc.  Toulouse,  (10),  3,  1903,  36-8. 


438  History  of  the  Theory  of  Numbers.  [Chap,  xviii 

0.  !Meissner^^^  stated  that,  if  n+1  successive  integers  m,.  . .,  m+n  are 
given,  we  can  not  in  general  find  another  set  mi,.  .  .,  mi-\-n  containing  a 
prime  7ni-\-v  corresponding  to  every  prime  m+v  of  the  first  set.  But 
for  n  =  2,  it  is  supposed  true  that  there  exist  an  infinitude  of  prime  pairs. 

G.  H.  Hardy ^^^  noted  that  the  largest  prime  dividing  a  positive  integer x  is 

m 

lim  lim  Hm  i:[l-{cos{{v\yir/x]n 

r=oo    TO=oo    n  =  oo  v=0 

C.  F.  Gauss,^^^  in  a  manuscript  of  1796,  stated  empirically  that  the 
number  T2ix)  of  integers  ^x  which  are  products  of  two  distinct  primes,  is 
approximately  x  log  log  x/log  x. 

E.  Landau^^^  proved  this  result  and  the  generalization 


7r,(x)  = 


1        a^qoglogx)"-^        ''-^i--  1 ^"-2' 

{v  —  l)\        log  a; 


^  ^ |x(log  log  x)"  ^\ 
1      log  X  J 


where  irXx)  is  the  number  of  integers  ^  x  which  are  products  of  v  distinct 
primes;  also  related  formulas  for  ir^x). 

Several  writers-^^  gave  numerous  examples  of  a  sum  of  consecutive  primes 
equal  to  an  exact  power. 

E.  Landau^^^  proved  that  the  probability  that  a  number  of  n  digits  be  a 
prime,  when  n  increases  indefinitely,  is  asymptotically  equal  to  l/(n  log  10). 

J.  Barinaga^^^  expressed  the  sum  of  the  first  n  primes  as  a  product  of 
distinct  primes  for  n  =  3,  7,  9,  11,  12,  16,  22,  27,  28,  and  asked  if  there  is  a 
general  law. 

Coblyn^^^  noted  as  to  prime  pairs  that,  when  4(6p  — 2)!  is  divided  by 
36p2  — 1,  the  remainder  is  —6p  —  S  if  6p  — 1  and  6p-i-l  are  both  primes, 
zero  if  both  are  composite,  —  2(6p+l)  if  only  Qp  —  1  is  prime,  and  Qp  —  1  if 
only  6p4-l  is  prime. 

J.  Hammond^^^  gave  formulas  connecting  the  number  of  odd  primes 
<2n,  and  the  number  of  partitions  of  2n  into  two  distinct  prunes  or  into 
two  relatively  prime  composite  numbers. 

V.  Brun^^*^  proved  that,  however  great  a  is,  there  exist  a  successive  com- 
posite numbers  of  the  form  l-\-u^.  There  exist  a  successive  primes  no  two 
of  which  differ  by  2.  He  determined  a  superior  limit  for  the  number  of 
primes  <  x  of  a  given  class. 

"•Archiv  Math.  Phys.,  9,  1905,  97. 

'^^MessenRer  Math.,  35,  1906,  145. 

"Cf.  F.  Klein,  Nachrichten  Ge.sell.  Wiss.  Gottingen,  1911,  26-32. 

'^Ibid.,  361-381;  Handbuch. .  .der  Primzahlen,  I,  1909,  205-211;  Bull.  Soc.  Math.  France,  28, 

1900,  25-38. 
""L'interm^diaire  des  math.,  18,  1911,  85-6. 
^*^Ibid.,  20,  1913,  180. 

"'L'intermddiaire  des  math.,  20,  1913,  218. 
"«Soc.  Math,  de  France,  C.  R.  des  Stances,  1913,  55. 

»»Proo.  London  Math.  Soc,  (2),  15,  1916-7,  Records  of  Meetings,  Feb.  1916,  xxvii. 
"""Nyt  Tidsskrift  for  Matematik,  B,  27,  1916,  45-58. 


Chap.  XVIII]  ASYMPTOTIC   DiSTKIBUTION   OF  PRIMES.  439 

Diatomic  Series. 

A.  de  Polignac^°^  crossed  out  the  multiples  of  2  and  3  from  the  series  of 
natural  numbers  and  obtained  the  "table  02": 

(0)    1    (2)    (3)    (4)    5    (6)    7     (8)     (9)     (10)     11.... 

The  numbers  of  terms  in  the  successive  sets  of  consecutive  deleted  numbers 
are  1,3,1,3,1,...,  which  form  the  "diatomic  series  of  3."  Similarly,  after 
deleting  the  multiples  of  the  first  n  primes,  we  get  a  table  a„  and  the  dia- 
tomic series  of  the  nth  prime  P„.  That  series  is  periodic  and  the  terms  after 
1  of  the  period  are  symmetrically  distributed  (two  terms  equidistant  from 
the  ends  are  equal),  while  the  middle  term  is  3.  Let  7r„  denote  the  product 
of  the  primes  2,  3, . .  . ,  P^-  Then  the  number  of  terms  in  the  period  is 
0(7r„).  The  sum  of  the  terms  in  the  period  is  x„— 0(7r„)  and  hence  is  the 
number  of  integers  <7r„  which  are  divisible  by  one  or  more  primes  ^Pn- 
As  applications  he  stated  that  there  exists  a  prime  between  P„  and  P^,  also 
between  o"  and  a""^\  He^°^  stated  that  the  middle  terms  other  than  3  of  a 
diatomic  series  tend  as  n  increases  to  become  1,  3,  7,  15, . . . ,  2"*  —  1, .  .  . . 

J.  Deschamps^°^  noted  that,  after  suppressing  from  the  series  of  natural 
numbers  the  multiples  of  the  successive  primes  2,  3, .  .  . ,  p,  the  numbers  left 
form  a  periodic  series  of  period  2-3 .  .  .  p ;  and  similar  theorems.  Like 
remarks  had  been  made  previously  by  H.  J.  S.  Smith.^°^ 

Asymptotic  Distribution  of  Primes. 

P.  L.  Tchebychef's^^^  investigation  shows  that  for  x  sufficiently  large 
the  number  7r(a;)  of  primes  ^a;  is  between  0-921Q  and  1-106Q,  where 
Q  =  a:/log  X.  He^^^  proved  that  the  limit,  if  existent,  of  7r(x)/Q  for  x=  00  is 
unity.  J.  J.  Sylvester^^^  obtained  by  the  same  methods  the  limits  0-95Q 
and  1-05Q. 

By  use  of  the  function  f (s)  =2"zfn~*  of  Riemann,  J.  Hadamard^^^ 
and  Ch.  de  la  Vallee-Poussin^^^  independently  proved  that  the  sum  of  the 
natural  logarithms  of  all  primes  ^x  equals  x  asymptotically.  Hence 
follows  the  fundamental  theorem  that  -wix)  is  asymptotic  to  Q,  i.  e., 

.      log  a: 
lim  t:{x) — ^—  =1. 


'°*Recherches  nouvelles  sur  les  nombres  premiers,  Paris,  1851,  28  pp.  Abstract  in  Comptes 
Rendus  Paris,  29,  1849,  397-401,  738-9;  same  in  Nouv.  Ann.  Math.,  8,  1849,  423-9. 
Jour,  de  Math.,  19,  1854,  305-333. 

»«Nouv.  Ann.  Math.,  10,  1851,  308-12. 

8"Bull.  Soc.  Philomathique  de  Paris,  (9),  9,  1907,  102-112 

3»«Proc.  Ashmolean  Soc,  3,  1857,  128-131;  Coll.  Math.  Papers,  1,  36. 

3"M6m.  Ac.  Sc.  St.  P^tersbourg,  6,  1851,  146;  Jour,  de  Math.,  17,  1852,  348;  Oeuvres,  1,  34. 

"6BuU.  Soc.  Math,  de  France,  24,  1896,  199-220. 

"«Annales  de  la  Soc.  Sc.  de  Bruxelles,  20,  II,  1896,  183-256. 


440  History  of  the  Theory  of  Numbers.  [Chap.  XVIII 

Now  Q  is  asjinptotic  to  the  "integral  logarithm  of  x": 

4=0  V/0       log  U     Jl+i  log  u/ 

so  that  the  latter  is  asymptotic  to  irix).  De  la  Vall^e-Poussin^^'  proved  that 
Lix  represents  t{x)  more  exactly  than  x/log  x  and  its  remaining  approxi- 
mations .       ,  ^ , 

x      ,     a:      ,  .{m  —  l)]x 

log  X    log^x  log'"a: 

The  historj^  of  this  extensive  subject  is  adequately  presented  in  the 
luminous  and  exhaustive  text  by  E.  Landau,"*^  in  which  is  given  (pp.  908- 
961)  a  complete  list  of  references.  The  reader  may  consult  the  article  by 
J.  Hadamard,'^^  the  extensive  report  by  G.  Torelli,^^^  the  summaries  by 
Landau,^^°  also  G.  H.  Hardy  and  J.  E.  Littlewood,^^^  and  the  recent  papers 
42-44  of  Chapter  XIX. 

"'M^m.  Couronn^s  Acad.  Roy.  Belgique,  59,  1889,  1-74. 

"^Encyclopedic  des  sc.  math.,  tome  I,  vol.  3,  pp.  310-345. 

"Utti  R.  Accad.  Sc.  Fis.  Mat.,  XapoU,  (2),  11,  1902,  No.  1,  222  pp. 

^oProc.  Fifth  Internat.  Congress,  Cambridge,  1,  1913,  93-108.     Math.  Zeitschrift,  1,  1918,  1-24, 

213-9. 
«»Acta  Math.,  41,  1917,  119-196. 


* 


I 


i 


CHAPTER  XIX. 

INVERSION  OF  FUNCTIONS;  MOBIUS'  FUNCTION  fi(n);  NUMERICAL 
INTEGRALS  AND  DERIVATIVES. 

Inversion;  Function  fx{n). 

A.  F.  Mobius^  defined  the  function  /.i(n)  to  be  zero  if  n  is  divisible  by  a 
square  >1,  but  to  be  (  —  1)'^  if  n  is  a  product  of  k  distinct  primes  >1, 
while  ^i(l)  =  1.     He  employed  the  function  in  the  reversion  of  series: 

Fix)  =  S  '-^  imphes  fix)  =  S  fxis)  -^' 

s=l      "  s=l  " 

His  results  were  expressed  in  more  general  form  by  Glaisher^^  and  cited  in 
Chapter  X.     See  also  E.  Meissel,^  who^  noted  that 

R.  Dedekind"*  proved  that,  if  F{m)  =2/(d),  where  d  ranges  over  all  the 
divisors  of  m,  then 


+  . 


(2)    /w=.w-z.o+..e)-..(^) 

where  the  summations  extend  over  all  the  combinations  1,  2,  3, ...  at  a  time 
of  the  distinct  prime  factors  a,  b,.  .  .,  k  oi  m.  The  proof  follows  from  a 
distribution  of  all  the  factors  of  m  into  two  sets  S  and  T.  Put  all  the  divisors 
of  m  into  set  S]  all  divisors  of  m/a  into  set  T,  all  of  m/b  into  T,  etc. ;  all  divi- 
sors of  m/iab)  into  S,  all  of  m/iac)  into  S,  etc.;  all  of  m/{abc)  into  T,  etc. 
Then,  with  the  exception  of  m  itself,  every  divisor  of  m  occurs  as  often  in  the 
set  S  as  in  the  set  T.     In  particular,  for  Euler's  0(m),  m=  20(d),  whence 

For  another  example,  see  Dedekind''^  of  Ch.  VIII.     Similarly,  F(w)  = 
n/(d)  implies 

F(m)  UF  g)  . . . 


/(»)  = 


H^Mi-:)- 


J.  Liouville^  stated  simultaneously  with  Dedekind  the  inversion  theorem 
for  sums  and  made  the  same  appHcation  to  ({)im). 

Liouville^  stated  the  theorem  for  sums  as  a  problem. 

iJour.  fiir  Math.,  9,  1832,  105;  Werke,  4,  591.     He  wrote  a„  for  /x(n). 

Hbid.,  48,  1854,  301-316. 

'Observationes  quaedam  in  theoria  numerorum,  Berlin,  1850,  pp.  3-6. 

*Jour.  fiir  Math.,  54,  1857,  pp.  21,  25. 

6Jour.  de  Math^matiques,  (2),  2,  1857,  110-2. 

«Nouv.  Ann.  Math.,  16,  1857,  181-2. 

441 


442  History  of  the  Theory  of  Numbers.  [Chap.  XIX 

B.  Merry'  gave  a  proof  by  noting  that,  if  d  is  any  divisor  of  m,  and  if  q 
of  the  prime  factors  of  m  occur  to  the  same  power  in  c?  as  in  ???,  then  f{d) 
occurs  once  in  F{m),  q  times  in  ZF{m/a),  q{q  —  l)/2  times  in  'LFim/ab),  etc. 
Thus  the  coefficient  of  f{d)  in  (2)  is 

if  5 > 0,  but  is  unity  ii  q  =  0,i.  e.,ii  d  =  m.    This  proof  is  only  another  way  of 
stating  Dedekind's  proof.  I 

R.  Dedekind^  gave  another  form  and  proof  of  his  theorems.     Let 


(-i)(-i) 


m(l--)(l-T  1  .  .    =2j/i-2j/2, 


where  vi  ranges  over  the  positive  terms  of  the  expanded  product  and  —V2 
over  the  negative  terms.  A  simple  proof  shows  that,  if  v  is  any  di\'isor 
<m  of  m,  there  are  as  many  terms  vi  di\'isible  by  v  as  terms  V2  divisible  by  v. 
Thus 

i:f{v)=F{m),  Uf{p)=F{m) 

imply,  respectively, 

fim)  =2F(.,)  -2F(^2),  Km)  -^^y 

Liou\'ille^  wTote  F(n)  =Tif{n/D''),  where  D  ranges  over  those  di\4sors  of 
n  =  a"}/ ...  for  which  D"  divides  n.    Then 

f{n)  =F{n)  -XFin/a")  +lF{n/a''b'')  -.... 

E.  Laguerre^  expressed  (2)  in  the  form 

(3)  /(m)=2M(|)F(d), 

where  d  ranges  over  the  divisors  of  m.     Let 

2/(n)T^=SF(n)x^ 

n  =  l  L—X  n=l 

whence  F{m)  ='Zf(d).  For  m^Up",  where  the  p's  are  distinct  primes,  let 
f{m)  =n/(p»),  and/(p")  =p''-np-l).    Then 

F{m)  =njl+/(p)  +  .  .  .  +f{p''-')]  =Up''  =  m. 

The  hypotheses  are  satisfied  if  /  is  Euler's  function  4>.     This  discussion 
deduces  l,(t){d)=m  from  the  usual  expression  of  type  (3)  for  <j){m),  rather 
than  the  reverse  as  claimed. 
N.  V.  Bougaief^°  proved  (1). 

F.  Mertens"  defined  /x(n)  and  noted  that  2/x(d)=0  if  n>l,  where  d 
ranges  over  the  divisors  of  n. 

'Nouv.  Ann.  Math.  16,  1857,  434. 

8Dirichlet's  Zahlentheorie,  mit  Zusatzen  von  Dedekind,  1863,  §138;  ed.  2,  1871,  p.  356;  ed.  4, 

1894,  p.  360. 
'"Jour,  de  Math.,  (2),  8,  1863,  349. 
•BuU.  Soc.  Math.  France,  1,  1872-3,  77-81. 

'"Mat.  Sbomik  (Math.  Soc.  Moscow),  6,  1872-3,  179.     Cf.  Sterneck." 
"Jour,  fur  Math.,  77,  1874,  289;  78,  1874,  53. 


I 


Chap.  XIX] 


Inversion;  Function  fj>{n). 


443 


E.  Cesaro^^  proved  formulas,  quoted  in  Ch.  X,  which  include  (3)  as  a 
special  case.     His  erroneous  evaluation  of  the  mean  of  /x(n)  is  cited  there. 

Cesaro^^  reproduced  the  general  formula  just  cited  and  extended  it  to 
three  pairs  of  functions: 

S/i(d)i^i(^)  =l^f2(d)F,(^  =S/3(rf)/^3(^), 

FM  =S/2(rf)/3  (^) ,  F,  =2/3/1,  Fs  =2/^2. 

where,  in  each,  d  ranges  over  the  divisors  of  n. 
Cesaro^^  noted  that,  if  h(ri)-\-k{n)  =  1  and 

H{n)=h{p)Hq)...,  Kin)=k(p)kiq)..., 

where  p,  q,. . .  are  the  prime  factors  of  n,  then 

Hin)  =i:fjiid)K{d),  Kin)  =Xfx{d)H{d). 

For  h{n)=k{n)  =  1/2,  then  H{n)=K{n)  is  the  reciprocal  of  the  number  of 
divisors,  without  square  factors,  of  n. 

Cesaro^^  treated  the  inversion  of  series.  Let  Q{x)  =  1  or  0,  according  as  x 
is  or  is  not  in  a  given  set  Q,  of  integers.  Let  Q{x)Q,{y)  =U(xy).  Let  €i{x)  be 
functions  such  that  €„{e^(x)}  =€„^(x)  for  every  pair  of  indices  a,  j8.     Then 

F{x)=Xh{o,)fleM], 
where  co  ranges  over  all  the  numbers  of  Q,,  implies  that 

fix)=i:H{c^)F{eM}, 

if  the  sum  Xh{d)H{n/d),  for  d  ranging  over  the  divisors  of  n,  equals  1  or  0 
according  as  n  =  1  orn>  1.     Cf .  Mobius\ 

N.  V.  Bougaief^^  considered  the  function  v{x)  with  the  value  log  p  if  x 
is  a  power  of  a  prime  p,  the  value  0  in  all  other  cases.  Then,  if  d  ranges 
over  the  divisors  of  n,  St'(d)  =log  n  implies  2ju(d)  log  d=  —v{n). 

H.  F.  Baker^'^  gave  a  generalization  of  the  inversion  formula,  the  state- 
ment of  which  will  be  clearer  after  the  consideration  of  one  of  his  appli- 
cations of  it.  Let  fli,.  . .,  a„  be  distinct  primes  and  S  any  set  of  positive 
integers.  For  k-^n,  let  F{ai,.  .  .,  a^)  denote  the  set  of  all  the  numbers 
in  S  which  are  divisible  by  each  of  the  primes  a^t+i,  o,k+2,-  •  ■,  cin,  so  that 
F(ai, .  . . ,  aj  =*S.  For  A;  =  0,  write  F{0)  for  F,  so  that  i^(0)  consists  of  the 
numbers  of  S  which  are  divisible  by  ai,.  .  .,  a„.  Returning  to  the  general 
F{ai,.  .  .,  ttk),  we  divide  it  into  sub-sets.  Those  of  its  numbers  which  are 
divisible  by  no  one  of  ai, .  .  .,  a^  form  the  sub-set  /(ai, .  .  .,  %)•  Those 
divisible  by  ai,  but  by  no  one  of  02, ... ,  ak,  form  the  sub-set /(a2,  Os,  ■  •  • ,  dk)- 

i^Mem.  soc.  roy.  sc.  de  Liege,  (2),  10,  1883,  No.  6,  pp.  26,  47,  56-8. 

I'Giornale  di  Mat.,  23,  1885,  168  (175). 

"/bid.,  25,  1887,  14-19.     Cf.  1-13  for  a  type  of  inversion  formulas. 

"AnnaU  di  Mat.,  (2),  13,  1885,  339;  14,  1886-7,  141-158. 

isComptes  Rendus  Paris,  106,  1888,  652-3.  Cf.  Cesaro,  ihid.,  1340-3;  Cesa,ro,i2  pp.  315-320; 
Bougaief,  Mat.  Sbornik  (Math.  Soc.  Moscow),  13,  1886-8,  757-77;  14,  1888-90,  1-44, 
169-201;  18,  1896,  1-54;  Kronecker3«  (p.  276);  Berger"''  (pp.  106-115);  Gegenbauer^^  of 
Ch.  XI— all  on  -Ln^d)  log  d. 

"Proc.  London  Math.  Soc,  21,  1889-90,  30-32. 


444  History  of  the  Theory  of  Numbers.  [Chap,  xix 

Those  di\'isible  by  Oi  and  02,  but  by  no  one  of  03, ... ,  a^,  form  the  sub-set 
/(oa,. .  .,  a  J.  Finally,  those  divisible  by  fli.  .  .,  O;.  form  the  sub-set  desig- 
nated /(O).    Thus 

F{ai,a-2, .  .  .,  Ok)  =/(ai,  Oo, .  .  . ,  a^)  +2/(a2,  as,  ••  • ,  Oa)  +|/(a3,  ^4,  •  •  • ,  a^t) 

+  ...-f2/(a,)H-/(0), 

7»— 1 

where  S  indicates  that  the  summation  extends  over  all  combinations  of 
Oi,.  .  . ,  flr  taken*  /c  — r  at  a  time. 

WTien  we  have  any  such  set  or  function /(oi, .  .  . ,  a^t),  uniquely  determined 
by  Oi, . . . ,  Qk,  independently  of  their  order,  and  we  define  F  by  the  foregoing 
formula,  then  we  have  the  inverse  formula 
/(oi,  02, .  .  . ,  a„)  =F(ai,  03, ... ,  aj  -2/^(a2,  03, ... ,  aj  +2/^(03,  04, .  .  .  ,aj 

-...+(-ir-^SF(aO  +  (-irF(0),  ' 

n-l 

where  S  now  indicates  that  the  summation  extends  over  all  the  combina- 

r 

tions  of  fli,.  .  .,  a„  taken  n  —  r  at  a  time.  The  proof  is  just  like  that  by 
B.  Merry  for  Dedekind's  formula. 

To  give  an  example,  let  n  =  2,  ai  =  2,  02  =  3,  S  =  S,  4,  6,  8.     Then  F{ai) 

=  3,  6;/(oi)=3, /(0)=6;   F(a2)=4,  6,  8;    /(o2)=4,  8.     Thus 

F(oi,O2)-/^(o0-F(o2)+F(0)=5-(3,6)-(4,6,8)+6=0=/(oi,O2). 

A.  Berger^^''  called  /i  conjugate  to  /a  if  2/i  id)f2{d)  =  1  f or  ^•  =  1 ,  0  f or  A:>  1 , 
when  d  ranges  over  the  divisors  of  k.  Let  g{mn)=g{m)g{n),  ^(1)  =  1. 
Write  h{k)=Zf{d)Md)g{8),  where  d8  =  k.  Then  fik)=2f.2{d)g{d)h(8). 
Dedekind's  inversion  formula  is  a  special  case.  For,  if  /i(n)  =  l,  then 
/2(n)=M(w). 

K.  Zsigmondy^^  stated  that  if,  for  every  positive  integer  r, 

?f{r;)=F{r), 

where  c  ranges  over  all  combinations  of  powers  ^r  of  the  relatively  prime 
positive  integers  ni, .  .  .,  n^,  while  r,.  denotes  the  greatest  integer  ^r/c,  then 

/(r)=F(r)-2F(r„)+SF(r„„0-..., 

n  n,n 

where  the  summation  indices  n,  n' ,.  . .  range  over  the  combinations  of 
ni, . . . ,  rip  taken  1,2,...  at  a  time. 

R.  D.  von  Sterneck^^  noted  that,  if  d  ranges  over  the  divisors  of  n 
l^B{d)  =^{n)  implies  that 

Taking  m=l,.  . .,  n  and  solving,  we  get  Q{n)  expressed  as  a  determinant 
of  order  n,  whence 

^(n)=,A(^)-S,A(di)+2:V'W----+(-l)VR), 
if  n  =  pi°'.  .  .p/"  and  d^  is  derived  from  n  by  reducing  p  exponents  by  unity. 

*Here  and  in  the  statement  of  the  theorem  occur  confusing  misprints  for  k  and  n. 
""Nova  Acta  Regiae  Soc.  Sc.  Upsaliensis,  (3),  14,  1891,  No.  2,  46,  104. 
"Jour,  fur  Math.,  Ill,  1893,  346.     Apphed  in  Ch.  V,  Zsigmondy." 
"Monatshefte  Math.  Phys.,  4,  1893,  53-6. 


Chap.  XIX] 


Inversion;  Function  fx{n). 


445 


P.  Bachmann^°  proved  that  f{n)  ='Zlz^F{kn)  implies  that 


F(l) 


S  ii{n)f{n). 

n=l 


Write  X  =  [x/n\.     Taking  F{n)  =X,  nX,  <J>(Z),  whence  /(n)  =  T{X),  no-(X), 
D{X),  respectively,  we  obtain  Lipschitz's^°  (Ch.  X)  formulas: 


ti 


[x]=  2  iJi{n)T\-\=2fji{n)na 


n 


*[a:]=2/x(??)i) 


[Q 


Let  F{n)  be  zero  if  n  is  not  a  divisor  of  P  and  write  x{/{P/n)  for  /^(n). 
Hence  if  d  divides  P,  f{d)=7:4^{P/kd)  implies  iA(P)  =Sm(^)/(^),  where 
A;  ranges  over  the  divisors  of  P/d,  and  d  over  those  of  P. 

D.  von  Sterneck^^  considered  a  function  f(n)  with  the  properties: 
(i)  /(I)  =  1 ;  (ii)  the  g.  c.  d.  of /(m)  and/(n)  isf{d)  if  rf  is  the  g.  c.  d.  of  m  and  n; 
(iii)  for  primes  p,  other  than  specified  ones,  one  of  the  numbers /(p±l)  is 
divisible  by  p;  (iv)  the  g.  c.  d.  oi  f{pn)/f{n)  and /(n)  divides  p.  Then  if 
L{n)  is  the  1.  c.  m.  of  the  values  of  /  for  all  the  divisors  <n  of  n,  F{n)  =f{n) 
■i-L{n)  is  an  integer  which  can  be  given  the  form 


Fin)  = 


^\PrpJ       ^\p,PiPkVl)' 

W    \pmpk}'" 


n=Jlpi 


The  four  properties  hold  for  the  function  defined  by  the  recursion 
formula  /(n)  =a/(n  — l)+i3/(n  — 2),  where  a  and  jS  are  relatively  prime, 
with  the  initial  conditions  /(I)  =  1 ,  /(2)  =  a.     For  a  =  2x,p  =  b  —  x^,  we  have^^ 

(x+Vbr-ix-VbT 


m= 


2\/b 


The  case  a  =  j3  =  l  was  discussed  by  Lucas-^  of  Ch.  XVII,  and  his  test  for 
primality  holds  for  the  present  generalization. 
The  four  properties  hold  also  for 


Kn)  = 


aJ'-lf 


if  a,  h  are  relatively  prime ;^^  then  fip  —  l)  is  divisible  by  p  if  p  is  a  prime 
not  dividing  a,  b  or  a  —  b. 

K.  Zsigmondy-^  gave  a  generalized  inversion  formula.     Let  A^  be  any 
multiple  of  the  relatively  prime  integers  ni, .  .  . ,  n^.     Set 

where  d  ranges  over  those  divisors  ^m  oi  N  which  are  products  of  powers 

2"Die  Analytische  Zahlentheorie,  1894,  310. 
"Monatshefte  Math.  Phys.,  7,  1896,  37,  342. 
22DirichIet,  Werke,  1,  47-62.     See  Dirichlet,^  Ch.  XVII. 
wZsigmondy,  Monatshefte  Math.  Phys.,  3,  1892,  265. 
^Ibid.,  7,  1896,  190-3. 


446  History  of  the  Theory  of  Numbers.  [Chap,  xix 

of  rii, . .  . ,  7J,.    Then,  if  a  ranges  over  those  divisors  d  which  are  divisible  by 
no  one  of  ^i, . .  . ,  v^',  chosen  from  nj, .  .  . ,  n^, 

The  left  member  equals  F{m,  N,  e)  constructed  for  the  numbers  other 
than  Vi,...,  V,'  of  the  set  ni, . . . ,  /z^.     For  s'  =  s,  we  have 


I 


f(m,  N,  e)  =F(m,  N,  e)  -Si^([f  ]'  ^'  ^n^ 


+  . 


The  latter  becomes  the  series  in  Bachmann^°  when  m=cD,  N  =  0,  €  =  1, 
while  rii,  n2, .  .  .  are  primes. 

F.  Mertens^^  considered  (7(n)  =ju(l)H-/i(2)  + . .  . +M(n)  and  proved  that 

l.o)[f]=.g)+.(l)+.. .+.©-..(.). 

<rin)  =  2a{g)  -  X  n{rMs)  f^l ,  g  =  [V^]. 

r.  s=i  LrsJ 

By  means  of  a  table  (pp.  781-830)  of  the  values  of  a{n)  and /x(n)  forn<  10000, 
it  is  verified  that  \(7{n)\<  \/n  for  1<  n<  10000. 

D.  von  Sterneck^^  verified  the  last  result  up  to  500  000,  and  for  16  larger 
values  under  5  million. 

A.  Berger"  noted  that,  if  g{m)g{n)=g{mn),  ^(1)  =  1, 
i:n{d)gid)=U{l-gip)]  (n>l), 

where  d  ranges  over  all  divisors  of  n,  p  over  the  prime  divisors  of  n.     If 
Xg{m)  is  absolutely  convergent, 

sM%W=n{i-^(p)), 

where  p  ranges  over  all  primes. 

D.  von  Sterneck^^  noted  that,  if  6ix)  ^  1  for  every  x  and  if 

|j.w|<^6-H|/(.)-/[g-/[=]-/[g|. 

In  particular,  |2iu(/c)|<8+?z/9. 

D.  F.  Seliwanov'-^  gave  Dedekind's  formula  with  appHcation  to  <t>{n). 
H.  von  Koch^^"  defined  n{k)  by  use  of  infinite  determinants. 

"Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  106,  II  a,  1897,  761-830. 

»Ibid.,  835-1024;  110,  Ila,  1901,  1053-1102;  121,  Ila,  1912,  1083-96;  Proc.  Fifth  Intern.  Con- 
gress Math.,  1912,  I,  341-3. 
"Ofversigt  Vctenskap.s-Akad.  Forhand.,  Stockholm,  55,  1898,  579-618. 
"Monatshcfte  Math.  Phys.,  9,  1898,  43-5. 
"Math.  Soc.  St.  Paersbourg,  1899,  120. 
""Ofversigt  K.  Vetensk.-Akad.  Forhand.,  Stockholm,  57,  1900,  659-68. 


Chap.  XIX]  INVERSION)  FUNCTION   fx{n).  447 

E.  B.  EUiott^^  of  Ch.  V  gave  a  generalization  of  fx{n). 

L.  Kronecker^"  defined  the  function  p{n,  k)  of  the  g.  c.  d.  (n,  k)  of  n,  k 
to  be  1  if  (n,  A:)  =  1,  0  if  (n,  A;)>1,  and  proved  for  any  function /(n,  k)  of 
(n,  k)  the  identity 

S  p{n,  k)f{n,  k)  =X  S  [x{d)fin,  kd), 

k=l  d  k=\ 

where  d  ranges  over  the  divisors  of  n.  The  left  member  is  thus  the  sum  of 
the  values  oif{n,  k)  ior  k<n  and  prime  to  n.    Set 

Fin,  rf)  =  S  f(n,  kd) ,  $(n,  d)  =  "s  p (^,  k)fin,  kd) . 

k=i  k=i  \tt     / 

Thus  when  d  ranges  over  the  divisors  of  n, 

Fin,  1)  =S$(n,  d),  *(n,  1)  =2M(d)F(n,  d) 

d 

are  consequences  of  each  other.    The  same  is  true  (p.  274)  for 
/i(n)=S/(d)^(^),  fin)  =i:ixid)gid)h{^, 

if  girs)  =gir)gis).     Application  is  made  (p.  335)  to  mean  values. 

E.  M.  L^meray^^  gave  a  generalized  inversion  theorem.  Let  i/'2(a,  b)  be 
symmetrical  in  a,  b  and  such  that  the  function  ^3  defined  by 

\f/sia,  b,  c)=\l/2{a,-ip2ib,  c)\ 

is  symmetrical  in  a,  b,  c.    Then  the  function 

\pM,  b,  c,  d)  =\l/3[a,  b,  yPiic,  d)} 

will  be  symmetrical  in  a,  b,  c,  d  and  similarly  for  t^fc(ai, .  . . ,  0;^) .     For  example, 

\l/2ia,  b)=aVl-\-b^+bVT+c^,  ^3  =  a&c+SaVl+6Vl+c2. 

Let  v=^iy,  u)  be  the  solution  of  y=yp2i'^j  ^)  for  ^-  The  theorem  states  that, 
li  di,. .  .,dk  are  the  divisors  of  m  =  p°gV.  . ,  and  if  Fim)  be  defined  by 

Fim)=Mfid,),...Jidj:)], 

we  have  inversely /(m)  =Q(G,  H),  where 

G=4.w,.£),.g),... ,.(-!-),...}, 

^=4K?)-(f)'-<^)'-;} 

where  ix  is  the  number  of  combinations  of  the  distinct  prime  factors  p, 
q,...  of  m  taken  0,  2,  4, ...  at  a  time,  and  v  the  number  taken  1,  3,  5, ...  at  a 
time. 

L.  Gegenbauer^^  defined  ^lix)  to  be  +1  if  a:  is  a  unit  of  the  field  Rii)  of 
complex  integers  or  a  product  of  an  even  number  of  distinct  primes  of 

soVorlesungen  iiber  Zahlentheorie,  I,  1901,  246-257.     His  €„  is  /i(n). 
31N0UV.  Ann.  Math.,  (4),  1,  1901,  163-7. 

32Verslag.  Wiss.  Ak.  Wetenschappen,  Amsterdam,  10,  1901-2,  195-207  (German.)     English 
transl.  in  Proc.  Sect.  Sc.  Ak.  Wet.,  4,  1902,  169-181. 


448  History  of  the  Theory  of  Numbers.  [Chap,  xix 

R{i),  —  1  if  a  product  of  an  odd  number,  0  if  x  is  di\'isible  by  the  square  of  a 
prime  of  R{i).  Let  [771]  denote  a  complete  set  of  residues  ?^0  of  complex 
integers  modulo  m.  Then  the  sum  of  the  values  of  fix)  for  all  complex 
integers  x  relatively  prime  to  a  given  one  n,  which  are  in  [m],  equals 
2jLi(c?)2/(d'x),  where  d  ranges  over  all  divisors  of  n  in  [m],  and  x  ranges 
over  {rn/d}.  This  is  due,  for  the  case  of  real  numbers,  to  Nazimov^"  of 
Chapter  V.  Again,  2/i((i)  =  l  or  0  according  as  norm  n  is  1  or  >1.  Also 
2/(d)  =F(7?)  miplies  Xfi{d)F(n/d)  =f{n). 

J.  C.  Kluyver^  employed  Kronecker's^"  identity  for  special  functions  / 
and  obtained  known  results  like 

2  cos  — =/x(n),  n  2sin  —  =  e^^''\ 

n  n 

where  v  ranges  over  the  integers  <Ji  and  prime  to  n,  while  y{n)  is  Bouga- 
ief's^^  function  p{n). 

P.  Fatou^  noted  that  Merten's  a{n)  does  not  oscillate  between  finite 
limits.  E.  Landau^^  proved  that  it  is  at  most  of  the  order  of  ne',  where 
t  =  —  aVlog  n.  Landau^®  noted  that  Furlan"  made  a  false  use  of  analysis 
and  ideal  theory  to  obtain  a  result  of  Landau's  on  Merten's^^  (r(n). 

0.  ■Meissner^^"  emploj-ed  primes  p,,  q,.  For  n=Up'i  set  Z(n)=nei''> 
and  Z2{n)  =  Z\Z{n)\.  Then  Z{?i)  =  n  only  if  n  is  Up.^i  or  16  or  Hjp'iqii. 
Next,  Z2(n)=n  in  these  three  cases  and  when  the  exponents  e,  in  n  are 
distinct  primes;  otherwise,  Z2(n)<n.     We  have  [1/Z{n)]=fx'^{n). 

R.  HackeP^  extended  the  method  of  von  Sterneck^^  and  obtained  vari- 
ous closer  approximations,  one^^  being 


xe{k) 


<^+io2+|2/g]-2/[g|, 


26 


where  a  =  l,  6,  10,  14,  105;  6  =  2,  3,  5,  7,  11,  13,  385,  1001. 
W.  Kusnetzov-^^  gave  an  analytic  expression  for  ;u(n). 
K.  Knopp^^°  of  Ch.  X  gave  many  formulas  involving  n{n). 
A.  Fleck''""  generahzed  n{m)=iJLi{m)  by  setting 

M*(m)=n(-l)»'(f),  m=Upr. 

,=1  \a,/  ,=1 

Using  the  zeta  function  (12)  of  Ch.  X,  and  ^^  of  Fleck^-^  of  Ch.  V,  we  have 

y     r^\  r    \  V  ^k-\{m)       .  .    *  ^t(m)  ^  (m\ 

i:Hk{d)=fXk-i{m),  S — — —  =f(s)  2  — — -,         <t)k{m)  =  Xdfik+A-T)- 

dim  fn  =  l        ^  m  =  l       ^  dm  \"/ 

"Verslag.  Wiss.  Ak.  Wetenschappen,  Amsterdam,  15,  1906,  423-9.     Proc.  Sect.  Sc.  Ak.  Wet., 

9,  1906,  408-14.  "Acta  Math.,  30,  1906,  392. 

«Rend.  Circ.  Mat.  Palermo,  26,  1908,  250.  "Rend.  Circ.  Mat.  Palermo,  23, 1907,  367-373- 

"Monatshefte  Math.  Phys.,  18,  1907,  235-240. 
»"Math.  Xaturw.  Blatter,  4,  1907,  85-6. 
"SitzunKsber.  Ak.  Wiss.  Wien  (Math.),  118,  1909,  II a,  1019-34. 
'•Sylvester,  Messenger  Math.,  (2),  21,  1891-2,  113-120. 
«°Mat.  Sbomik  (Math.  Soc.  Moscow),  27,  1910,  335-9. 
'■wSitzungsber.  Berlin  Math.  GeseU.,  15,  1915,  3-8. 


Chap.  XIX]  NUMERICAL   INTEGRALS   AND   DERIVATIVES.  449 

The  theorem  Sn=iM(^)A  =  0  and  other  results  on  sums  involving  /i(n) 
play  an  important  role  in  the  theory  of  the  asymptotic  distribution  of 
primes.  In  accord  with  the  plan  of  not  entering  into  details  on  that  topic 
(Ch.  XVIII),  the  reader  is  referred  for  the  former  topic  to  the  history  and 
exposition  by  E.  Landau,^^  and  to  the  subsequent  papers  by  A.  Axer,^^ 
E.  Landau/3  and  J.  F.  Steffensen.^'' 

Proofs  of  (2)  or  (3)  are  given  in  the  following  texts : 

P.  Bachmann,  Die  Lehre  von  der  Kreistheilung,  1872,  8-11;   Die  Elemente  der 
Zahlentheorie,  1892,  40-4;  Grundlehren  der  Neueren  Zahlentheorie,  1907,  26-9. 
T.  J.  Stieltjes,  Theorie  des  nombres,  Ann.  fac.  Toulouse,  4,  1890,  21. 
Borel  and  Drach,  Introd.  theorie  des  nombres,  1895,  24-6. 
E.  Cahen,  ;Sl6ments  de  la  theorie  des  nombres,  1900,  346-350. 
E.  Landau,"  577-9. 

Numerical  Integrals  and  Derivatives. 

N.  V.  Bougaief^^  (Bugaiev)  called  F{n)  the  numerical  integral  of  /(n)  if 
F(m)  =2/(5),  summed  for  all  the  divisors  5  of  m,  and  called  /(n)  the  numer- 
ical derivative  function  of  F{n),  denoted  by  DF(n)  symbolically. 

Granting  that  there  is,  for  every  n,  the  development 


F{n)  =  a,[n]+a2y^j  '^^Hjj 


+ 


where  [x]  is  the  largest  integer  ^x,  then  a^  is  the  numerical  derivative  of 
F{k)  -F{k-1).     He  developed  [n'^%  [n'%  etc. 

N.  V.  Bougaief,^^  after  amplifying  the  preceding  remarks,  proved  that 

S  e{8)x{d)  =yp{n),  d(n)d(m)  =d{nm) 

d6=n 

imply 

Writing  D~^d{d)  for  '29(d),  summed  for  the  divisors  d^  of  n,  we  have 

D'^2x{md)=^2xi^)D''d{d), 
for  any  integer  /x,  positive  or  negative.    There  are  formulas  like 

"Handbuch. .  .Verteilung  der  Primzahlen,  II,  1909,  567-637,  676-96,  901-2. 

«Prace  mat.  fiz.,  Warsaw,  21,  1910,  65-95;  Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  120,  1911, 

II  a,  1253-98. 
«Sitzungsber.  Ak.  Wiss.  Wien.  (Math.),  120,  1911,  Ila,  973-88;  Rend.  Circ.  Mat.  Palermo,  34, 

1912,  121-31. 
"Analytiske  Studier. . .,  Diss.,  Kjobenhavn,  1912,  148  pp.     Fortschritte,  43,  262-3.     Extract 

in  Acta  Math.,  37,  1914,  75-112. 
*^ Journal  de  la  Soc.  Philomatique  de  Moscou,  5,  1871. 
**Theory  of  numerical  derivatives,  Moscow,  1870-3,  222  pp.     Extracts  from   Mat.  Sbomik 

(Math.  Soc.  Moscow),  5,  I,  1870-2,  1-63;  6,  1872-3,   I,  133-180,  199-254,  309-360 

(reviewed  in  BuU.  Sc.  Math.  Astr.,  3,  1872,  200-2;  5,  1873,  296-8;  6,  1874,  314-6). 

R^8um6  by  Bougaief,  BuU.  Sc.  Math.  Astr.,  10,  I,  1876,  13-32. 


450  History  of  the  Theory  of  Numbers.  [Chap,  xix 

u  =  l  u  =  l  L"J 

ld(^fj^e(x^  (mod  2),  20^^)  -2d(£j^d{^n)  (mod  3), 

where  6{n)  is  the  number  of  primes  a^n.  Other  special  results  were  cited 
under  155,  Ch.  V;  6,  Ch.  XI;  217,  Ch.  XVIIL 

E.  Ces^ro^^"  treated  2/(5)  in  connection  with  median  and  asymptotic 
formulas. 

Bougaief^^  treated  numerical  integrals,  noting  formulas  like 


liP'"' 


where  ^(n)  is  the  number  of  prime  factors  a,  b,.  .  .  of  n  =  a°6^ .  .  . , 
2  rPid)  =  2  \l/(d)+  2  rPia'd)  =  2  yp{ad)+  2  yp{d). 

'^i"      d\i      dr       d\i       dL^ 

Bougaief^^  gave  a  large  number  of  formulas  of  the  type 

2V^(d)[^]  =2V(rf)+2"*/^V(^)+Sf'»/^V(rf)+  •  • ., 

where,  on  the  left,  d  ranges  over  all  the  di\isors  of  m;  while,  on  the  right,  d 
ranges  over  those  divisors  of  m  which  do  not  exceed  n,  [n/2], .  . .,  respectively. 
Bougaief°^  gave  the  relation 


Xd(Vd)=^U^2'n): 
d\n  P        \P  / 


where  p  ranges  over  all  primes  ^  -y/n,  and  ^^(t??,  n)  is  the  sum  of  the  kth. 
powers  of  all  di\'isors  ^m  of  n,  so  that  ^o  is  their  number,  and  ^(0  is  the 
number  of  primes  ^t. 

L.  Gegenbauer^'^  noted  that  the  preceding  result  is  a  case  of 

S,(d).(JJ/(X)=J/(X){2,W.(3}, 

where  dx  ranges  over  the  divisors  ^X  of  n.     Special  cases  are 

Y%(<f)  =yUa'. n),  ^d%{(^)^^2a%{l,  n), 

where  |p  {m,  n)  is  the  sum  of  the  pth  powers  of  the  di\'isors  ^m  of  n. 

«^Giornale  di  Mat.,  2.5,  18S7,  1-13. 

"Mat.  Sbornik  (Math.  Soc.  Moscow),  14,  1888-90,  169-197;  16,  1891,  169-197  (Russian). 

"Ibid.,  17,  1893-5,  720-59. 

"Comptes  Rendus  Paris,  119,  1894,  1259. 

•oMonatshefte  Math.  Phys.,  6,  1895,  208. 


Chap.  XIX]  NUMERICAL   INTEGRALS    AND    DERIVATIVES.  451 

Bougaief^""  noted  that,  for  an  arbitrary  function  \p, 

<*       J       M=l      n  =  l  M=l  n=l  u  =  ln=l 

N.  V.  Bervi"  treated  numerical  integrals  extended  over  solutions  of 
indeterminate  equations,  in  particular  for  n  =  a-\-h{x-{-y)-\-cxy,  h^  =  b+ac. 

Bougaief^"  considered  definite  numerical  integrals,  viz.,  sums  over  all 
divisors,  between  a  and  b,  oi  n.  He  expressed  sums  of  [x],  the  greatest 
integer  ^x,  as  sums  of  values  of  f  (n,  m),  viz.,  the  number  of  divisors  ^n 
of  m.  Also  sums  of  ^s  expressed  as  ^i{l)+^i{2)-\- .  .  .-{-^i{n),  where 
fi(n)  is  the  number  of  the  divisors  of  n  which  are  ith.  powers. 

1. 1.  Cistiakov^^"  (Tschistiakow)  treated  the  second  numerical  derivative. 

Bougaief^^^  gave  13  general  formulas  on  numerical  integrals. 

Bougaief^^  gave  a  method  of  transforming  a  sum  taken  over  1,  2, . . .,  n 
into  a  sum  taken  over  all  the  divisors  of  n.  He  obtains  various  identities 
between  functions. 

D.  J.  M.  Shelly, ^^  using  distinct  primes  a,h,.  .  .,  called 


^'=K^^•••) 


the  derivative  of  iV  =  a^b^ ....  Similar  definitions  are  given  for  derivatives 
of  fractions  and  for  the  case  of  fractional  exponents  a,  j8, .  .  . .  The  primes 
are  the  only  integers  whose  derivatives  are  unity. 

ec^Comptes  Rendus  Paris,  120,  1895,  432-4. 

«iMat.  Sbornik  (Math.  Soc.  Moscow),  18,  1896,  519;  19,  1897,  182. 

^mid.,  18,  1896,  1-54  (Russian);  see  Jahrb.  Fortschritte  Math.,  27,  1896,  p.  158. 

«2o/6id.,  20,  1899,  595;  see  Fortschritte,  1899,  194. 

^^^Ibid.,  549-595.     Two  of  the  formulas  are  given  in  Fortschritte,  1899,  194. 

«/6id.,  21,  1900,  335,  499;  see  Fortschritte,  31,  1900,  197. 

"Asociaci6n  espanola,  Granada,  1911,  1-12. 


CHAPTER  XX. 

PROPERTIES  OF  THE  DIGITS  OF  NUMBERS. 

John  HilP  noted  that  139854276=118262  is  formed  of  the  nine  digits 
permuted  and  believed  erroneously  that  it  is  the  only  such  square. 

N.  BrownelP"  found  169  and  961  as  the  squares  whose  three  digits  are 
in  reverse  order  and  whose  roots  are  composed  of  the  same  digits  in  reverse 
order.  The  least  digit  in  the  roots  is  also  the  least  in  the  squares,  while  the 
greatest  digit  in  the  roots  is  one-third  of  the  greatest  in  the  squares  and 
one-half  of  the  digit  in  the  tens  place. 

W.  Saint^''  proved  that  every  odd  number  N  not  divisible  by  5  is  a  divisor 
of  a  number  11 .  .  .1  oi  D^N  digits  [by  a  proof  holding  only  for  N  prime 
also  to  3].  For,  let  1 ...  1  (to  D  digits)  have  the  quotient  q  and  remainder 
r  when  divided  by  D.  This  remainder  r  must  recur  if  the  number  of  digits 
1  be  increased  sufficiently.  Hence  let  1...1  (to  D+d  digits)  give  the 
remainder  r  and  quotient  Q  when  divided  by  D.  By  subtraction,  D{Q  —  q) 
=  1.  .  .10. .  .0  (with  d  units  followed  by  D  zeros).  Hence  if  1.  .  .1  (to  d 
digits)  were  not  divisible  by  every  odd  number  ^  D  and  prime  to  5  [and  to 
3],  there  would  be  a  remainder  R;  then  RO.  .  .0  (with  D  zeros)  would  be 
divisible  by  an  odd  number  prime  to  5  [and  to  3],  which  is  impossible. 

P.  Barlow^"  stated,  and  several  gave  inadequate  proofs,  that  no  square 
has  all  its  digits  alike.  He^'^  stated  and  proved  that  111111111^=1 23456- 
78987654321  is  the  largest  square  such  that  if  unity  be  subtracted  from  each 
of  its  digits  and  again  from  each  digit  of  the  remainder,  etc.,  all  zeros  being 
suppressed,  each  remainder  is  a  square.  Denote  (10^  — 1)/(10  — 1)  by  [k]. 
Then  |^(x+l)p  has  x  digits  and  exceeds  [x]  by  10{|(a:  — l)p.  Since 
zeros  are  suppressed  we  have  a  square  as  remainder,  and  the  process  can 
be  repeated.  It  is  stated  that  therefore  the  property  holds  only  for  V, 
IV,  IIP,  .... 

Several^*  found  that  135  is  the  only  number  N  composed  of  three  digits 
in  arithmetical  progression  such  that  the  digits  will  be  reversed  if  132  times 
the  middle  digit  be  added  to  N. 

W.  Saint  ^•'^  found  the  least  integral  square  ending  with  the  greatest  num- 
ber of  equal  digits.  The  possible  final  digits  are  1,  4,  5,  6,  9.  Any  square  is 
of  the  form  4n  or  4n+l.  Hence  the  final  digit  is  4.  If  the  square  termi- 
nated with  more  than  three  4's,  its  quotient  by  4  would  be  a  square  ending 
with  two  I's,  just  proved  to  be  impossible.     Of  the  numbers  ending  with 

^Arithmetic,  both  ia  Theory  and  Practice,  ed.  4,  London,  1727,  322. 
i«The  Gentleman's  Diary,  or  Math.  Repository,  London,  1767;  Davis'  ed.,  2,  1814,  123. 
ifcJour.  Nat.  Phil.  Chem.  Arts  (ed.,  Nicholson),  London,  24,  1809,  124-6. 
I'^The  Gentleman's  Diary,  or  Math.  Repository,  London,  1810,  38-9,  Quest.  952. 
^^Ibid.,  1810,  39-40,  Quset.  953. 
^^Ibid.,  1811,  33-4,  Quest.  960. 

I/Ladies'  Diary,  1810-11,  Quest.,  1218;  Leybourn's  M.  Quest.  L.  D.,  4,  1817,  139^1. 

453 


454  History  of  the  Theory  of  Numbers.  [Chap.  XX 

three  4's,  the  least  is  1444.  J.  Davey  discussed  only  numbers  of  3  or  4 
digits  of  which  the  last  2  or  3  are  equal,  respectively. 

Several^"  found  that  the  squares  169  and  961  are  composed  of  the  same 
digits  in  reverse  order,  have  roots  of  two  digits  in  reverse  order,  while 
the  sum  of  the  digits  in  each  square  equals  the  square  of  the  sum  of  the 
digits  in  each  root;  finally,  the  sum  of  the  digits  in  each  root  equals  the 
square  of  their  difference. 

An  anonymous  writer^  proposed  the  problem  to  find  a  number  n  given 
the  product  of  n  by  the  number  obtained  from  n  by  writing  its  digits  in 
reverse  order  [Laisant*^]. 

P.  T^denat^  considered  the  problem  to  find  a  number  of  n  digits  whose 
square  ends  with  the  same  n  digits  in  the  same  order.  If  a  is  such  a  number 
of  n  — 1  digits,  so  that  a^  =  10'*~^64-a,  we  can  find  a  digit  A  to  annex  at  the 
left  of  a  to  obtain  a  desired  number  10"~^A  +  a  of  n  digits.  Squaring  the 
latter,  we  obtain  the  condition  {2a  —  l)A=  —b  (mod  10). 

J.  F.  Frangais^  noted  the  solutions 

x  =  2"p  =  5"5  +  l,  x2  =  10"p5+a:, 

2/  =  5«r  =  2"s  +  l,  y^  =  lO"rs-\-y, 

in  which  the  resulting  condition  2"p  — 5"^  =  1  or  5"r  —  2"s=  1  is  to  be  satisfied. 
Special  solutions  are  given  by  n  =  1,  p  =  3;  n  =  2,  p  =  19;  n  =  3,  p  =  47;  n  =  4, 
p  =  586;  etc.,  to  n  =  7. 

J.  D.  Gergonne^  generalized  the  problem  to  base  B.     Then 

x(rc-l)=5"t/. 

Let  p,  q  be  relatively  prime  and  set  jB"  =  pq.  Then  x  =  pt,  x  —  l=qu,  or  vice 
versa.  The  condition  pt  —  qu  =  l  is  solved  for  t,  u.  When  B  =  10,  n  =  20, 
the  least  u  is  81199. 

Anonymous  writers^  stated  and  proved  by  use  of  the  decimal  fraction  for 
1/n  that  every  number  divides  a  number  of  the  form  9 ...  90 ...  0. 

A.  L.  Crelle^  proved  the  generalization :  Every  number  divides  a  number 
obtained  by  repeating  any  given  set  of  digits  and  affixing  a  certain  number  of 
zeros,  as  23..  .230..  .0. 

Several^"  found  a  square  whose  root  has  two  digits,  their  quotient  be- 
ing equal  to  their  difference.  By  x/y=x—y,  x=i/+l  +  l/(2/  — 1),  an  inte- 
ger, whence  y=2,  x=4t.     Thus  the  squares  are  24^  or  42^. 

The^^  three  digits  of  a  number  are  in  geometrical  progression;  the  prod- 
uct of  the  sum  of  their  cubes  by  the  cube  of  their  sum  is  1663129;  if  the 
number  obtained  by  reversing  the  digit  be  divided  by  the  middle  digit,  the 

^ffLadies'  Diary,  1811-12,  Quest.  12.31;  Leybourn,  I.  c,  153-4. 

»Annales  de  Math,  (ed.,  Gergonne),  3,  1812-3,  384. 

*Ibid.,  5,  1814-5,  309-321.     Problem  proposed  on  p.  220. 

*Ibid.,  321-2. 

^Ibid.,  322-7. 

*Ibid.,  19,  1828-9,  256;  20,  1829-30,  304-5. 

Ubid.,  20,  1829-30,  349-3.52;  Jour,  fiir  Math.,  5,  1830,  296. 

^"Ladies'  Diary,  1820,  36,  Quest.  1347. 

'I'Ibid.,  1822,  33,  Quest.  1374. 


i 


Chap.  XX]  PROPERTIES   OF  THE   DiGITS   OF  NXJMBERS.  455 

quotient  is  46|.     By  the  last  condition,  the  middle  digit  must  be  3,  since 
not  a  higher  multiple  of  3.     Hence  the  number  is  931. 

To  find  a  synunetrical  number  abcba  of  five  digits  whose  square  exhibits 
all  ten  digits,  W.  Rutherford^"  noted  that  the  square  is  divisible  by  9  since 
the  sum  of  the  digits  is  divisible  by  9.  Hence  the  sum  of  the  digits  of  the 
number  is  divisible  by  3.  Also  a  ^  3.  Taking  c  =  a-{-b,  c=S,he  got  35853. 
J.  Sampson  noted  also  the  answers  84648,  97779, 

J.  A.  Grunert^  proved  by  use  of  Euler's  generahzation  of  Fermat's  the- 
orem that^  every  number  divides  9 .  .  .  90 .  .  .0.   ---^ 

Drot^"  asked  for  the  values  of  x  for  which  N""  has  the  same  final  k  digits 
as  N,  when  ^  =  1,  2  or  3. 

J.  Bertrand^''  discussed  the  numbers  of  digits  of  certain  numbers. 

A.  G.  Emsmann^  treated  a  number  6  of  n  digits  to  base  10  equal  to  the 
product  of  the  sum  of  its  digits  by  a,  and  such  that  if  another  number  of  n 
digits  be  subtracted  from  b  the  remainder  shall  equal  the  number  obtained 
by  writing  the  digits  of  b  in  reverse  order. 

J.  Booth^°  noted  that  a  number  of  six  digits  formed  by  repeating  any  set 
of  three  digits  is  divisible  by  7,  11,  13  [since  by  1001]. 

.  G.  Bianchi^^"  noted  various  numerical  relations  like  10^  =  11111111  + 
8.1111111  + 8.9.111111+...  +  8.9M  + 9^  =  2222222 +  ...+7.8^2+8^  98  = 
(12-1-0)9-1,  987  =  (123-12-l)9-3,  9876  =  (1234-123-13)9-6. 

C.  M.  Ingleby^^  added  the  digits  of  a  number  N  written  to  base  r,  then 
added  the  digits  of  this  sum,  etc.,  finally  obtaining  a  number,  designated 
SN,  of  a  single  digit;  and  proved  that  S{MN)=S{SM-SN). 

P.  W.  Flood^^"  proved  that  64  is  the  only  square  the  sum  of  whose  digits 
less  unity  and  product  plus  unity  are  squares. 

G.  Cantor^^  employed  any  distinct  positive  integers  a,  6, ... ,  considered 
the  system  of  integers  in  which  a  occurs  a  times,  b  occurs  b  times,  etc.,  and 
called  a  system  simple  if  every  number  can  be  expressed  in  a  single  way  in  the 
form  aa+/36+  .  .  .,  where  a  =  0,  1,.  .  .,  a;  /3  =  0,  1, .  .  .,  6; .  .  ..  A  system  is 
simple  if  and  only  if  each  basal  number  k  divides  the  next  one  I  and  if  k 
occurs  k  =  (l/k)  —  1  times. 

G.  Barillari^^  noted  that,  if  10  belongs  to  the  exponent  m  modulo  b, 
the  number  P  =  a/3 .  .  .  Xa/3 ...  X ... ,  obtained  by  repeating  h  times  (/i  >  1)  any 
set  of  n  digits,  is  divisible  by  6  if  6  is  prime  to  10^-1  and  if  nh  is  a  multiple 

'^Ladies'  Diary,  1835,  38,  Quest.  1576. 

8Jour.  fur  Math.,  5,  1830,  185-6. 

^'^Nouv.  Ann.  Math.,  4  1845,  637-44;  5,  1846,  25.     For  references  to  tables  of  powers,  13, 

1854,  424-5. 
oblbid.,  8,  1849,  354. 

"Abhandlung  liber  eine  Aufgabe  aus  der  Zahlentheorie,  Progr.  Frankfurt,  1850,  36  pp. 
loProc.  Roy.  Soc.  London,  7,  1854-5,  42-3. 
^""Proprieta  e  rapporti  de'  numeri  interi  e  composti  coUe  cifre  semplici  .  .  .  ,  Modena,  1856. 

Same  in  Mem.  di  Mat.  e  di  Fis.  Soc.  Ital.  Sc,  Modena,  (2),  1,  1862,  1-36,  207. 
"Oxford,  Cambr.  and  DubUn  Messenger  Math.,  3,  1866,  30-31. 
"''Math.  Quest.  Educ.  Times,  7,  1867,  30. 
"Zeitschrift  Math.  Phys.,  14,  1869,  121-8. 
"Giomale  di  Mat.,  9,  1871,  125-135. 


456  History  of  the  Theory  of  Numbers.  [Chap.  XX 

of  w,  but  P  is  not  divisible  by  h  if  nh  is  not  a  multiple  of  m.  If  h  divides 
10"—  1,  P  is  divisible  by  b  when  h  =  b,  but  not  divisible  by  b  when  h  is  not  a 
multiple  of  b. 

A.  MoreP*  proved  that  the  numbers  ending  with  12,  38,  62  or  88  are  the 
only  ones  whose  squares  end  with  two  equal  digits. 

H.  Hoskins^^'*  found  the  sum  of  the  117852  numbers  of  7  digits  which 
can  be  formed  with  the  digits  1,  1,  2,  2,  2,  2,  2,  3,  3,  4,  5,  6,  7. 

J.  Plateau^^  noted  that  every  odd  number  not  ending  with  5  has  a  multiple 
of  the  form  11...  1  [Saint^^. 

P.  Mansion^^  proved  the  theorem  of  Plateau. 

J.  W.  L.  Glaisher^^  deduced  Crelle's^  theorem  from  Plateau's.^^ 

C.  A.  Laisant^^  treated  a  problem^  on  reversing  digits. 

G.  R.  Perkins^^"  and  A.  Martin^^  stated  that  all  powers  of  numbers  end- 
ing with  12890625  end  with  the  same  digits. 

E.  Catalan^"  noted  that  the  g.  c.  d.  of  two  numbers  of  the  form  1 ...  1 
of  n  and  n'  digits  is  of  Uke  form  and  has  A  digits,  where  A  is  the  g.  c.  d.  of  n 
and  n'. 

Lloyd  Tanner,^""  generalizing  Martin's^^  question,  found  how  many 
numbers  N  oin  digits  to  the  base  r  end  with  the  same  digits  as  their  squares, 
i.  e.,  N^—N  =  Kr''.  If  r"  is  the  product  of  q  powers  of  primes,  there  are 
2^  —  2  values  of  N.  He^^^  found  numbers  M  and  N  with  n  digits  to  the 
base  r  such  that  the  numbers  formed  by  prefixing  M  to  N  and  iV  to  M 
have  a  given  ratio. 

J.  Plateau^ ^  proposed  the  problem  to  find  two  numbers  whose  product 
has  all  its  digits  alike.    Angenot  noted  that 

6^-1'  6-1 

give  a  solution  for  base  b.    Catalan^^  noted  that  Euler's  theorem 


6-1 


nm 


for  n  prime  to  6,  furnishes  a  solution  n,  m. 

Lloyd  Tanner"  stated  and  Laisant  proved  that  87109376  and  12890625 
are  the  only  numbers  of  8  digits  whose  squares  end  with  the  same  8  digits. 

"Nouv.  Ann.  Math.,  (2),  10,  1871,  44-6,  187-8. 

i^Math.  Quest.  Educ.  Times,  15,  1871,  89-91. 

"BuU.  Acad.  Roy.  de  Belgique,  (2),  16,  1863,  62;  28,  1874,  468-476. 

"Nouv.  Corresp.  Math.,  1,  1874-5,  8-12;  Mathesis,  3,  1883,  196-7.     Bull.  Bibl.  Storia  Sc. 

Mat.,  10,  1877,  476-7. 
"Messenger  Math.,  5,  1875-6,  3-5. 

**M6m.  80C.  sc.  phys.  et  nat.  de  Bordeaux,  (2),  1,  1876,  403-11. 
i«»Math.  Miscellany,  Flushing,  N.  Y.,  2,  1839,  92. 
"Math.  Quest.  Educat.  Times,  26,  1876,  28. 
»oM6m.  Society  Sc.  Li^ge,  (2),  6,  1877,  No.  4. 

"•^Messenger  Math.,  7,  1877-8,  63-4.    Cases  r^  12,  Math.  Quest.  Educ.  Times,  28,  1878, 32-4. 
Jo'-Math.  Quest.  Educ.  Times,  29,  1878,  94-5. 
«Nouv.  Corresp.  Math.,  4,  1878,  61-63. 
«/Wd.,  5,  1879,  217;  6,  1880,  43. 


Chap.  XX]  PROPERTIES  OF  THE   DiGITS   OF  NuMBERS.  457 

Moret-Blanc^^  proved  that  1,  8,  17,  18,  26,  27  are  the  only  numbers  equal 
to  the  sum  of  the  digits  of  their  cubes. 

C.  Berdelle^^"  considered  the  last  n  digits  of  numbers,  in  particular  of  5*. 

E.  Cesaro^^  noted  that  the  sum  of  the  pth  powers  of  ten  consecutive 
integers  ends  with  5  unless  p  is  a  multiple  of  4,  when  it  ends  with  3. 

F.  de  Rocquigny^^  noted  that  if  a  number  of  n  digits  equals  the  sum  of  the 
2"  — 1  products  of  its  digits  taken  1,  2, . .  .,  n  at  a  time,  its  final  n  —  1  digits 
are  all  9. 

E.  Cesaro^®  considered  the  period  of  the  digits  of  rank  n  in  powers  of  5. 

Lists^"  have  been  given  of  squares  formed  by  the  nine  digits  >  0,  or  the 
ten  digits,  not  repeated. 

0.  Kessler^^  gave  a  table  of  divisors  of  numbers  formed  by  repeating  a 
given  set  of  digits  a  small  number  of  times. 

T.  C.  Simmons""  noted  that,  if  the  sum  of  the  digits  of  n  is  10,  that  of 
2n  is  11  unless  each  digit  of  n  is  <5  or  two  are  5.  For  4  digits  the  numbers 
of  each  type  are  counted. 

J.  S.  Mackay^^  treated  the  last  subject. 

E.  Lemoine^^  considered  numbers  like  A  =  8607004053  such  that,  if  a  is 
the  number  derived  by  reversing  the  digits  of  A,  the  sum  A+a  =  12111011121 
reverses  into  itself. 

M.  d'Ocagne^°  considered  the  sum  a{N)  of  the  digits  of  the  first  N  integers. 
If  iVp  =  ap-10^+  .  .  .  +ai-10+ao  and  d  =  ap-10^-l,  then 

aid)  =  10'-'-5a^{a,-l+9p),        (7{N,)  =(T{d)  +  {N,.,  +  l)a^+a{N,.,). 

Hence 

^(iVp)=|ao(ao+l)  +  ia,{lO'-^-5(ai-l+97;)+iV,_i  +  lj- 

The  number  of  digits  in  1,. . .,  AT  is  (p+l)(iV+l)-(10^+'-l)/9.  See  the 
next  paper. 

M.  d'Ocagne^^  noted  that,  in  writing  down  the  natural  numbers  1,  . .  .,N, 
where  N  is  composed  of  n  digits,  the  total  number  of  digits  written  is 
n(iV+l)  —  In,  where  1„  =  1 . . .  1  (to  n  digits). 

E.  Barbier^i"  asked  what  is  the  W^^%h.  digit  written  if  the  series  of 
natural  numbers  be  written  down. 

23Nouv.  Ann.  Math.,  (2),  18,  1879,  329;  proposed  by  Laisant,  17,  1878,  480. 
23aAssoc.  franQ.,  8,  1879,  176-9. 
\  24Nouv.  Corresp.  Math.,  6,  1880,  519;  Mathesis,  1888,  103. 

N  s^Les  Mondes,  53,  1880,  410-2. 

26NOUV.  Corresp.  Math.,  4,  1878,  387;  Nouv.  Ann.  Math.,  (3),  2,  1883,  144,  287;  1884,  160. 
»8"Math.  Magazine,  1,  1882-4;  69-70;  I'intermediaire  des  math.,  4,  1897,  168;  14,  1907,  135; 

Sphinx-Oedipe,  1908-9,  35;  5,  1910,  64;  Educ.  Times,  March,  1905.     Math.  Quest.  Educ. 

Times,  52,  1890,  61;  (2),  8,  1905,  83-6  (with  history). 
"Zeitschrift  Math.  Phys.,  28,  1883,  60-64. 
"«Math.  Quest.  Educ.  Times,  41,  1884,  28-9,  64-5. 
"Proc.  Edinburgh  Math.  Soc,  4,  1885-6,  55-56. 
"Nouv.  Ann.  Math.,  (3),  4,  1885,  150-1. 
^ojornal  de  so.  math,  e  ast.,  7,  1886,  117-128. 
"/bid.,  8,  1887,  101-3;  Comptes  Rendus  Paris,  106,  1888,  190. 
""Comptes  Rendus  Paris,  105,  1887,  795,  1238. 


458  History  of  the  Theory  of  Numbers.  [Chap,  xx 

L.  Gegenbauer^^''  proved  generalizations  of  Cantor's^-  theorems,  allowing 
negative  coefficients.  Given  the  distinct  positive  integers  ai,  a2, .  .  .,  every 
positive  integer  is  representable  in  a  single  way  as  a  linear  homogeneous 
function  of  ai,  a2, .  .  .  with  integral  coefficients  if  each  a^  is  di\isible  by  a^-i 
and  the  quotient  equals  the  number  of  permissible  values  of  the  coefficients 
of  the  smaller  of  the  two. 

R.  S.  .Aiyar  and  G.  G.  Storr^^*"  found  the  number  p„  of  integers  the  sum 
of  whose  digits  (each  >0)  is  n,  by  use  of  Pn=  p„_i+  .  .  .  +p„_9. 

E.  Strauss''^  proved  that,  if  ai,  ao, .  .  .  are  any  integers  >  1,  every  positive 
rational  or  irrational  number  <  1  can  be  written  in  the  form 

— f- 1 r...  (ai<ai,  a2<a2,  •■  •), 

the  a's  being  integers,  and  in  a  single  way  except  in  the  case  in  which  all  the 
a„  beginning  with  a  certain  one,  have  their  maximum  values,  when  also  a 
finite  representation  exists. 

E.  Lucas^  noted  that  the  only  numbers  having  the  same  final  ten  digits 
as  their  squares  are  those  ending  with  ten  zeros,  nine  zeros  followed  by  1, 
8212890625  and  1787109376.  He  gave  (ex.  4)  the  possible  final  nine  digits* 
of  numbers  whose  squares  end  with  224406889.  He  gave  (p.  45,  exs.  2,  3) 
all  the  numbers  of  ten  digits  to  base  6  or  12  whose  squares  end  with  the  same 
ten  digits.  Similar  special  problems  were  proposed  by  Escott  and  Palm- 
strom  in  Tlntermediaire  des  Mathematiciens,  1896,  1897. 

J.  Kraus^  discussed  the  relations  between  the  digits  of  a  number 
expressed  to  two  different  bases. 

A.  Cunningham**"  called  N  an  agreeable  number  of  the  mth  order  and 
nth  degree  in  the  r-ary  scale  if  the  m  digits  at  the  right  of  'N  are  the 
same  as  the  m  digits  at  the  right  of  A^"  when  each  is  expressed  to  base  r; 
and  tabulated  all  agreeable  numbers  to  the  fifth  order  and  in  some  cases 
to  the  tenth.  A  number  A  of  m  digits  is  completely  agreeable  if  the  agree- 
ment of  A  with  its  nth  power  extends  throughout  its  m  digits,  the  condition 
being  A"=A  (mod  r'"). 

E.  H.  Johnson^''  noted  that,  if  a  and  r  —  \  are  relatively  prime  and 
aa.  .  a  (to  r  —  1  digits  to  base  r)  is  divided  by  r  —  1 ,  there  appear  in  the 
quotient  all  the  digits  1,  2, .  . .,  r— 1  except  one,  which  can  be  found  by 
dividing  the  sum  of  its  digits  by  r  — 1. 

C.  A.  Laisant^"  stated  that,  if  A  =  123.  .  .n,  written  to  base  n+1,  be 
multiplied  by  any  integer  <n  and  prime  to  n,  the  product  has  the  digits 
of  A  permuted. 

^"'Sitzungsber.  Ak.  Wiss.  Wien  (Math.),  95,  1887,  II,  618-27. 
'x^Math.  Quest.  Educ.  Times,  47,  1887,  64.  »*Acta  Math.,  11,  1887-8,  13-18. 

»'Th6orie  des  nombres,  1891,  p.  38.     Cf.  Math.  Quest.  Educ.  Times,  (2),  6,  1904,  71-2. 
*Same  by  Kraitchik,  Sphinx-Oedipe,  6,  1911,  141. 
"Zeitschr.  Math.  Phys.,  37,  1892,  321-339;  39,  1894,  11-37. 

»*'British  Assoc.  Report,  1893,  699.  »♦* Annals  of  Math.,  8,  1893^,  160-2. 

»*«L'interm6diaire  des  math.,  1894,  236;  1895,  262.     Proof  by  "Nauticus,"  Mathesis,  (2),  5, 
1895,  37-42. 


Chap.  XX]  PROPERTIES   OF  THE   DiGITS   OF   NUMBERS.  459 

Tables  of  primes  to  the  base  2  are  cited  under  Suchanek^"  of  Ch.  XIII. 
There  is  a  eollection^^'^  of  eleven  problems  relating  to  digits. 
To  find^*'  the  number  <90  which  a  person  has  in  mind,  ask  him  to 
annex  a  declared  digit  and  to  tell  the  remainder  on  division  by  3,  etc. 
T.  Hayashi^^  gave  relations  between  numbers  to  the  base  r: 

123.  .  .  {r-l}-(r-l)+r  =  l .  .  .1  (to  r  digits), 

{r-ljjr-2)  ...321-(r-l)-l  =  {r-2}  jr-2)  .  .  .  (to  r  digits). 

Several  writers^*^  proved  that 

123.  .  .  {r-l}-(r-2)+r-l  =  {r-l}  .  .  .321. 

T.  Hayashi"  noted  that  if  A  =  10+r(10)Hr2(10)^+  ...  be  multiplied  or 
divided  by  any  number,  the  digits  of  each  period  of  A  are  permuted  cyclically. 

A.  L.  Andreini^^"  found  pairs  of  numbers  N  and  p  (as  37  and  3)  such  that 
the  products  of  N  by  all  multiples  ^  (J5  — l)p  of  p  are  composed  of  p  equal 
digits  to  the  base  5^  12,  whose  sum  equals  the  multiplier. 

P.  de  Sanctis^^  gave  theorems  on  the  product  of  the  significant  digits  of, 
or  the  sum  of,  all  numbers  of  n  digits  to  a  general  base,  or  the  numbers 
beginning  with  given  digits  or  with  certain  digits  fixed,  or  those  of  other 
types. 

A.  Palmstrom^^  treated  the  problem  to  find  all  numbers  with  the  same 
final  n  digits  as  their  squares.  Two  such  numbers  ending  in  5  and  6, 
respectively,  have  the  sum  10"+ 1.  If  the  problem  is  solved  for  n  digits, 
the  (n+l)th  digit  can  be  found  by  recursion  formulae.  There  is  a  unique 
solution  if  the  final  digit  (0,  1,  5  or  6)  is  given. 

A.  Hauke^°  discussed  obscurely  x'^^x  (mod  s'')  for  x  with  r  digits  to  base 
s.  If  m  =  2,  while  r  and  s  are  arbitrary,  there  are  2"  solutions,  v  being  the 
number  of  distinct  prime  factors  of  s. 

G.  Valentin  and  A.  Palmstrom^^  discussed  x'^^x  (mod  10"),  for  k  =  2,  3, 
4,  5. 

G.  Wertheim^^  determined  the  numbers  with  seven  or  fewer  digits  whose 
squares  end  with  the  same  digits  as  the  numbers,  and  treated  simple  prob- 
lems about  numbers  of  three  digits  with  prescribed  endings  when  written  to 
two  bases. 

"'iSammlung  der  Aufgaben . .  .  Zeitschr.  Math.  Naturw.  Unterricht,  1898,  35-6. 

«^«Math.  Quest.  Educ.  Times,  6.3,  1895,  92-3. 

«Jour.  of  the  Physics  School  in  Tokio,  5,  1896,  153-6,  266-7;  Abhand.  Geschichte  der  Math. 

Wiss.,  28,  1910,  18-20. 
»Jour.  of  the  Physics  School  in  Tokio,  5,  1896,  82,  99-103;  Abhand.,  16-18. 
"Ibid.,  6,  1897,  148-9;  Abhand.,  21. 
"oPeriodico  di  Mat.,  14,  1898-9,  243-8. 
»8Atti  Accad.  Pont.  Nuovi  Lincei,  52,  1899,  58-62;  53,  1900,  57-66;  54,  1901,  18-28;  Memorie 

Accad.  Pont.  Nuovi  Lincei,  19,  1902,  283-300;  26,  1908,  97-107;  27,  1909,  9-23;  28,  1910, 

17-31. 
''Skrifter  udgivne  af  Videnskabs,  Kristiania,  1900,  No.  3,  16  pp. 
"Archiv  Math.  Phys.,  (2),  17,  1900,  156-9. 
^iForhandlinger  Videnskabs,  Kristiania,  1900-1,  3-9,  9-13. 
^^Anfangsgninde  der  Zahlenlehre,  1902,  151-3. 


460  History  of  the  Theory  of  Numbers.  [Chap,  xx 

C.  L.  Bouton^  discussed  the  game  nim  by  means  of  congruences  between 
sums  of  digits  of  numbers  to  base  2. 

H.  Piccioli^^"  employed  N  =  ai.  .  .a„  of  n^3  digits  and  numbers  a,^.  .  .a«„ 
and  Qj^ . .  .  aj„  obtained  from  N  by  an  even  and  odd  number  of  transpositions 
of  digits.     Then  2o,, .  .  .  a«„  =  laj^ .  .  .  aj^, 

W^^  a  number  of  n  digits  to  base  R  has  r  fixed  digits,  including  the  first, 
and  the  sum  of  these  r  is  =  —  a  (mod  R  —  l),  the  number  of  ways  of  choosing 
the  remaining  digits  so  that  the  resulting  number  shall  be  divisible  by  R  —  l 
is  the  number  of  integers  of  n — r  or  fewer  digits  whose  sum  is  =  a  (mod  R  —  l) 
and  hence  is  A^'+l  or  N,  according  as  a=0  or  a>0,  where  N=  (/2"~''  — 1) 

G.  Metcalfe^^''  noted  that  19  and  28  are  the  only  integers  which  exceed 
by  unity  9  times  the  integral  parts  of  their  cube  roots. 

A.  Tagiuri^  proved  that  every  number  prime  to  the  base  g  divides  a 
number  1 ...  1  to  base  g  (generalization  of  Plateau's^^  theorem) . 

If"^"  A,  B,  C  have  2,  3,  4  digits  respectively  and  A  becomes  A'  on  re- 
versmg  its  digits,  and  2A-1  =  A',  SB-2A  +  10=B',  4C-B-\-l  +  [B/10] 
=  C,  then  A  =  37,  B=  329,  C=  2118. 

P.  F.  Teilhet^^  proved  that  we  can  form  any  assigned  number  of  sets, 
each  including  any  assigned  number  of  consecutive  integers,  such  that  with 
the  digits  of  the  ^th  power  of  any  one  of  these  integers  we  can  form  an 
infinitude  of  different  qth.  powers,  provided  q<m,  where  m  is  any  given 
integer. 

L.  E.  Dickson^*"  determined  all  pairs  of  numbers  of  five  digits  such  that 
their  ten  digits  form  a  permutation  of  0,  1,. . .,  9  and  such  that  the  sum 
of  the  two  numbers  is  93951. 

A.  Cunningham^^''  found  cases  of  a  number  expressible  to  two  bases  by  a 
single  digit  repeated  three  or  more  times.  He^^"  noted  that  all  10  digits  or 
all  >0  occur  in  the  square  of  10101010101010101  or  of  1 .  .  .1  (to  9  digits), 
each  square  being  unaltered  on  reversing  its  digits. 

He^^*^  and  T.  Wiggins  expressed  each  integer  ^  140  by  use  of  four  nines, 
as  13  =  9+ V9+9/9,  allowing  also  .9  =  1,  ( V9) !,  and  the  exponent  V9,  and 
cited  a  like  table  using  four  fours. 

j£45c  ^=  I  (mod  q),  1 ...  1  (with  q^  digits  to  base  r)  is  divisible  by  q"". 

W^^  the  square  of  a  number  n  of  r  digits  ends  with  those  r  digits,  then 
10''+1  — n  has  the  same  property.     Also,  {n  —  iy  ends  with  the  same  r  digits 

«Annals  of  Math.,  (2),  3,  1901-2,  35-9.     GeneraUzed  by  E.  H.  Moore,  11,  1910,  93-4. 

^^'Nouv.  Ann.  Math.,  (4),  2,  1902,  46-7. 

"''Math.  Quest.  Educ.  Times,  (2),  1,  1902,  119-120. 

*'^Ibid,  63-4. 

"Periodico  di  Mat.,  18,  1903,  45. 

«*<»Math.  Quest.  Educ.  Times,  (2),  5,  1904,  82-3. 

**L'interm6diaire  dea  math.,  il,  1904,  14-6. 

""Amer.  Math.  Monthly,  12,  1905,  94-5. 

"''Math.  Quest.  Educ.  Times,  (2),  8,  1905,  78. 

*^Ibid,  10,  1906,  20.  «<iMath.  Quest.  Educ.  Times,  7,  1905,  43-46. 

««7Wd.,  7,  1905,  49-50.  *^flbid.,  7,  1905,  60-61. 


Chap.  XX]  PROPERTIES    OF   THE    DiGITS   OF   NXH^BERS.  461 

as  71  — 1.  If  the  cube  of  a  number  n  of  r  digits  ends  with  those  r  digits, 
W—n  has  the  same  property. 

P.  Ziihlke^®  proved  the  three  theorems  of  Palmstrom^^  and  gave  all  solu- 
tions of  x^^x  (mod  10^)  for  p  =  3,. .  .,  12. 

M.  Koppe^^  noted  that  by  prefixing  a  digit  to  a  solution  0,  1,  5  or  6 
of  x^=x  (mod  10)  we  get  solutions  of  a;^=a;  (mod  10^),  then  for  10^,  etc.  We 
can  pass  from  a  solution  with  n  digits  for  10"  to  solutions  with  2n  digits 
for  10^".     He  treated  also  x^=x  (mod  10"). 

G.  Calvitti^^  treated  the  problem:  Given  a  number  A,  a  set  C  of  7  digits, 
and  a  number  p  prime  to  the  base  g,  to  find  the  least  number  x  of  times  the 
set  C  must  be  repeated  at  the  right  of  A  to  give  a  number  Nx=A  (mod  p). 
The  condition  is  G{Ni—No)=0  (mod  p),  where 

o^"^  — 1 

If  iVi— iVo=0,  any  x  is  a  solution.  If  not,  the  least  value  X  of  a:  makes 
G=0  (mod  p/p),  where  p  is  the  g.  c.  d.  of  Ni  —  Nq  and  p.  Then  X  is  the 
1.  c.  m.  of  Xi, . . . ,  X;t,  where  X^  is  the  least  root  of  G=0  (mod  Pi),  if  p/p  is  the 
product  of  pi, .  . . ,  Pk,  relatively  prime  in  pairs.  Hence  the  problem  reduces 
to  the  case  of  a  power  of  a  prime  p.  Write  (a)^  for  (a^  — l)/(a  — 1).  It  is 
shown  that  the  least  root  of  (a)j.=  0  (mod  p'')  is  mp^~\  where  m  is  the  least 
root  of  (a)^=0  (mod  p),  and  p'  is  the  highest  power  of  p  dividing  (a)^. 
Given  any  set  C  of  digits  and  any  number  p  prime  to  the  base  g,  there  exist 
an  infinitude  of  numbers  C . .  .C  divisible  by  p. 

A.  G^rardin^^''  added  220  to  the  sum  of  its  digits,  repeated  the  operation 
18  times  and  obtained  418;  9  such  operations  on  284  gave  418.  A.  Boutin 
stated  that  if  a  and  b  lead  finally  to  the  same  number,  neither  a  nor  6  is 
divisible  by  3,  or  both  are  divisible  by  3  and  not  by  9,  or  both  are  divi- 
sible by  9. 

E.  Malo^^  considered  periodicity  properties  of  A  and  a  in 

5'  =  10'"A„.p+ap  (ap<10^    k=n-2'^-^+p,    0^p^2"'-^-l) , 

and  solved  Cesaro's^^  three  problems  on  the  digits  of  powers  of  5. 

A.  L.  Andreini^°  noted  that  the  squares  of  A  and  B  end  with  the  same 
p  digits  if  and  only  if  the  smaller  of  r+s  and  u-\-v  equals  p,  where 

iH-J5  =  a-2'^-5",        A-B=^-T-5'' 

«Sitz.  Berlin  Math.  Gesell.,  4,  1905,  10-11  (Suppl.,  Archiv  Math.  Phys.,  (3),  8,  1905). 

"Ibid.,  5,  1906,  74-8.     (Suppl.,  Archiv,  (3),  11,  1907.) 

"Periodico  di  Mat.,  21,  1906,  130-142. 

""Sphinx-Oedipe,  1,  1906,  19,  47-8.     Cf.  I'interm^d.  math.,  22,  1915,  134,  215. 

*'Sur  certaines  propri6t6s  arith.  du  tableau  des  puissances  de  5,  Sphinx-Oedipe,  1906-7, 97-107; 

reprinted,  Nancy,  1907,  13  pp.,  and  in  Nouv.  Ann.  Math.,  (4),  7,  1907,  419-431. 
"II  Pitagora,  Palermo,  14,  1907-8,  39-^7. 


462  History  of  the  Theory  of  Numbers.  [Chap,  xx 

W.  Janichen^"'*  stated  that,  if  qp{x)  denotes  the  sum  of  the  digits  of  x  to 
the  base  p  and  if  p  is  a  prime  divisor  of  n,  then,  for  /j,  as  in  Ch.  XIX, 


s.W,.6)  = 


0. 


E.  N.  Barisien^"'"  noted  that  the  sum  of  all  numbers  of  n  digits  formed 
with  p  distinct  digits  f^O,  of  sum  s,  is 

s(p  +  1)"-Mp(10"-^-1)/9  +  (p  +  1)10"-^). 

A.  G^rardin^"''  listed  all  the  124  squares  formed  of  7  distinct  digits. 

Several  writers^^  treated  the  problem  to  find  four  consecutive  numbers 
a^  6  =  a  +  l,  c  =  a+2,  d  =  a-\-3,  such  that  (a)i  =  ll .  .  .1  (to  a  digits)  is  divisible 
by  a  +  1,  (6)i  by  26  +  1,  (c)i  by  3c+l,  (d),  by  4d+l. 

A.  Cunningham  and  E.  B.  Escott^-  treated  the  problems  to  find  integers 
whose  squares  end  with  the  same  n  digits  or  all  with  n  given  digits;  to  find 
numbers  having  common  factors  with  the  numbers  obtained  by  permuting 
the  digits  cyclically,  as 

259  =  7-37,  592  =  16-37,  925  =  25-37. 

E.  N.  Barisien^^  noted  that  the  squares  of  625,  9376,  8212890625  end 
with  the  same  digits,  respectively.     R.  Vercellin^^  treated  the  same  topic. 

E.  Nannei^^  discussed  a  problem  by  E.  N.  Barisien:  Take  a  number  of 
six  digits,  reverse  the  digits  and  subtract;  to  the  difference  add  the  number 
with  its  digits  reversed;  we  obtain  one  of  13  numbers  0,  9900, .  .  . ,  1099989. 
The  problem  is  to  find  which  numbers  of  six  digits  leads  to  a  particular  one 
of  these  13,  and  to  generalize  to  n  digits. 

Several  writers^^  examined  numbers  of  6  digits  which  become  divisible 
by  7  after  a  suitable  permutation  of  the  digits ;  also^^  couples  of  numbers, 
as  18  and  36,  36  and  54,  whose  g.  c.  d.  18  is  the  sum  of  their  digits. 

E.  N.  Barisien^^  gave  ten  squares  not  changed  by  reversing  the  digits, 
as676  =  26^ 

A.  Witting^^  noted  that,  besides  the  evident  ones  11  and  22,  the  only 
numbers  of  two  digits  whose  squares  are  derived  from  the  squares  of  the 
numbers  with  the  digits  interchanged  by  reversing  the  digits  are  12  and  13. 
Similarly  for  the  squares  of  102  and  201,  etc.     Also, 

102-402  =  201  -204,         213-936  =  312-639,         213-624  =  312-426. 

A.  Cunningham^"  treated  three  numbers  L,  M,  N  of  I,  m,  n  digits, 
respectively,  such  that  N  =  LM,  and  A^  has  all  the  digits  of  L  and  M  and  no 
others. 

"^Archiv  Math.  Phys.,  (3),  13,  1908,  361.     Proof  by  G.  Szego,  24,  1916,  85-6. 

'ocSphinx-Oedipc,  1907-8,  84-86.     For  p  =  n.  Math.  Quest.  Educ.  Times,  72,  1900,  126-8. 

•tx^/bid.,  1908-9,  84-5. 

"L'intermddiaire  des  math.,  16,  1909,  219;  17,  1910,  71,  203,  228,  286  [136]. 

"Math.  Quest.  Educat.  Times,  (2),  1.5,  1909,  27-8,  93-4. 

"Suppl.alPeriodicodiMat.,  13, 1909,20-21.         "Suppl.alPeriodicodiMat.,  14, 1910-11, 17-20. 

"/bid.,  13,  1909,  84-88.  ^«L'interm6diaire  des  math.,  17,  1910,  122,  214-6,  233-5. 

"Ibid.,  170,261-4;  18, 1911,207.  "Mathesis,  (3),  10,  1910,  65. 

"Zeitschrift  Math.-Naturw.  Unterricht,  41,  1910,  45-50. 

"Math.  Quest.  Educat.  Times,  (2),  18,  1910,  23-24. 


Chap.  XX]  PROPERTIES  OF   THE  DiGITS    OF  NUMBERS.  463 

D.  Biddle®^  applied  congruences  to  find  numbers  like  15  and  93  whose 
product  1395  has  the  same  digits  as  the  factors. 

P.  Cattaneo^^  considered  numbers  Q  (and  C)  whose  square  (cube)  ends 
with  the  same  digits  as  the  number  itself.  No  Q>\  ends  with  1.  No  two 
Q's  with  the  same  number  of  digits  end  with  5  or  with  6.  All  Q's  <  10^*  are 
found.  A  single  C  of  n  digits  ends  with  4  or  6.  Any  Q  is  a  C  Any  Q  — 1 
is  a  C.     If  A?"  is  a  Q  with  n  digits  and  if  2N —  1  has  n  digits,  it  is  a  C. 

M.  Thie,^^"  using  all  nine  digits  >0,  found  numbers  of  2,  3  or  4  digits 
with  properties  like  12483  =  5796. 

Pairs^^*^  of  cubes  3^  6^  and  375^  387^  whose  sums  of  digits  are  squares, 
32  and  61 

T.  C.  Lewis^^  discussed  changes  in  the  digits  of  a  number  to  base  r  not 
affecting  its  divisibiUty  by  p. 

Numbers^'*  B  and  5"  having  the  same  sum  of  digits. 

Pairs^^  of  primes  like  23-89  =  29-83. 

Cases<^«  like  7-9403  =  65821  and  3-1458  =  6-0729,  where  the  digits  0, 
1,.  .  .,  9  occur  without  repetition. 

jy-pn+i  ending"  with  the  same  digits  as  A^. 

Numbers^^  like  512  =  (5+l+2)^  47045881000000  =  (47+4+58+81)^ 

AlP  numbers  like  2-5-27  =  M8-15,  2+5+27  =  1  +  15  +  18. 

Number^°  divisible  by  the  same  number  reversed. 

Number^^  an  exact  power  of  the  sum  of  its  digits;  two  numbers  each 
an  exact  power  of  the  sum  of  the  digits  of  the  other. 

Solve'^^  KN-{-P  =  N',  N'  derived  from  N  by  reversing  the  digits. 

Symmetrical  numbers  (ibid.,  p.  195). 

F.  Stasi^^  proved  that,  if  a,  h  are  given  integers  and  a  has  m  digits,  we 
can  find  a  multiple  of  b  of  the  form 

10''(a-10'"'+a-10"^^'-^^+  .  . .  +a),        p^O. 

Taking  b  prime  to  a  and  to  10,  we  see  that  b  divides  10""'+ ...  +1.     The 
case  m=  1  gives  the  result  of  Plateau. ^^ 

Cunningham'^^"  and  others  wrote  iVi  for  the  sum  of  A^  and  its  digits  to 
base  r,  iV2  for  the  sum  of  A^i  and  its  digits,  etc.,  and  found  when  N^  is 
divisible  by  r  —  1 .    • 

"Math.  Quest.  Educ.  Times,  (2),  19,  1911,  60-2.     Cf.  (2),  17,  1900,  44. 

«2Periodico  di  Mat.,  26,  1911,  203-7. 

fi^^Nouv.  Ann.  Math.,  (4),  11,  1911,  46. 

62''Sphinx-Oedipe,  6,  1911,  62. 

"Messenger  Math.,  41,  1911-12,  185-192. 

6*L'intermMiaire  des  math.,  18,  1911,  90-91;  19,  1912,  267-8. 

^Ubid.,  1911,  121,  239.  ««/6id.,  19,  1912,  26-7,  187. 

"Ibid.,  50-1,  274-9. 

^^Ibid.,  77-8,  97. 

«9/6id.,  125,  211. 

''"Ibid.,  128. 

''Ibid.,  137-9,  202;  20,  1913,  80-81. 

"76id.,  221. 

"II  Boll.  Matematica  Gior.  Sc.-Didat.,  11,  1912,  233-5. 

73«Math.  Quest.  Educ.  Times,  (2),  21,  1912,  52-3. 


"L'interm^diaire  des  math.,  20,  1913,  42-44. 

^^Ibid.,  80. 

'*Ibid.,  124,  262,  283-4. 

"Atti  Accad.  Romana  Nuovi  Lincei,  66,  1912-3,  43-5. 

"Archiv  Math.  Phys.,  (3),  22,  1914,  365-6. 

"Giomale  di  Mat.,  52,  1914,  53-7. 

'"L'interm^diaire  des  math.,  21,  1914,  23^,  58. 

"/6id.,  22,  1915,  110-1.     Objections  by  MaUlet,  23,  1916,  10-12. 

"Suppl.  al  Periodico  di  Mat.,  19,  1915,  17-23. 

"''Sphinx-Oedipe,  9,  1914,  42. 

"Sitzungsber.  Berhn  Math.  Gesell.,  15,  1915,  8-18. 

"Nouv.  Ann.  Math.,  (4),  17,  1917,  234. 

"L'intermddiaire  des  math.,  24,  1917,  31-2. 

»/Wd.,  96.     Cf.  H.  Brocard,  25,  1918,  35-8,  112-3. 


I 


464  History  of  the  Theory  of  Numbers.  [Chap,  xx 

A.  Cunningham"^  listed  63  symmetrical  numbers  aoaia2aiao  each  a 
product  of  two  symmetrical  numbers  of  3  digits,  and  all  numbers  n^, 
r2<  10000,  and  all  n'\  n\  n^,  n^\  7i<1000,  ending  with  2,  7,  8,  symmetrical 
with  respect  to  2  or  3  digits,  as  618'  =  236029032.  f    ie^' 

Pairs'^  of  numbers  whose  1.  c.  m.  equals  the  product  of  the  digits. 

Pairs'^  of  biquadrates,  cubes  and  squares  having  the  same  digits. 

*P.  de  Sanctis'^  noted  a  property  of  numbers  to  the  base  h^+1. 

L.  von  Schrutka^^  noted  that  15,  18,  45  in  7-15  =  105,  6-18  =  108  and 
9-45  =  405  are  the  only  numbers  of  two  digits  which  by  the  insertion  of 
zero  become  multiples. 

G.  Andreoli^^  considered  numbers  N  of  n  digits  to  the  base  k  whose  rth 
powers  end  with  the  same  ?i  digits  as  N.  Each  decomposition  of  k  into 
two  relatively  prime  factors  gives  at  most  two  such  N's.  If  the  base  is  a 
power  of  a  prime,  there  is  no  number  >  1  whose  square  ends  with  the  same 
digits. 

Welsch^°  discussed  the  final  digits  of  pth  powers. 

H.  Brocard^ ^  discussed  various  powers  of  a  number  with  the  same  sum 
of  digits.  _ 

A.  Agronomof®^  wrote  N  for  the  number  obtained  by  reversing  the  digits 
of  N  to  base  10  and  gave  several  long  formulas  for  2ji  J  j. 

The^-''  only  case  in  which  N^—N"^  is  a  square  for  two  digits  is  65-  — 
56^=33".     There  is  ^o  case  for  three  digits. 

R.  Burg^  found  the  numbers  N  to  base  10  such  that  the  number 
obtained  by  reversing  its  digits  is  a  multiple  kN  of  A^,  in  particular  for 
A:  =  9,  4. 

E.  Lemoine^  asked  a  question  on  sjinmetrical  numbers  to  base  b. 

H.  Sebban^^  noted  that  2025  is  the  only  square  of  four  digits  which  yields 
a  square  3136  when  each  digit  is  increased  by  unity.  Similarly,  25  is  the 
only  one  of  two  digits. 

R.  Goormaghtigh^^  noted  that  this  property  of  the  squares  of  5,  6  and 
45,  56  is  a  special  case  of  A^—B"^  —  !. .  .1  (to  2p  digits),  where  A  =  5. .  .56, 
5  =  4.  .  .45  (to  p  digits).  Again,  the  factorizations  11111=41-271,  1111111 
=  239-4649  yield  the  answers  115^,  156^  and  2205^,  2444^. 


I 

1 


Chap.  XX]  PROPERTIES    OF   THE   DiGITS    OF   NUMBERS.  465 

SeveraP^"  gave  9'w!+n+l=  1 . .  .1  for  n^9,  with  generalization  to  any 
base. 

E.  J.  Moulton^^  found  the  number  of  positive  integers  with  r+1  digits 
fewer  than  p  of  which  are  unity  (or  zero).  L.  O'Shaughnessy^^  found  the 
number  of  positive  integers  <  10*  which  contain  the  digit  9  exactly  r  times. 

Books^^  on  mathematical  recreations  may  be  consulted. 

F.  A.  Halliday^^  considered  numbers  N  formed  by  annexing  the  digits 
of  B  to  the  right  of  A,  such  that  N=  (A+B^,  as  for  81=  (8  +  1)1  Set 
N=A-W+B.  Then  A{W-1)=  iA+B)(A-}-B-l),  so  that  it  is  a  ques- 
tion of  the  factors  of  lO''  — 1. 

*J.  J.  Osana^^  discussed  numbers  of  two  and  three  digits. 
E.  Gelin^^  listed  450  problems,  many  being  on  digits. 

*^"L'intermediaire  des  math.,  2.5,  1918,  44-5. 

"Amer.  Math.  Monthly,  24,  1917,  340-1. 

8«/6id.,  25,  1918,  27, 

*^E.  Lucas,  Arithmetique  amusante,  1895.     E.  Fourrey,  Recreations  Arithmetiques,   1899. 

W.  F.  White,  Scrap-Book  of  Elem.  Math.,  etc. 
90Math.  Quest,  and  Solutions,  3,  1917,  70-3. 
siRevista  Soc.  Mat.  Espafiola,  5,  1916,  156-160. 
92Mathesis,  (2),  6,  1896,  Suppl.  of  34  pp. 


AUTHOR  INDEX. 


The  numbers  refer  to  pages.     Those  in  parenthesis  relate  to  cross-references, 
brackets  refer  to  editors  or  translators.     The  other  numbers  refer  to  actual  reports. 


Those  in 


Ch.  I.  Perfect,  Multiply  Perfect,  and  Amicable  Numbers. 


Algafd,  39 
Alcuin,  4 
Alkalacadi,  40 
Allemanno,  39 
Ankin,  5 
Anonymous,  47 
Aristotle,  38 
Astius,  [3] 
Aubry,  31,  33 
Augustinus,  4 
Azulai,  39 

Bachet,  10 
Bachmann,  33 
Baeza,  9 
BaU,  12,  32 
Ben  Kalonymos,  39 
Ben  Korrah,  5,  39 
Bezdi^ek,  32,  48 
Bickmore,  28 
Boethius,  4  (5-7,  11) 
Bourlet,  28  (29) 
BoviUus,  7  (6,  10,  32) 
Bradwardin,  6 
Brassinne,  [12,  36] 
Brocki,  11 
Bronckhorst,  8 
Broscius,  11,  13,  36  (41) 
Bulhaldo,  [3] 
Bungus,  9,  40  (10-15) 

Capella,  11 

Cardan,  8,  40  (11,  14) 
Carmichael,  29,  37,  38  (35) 
Carvallo,  22,  24  (26) 
Catalan,  22,  24,  26,  27,  32,  48 

(28,  49) 
Cataldi,  10  (7) 
Cesaro,  26 
Chevrel,  27,  48 
Christie,  27 
Chuquet,  6,  40 
Ciamberlini,  29 
Cipolla,  29,  33 
Cole,  29  (13,  22,  25,  27,  32) 
Cunningham,  27,  28,  29,  30, 

31,  37,  48  (21,  32) 
Curtze,  [6] 

De  Backer,  [11] 
De  la  Roche,  8,  40 


De  Longchamps,  22 
De  Neuveghse,  15 
Desboves,  23,  37  (21,  25) 
Descartes,     12,     33,     34-36, 

40-42  (37,  46) 
De  Slane,  [39] 
De  Tovmes,  [8] 
Dickson,  30-32,  49,  50 
Dombart,  [4] 
Dupuis,  48 

El  Madschritt,  39 
El  Magritl,  39 
Enestrom,  46 
Ens,  11  (16,  19) 
Escott,  50 
EucHd,  3 

Euler,  17-19,  41-46  (12,  14, 
16,  22-28,  31,  47-50) 

Faber  Stapulensis,  6  [5] 
Fauquembergue,    27,    30-32 

(28,  29) 
Feliciano,  7 
Fermat,  12,  33,  34,  36,  37,  40 

(18) 
Ferrari,  33 

Fibonacci  (see  Leonardo) 
Fitz-Patrick,  27,  33,  48  [28] 
Fontana,  20  (17) 
Font^s,  32  [9] 
Forcadel,  9 
Frenicle,  12,  14,  35  (13,  19, 

28,  31) 
Fried  lein,  [4] 
Frizzo,  (4)  [8] 
Fuss,  46  [15,  18] 

Genaille,  27 

G^rardin,  29-32,  48-50  (21) 

Gerhardt,  [6] 

Ginsburg,  [5,  39] 

Giraud,  33 

Goldbach,  15 

GosseUn,  [9] 

Gough,  47 

Goulard,  28 

Graevius,  [11] 

Griison,  20,  47  (16) 

Giideman,  [5] 


Haas,  48 

Halcke,  41 

Hammond,  30 

Hankel,  [4] 

Hansch,  17  (18,  19) 

Harris,  16 

Hebrews,  3 

Heilbronner,  18 

Heinlin,  14 

Henischiib,  10 

Henry,  [10,  12,  36,  40] 

HiU,  16  [20] 

HiUer,  [3] 

Hoche,  [3] 

Hrotsvitha,  5 

Hudelot,  25 

Hultsch,  32 

Himrath,  43 

Button,  [16,  18,  19,  36,  47] 

Huygens,  [14] 

lamblichus,  4,  38  (15,  32) 

Ibn  Albanna,  40 

Ibn  el-Hasan,  39 

Ibn  Ezra,  5 

Ibn  KJialdoun,  39 

Ibn  Motot,  5 

Isidorus,  4 

Jacob,  39 
Jordanus,  5  (6,  7) 
Jumeau  (see  St.  Croix) 

Kastner,  16 

Kiseljak,  33 

Klugel,  47 

Kraft,  17-19,  41,  50  (8,  9,  20, 

46,  47) 
Kraitchik,  22,  25,  47  (32) 
Kummer,  [21] 

Landen,  18  (41) 
Landry,  21-23  (25) 
Lantz,  11 
Lax,  7 

Lazzarini,  29 
Lebesgue,  20  (23) 
Ijcfevre  (see  Faber) 
Legendre,  36,  47  (27,  35) 
Lehmer,  37  (36) 
Leibniz,  15 

467 


468 


Author  Index 


Le  Lasseur,    (21,   23-25,   28, 

48) 
Leonardo  Pisano,  5 
Leuneschlos,  14 
Leurechon,  11 

Lcybourn,  15,  41  [16,  18,  47] 
Libri,  [10] 
Liebnecht,  16 
Lionnet,  23  (19) 
LiouviUe,  (19) 
Lucas,  14,  15,  22-25,  27,  28, 

37,40(12,  17,  19,  21,  29- 

32,  35,  36,  48) 

Magnin,  [5] 

Mahnke,  [15] 

Maier,  17  (20) 

Malcolm,  16 

Mandey,  [14] 

Mansion,  24,  26 

Manuscript,  6 

Marre,  [6] 

Martinus,  7 

Maser,  [36,  47] 

Mason,  32,  38 

Maupin,  [7] 

Maurolycus,  9  (20) 

McDonnell,  32 

Meissner,  49 

Mersenne,  12,  13,  33,  35,  36, 

40,  41  (14,  15,  18,  20,  22, 

25,  27,  28,  31,  32) 
Migne,  [4] 

Montucla,  19  (16,  36) 
Moret-Blanc,  23 
Munyos,  4 

Nachshon,  39 
Nassd,  32 
Neomagus,  8 
Nicomachus,  3  (4) 
Niewiadomski,  32 
Nocco,  21  (26) 
Novarese,  25  (26) 
Noviomagus,  8 


Oughtred  [11] 

Ozanam,  15-17, 36, 41(19, 20) 

Paciuolo,  6,  (10,  19) 

Paganini,  47  (49) 

Pauli,  15 

Peacock,  50 

Peletier,  8 

Pellet,  25 

Pepin,  23,  28,  47 

Perrott,  48 

Pervnisin,  25 

Philiatrus,  15 

Philo,  3 

Pistelli,  [4] 

Plana,  21  (17,  22,  24,  29) 

Poggendorff,  [11] 

Postello,  9 

Poulet,  50 

Powers,  30,  32  (22) 

Prestet,  15 

Pudlowski,  12 

Puteanus,  11  (14) 

Putnam,  30 

Pythagoreans,  4,  5,  38 

Ramesam,  31 
Ramus,  9 
Recorde,  9,  33 
Regius,  7 

Reuschle,  21  (24,  31) 
Ricalde,  32 
Rubin,  [3] 
Rudio,  45 
Ruffus,  7 

Saverien,  19 
Schonero,  [9] 
Schooten  (see  Van) 
Schubert,  32 
Schwenter,  11,  40,  50 
Seelhoflf,  25,  48  (29) 
Sempilius,  11 
Semple,  11 
Servais,  26  (25) 
Spoletanus,  7 


St.  Croix,  34 
Steinschneider,  [5,  39] 
Stern,  25  (19) 
Stifel,8,  40(11,  17,  41) 
Stone,  41 
Struve,  20,  47 
Studni^ka,  28 
Stuj'vaert,  28 
Suter,  [39] 
Sylvester,  26,  27  (19,  31) 

Tacquet,  14 

Tannery,  28 

Tarry,  30 

TartagUa,  9,  40  (11,  17) 

Tassius,  15 

Taylor,  20,  50 

Tchebychef,  47 

Tennulius,  15  [4] 

Terquem,  20 

Thabit  (see  Ben  Korrah) 

Theon  of  Smyrna,  3 

Turcaninov,  29 

Turtschaninov,  29 

Unicornus,  10 

Vaes,  33 

Valentin,  25 

Valla,  6 

Van  Etten,  11 

Van  Schooten,  14,  41  (42,  47) 

Von  Graffenried,  10 

Wantzel,  20  (17) 
Waring,  46 
Wertheim,  33  [10] 
Westerberg,  20 
Westlund,  37,  38 
WiUibaldus,  11 
WiUichius,  8 
Winsheim,  18  (17) 
Woepcke,  [5,  39] 
Wolf,  16  (17) 
Woodall,  30  (28,  32) 

Young,  [3] 


Ch.  II.  Formulas  for  the  Number  and  Sum  of  Divisors,  Problems  of  Fermat 

AND  Wallis. 


Bang,  56 
BernouUi,  55 
Blaikie,  58 
Brocard,  57 
Brouncker,  54 

Cantor,  53 
Cardan,  51 
Castillionei,  51 


Chalde,  52 
Cunningham,  58 

Deidier,  52 

De  Montmort,  52 

Descartes,  52,  53 

Escott,  54 
Euler,  54 


Fauquembergue,  57 
Fermat,  53,  54,  56  (57) 
Fontend,  52 
Frenicle,  53,  55  (56) 

Genocchi,  57 
G^rardin,  57,  58 
Gerono,  56 


Author  Index 


469 


Hain,  52 
Halcke,  58 
Henry,  [53,  54,  56] 
H^v^lius,  54 
Hoppe,  51 

Kersey,  51 
Kraft,  53 
Kronecker,  54 

Landau,  57 
Lionnet,  52,  58 
Lucas,  55-58 


Mersenne,  51,  53 
Minin,  52 
Moreau,  57 
Moret-Blanc,  57 

Newton,  51  (53) 

Ozanam,  56 

Peano,  53 
Plato,  51 
Prestet,  52 
Pujo,  52 


Rudio,  54 

Stifel,  51 

Vacca,  57 

Van  Schooten,  51,  55 

Wallis,  51,  53-56 
Waring,  52,  54,  55 
Wertheim,  54 
Winsheim,  51 
Wolff,  54 


Ch.  III.  Fermat's  and  Wilson's  Theorems,  Generalizations  and  Converses; 
Symmetric  Functions  of  1,  2,...,  p-1  Modulo  p. 


Allardice,  96 

Anton,  75 

Anonymous,  67,  70,  92 

Arevalo,  83 

Arndt,  71,  72  (77,  81,  82) 

Arnoux,  81,  94 

Aubry,  81,  101  (71,  103) 

Axer,  86 

Bachmann,  72,  81,  82,  95  (78) 

Banachiewicz,  94 

Bauer,  88,  89 

Beaufort,  62 

Beaujeux,  75 

Binet,  67 

Birkenmajer,  96 

Blissard,  74 

Borel  and  Drach,  85,  87  (89) 

Bossut,  [63] 

Bottari,  83 

Bouniakowsky,  67,  73,  92,  95 

(89,  93) 
Brennecke,  69,  73  (80) 
Bricard,  81  (75,  82) 

Cahen,  80,  103 

Candido,  80 

Cantor,  62 

Capelli,  80,  89 

Caraffa,  69 

Carmichael,  82,  94  (78,  86, 

95) 
Carre,  60 

Catalan,  71,  77,  98 
Cauchy,  67,  69,  70,  95  (86) 
Cayley,  76  (75,  83) 
Cesaro,  98 
Chinese,  59,  91 
CipoUa,  93,  94 
Concina,  101 
Cordone,  85 
Crelle,  68,  70,  71,  (72) 


Cunningham,  94 

D'Alembert,  63 

Daniels,  78 

Dedekind,  [74] 

De  la  Hire,  60 

Del  Beccaro,  79 

De  Paoli,  67  (79,  82) 

D'Escamard,  81 

Desmarest,  73 

Dickson,  80,  85,  89 

Dirichlet,  66,  74  (65,  68,  71, 

73,  77,  80,  81,  84) 
D'Ocagne,  79,  98 
Donaldson,  82 
Diu-ege,  73 

Earnshaw,  70 
Eisenstein,  95 
Epstein,  (86) 
Escott,  93,  94,  103 
Euler,  60-65,  92  (66,  67,  69, 
72-74,  76,  77,  81-83) 

Fergola,  96  (97,  98) 
Fermat,  59  (60,  94) 
Ferrers,  79,  99 
Fontebasso,  74 
Franel,  93 
Frattini,  84 
Frost,  96  (100) 

Garibaldi,  77 

Gauss,  60,  65,  84  (67,  69,  71, 

72,  74,  75,  82,  83,  101) 
Gegenbauer,  86,  93,  99 
Genese,  78 
Genty,  64 
Gerhardt,  [59] 
Glaisher,  99,  100  (102,  103) 
Goldbach,  92 
Gorini,  70 
Grandi,  85 


Graves,  73 
Gruber,  87 

Grunert,  66-68,  71,  72   (65, 
73,  81) 

Harris,  81 
Hayashi,  93 
Heal,  77 
Heather,  72 
Hensel,  91,  101 
Horner,  66,  69  (68,  71,  73,  74, 
82) 

lllgner,  83 
Irwin,  103 
Ivory,  65  (66-69,  71,  74) 

Jacobi,  90  (91) 
Janssen  van  Raay,  101 
Jeans,  93  (59,  91) 
Jolivald,  93 
Jorcke,  76 

Kantor,  84 
Klugel,  [67] 
Koenigs,  85 
Korselt,  93 
Kossett,  93 
Kraft,  60 
Kronecker,  80,  88 

Lacroix,  65,  [63] 

Lagi-ange,  62  (59,  64,  73,  74, 

97,  99) 
Laisant,  75  (76,  78) 
Lambert,  61,  92 
Laplace,  63  (67,  73,  82) 
Lebesgue,  74,  96 
Legendre,  64  (80) 
Leibniz,  59,  60,  91  (65,  70,  94) 
Leudesdorf,  96 
LeVavasseur,  88 


470 


Author  Index 


Levi,  79,  93 
Libri,  69  (101) 
Lindsay,  82 
Lionnet,  71,  98 
Lista,  73 
Lottner,  73 

Lucas,  77,  78,  85,  92,  99  (81, 
82,  86,  93,  94) 

MacMahon,  78,  85,  86 
Mahnke,  59,  94  (91) 
Maillct,  78,  85 
Malo.  88,  93 
Mansion,  76 
Maser,  [65] 
Mason,  102 
Meissner,  102  (101) 
Meyer,  101 
Midy,  69 
MUler,  81 

Minding,  68  (71,  77) 
Minetola,  (66) 
MitcheU,  98,  (86,  87) 
Monteiro,  100 
Moore,  87  (88,  89) 
Moreau,  80  (67,  81,  86) 
Morehead,  94 
Moret-Blanc,  98 

Nicholson,'81,  101 

Nielsen,  62,  89,  96,  99,  102, 

103  (87)    ' 
Niewenplowski,  96 

Osbom,  96,  99 
Ottinger,  75 

Peano,  [59] 
Pellet,  85 


Perott,  80,  88,  99  (75) 

Petersen,  75  (76,  80,  81) 

Petr,  81 

Piccioli,  84 

Picquet,  85 

Pincherle,  80 

Plana,  74 

Pocklington,  82 

Poinsot,  71,  95  (72,  76,  101) 

Pollock,  73 

Poselger,  66 

Prompt,  81,  83 

Proth,  92 

Prouhet,  72 

Putnam,  102 

Rawson,  99 
Ricalde,  93 
Rieke,  96 
Rogel,  87 

Sardi,  74,  97 
Sarrus,  92 
Sauer,  89 
Scarpis,  81,  82 
Schaflfgotsch,  64  (65) 
Schapira,  93 
Schering,  76  (80, 81) 
Scherk,  90,  91 
Schlomilch,  73 
Schmidt,  79 
Schonemann,  69,  84 
Schubert,  64 
Schuh,  83 
Schumacher,  83 
Serret,  73,  84,  96  (100) 
Sharp,  96 
Sibirani,  100 
Steiner,  90  (91) 


Stern,  74 

Sylvester,  69,  73,  96,  98  (79, 

81,  97) 

Talbot,  74 
Tarrj',  65 

Tchebychef,  73  (77,  100) 
Terquem,  71 
Thaarup,  92 
Thue,  79,  82 
ToepUtz,  74 
Torelli,  97  (98) 
Tschistjakov,  82 

Unferdinger,  (75) 

Vacca,  59 
Valentiner,  96 
V%i,  86 
Vandiver,  89,  93 
Verhulst,  66 
Von  Schaewen,  76 
Von  Staudt,  95 
Von  Stemeck,  80 

Waring,  62,  64,  95  (81) 
Weber,  80 
Welsch,  95 
Wertheim,  77,  101 
Western,  94 
Westlund,  86,  100 
Weyr,  85 

Wildschiitz-Jessen,  84 
Wilson,  62  (59) 
Wolstenhohne,  96  (99,  103) 

ZeUer,  76 
Zsigmondy,  93 


Ch.  IV.  Residue  of  {vP-'^  —  l)/p  Modulo  p. 


Abel,  105 
Aladow,  107 

Bachmann,  109,  111 
Baker,  110 
Bastien,  111 
Beeger,  111 
Brocard,  111 

Cunningham,  107,110 

De  Romilly,  108 
Desmarest,  105  (106) 

Eisenstein,  105  (106) 
Euler,  112  (106) 

Fauquembergue,  111 


Friedmann,  110 
Frobenius,  110  (HI) 

Gegenbauer,  106 
G^rardin,  112 
Glaisher,  108,  109 
Grave,  110  (111) 

Hertzer,  110 

Jacobi,  105  (111) 
Janssen  van  Raay,  110 

Lerch,  109,  110  (112) 
Lucas,  106 

Meissner,  111,  112  (106, 110) 
Meyer,  107 

Mirimanoff,    107,    110    (109, 
111) 


Nielsen,  112 

Palmstrom,  108 
Panizza,  106 
Plana,  106  (109,  112) 
Pleskot,  109 
Pollak,  108 
Proth,  106  (111) 

Stem,  106  (109) 
Sylvester,  105  (106-110) 

Tamarkine,  110 
Tarry,  111 
Thibault,  (105) 

Vandiver,  112  (111) 
Verkaart,  111 

Wieferich,  110  (111) 


Author  Index 


471 


Ch.  V.  Euler's  ^-Function,  Generalizations,  Farey  Series. 


Airy,  157 
Alasia,  119 
Anonymous,  158 
Arndt,  116,  118  (140) 
Arnoux,  151 
Axer,  138  (157) 

Bachmann,  124,  132, 147, 158 

Bauer,  134 

Berger,  131 

BernoulU,  140 

Betti,  120 

Binet,  141  (142,  146) 

Blind,  142,  148 

Bonse,  137  (132) 

Borel  and  Drach,  120,  133 

Bougaief,  142  (145) 

Brennecke,  141  (143) 

Brocot,  156 

Busche,  130, 137, 155, 158 

Cahen,  140,  155,  158 
Cantor,  125  (122,  149) 
Carlini,  136, 150, 153 
Carmichael,  137,  155 
Catalan,  116,  124,  130,  143 

(117,  118) 
Cauchy,  116,  140,  156 
Cayley,  121 
Ces^ro,     127-130,     143-145, 

148-150  (126, 138, 140, 153, 

157) 
Chrystal,  120 
Cistiakov,  151 
Composto,  138 
Concina,  131 
Cordone,  155 
Crelle,    115,    117,    140   (118, 

119,    121,    132,   137,   139, 

141,  143) 
Crone,  125 
Cunningham,  140 
Curjel,  126 

Da  Silva,  119,  141 

Davis,  132 

Dedekind,  120,  123  (140) 

Del  Beccaro,  146 

De  Rocquigny,  124,  143 

Desmarest,  116 

De  Vries,  134 

Dirichlet,  119,  122  (121,  126, 

127, 130, 132-134, 136, 150, 

152,  154) 
D'Ocagne,  157 
Druckenmiiller,  116 

Eisenstein,  156 


Elliott,  135 

Euler,  113, 114  (115-118, 122, 
138, 146) 

Farey,  156 
Fekete,  137 
Fleck,  139 
Flitcon,  155 
Fontebasso,  122  (120) 

Gauss,    114    (116-118,    122, 

133,  142,  147) 
Gegenbauer,    122,    129,    145, 

146,    149,    151,    152    (142, 

150,  155) 
Glaisher,  146,  148,  157  (150, 

158) 
Goldschmidt,  132,  147 
Goodwyn,  156 
Goormaghtigh,  140 
Grunert,  117  (134) 
Guilmin,  119  (118) 

Halphen,  126,  156  (132) 
Hammond,  131,  139,  140,  149 

(133) 
Hancock,  139 
Haros,  156 
Harris,  143 
Hensel,  135,  139 
Hermes,  158 
Herzer,  156 
Horta,  119 
Hrabak,  156 
Humbert,  158 
Hurwitz,  158 

Jablonski,  131,  151 

Jensen,  130 

Jordan,  147  (123,  132) 

Kaplan,  134 
Klein,  158 
Knopp,  (140) 
Kronecker,  135  (155) 
Kuyver,  139 

Laguerre,  122  (129,  133,  152) 
Landau,  134,  136,  138  (132) 
Landry,  119 
Landsberg,  136 
Lebesgue,  118,  121 
Legendre,  114  (116,  118,  121, 

132, 143,  145) 
Lehmer,  153  (157) 
Le  Paige,  124 


Lerch,  136 
Leudesdorf,  145 
LiouviUe,  120,  142  (127,  144) 
Lipschitz,  (140) 
Lucas,  125, 131, 142, 147, 157, 
158  (123, 126, 137) 

MacMahon,  131 

Made,  158 

Maillet,  134  (132,  137,  138) 

Mansion,  123,  124,  128  (127) 

Mathews,  120 

Mennesson,  143 

Merrifield,  156 

Mertens,  122  (126,  127,  132, 

154) 
Metrod,  155 
Miller,  137,  138,  155 
Minding,  116  (118) 
Minin,  133,  155  (140) 
Minine,   124,   143,   144,   157 

(145) 
MitcheU,  125 
Moreau,  131,  134  (136) 
Moret-Blanc,  122,  144 

Nasimof,  145  (146) 
Nazimov  (Nasimof) 
Nielsen,  146 
Nordlund,  137 

Occhipinti,  136 
Oltramare,  142 
Orlandi,  138 
Orlando,  137 

Pepin,  122  (151) 
Perott,  126  (127,  131) 
Pichler,  134  (137) 
Poinsot,  117  (114) 
Poretzky,  130 
Postula,  143 

Prouhet,  118,  140  (131, 150) 
Pullich,  157 

Radicke,  125 
Ranum,  137 
Ratat,  140 
Remak,  138  (132) 
Rogel,  126,  133,  134  (140) 

Sanderson,  155 
Sang,  157 
Scar  pis,  139 

Schatunowsky,  132  (134) 
Schemmel,  147 
Sierpinski,  158 


472' 


Author  Index 


Smith,    122   (123,    124,    127, 

128,  130,  136) 
Sommer,  137 
Steggall,  132  (118) 
Stem,  156  (157) 
Stor>',  148 
Stouvencl,  156 
Sturm,  120 
Suzuki,  137 
Sylvester,  121,  124,  126,  133, 

157    (115,    129,    132,    154, 

155) 


Tanner,  131 
Tchebychef,  119  (132) 
Thacker,  140  (123,  142,  144- 

147) 
Tschistiakow,  151 

Vahlen,  133,  158  (157) 

Vdlyi,  150 

Van  der  Corput,  139 

Von  Ettingshausen,  115  (121) 

Von  Schrutka,  158 

Von  Sterneck,  151  (153) 


Walla,  126 

Weber,  123,  133,  150 

Wevr,  151 

Wolfskehl,  134  (132, 138) 

Zerr,  140 

Zsigmondy,    132,    152    (145, 
146, 155) 


Ch.  VI.  Periodic  Decimal  Fractions;  Factors  of  10"±1 


Adams,  179 
Akerlund,  176 
Albanna,  159 
Anonymous,  163 
Arndt,  179 

Bachmann,  174,  176 
Barillari,  (167) 
Beaujeux,  167  (171) 
Bella\-itis,  170 
Bennett,  177 

Bernoulli   160  (159,  161,  166) 
Bertram,  165  (168) 
Bettini,  175 

Bickmore,  175  (176,  179) 
Biddle,  176 
Bork,  174  (165) 
Bouniakowsky,  171 
Bredow,  163 

Brocard,  165,  172,  174  <159) 
Broda,  169,  172 
Brogtrop,  170 

Burckhardt,    161    (168,    172, 
173,  177) 

Carra  de  Vaux,  159 
Catalan,  164 
Cicioni,  177 
Clarke,  160 
Collins,  166 
Contejean,  173 
Cullen,  176  (179) 
Cunningham,    168,    175-177, 

179    (161,    165,    169,    172, 

174) 

De  Coninck,  168 
Desmarest,    165    (168,    170, 

177) 
Dickson,  174 
Dienger,  179 
Druckenmiiller,  163 

El-M&ridini,  159 
Escott,  176 
Eulor,  160  (165) 

Farey,  162 
Felkel,  161 


Filippov,  177,  179 
Fujimaki,  176 

Gauss,  161  (167,  170) 

Genese,  173 

Genocchi,  165  (160) 

G^rardin,  177  (161,  165) 

Ghezzi,  178 

Glaisher,  162,  166,  168,  170, 

171  (173,  175) 
Goodwyn,  161,  162  (170) 
Gosset,  177 

Hartmarm,  168 
Hausted,  170 
Heal,  173 
Heime,  166 

Hertzer,  176,  177  (165) 
Hoppe,  179 
Howarth,  179 
Hudson,  166  (167) 

Ibn-el-Banna,  159 

Jackson,  177 
Jenkins,  179 
Johnson,  172 

Kessler,  172,  174  (161,  176) 
Kraitchik,  179 
Kraus,  174 
Kronecker,  176 

Lafitte,  164 
Lagrange,  179 
Laisant,  167,  171,  173 
Lambert,     159     (160,     168) 

[161] 
Law,  161 
Lawrence,  175 
Lebesgue,  167 
Lehmann,  167 
Leibniz,  159 
LeLasseur,  172 
Leman,  179 
Levanen,  179 
Lichtenecker,  178 
Lionnet,  167 
Loof,  165,  171,  172,  175  (168) 


Lucas,  159,  171,  172,  175,  177 

(176) 
Lugh,  172 

Mahnke,  159 
Maillet,  178 
Mansion,  169 
Maver,  173,  174 
Mever,  167  (176) 
Midy,  163  (164,  166) 
Mignosi,  177 
Miller,  176 
Morck,  179 
Morel,  167 
Moret-Blanc,  168 
Muir,  168,  169 
Murer,  175 

Nordlund,  176 

Oberreit,  161 
OUver,  166 

Pasternak,  178 
PeUet,  167 
Perkins,  163  (176) 
Pokorny,  166 
Poselger,  162 
Prouhet,  164 

Reuschle,  165,  169  (172,  174, 

176) 
Reyer,  165 
Rej-nolds,  175 
Rieke,  173 
Robertson,  159  (160) 

Sachs,  175 
Sahnon,  168 
Sanio,  167 
Sardi,  167  (175) 
Schlomilch,  171  (172) 
Schroder,  168 
Schroter,  161 
Schuh,  177 
Seelhoff,  160 
Sensenig,  169 

Shanks,    168,  170  (161,  165, 
169,  176) 


Author  Index 


473 


Sornin,  164 
Stammer,  166 
Stasi,  178 
Sturm,  166 
Suffield,  166 

Tagiuri,  176 


Telosius,  179 
Thibault,  164  (168) 

Van  den  Broeck,  172 
Van  Henekeler,  165 

WaUis,  159  (160) 
Weixer,  179 


Welsch,  179 
Wertheim,  161 
Westerberg,  163 
Wiley,  179 
Workman,  176  (168) 
Wucherer,    161 
Young,  165 


Ch.  VII.  Primitive  Roots,  Binomial  Congruences. 


Alagna,  217  (218) 

Alasia,  190  (210) 

AUegret,  190 

Amici,  197,  216,  217 

Anonymous,  204 

Arndt,    187,    188,    208,    209 

(193) 
Arnoux,  199,  218 

Bachmann,  194,  199,  218 

Barillari,  192  (193) 

Barinaga,  203 

Bellavitis,  193 

Bennett,  195 

Berger,  214  (215) 

Besant,  217 

Bhdscara,  204 

Bindoni,  199 

Bougaief,  213 

Bouniakowsky,  191,  192,  212 

(204) 
Brennecke,  (208) 
Bukaty,  212  (210) 
Burckhardt,  (185,  201) 
Buttel,  190 

Cahen,  198 
Calvitti,  (204) 
Carmichael,  200,  202 
Cauchy,   184,   186,   187,  209 

(188,    190,    194,    195,    198, 

200,  212,  213) 
Cayley,  191 
Chabanel,  202 
Christie,  199 
CipoUa,  200,  218-221 
Colebrooke,  [204] 
Concina,  222 
Contejean,  194 
Creak,  222 
Crelle,    185,   209   (186,    188, 

190,  208) 
Cunningham,   198-204,  217- 

222  (185,  189,  190,  213) 

Daniels,  194 
Da  Silva,  190,  210 
De  Jonquieres,  197 
Demeczky,  201 


Desmarest,   188,   189,  210 

(214) 
Dickstein,  210,  215 
Dirichlet,  185,  191,  211  (198, 

214) 
Dittmar,  212 
Dupain,  192 

Epstein,  200 

Erlerus,  186,  208  (196) 

Euler,  181,  204,  205  (222) 

Foghni,  199 
Fontene,  201 
Forsyth,  193 
Frattini,  193 
Fregier,  183 
Friedmann,  219 
Frolov,  196 

Gauss,  182,  194,  195,  207 
(183-185,  187,  188,  193. 
194, 197, 198,  209,  210,  213, 
214) 

Gazzaniga,  213 

Gegenbauer,  194,  196,  215 

G6rardin,  222 

Goldberg,  192 

Gorgas,  211 

Grave,  201 

Grigoriev,  199 

Grosschmid,  221 

Hacken,  194 
Hanegraeff,  210 
Heime,  191 
Hill,  192 
Hofmann,  193 
Houel,  191 
Hurwitz,  (203) 

Ivory,  184  (190) 

Jacobi,    185    (188,    190-192, 

198,  201,  203,  211) 
Japanese,  204 

Keferstein,  194 
Korkine,  201  (203,  221) 


Kraitchik,  202 
Krediet,  203 
Kronecker,  192,  198 
KuUk,  189 
Kunerth,  213 

Lacroix,  183  [208] 

Ladrasch,  212 

Lagrange,  181,  205  (182,  206, 

207,  214,  216) 
Laisant,  193 
Lambert,  181 
Landau,  201 
Landry,  190 
Laplace,  208 
Lazzarini,  (222) 
Lebesgue,  184,  188-192,  208, 

211  (196,  204) 
Legendre,  182,  205-207  (185, 

187,  188,  194,    208,    213- 

215,  219,  222) 
Leibniz,  215 
Libri,  208 
Lucas,    194,   213    (198,   200, 

202,  203,  218) 

Maillet,  202 
Mann,  215 
Marcolongo,  214 
Maser,  [182,  206,  207] 
Massarini,  188 
Mathews,  195,  215 
Matsunaga,  204 
Maximoff,  222 
Mayer,  216 
Meissner,  219 
Mertens,  198  (192) 
Meyer,  211 
Miller,  198,  201,  203 
Minding,  185 
Moreau,  198 
Murphy,  186 

Nordlund,  200 

Oltramare,  189,  190  (191) 
Ostrogradsky,  185  (186,  188) 

Pepin,  197,  213 


474 


Author  Index 


Perott,  194,  196,  197 

Picou,  218 

Pocklington,  222 

Poinsot,  183,    184,  187,   208, 

209   (190,    194,    199,   207, 

208) 
Posse,  201,  202,  203,  221 
Prouhet,  188  (194,  208) 

Rados,  222 

Reuschle,  190,  213  (200,  204) 
Richelot,  185  (194) 
Rochette,  188 

Sancery,  193,  213  (194,  195) 
Schapira,  188 


Scheffler,  190,  194 

Schering,  (208) 

Schuh,  202  (221) 

Schumacher,  202 

Schwartz,  194 

Seelhoff,  214 

Serret,  194,  214 

Smith,    191,   210   (184,    190, 

207) 
Speckmann,  216,  217 
Stankewitsch,  213 
Stasi,  221 
Stern,  184,  208  (187, 194, 199, 

203) 
Studni6ka,  215 
Szily,  194 


Tamarkine,  219 
Tchebychef,    188    (185,    191, 

195,  196,  198,  207) 
Thiele,  212 

Tonelli,  215,  216  (217,  218) 
Traub,  192 

Vab-off,  222 

Von  Schrutka,  202,  221 

Wcrtheim,  194,  196,  197,  199, 

214  (201,  202) 
Woodall,  203  (202) 
Wronski,  210  (211,  212,  215, 

218) 


Abel,  259 

Alasia,  224 

Amoux,  250,  251,  254  (255) 

Bachmann,  250,  251 
Bauer,  249,  251 
Bellavitis,  244 
Biase,  260 

Borel  and  Drach,  247 
Bunickij,  260 
Bunitzky,  260 
Bussey,  251 

Cailler,  255,  256 

Carey,  249 

Cauchy,  223,  225,  238,  252, 

258  (243) 
Ch&telet,  262 
Christie,  262 
Cipolla,  232 
Cordone,  247,  254,  259 
Creak,  262 
CreUe,  223  (224) 
Cunningham,  262 

Damm,  247,  254 

Da  Silva,  224 

Dedekind,  240,  245  (242) 

Demeczky,  228 

Dickson,  232,  248-250,  252, 

254,  256  (244,  249) 
Dina,  246 

Earnshaw,  223 
Eisenstein,  239 
Epstoen,  250  (249) 
Escott,  256 
Euler,  223  (259) 

Fermat,  257  (260) 
Frattini,  259,  261 

Galois,  235   (232,   239,   242, 
243,  247,  252) 


Zsigmondy,  195, 197,  216 

Ch.  VIII.  Higher  Congbuences. 

Neikirk,  251 
Oltramare,  253  (254) 


Gauss,   223,   233    (235,   238, 

240,  256) 
Gegenbauer,  226-229,  231 
Genocchi,  258 
Giudice,  (259) 
Grotzsch,  261 
Grunert,  224 
Guldberg,  248,  250 

Hathaway,  259 

Hayashi,  256 

Hensel,  226,  249 

Hermite,  225 

Hurwitz,  231,  259  (232,  233) 

Iwanow,  254 

Jacobi,  235 

Jenkins,  259 

Jordan,  243,  244,  261  (252) 

Kantor,  255,  262 

Konig,  225  (226,  229) 

Krediet,  261 

Kronecker,     226,     249,     260 

(228,  229,  251) 
Kiihne,  232 

Lagrange,  223 

Landau,  261 

Lebesgue,  224,  235,  258  (245) 

Legendre,  223,  257 

Lerch,  (227,  228) 

Le  Vavasseur,  248 

Libri,  224  (225,  258) 

Lipschitz,  260  (257) 

Maser,  [223,  233,  235] 
Mathieu,  241  (248) 
Miller,  251 
Mirimanoff,  255 
Mitchell,  246 
Moore,  247 


Pellet,  243,  245,  246 
Pepin,  244 
PiccioU,  261 
Pierce,  262 
Poinsot,  224  (259) 

Rados,    226,    233,    261,    262 

(225) 
Raussnitz,  226 

Sanderson,^252 
Satunovskij,  231 
Scarpis,  252 
Schonemann,  225,  236,  238, 

239  (251) 
Schiitz,  243 
Schwacha,  233 
Serret,    239,    241,    244,    258 

(233,  245,  246,  248,  256) 
Smith,  241,  259 
Snopek,  229 
Stephan,  232 
Stickelberger,  249  (251) 
Sylvester,  259  (245) 

Tarry,  252 
Tchebychef,  225 
Tihanyi,  262 

VonSterneck,  255,  260,  261 

(262) 
Voronoi,  251,  253  (255) 

Weber,  247 
Wertheim,  260 
WoodaU,  262 
Woronoj,  253,  254 
Wronski,  257 

Zsigmondy,  230,  247 


Author  Index 


475 


Ch.  IX.  Divisibility  of  Factorials,  Multinomial  Coefficients. 


Adams,  278 
Andr6,  263,  265-6 
Anonymous,  275 
Anton,  263,  271 
Arlvalo,  278 
Arndt,  276  (277) 

Babbage,  270 
Bachmann,  266,  274 
Bauer,  268 
Beaujeux,  277 
Beeger,  278 
Bernoulli  (268) 
Bertram,  263 
Birkeland,  269 
Bouniakowsky,  276-7 
Bourguet,  266 

Carmichael,  264,  276,  278 
Catalan,  265-7,  271-2 
Cauchy,  265 
Cayley,  269,  270 
Cesltro,  266,  272 
Child,  278 
Cunningham,  265,  274 

De  Brun,  263 
De  Jonquieres,  270 
De  Polignac,  266,  269 
De  Presle,  267 
Dickson,  273  (272) 
Dirichlet,  275  (276) 

ElUott,  270 

Fleck,  274-5 
Fonten6,  274 
Franel,  276 

Gauss,  269 

Gegenbauer,  267,  272 
Genocchi,  271 
Genty,  263 


Gerhardt,  [269] 
Glaisher,  268,  273-4 
Gmeiner,  267 
Greatheed,  269 
Grosschmid,  274 
Guerin,  275 

Hayashi,  274 
Heine,  267 
Hensel,  263 

Hermite,    266,    271-2    (267, 
269,  274) 

Jacobi,  275  (276) 
Janichen,  265 
Jenkins,  269,  271 

Kapferer,  274 
Kempner,  263 
Korkine,  276 
KJronecker,  276 
Kummer,  270  (272-3) 

Lagrange,  275 
Laisant,  277 
Landau,  267-9 
Lebesgue,  269 
Legendre,  263  (264) 
Leibniz,  269  (270) 
Lerch,  276 
Libri,  270 
Li^nard,  266 
Lionnet,  269 
Liouville,  276 

Lucas,  266,  271-2,  275  (274, 
278) 

MacMahon,  268 
MaiUet,  265,  268 
Maitra,  269 
Malo,  276 

Marks,  278 
Mason,  278 


Mathews,  272-3 
Mertens,  273 
MUler,  269 
Morley,  273 

Neuberg,  263,  266 
Nielsen,  274,  278 

Oltramare,  277 
Ouspensky,  276 

Pascal,  269 
Pessuti,  269 
Petersen,  272 
Pincherle,  267 

Ram,  274 
Rogel,  267,  272 

Schlomilch,  272 
Schonbaum,  269 
Schonemann,  265 
Segar,  269  (270) 
Sharp,  272 
Stern,  270 
Stickelberger,  263 
Stridsberg,  264,  269 
Studnicka,  270 
Szego,  265 
Szily,  273 

Tanner,  267 
Teixeira,  267,  277 
Thue,  267 

Van  den  Broeck,  272 
Vandiver,  276 
Vecchi,  277 

Waring,  275 
WeiU,  266  (267-8) 
Wolstenhohne,  271-2  (275) 
WoodaU,  272 


Ch.  X.  Sum  and  Number  of  Divisors. 


Ahlborn,  291 
Andreievsky,  288 


Bachmann,    279,    315,    321, 

323  (281,  291,  319) 
BeU,  323,  325 
Berger,  291,  312,  317   (292, 

295  299) 
Bougaief,  303,  315  (301,  312, 

316,  325) 
Bouniakowsky,  283,  284,  287 

(281,  286,  302) 


Burhenne,  283 

Busche,  306,   308,   314,   319 
(315,  323) 

Cantor,  291 

Catalan,   289-291,   302,   306 

(292,  295) 
Ces^ro,  290-4,  302,  306,  308 

(295,     298-9,     312,     315, 

320) 
Cunningham,  325 

De  Vries,  317 


Dirichlet,  281-2,  301  (284r-5, 
289,  291,  298-9,  305,  307, 
314-5,  318,  322,  325) 

Egorov,  312 
Elliott,  (318) 

Euler,  279  (284,  290,  303, 
312,  317-8,  321,  323) 

Fekete,  321 
Fergola,  (302) 
Franel,  317-8 


476 


Author  Index 


Gauss,  308 

Gegenbauer,    298-9,    301-5, 

307-8,  316  (288,  315,  318, 

325) 
Giulini,  319 
Glaisher,     289-292,     294-6, 

300,  303-4,  308-310,  312, 

318,   320,   322    (280,   302, 

321) 
Gram,  295,  308  (291) 
Gronwall,  322 

Hacks,  306-7  (303,  322) 
Halphen,    289,    290    (294-5, 

309,  312) 
Hammond,  311  (325) 
Hansen,  319 
Hardy,  325  (319) 
Hemming,  284 
Hermite,  292,  295,  297,  306 

(304-5,  315) 

Jacobi,  281-2  (283,  288,  290, 
295.  300-1,  313,  317-8) 

Knopp,  321,  323 
Kronecker,  297,  318 

Laisant,  308 


Lambert,  280  (306,  323) 
Landau,    294,    305,    317-8, 

321-4 
Lebesgue,  284 
Legendre,  281 
Lerch,  307,  313,  316  (314-5, 

317) 
Lionnet,  288 
LiouviUe,  284-8  (291-4,  298- 

9,  312-3,  321,  323) 
Lipschitz,    291-2,    298,    302 

(299,  307,  313,  315) 
Lucas,  291,  312 

Meissel,  284  (299,  314,  317- 

8,  322) 
Meissner,  320 
Mellin,  319 

Mertens,  289  (294,  315) 
Minetola,  322 
Minin,  313 
Mobius,  296 

Nachtikal,  316 

Pexider,  320 
Pfeiffer,  305  (322) 
PUtz,  291  (317,  322,  325) 
Plana,  (281) 


Radicke,  291 
Ramanujan,  323-5 
Roberts,  303 
Rogel,  316,  325 
Runge,  302 

Sardi,  288 

Schroder,  314,  319,  321  (315) 

Sierpinski,  320 

Smith,  289  (296) 

Sokolov,  312 

Steffensen,  323 

Stern,  281,  303  (321) 

Stieltjes,  292 

Strnad,  307 

Traub,  287 

Vahlen,  313 
Van  der  Corput,  323-4 
Von  Mangoldt,  294 
Von  Sterneck,  317 
Voronoi,  318-9  (322,  325) 

Waring,  280  (303) 
Wigert,  320,  323,  325 

ZeUer,   291,   295,   313    (299, 
303,  312,  321) 


Ch.  XI.   Miscellaneous  Theorems  on  Divisibility,  Greatest  Common 
Divisor,  Least  Common  Multiple. 


Anonymous,  327 
Avery,  332 
Axer,  331 

Bachmann,  335 
Barinaga,  336 
Berger,  328 
Binet,  (332) 
Birkeland,  331 
Borel,  334  (333) 
Bougaief,  327  (328,  330) 
Bouniakowsky,  332 
Brown,  336 

Ces^o,  328,  333  (336) 

Darbi,  335 

Dedekind,  334 

De  Jough,  335 

De   la   Valine   Poussin,   330 

(331) 
De  Polignac,  336 
Dickson,  331 
Dienger,  327 
Dintzl,  335 

Dirichlet,  327,  335  (328-331) 
Dupr^,  (332) 


Gegenbauer,    328-330,    333, 

336 
Gelin,  335 
Grolous,  328 
Guzel,  331 

Hacks,  330,  333 
Hammond,  333 
Hensel,  334  (333) 

Klein,  334 
Kluyver,  335 
Kronecker,  334  (336) 

Lam6,  (332) 
Landau,  328,  331 
Lebesgue,  332  (335) 
Lecat,  336 
Lucas,  333 

Mertens,  334-5 
Mitchell,  336 
Moschietti,  332 

Neuberg,  333 

Pichler,  335 


Rogel,  331 
Ross,  336 
Rothe,  332 

Saint-Loup,  330 

Sierpinski,  335 

Stern,  335 

Stieltjes,  333 

Stifel,  327 

Sylvester,  327,  333,  336 

Terquem,  [327] 

Van  der  Corput,  332 
Vandiver,  331 
Verhagen,  336 
Verson,  332 

Weitbrecht,  332 
Wijthoff,  336 
WUlaert,  336 

Yanney,  335 

Zeller,  328 


Author  Index 


477 


^■Alkarkhi,  337 
HpAnonymous,  339,  344 
K  Anton,  345 
^    Apianus,  337 

Argardh,  338 

Avicenna,  337 

Ayza,  346 

Badoureau,  345 
Barlow,  (346) 
Belohldvek,  342 
Biase,  343 
Biddle,  345 
Bindoni,  343 
Borgen,  342 
Bougaief,  340  (342) 
Bougon,  341 
Bouniakowsky,  344 
Breton,  341 
Broda,  345 
Brooks,  345 
Bruzzone,  344 
Burgess,  345 
Battel,  344 

Calvitti,  345 
Cantor,  [337] 
Carra  de  Vaux,  337 
Castelvetri,  338 
Catalan,  340 
Cattaneo,  346 
Cazes,  345 
Cesaro,  345 
Chiari,  344 
Christie,  341,  345 
Chuquet,  337 
Church,  345 
Cicero,  346 
Collins,  345 
Conti,  345 
Crelle,  339 
Csada,  346 
Cunningham,  345 

D'Alembert,  338  (344) 
Da  Ponte  Horta,  345 
De  Fontenelle,  338 
De  Lapparent,  344 
Delboeuf ,  340 
De  Montferrier,  344 
Dickstein,  341,  345 
Dietrichkeit,  341 
Dietz,  344 
Dodgson,  342 
Dorr,  345 
Dorsten,  345 


Ch.  XII.   Criteria  for  Divisibility  by  a  Given  Number. 


Dostor,  340 
Drach,  345 
Dupain,  339 

Elefanti,  344 
EUiott,  339 
Evans,  345 

Fazio,  345 
Filippov,  346 
Flohr,  344 
Folie,  339 
Fontebasso,  346 
Fontes,  341-2  (337) 
Forcadel,  337 

Gale,  346 
GeUn,  343,  345 
Gerardin,  345 
Gergonne,  338  (345) 
Ghezzi,  345 
Gorini,  344 
Greenfield,  345 
Greenstreet,  345 
Grunert,  344 

Haas,  345 
Harmuth,  (346) 
Heal,  342 
Heilmann,  341 
Herter,  338  (344) 
Hill,  338 
Hippolytos,  337 
Hocevar,  340 
Holten,  340 
Hommel,  340 

Ibn  Albanna,  337 

Ibn  Mtisa  Alchwarizmi,  337 

Ibn  Sina,  337 

Ingleby,  345 

lodi,  346 

Jenkins,  345 
Jorcke,  345 
Joubin,  346 

Karwowski,  344 
Kraft,  338 
Krahl,  346 
Kroupa,  346 
Kylla,  346 

Lagrange,  338 
Lalbaletrier,  345 
La  Marca,  346 
Lange,  345 


La  Paglia,  346 
Lebesgue,  344 
Lebon,  (346) 
Lenth6ric,  346 
Lenzi,  345 

Leonardo  Pisano,  337 
Levanen,  341 
Lichtenecker,  346 
Liljevalch,  338  (340) 
Lindman,  344 
Loir,  341-2,  345 
Loria,  343 
Lubin,  345 

Malengreau,  343 
Mantel,  340 
Mariantoni,  345 
Marre,  [337] 
Mason,  344 
Meissner,  343 
Mennesson,  345 
MiceU,  346 
MoUer,  340 
Morale,  345-6 

Nannei,  344 
Nasso,  345 
Niegemann,  339,  344 
Niewenglowski,  345 
Noel,  341 

Oskamp,  340 
Otto,  340 

Paciuolo,  337 

Paoletti,  346 

Pascal,  337  (338,  342j  344-5) 

Perisco,  346 

Perrin,  341 

Pick,  345 

Pietzker,  345 

Pinaud,  344 

Plakhowo,  342 

Polpi,  346 

Recorde,  337 
Reyer,  344 
Riess,  342 
Ripert,  343 
Romm,  [337] 

Sanvitali,  338 
Schlegel,  340  (345) 
Schobbens,  341 
Schroder,  346 
Schuh,  344 
Sibt  el-MAridini,  337 
Speckmann,  345 


478 


Author  Index 


StoufF,  345 
Stujrvaert,  344 
Sylvester,  342 
Szenic,  345 

Tagiuri,  343 
Tarry,  (346) 
Terquem,  344 
Tiberi,  346 
TirelU,  346 


Transon,  339 
Tucker,  341 


Unferdinger,  345 

Valerio,  342 
Van  Langeraad,  344 
Vincenot,  344 
Volterrani,  346 


Walenn,  345 
Wertheim,  345 
Widmann,  337 
Wilbraham,  339 
Wronski,  344 

Young,  344 

Zbikowski,  339 
Zeipel,  339 
Zuccagni,  345 


Ch.  XIII.   Factor  Tables,  Lists  of  Primes. 


Akerlund,  355 
Alliston,  356 
Anjema,  348 
Aratus,  347 
Aubry,  355 

Barlow,  351  (355) 
Beguelin,  349 
Bernhardy,  347 
Bernoulli,  349  [350] 
Berteken,  354 
Bertrand,  349 
Boethius,  347 
Boulogne,  356 
Bouniakowsky,  353 
Bourgerel,  354 
Brancker,  347  (348,  350) 
Burckhardt,  350  (352-5) 

Camerarius,  347 
Cataldi,  347 
Cayley,  353 
Chernac,  350  (351) 
Colombier,  351 
CreUe,  351-2 

Cunningham,  354-5  (350-2, 
356) 

D'Alembert,  349 

Dase,  352  (353-5) 

Davis,  352 

De  Polignac,  (356) 

Deschamps,  355 

Desfaviaae,  350 

De  Traytorens,  348  (350) 

Di  Girio,  354 

Dines,  355 

Dodson,  348 

Du  Tour,  348 

Eratosthenes,  347  (348,  353- 

6) 
Escott,  355  (356) 
Euler,  349  (356) 

Felkel,  349,  350 


Gauss,  350,  352  (356) 
G6rardin,  356 
Gill,  353 

Glaisher,  350,  353  (355) 
Goldberg,  352 
Gram,  354 
Groscurth,  353 
Griison,  350 
Gudila-Godlewski,  353 
Guyot,  351 

Hansen,  356 
Hantschl,  351 
Harris,  348 
Hindenburg,  349 
Hinkley,  351  (347) 
Horsley,  347 
Houel,  351-2 
Hiilsse,  [350] 
Hutton,  351 

Ibn  Albannd,  347 

Jager,  348 
Johnson,  353 
Jolivald,  354 

Kastner,  350 
Kempner,  (356) 
Kliigel,  348 
Kohler,  351 
Krause,  350 
Kronecker,  (354) 
Kriiger,  348 
KuUk,  351  (355-6) 

Laisant,  354-5 
Lambert,  348,  350  (349) 
Landry,  351 
Lebesgue,  352 
Lebon,  355-6 
Lehmer,  352-3,  355-6 
Leonardo  Pisano,  347 
Libri,  347 
Lidonne,  350 


Lionnet,  355 
Lucas,  353 

Marci,  349 
Marre,  [347] 
Maseres,  350 
Meissel,  352  (350) 
MerUn,  (356) 
Mobius,  351 
Morehead,  355 

Neumann,  350 
Nicomachus,  347 
Noviomagus,  (356) 

Oakes,  352 
Oberreit,  349 
Ozanam,  349 

FeU,  [347] 
Perott,  352 
Petzval,  352 
Pigri,  348 
Poetius,  348 
Poretzky,  (356) 

Rahn  (Rhonius),  347 
Rallier  des  Ourmes,  348 
Rees,  351 
Reymond,  (356) 
Rosenberg,  352  (355) 
Rosenthal,  349 

Saint-Loup,  353  (356) 
Salomon,  351 
Schaffgotsch,  349 
Schallen,  351 
Schapira,  354 
Schenmarck,  350 
Schwenter,  348 
Seelhoff,  353 
Simony,  353  (354) 
SneU,  350 
Speckmann,  354 
Stager,  356 
Struve,  350 
Suchanek,  354 


Author  Index 


479 


Tarry,  355 
Tennant,  354 
Tessanek,  349 
Tuxen,  353 

Valerio,  354 


Van  Schooten,  347 
Vega,  350 
VoUprecht,  353^ 
Von  Stamford,  349 
Von  Sterneck,  354 


Wallis,  348  (347) 
Wertheim,  [347] 
WilUgs  (Willich),  348 
Wolf,  348 
WoodaU,  354  (356) 


Ch.  XIV.  Methods  of  Factoring. 


Aubry,  373  (369) 

Ball,  368 

Barbette,  367,  373 
Bartl,  370 
Beguelin,  361,  366 
Bernoulli,  371 
Bickmore,  369 
Biddle,  359,  367,  369-374 
Birch,  368 
Bisman,  374 

Bouniakowsky,  369  (370) 
Burgwedel,  365 
Busk,  358  (359) 

Cahen,  364 
Canterzani,  366 
Cantor,  [366] 
Christie,  361,  367,  372 
Cole,  365 
Collins,  357 
Cullen,  365,  369 
Cunningham,     358-9,     361, 
365,  368-9,  373-4  (362) 

De  Bessy  (see  Frenicle) 
De  Montferrier,  358 
Deschamps,  367 
Dickson,  370  (360) 

Euler,  360-2  (363-5) 

Fermat,  357  (358,  367) 
Frenicle,  360 
Fuss,  362 

Gauss,  363,  369  (364-5,  370) 
GIrardin,  365-7,  370,  374 


Gmeiner,  374 
Gough,  371 
Grube,  363 

Hansen,  (371) 
Harmuth,  361 
Henry,  358 
Hudson,  358 

Johnsen,  369 
Joubin,  372 

Kausler,  357,  362 
Kempner,  374 
Kielsen,  372 
Klugel,  366 
Kraft,  370 
&aitchik,  359,  360 
KuUk,  361,  372 

Lagrange,  369 
Lambert,  371 
Landry,  358,  369,  371 
Laparewicz,  365 
Lawrence,  358-360 
Lebon,  359,  373 
Legendre,  361-2  (363) 
Lehmer,  368 
Levanen,  364 
Lucas,  363-4  (372) 

Marcker,  368 

Mathews,  364 

Matsunaga,  371 

Meissner,  372  (358,  364,  367) 

Mersenne,  357,  360  (367-8) 

Meyer,  365 

Minding,  363 


Mobius,  (374) 

Neimaann,  359 
Niegemann,  366 
Nordlund,  370-1  (369) 

Pepin,  364 
Petersen,  360 
Pockhngton,  370 

Rawson,  367 
Reymond,  374 

Schaffgotsch,  367 
Schatunovsky,  370 
Seelhoff,  363  (365) 
SeUwanoff,  364 
Speckmann,  367 
Studnicka,  366 

Tchebychef,  363 
Teilhet,  359 
Tessanek,  366 
Thaarup,  358 
Thiehnann,  368 

Vaes,  359,  360 
Vah-off,  365 
Von  Segner,  366 
Vuibert,  361 

Waring,  362 
Warner,  358 
Weber,  364 
Wertheim,  358,  361 
Winter,  372 
WoodaU,  369 


Ch.  XV.  Fermat  Numbers  Fn  =  22^^-1-1. 


Anonjmaous,  376 
Archibald,  380 

Bachmann,  379 
Ball,  378 
Baltzer,  375 
Beguelin,  375  (377) 
Bisman,  379 
Broda,  377 

Canterzani,  375 


Carmichael,  377,  380 
Catalan,  377 
Cipolla,  378 
Cullen,  378 
Cunningham,  378-380 

Eisentein,  376 
Euler,  375 

Fermat,  375  (376) 
Frenicle,  375 


Gauss,  375 
Gelin,  377 
Genocchi,  375 
G^rardin,  377,  380 
Goldbach,  375 
Gosset,  379 

Hadamard,  378 
Henry,  375,  380 
Hermes,  378 
Hurwitz,  378  (380) 


480 


Author  Index 


Joubin,  376 

Klein,  378  (379) 
Klugel,  375 

Landry,  376-7 
Legendre,  (378) 
Le  Lasseur,  377 
Lipschitz,  378 
Lucas,  376-8  (379) 


Malvy,  378  (376) 
Mansion,  375 
Mersenne,  375 
Morehead,  376,  379 

Nazarevsky,  378 

Pepin,  376  (377-380) 
Perv'ouchine,  376-8 
Pervusln,  376 


Proth,  377  (378) 

Scheffler,  378 
Seelhoff,  377 
Simerka,  377 
Studnidka,  377 

Western,  378-9 
Woodall,  379 


Ch.  XVI.   Factors  of  a"±b". 


AurifeuiUe,  383  (386) 

Bang,  385  (386) 
Bauer,  385 
Beguelin,  381  (385) 
Bickmore,  386  (385) 
Biddle,  387  (391) 
Birkhoff,  388  (386) 

Carmichael,  389,  390  (388) 
Catalan,  383-4  (386) 
Cunningham,  386-391   (384) 

Dedekind,  384 
Dickson,  388-9 
Dines,  (391) 
Dirichlet,  (391) 

Escott,  385,  387-9 
Euler,    381-2    (383,    388, 
390-1) 

Fauquembergue,  390 
Felkel,  382 
Fermat,  381 
Fontene,  390 
Foster,  385 


Gauss,  382 
Genese,  385 
Gt5rardin,  390 
Germain,  382  (383) 
Gianni,  385 
Glaisher,  386,  388 

Henry,  382 

Kannan,  389 
Kraitchik,  (391) 
Kronecker,  385,  387 
Kummer,  383 

Lawrence,  (391) 

Lebesgue,  382-3 

Lef^bure,  384 

Legendre,  382 

Le  Lasseur,  383  (384-5,  389) 

Lucas,  383^,  386 

Markoff,  386 
Minding,  382 
Miot,  389 
Morehead,  (391) 


Pepin,  384-6 
Plana,  383 
Pocklington,  390 

Realis,  384 
Reuschle,  382-3 

Sanjana,  389 
Scheffler,  385 
Schering,  382 
Seelhoff,  (391) 
Soons,  383 
Sylvester,  384-5 

Tchebychef,  382 
Teilhet,  388 

Vah-off,  390 

Van  der  Corput,  390 

Vandiver,  387-8 

Welsch,  389 
Wertheim,  388 
WoodaU,  386,  388,  391 
Workman,  386 

Zsigmondy,  386  (388) 


Ch.  XVII.   Recurring  Series;  Lucas'  u„,  v„. 


Agronomof,  406 
Amsler,  410 
Andr6,  408 
Archibald,  411 
Arista,  405 
Arndt,  (397) 
Aubry,  405 

Bachmann,  394,  405 
Bastien,  406 
Bernoulli,  407  (408) 
Betti,  408 
Bickmore,  404 
Binet,  394  (403) 
Boutin,  406 


Braun,  411 
Brocard,  402 

Candido,  405 
Cantor,  [407] 
Carmichael,   394,   406   (398, 

400) 
Cassini,  407 

Catalan,  395,  402-3  (404) 
Ces^ro,  401-2 
Christie,  404 
CipoUa,  405 
Cunningham,  397 

Damm,  405 


Degen,  407,  411 
De  Longchamps,  401,  408 
De  Moivre,  407 
Dickson,  405,  410 
Dienger,  394 
Dirichlet,  393  (402) 
D'Ocagne,  402,  409,  410 

Emmerich,  404 

Escott,  404-5 

Euler,  393,  407  (397-8,  400) 

Fermat,  (396) 
Fibonacci  (see  Leonardo) 
Fontte,  403 


Author  Index 


481 


Foster,  403 
Fourier,  408 
Fransen,  405 
Frisiani,  408 
Frolov,  403 

Galois,  (403) 

Gauss,  393  (397) 

Gegenbauer,  403,  409 

Gelin,  401 

Genocchi,  394,  397,  402  (405) 

G^rardin,  406 

Girard,  393 

Grosschmid,  394 

Hayashi,  410 
HiU,  394 

Kepler,  (411) 
Kronecker,  402  (393) 

Lagrange,   393,   407   (396-7, 

408) 
Laisant,  408,  410 
LamI,  394 
Landau,  404r-5 
Laparewicz,  405 
Laplace,  407 
Lattes,  410 
Legendre,  393 
Le  Lasseur,  400 
Leonardo  Pisano,  393  (394r- 

411) 


Liebetruth,  402 

Lionnet,  394 

Lucas,  394-403  (405-6) 

Magnon,  402 
Maillet,  403,  410 
Malfatti,  407 
Malo,  404,  406 
Mantel,  403 
Mathieu,  405 
Mersenne,  (397) 
Moret-Blanc,  397 
Murphy,  408 

Netto,  410 
Neuberg,  410 
Nicita,  411 
Niewiadomski,  406 

PaoU,  407 
PeUet,  406 
Pepin,  398,  401 
Perrin,  404,  410 
Piccioli,  407 
Pierce,  407 
Pincherle,  409 
Prompt,  406 
Proth,  (398) 

Ranum,  410 
R6aUs,  404 


Riccati,  407 
Ruggieri,  405 

Sancery,  (397) 
Scheibner,  408 
Scherk,  411 
Schlegel,  411 
Schonflies,  403 
Seliwanov,  403 
Serret,  394 
Siebeck,  394    ' 
Simson,  393 
Stirling,  407 
Study,  409 
Svanberg,  411 
Sylvester,  401,  403,  411 

Tagiuri,  404 
Tarn,  411 
Traverso,  410 

Valroff,  405 

V^sz,  411 

Vogt,  411 

Von  Sterneck,  (398) 

Wasteels,  405 
Weiss,  411 
White,  405 
Whitworth,  411 

Zeuthen,  405 


Ch.  XVIII.   Theory  of  Prime  Numbers. 


Andreoli,  434 
Aubry,  422 
Auric,  414 

Bachmann,  416,  418,  432 

Bang,  418-9 

Baranowski,  432 

Barinaga,  428,  438 

Bauer,  419,  420 

Berton,  416 

Bertrand,  435  (413,  425,  436) 

Bervi,  419 

Biddle,  (426) 

Bindoni,  427 

Birkhoflf,  418 

Boije  af  Gennas,  414 

Bonolis,  436 

Bougaief,  422,  429  (430-1) 

Bouniakowsky,  421 

Braun,  414,  421,  437 

Brocard,  425-6,  436 

Brun,  438 

Cahen,  414,  419,  436 


Cantor,  422,  425 
Carmichael,  420,  428 
Catalan,  421,  429,  431,  435 

(426) 
Cesaro,  430  (432,  435) 
Chabert,  420 
Chiari,  428 
CipoUa,  (426) 
Coblyn,  438 
Cole,  (426) 

Cunningham,  417,  423 
Curjel,  429,  431-2 

Dedekind,  415 

De  Jonqui^res,  429  (432) 

De  la  Vallee  Poussin,   416, 

418,  439-440  (417) 
De  Monddsir,  430 
De  Montferrier,  426 
DePohgnac,    A.,    424,    439 

(425) 
De  Pohgnac,  C,  436-7 
De  Rocquigny,  425 
Desboves,  415,  422,  435  (436) 


Descartes,  421 

Deschamps,  439 

Devignot,  426 

Dickson,  417 

Dirichlet,  415,  417  (416,  418) 

Dormoy,  437 

Dupr6,  415 

Durand,  415 

Enestrom,  [421] 
Eratosthenes,  (424) 
Escott,  420,  426,  428 
EucUd,  413 

Euler,  413,  415,  420-1,  424, 
426 

Fontebasso,  427 
Frobenius,  421 

Gambioli,  427 

Gauss,  438 

Gegenbauer,  413,  427,  431-3 

(426,  435) 
Genocchi,  418 


482 


Author  Index 


Genty,  (426) 
Giovannelli,  424 
Goldbach,  420-1,  424 
Graefe,  432 
Gram,  430 
Guibert,  425 

Hacks,  414,  427 
Hadamard,  424,  439,  440 
Hammond,  423,  429,  438 
Hardy,  438,  440 
Hargreave,  429 
Haussner,  422  (423) 
Hayashi,  433 
Heiberg,  [413] 
Heine,  415 
Hensel,  416,  418-9 
Hermite,  437 
Hoffmann,  430 
Hossfeld,  431 
Hurwitz,  (426) 

Isenkrahe,  437 
Iwanow,  419,  437 

Jaensch,  413 
Johnsen,  429 
Jolivald,  427  (426) 

Klein,  438 

Kossler,  435 

Kraft,  426 

Kraus,  418 

Kronecker,    413,    416,    418, 

425,  429  (414) 
Kummer,  413 


Labey,  [413] 
Lagrange,  424-5  (426) 
Lambert,  (426) 
Landau,    417-9,    423, 

43.5-8,  440 
Landry,  418 
Laurent,  427  (433) 
Lebesgue,  418-9  (426) 
Lef^bure,  418 


425, 


Legendre,  415,  420,  429,  (416, 

430-1,  434-5) 
Leibniz,  (426) 
Le  Lasseur,  420 
Lemaire,  420,  426 
Lemoine,  424 
Le  Vavasseur,  437 
Levi-Civita,  433 
L<5vy,  414,  421 

Lionnet,  422,  426,  429  (423) 
Lipschitz,  429  (432) 
Littlewood,  440 
Lorenz,  430 

Lucas,  418-9,  421  (426,  428) 
LugU,  431 

MaiUet,  423,  425,  436 
Marcker,  436 
Markow,  437 
Martin,  426 
Mathews,  429 
Mathieu,  425 
Meissel,  429,  431  (432) 
Meissner,  427,  438 
Merlin,  424 
Mertens  416-8 
M6trod,  415 
Meyer,  418 
Minetola,  427,  434 
Minin,  433 
Miot,  421 
Moreau,  416 

Oltramare,  420 
Oppermann,  435 

Paci,  430 
Pascal,  420 
Perott,  413-4 
Petrovitch,  434 
Pexider,  433 
Piltz,  416 

Pocklington,  419,  428 
Proth,  435 

Rados,  428 
Riemann,  439 


Ripert,  423,  425 
Rogel,  429,  431,  433-4 

Sardi,  (426) 

Schaffgotsch,  (426) 

Scheffler,  416,  431 

Schepp,  [420] 

Schering,  418 

Scherk,  436 

Schur,  419 

Serret,  418-9,  435 

Smith,  436,  439 

Speckmann,  416 

Stackel,  423  (422) 

Stasi,  428 

Stern,  424,  426 

Stieltjes,  414,  436 

Stormer,  437  (414) 

Studni6ka,  423 

Sylvester,  416,  418-9,  422-3, 

429,  431-2,  435,  437,  439 

(433) 

Tchebychef,   413,   435,   437, 

439  (426) 
Teege,  417 
Terquem,  421,  436 
Thue,  414 
Torelli,  440 

Vahlen,  419 

Van  der  Corput,  419 

Vandiver,  418 

Van  Laar,  431 

Vecchi,  424,  428 

Von  Koch,  427,  432  (433) 

Von  Sterneck,  419,  427,  435 

Warmg,  421,  425 
Weber,  417-8 
Wendt,  418 
Wertheim,  429 
Wigert,  432  (428,  434) 

Zignago,  416 
Zondadari,  428 
Zsigmondy,  418,  427  (426) 


Ch.  XIX.    Inversion  of  Functions;    Mobius'  Function  n{n);    Numerical 

Integrals  and  Derivatives. 


Axer,  449 

Bachmann,  445-6,  449 
Baker,  443 

Berger,  444,  446  (443) 
Bervi,  451 

Borel  and  Drach,  449 
Bougaief    (Bugaiev),    442-3, 
449-451  (448) 

Cahen,  449 


CeslLro,  443,  450 
Cistiakov,  451 

Dedekind,  441-2  (444,  446) 
Dirichlet,  (445) 

Elliott,  447 

Fatou,  448 
Fleck,  448 
Furlan,  448 


Gegenbauer,  447,  450  (443) 
Glaisher,  (441) 

Hackel,  448 

Kluyver,  448 
Knopp,  (448) 

Kronecker,  447  (443,  448) 
Kusnetzov,  448 

Laguerre,  442 


Author  Index 


483 


Landau,  448-9 
L^meray,  447 
Liouville,  441-2 
Lipschitz,  445 
Lucas,  445 

Meissel,  441 
Meissner,  448 
Merry,  442  (444) 


Mertens,  442,  446  (448) 
Mobius,  441  (443) 

Nazimov,  (448) 

Seliwanov,  446 
SheUy,  451 
Steffensen,  449 


Stieltjes,  449 

Tschistiakow,  451 

Von  Koch,  446 
Von    Sterneck,   444-6    (442, 
448) 

Zsigmondy,  444-5. 


Ch.  XX.  Properties  of  the  Digits  of  Numbers. 


Agronomof,  464 
Aiyar,  458 
Andreini,  459,  461 
Andreoli,  464 
Anonymous,  454,  458 

Barbier,  457 
Barillari,  455 
Barisien,  462 
Barlow,  453 
BerdeU^,  457 
Bertrand,  455 
Bianchi,  455 
Biddle,  463 
Booth,  455 
Boutin,  461 
Bouton,  460 
Brocard,  464 
Brownell,  453 
Burg,  464 

Calvitti,  461 

Cantor,  455  (458) 

Catalan,  456 

Cattaneo,  463 

Cesaro,  457  (461) 

Crelle,  454  (456) 

Cunningham,  458,  460,  462-4 

Davey,  454 
De  Rocquigny,  457 
De  Sanctis,  459,  464 
Dickson,  460 
D'Ocagne,  457 
Drot,  455 

Emsmann,  455 
Escott,  458,  462 

Flood,  455 


Fourrey,  465 
Fran^ais,  454 

Gegenbauer,  458 
Gelin,  465 
G^rardin,  461-2 
Gergonne,  454 
Glaisher,  456 
Goormaghtigh,  464 
Gnmert,  455 

HaUiday,  465 
Hauke,  459 
Hayashi,  459 
HiU,  453 
Hoskins,  456 

Ingleby,  455 

Janichen,  462 
Johnson,  458 

Kessler,  457 
Koppe,  461 
Ki-aitchik,  458 
Kraus,  458 

Laisant,  456-8  (454) 
Lemoine,  457,  464 
Lewis,  463 
Lucas,  458,  465 

Mackay,  457 
Maillet,  464 
Malo,  461 
Mansion,  456 
Martin,  456 
Metcalfe,  460 
Moore,  460 
Morel,  456 


Moret-BIanc,  457 
Moulton,  465 

Nannei,  462 

Osana,  465 
O'Shaughnessy,  465 

Pahnstrom,  458-9  (461) 
Perkins,  456 
PiccioU,  460 
Plateau,  456  (460,  463) 

Rutherford,  455 

Saint,  453  (456) 
Sampson,  455 
Sebban,  464 
Simmons,  457 
Stasi,  463 
Storr,  458 
Strauss,  458 
Suchanek,  (459) 
Szego,  462 

Tagiuri,  460 
Tanner,  456 
Tedenat,  454 
Teilhet,  460 
Thi6,  463 

Valentin,  459 
VerceUin,  462 
Von  Schrutka,  464 

Welsch,  464 
Wertheim,  459 
White,  465 
Wiggins,  460 
Witting,  462 

Ziihlke,  461 


SUBJECT  INDEX. 


Abundant,  3,  7, 11, 14, 15,  20, 
31-3 

Agreeable,  38,  458 

Algebraic  numbers,  86,  221, 
245,  251,  322,  379,  417, 
447-8 

Aliquot  parts,  3,  50-8 

Amantes,  39 

Amiable,  38,  41 

Amicable,  5,  38-50 

of  higher  order,  49,  50 

triple,  50 

Anatomiae  numeronmi,  348 

Approximation,  114-5,  158, 
281-3,  318,  330-1,  352, 354, 
411,  422-3,  430,  448  (see 
asymptotic,  mean) 

Arrangement  in  cycles,  269 

Arithmetical  progression, 
100-1,  114,  131,  336,  (see 
prime) 

Associated  numbers,  64-6,  73 

Asj-mptotic,  119,  122,  126-7, 
129-132,  134-6,  138,  144, 
154-5,  214-5,  289,  291, 
294,  301-2,  305-6,  308, 
317-325,  328,  333,  416-9, 
434-6,  438-440,  450  (see 
approximation,  mean) 

Aurifeuillian,  386,  390 

Base,  178,  182,  186,  199,  273, 
338,  340-1,  354-5,  369, 
373,  375,  379,  385,  398, 
454,  456,  458-460,  463^ 
(see  digits,  periodic) 

Befreundete,  38 

Belongs  (see  exponent) 

Bemoullian  numbers,  100, 
109,  110,  112,  140-1,  145- 
6,  220,  274,  278,  309,  311 

Bernoulli's  function,  268,  325 

Bertrand's  postulate,  132, 
413,  425,  435-6 

Bilinear  form,  409 

Binomial  coefficients,  59,  62, 
67,  77,  91,  97,  99,  266-278 

congruence,  92-5,  105, 

175,  177,  204-222,  388,  391 

,  identical,  78, 

82,  87-9,  94-5 

Casting  out  nines,  337-346 

Characters,  201,  415 

Circular  permutations,  75, 78, 
81,  131,  136 

(484) 


Combinations,  77,  90-1,  106, 

261,  281,  303,  410 
Complementary  fractions,  156 
Congeneres,  39 
Congruence  (see  binomial) 

,  cubic,  252-6,  262 

,  higher,  223-61 

,  identical,  73.  87-9 

,  involving    factorials, 

275-8,  428 

-,  irreducible,  84,  234- 


52 
,  quartic,    254-5,  259, 

260 
Congruent  form,  362 

fractions,  258-9 

series,  259 

Conjugate  functions,  444 
Consecutive    numbers,    147, 

332,  353,  355,  373,  457  (see 

product) 
Continued     fractions,      138, 

158,  210,  363,  367-8,  381, 

393,  399,  403,  408-9 
Crib  (see  sieve) 
Criteria  for  given  divisor,  337 
Cyclotomic  function,  199, 245, 

378,  383-5,  387-90,  418 


Decimal  (see  periodic) 
Defective,  3 
Deficient,  3 
Determinant,  77,  87,  97,  137, 

149,  150-1,  226,  228,  231, 

233,   261,   288,    295,    321, 

336,  368,  399,  410-1,  444, 

446 

=c  (modm),  155,  261 

of  Smith,  122-4,  127- 

130,  136 
Diatomic  series,  439 
Differences  of  order  m,  62— i, 

74,  78,  79,  204 
two  primes, 

424-5 


squares,  357 
Digits,  81,  343,  353-4,  358, 

360,  366,  438,  453-65 
of  perfect  number,  7, 

10,  17,  20 

permuted    in    multi- 


Digits,  sum  of,  263-4,  266, 
272, 337-8,  342-3, 367,  455, 
457-8,  461-4 

Diminute,  3,  4 

Equivalent  fractions,  135 
EucUdean  number,  28 
Euler's   constant,    122,    134, 

136.  281-3,  289,  294,  317- 

24,  328-30 

criterion,  67,  205 

generalization  of  Fer- 

mat's  theorem,  60-89,  398, 

400 

numbers,  363 

0-function,    82,    85, 


110,  113-58,  182,  285-6, 
293,  312,  333-6,  404,  434, 
441-2,  446 

,  gener- 
alized by  Schemmel,  147 

Jordan, 

123,  132,  147,  252,  298-9 

Exc^dant,  3 

Excess  E  of  divisors  4m-|-l 
over  divisors  4m-|-3,  281, 
289,  293,  295-6, 300-1,  308, 
318-9 

of     odd     over    even 

divisors,  290-1,  317-8 

Exclusion  method,  207,  369- 
70 

Exponent,  61,  112,  163,  169, 
181-204,  240,  242-3,  246, 
257,  259,  260  (see  Haupt) 

to  which  10  belongs, 

159-204,  339,  341-2 

2  be- 
longs, 111,  181, 190-1,  193, 
198,  200,  203,  369-70 


Factor  tables,  347  (see  graph- 
ical) 

Factorial,  62-3,  77,  263-78 

Factoring,  13,  25,  241,  248, 
252,  357  (see  graphical,  cri- 
teria, sieve) 

,  number  of  ways  of, 

52,  109,  282,  285,  298,  331 

Factors  of  10"  ±1,  159-179 
—  2''-l  (see  per- 


ples,  164-5, 170, 174, 176-7, 
458-9 


feet) 


381-91 


a^^b",   258, 


Subject  Index 


485 


Farey  series,  155-8 
Fermatian  function,  385 
Fermat's  numbers  22° +1,  94, 
140,  199,  375,  398,  401 

theorem,  12,  17,  18, 

59-89,  179 

— ,  converse  of, 

— ,  generaliza- 
tion, 84-9,  406  (see  Galois) 
Finite  algebra,  388 

differences,  250,  394, 


91-5 


407 


field,  247,  250 


Flachen  Zahlen,  4 
Frequency  of  a  divisor,  126 

Galois  field,  232,  247,  250 

imaginary,  233-55 

Galois'  generalization  of  Fer- 
mat's theorem,  235,  240, 
246-7,  249,  250,  252,  403 

Wil- 
son's theorem,  240,  246-7, 
252 

Gaussien,  194 

Graphical  factoring,  351, 353- 
4,  356,  365,  369,  372,  374 

representation  of  di- 
visors, 330,  351,  354 

Greatest  common  divisor, 
139,  147,  150,  252,  328, 
332-6,  394,  401-3,  447, 
456,  462  (see  determinant 
of  Smith) 

di\asor,  329,  331 

integer   in,    89,    119, 

121-2,  126,  130,  132,  138, 
144,  153,  158,  263,  282, 
293,  295,  297-9,  302-3, 
319,    427,    429-432,  450-1 

Goldbach's  theorem  and  anal- 
ogues, 421-5 

Golden  section,  411 

Ground  forms,  268 

Groups,  78,  80-1,  84-5,  131, 
137,  152,  155,  177,  194, 
196-8,  201,  203,  216,  221, 
248,  251,  268,  287,  332, 
356,  414-5 

Haupt-exponent,  190, 200, 203 

Hexagon,  9,  411 

Highest  prime  power  in  m.\, 
263,  272 

a  poly- 
nomial, 334 

Highly  composite  number, l323 

History,  32,  84,  157,  200, 
342,  353 

Hyper-even  number,  379 


Hyper-exponential     number, 
379 

Idoneal  (idoneus),  361-5 
Imperfectly  amicable,  50 
Index,    85,   182-3,    185,  188, 

190-4,  197-204,   211,  240, 

244-5,  249,  251 
Indian,  337 
Indicator,  118,  131,  155,  186, 

194,  200 
Indivisibilis,  6 
Integral  logarithm,  353,  417, 

440 
Invariant,  89,  232-3,  260,  364 
Inversion,  84,  120,  127,  129, 

132-3.  135,  140,  145,  150, 

153,    234,    296,    429,    430, 

441-8 
Irreducible  function,  234-252 
fraction,    126,    129, 

133,  138,  155-8;  162,  175 

Kerne,  334 
Korper  Zahlen,  4 
Kronecker's  plane,  155 

Lattice,  173 

Leaf  arrangement,  411 

Least  common  multiple,  82, 

328,  332-6,  445,  464 

residue,  341-2,344,369 

Legendre-Jacobi  symbol,  109, 

210,219,249,251,255,260, 

276,  288,  300,  308,  330,  364, 

382,  385,  394,  398 
Linear  differential  form,  248, 

250 
forms  of  divisors,  160, 

362-4,  370,  382,  386,  390, 

399 
function,  117-8,  134, 


204-5 


nimabers,  4 

Lucas'  un,  vn,  218,  395,  418 
Lucassian,  27 

Mangelhaft,  3 

Matrix,  137,  226,  228,  233 

Maximum  divisor,  332 

Mean,  281,  291-4,  301-2,  305, 
312,  318,  320,  328-331,  333, 
335,  447  (see  asymptotic) 

Mediation,  156 

Mersenne  number,  31 

Mobius'  (Merten's)  function 
M(n),  86,  122,  127-9,  144-5, 
148-9, 150-1,  265,  289, 322- 
3,  329,  335,  431,  441-9,  462 

gener- 
alized, 135-6 


Modular    system,    88,    249, 

251,  402 
Mosaic,  212 
Multinomial    coefficient,    59, 

266-78 
Multiply  perfect,  33 

Nim  (game),  460 

Nombres  associes,  50 

Norm,  236,  252,  322 

Normal  order,  325 

Nmnber  of  divisors,  51,  54, 
135-6,  142,  279-325,  328, 
443,  451 

integers  divis- 
ible by  nth  power,  327-32 

solutions    of 


ui  .  .  .  ujk  =  n,  125, 149,291, 
298,  308,  312,  317,  324 

n  =  x'^y^, 

318 

Numerical  integrals  and  de- 
rivatives, 152,  449 

Order  modulo  m,  138 
of  root,  189 

Partial  fraction,  73,  135,  161, 

198,  410 
Partition,  279,  290,  292,  303, 

312,  427,  438 
Patrone,  349 

Pedal  triangle,  86,  388,  402 
Pell  equation,  56,  367-8,  393 
Pentagonal  number,  279,  292, 

312 
Perfect  number,  3-33,  38 
of  second  kind, 

58 
Period,  133,  182,  202,  207 
Periodic  fraction,   75-6,   82, 

92,  159-179,  193, 202,  339- 

341,  371,  379,  386,  454 
Pei-mutations,  78-80, 131, 136 
Plateau's  theorem,  456,  460, 

463 
Pluperfect,  33 
Plus  quam-perfectus,  3 
Polygon,  cm-viUnear,  85 
,   inscribed    in    cubic 

curve,  85,  150 

regular,  71,  75,  133, 


139,  193,  375 

Polynomial,  divisors  of,  384, 
393^ 

in  X  divisible  by  m  for 

every  x,  87,  89,  336 

Primary  function,  240 

Prime  functions  (see  irredu- 
cible) 

pairs,  353,  425,  438 


486 


Subject  Index 


Primes  6n=fcl,  7  (see  differ- 
ence, highest) 

,  asymptotic  distribu- 
tion of,  439,  449 

,  density  of,  329,  416 

in  arith.  progression. 


425 


-,  infinity  of,  413 


arith.  progressions,  85,  395, 
415-20,  436 

-,  large,     352^,     362, 


365,  386,  388 

-,  law  of  apparition  of. 


396,  398,  406 
repeti- 
tion of,  396-8 

-,  miscellaneous  results 


on,  436-9 

-,  number    of,    352-4, 


429-35,  450 

,  product  of,  126 

represented  by  quad- 


ratic forms,  417 
poly- 
nomials,   333,    414,    418, 
420-1 

-,  simi  of  two,  421-4, 


435 
Primes,  tables  of,  347,  381 

,  test  for,  35,  276,  302, 

305,  360-65,  370,  374,  376- 
8,  380,  396-404,  426-8,  445 
-,  to  base  2,  22,  353-4 


Primitive  diN-isor  of  a'^-b^,  388 

X-root,  202 

non-deficient  number, 

31 

number,  327,  334 

root,  63,  65,  72,  103, 

117,  181-204,  222,  378-9 
-,    imaginarj'. 


235-252 


136 


of  unity,  133, 


ProbabUity,  138,  302,  308, 
328,  330,  333,  335,  407,  438 

Product  of  consecutive  inte- 
gers, 79,  263-4,  269,  331 

differences,  269 

divisors,   58, 

332 

Pronic,  357 

Quadratic  forms,  109,  130, 
158,  207,  210,  219,  276, 
318,  330,  361-5,  369-70, 
400,  415-8,  420-1 

residues,  23,  25,  29, 

65-8,  71,  76,  92,  109,  165, 


185,  189,  190,  196-8,  202, 
210,  213-4,  218,  221,  231, 
240,  245-6,  253-5,  275, 
277,  360,  363,  365,  373, 
382,  393,  395-6,  403 
Quasi-Mersenne  number,  390 
Quotient  (a«'")-l)/m,  102, 
10.5-112 

[{}^l)\  +  l]/p,  109, 

112 

Rank  (see  matrix) 
Recurring  series,  376-7,  393- 

411 
,  algebraic  the- 

or\-  of,  407 
Reducible  law  of  recurrence, 

409-10 
Redundantem,  3,  4 
Remainders  on  di\dding  n  by 

1, . .  . ,  n,  290,  313,  327-31 
Roots  of  unity,  133, 136,  183- 

4,  245,  250,  256,  419 

Secondary  nimaber,  327 

root,  191 

Series  of  composition,  332 

Lame,  411 

Leonardo  Pis- 

ano,  393 
Sieve    of    Eratosthenes,    8, 

347-8,  353-6,  424,  439 
Similar  modulo  k,  260 
Simple  sj'stem   of  numbers, 

455,  458 
Solution    of    alg.    equations, 

407-8 
Sous-double,  33 
Squares,  52,  54,  284-6,  358, 

361,  366,  453-464 
Stencil,  349,  356,  359 
Substitutions,  75,  78-80,  82, 

85,  158,  232,  262 
Sum   of  divisors,  5,   18,  19, 

22,  42,  48,  52-8,  135,  139, 

279-325,  445,  450 
A-th  powers  of 

divisors,  38,  123,  151,  286- 

325,  450 
integers 

<n,  95,  106,  121,  123,  126, 

140,  332 

four  squares,  283 

two  squares. 


247,  286,  340,  360,  381-2, 

390,  402-3 
Superfluos,  3,  4 
SymboUc,  99,  119,  124, 141-2, 

144-5,  148,  248,  250,  278, 

296,  395,  399,  402,  449 


Symbols,  Ein),  281;  Er{n), 
296;  F{a,  N),  M;  Fr,  375; 
H{m),  Hm,  264;  Jkin),  147; 
Mg,  31;  nin),  441;  0,305; 
Pm,  33;  <^(n),  61,  113; 
(pkin),  140;  qu,  105,  109; 
s^in),  48;  s„,  95;  5„,  „,,  96; 
a(n),  53,  279,  446;  akin), 
rin),T{n),279;  n(n),291; 
d(n),429;  f/„,u„,393;  ?(«), 
292;  [x],  115,  276;  /n,  42; 
*before  author,  not  avail- 
able. 

Symmetric  functions  mod. 
p,  70,  95,  106,  143 

number,     112,    455, 

463-4  •. 

Tables,  10,  14,  16,  18,  21-2, 
25,  27,  30-2,  37-8,  45, 
48-9,  54-5,  110-2, 126, 135, 
137,  140,  156-7,  160-79, 
181, 183, 185, 187-203,  213, 
217,  219,  222,  244-5,  248- 
51,  254,  262,  296,  308,  318, 
331,  339-41,  347-58,  361^, 
366-7,  379,  381-4,  386,  388, 
390-1,  399,  417,  422,  432, 
446,  457 

Tahnud,  337 

Totient,  124-5,  148,  153,  246 

point,  154 

Totitives,  98,  124,  130-1,  246 

all  primes,  132,  134 

Triangular  number,  7,  9,  20, 
59,  284,  290,  295,  302,  310, 
373,  425,  427 

Trinomials,  factors  of,  391 

Uberflussig,  3 
Uberschiessende,  3 
tJbervollstandig,  3 
Unvollkommen,  3 
Unvollstiindig,  3 

Verwandte,  38,  47 
Vollkommen,  3 
Vollstandig,  3 

Wilson's  theorem,  59-91,  99, 

103,  275 
,  converse    of, 

63,  427-8 
,  generalization 

of,  65,   68-74,  77-84,  87, 

90-1  (see  Galois) 

Zeta  function,  121,  125-7, 
134, 139, 149, 292-3,  298-9, 
310,318,322,324,328,331, 
439,  448 


486 


Subject  Index 


Primes  6n±l,  7  (see  differ- 
ence, highest) 

,  asymptotic  distribu- 
tion of,  439,  449 

,  density  of,  329,  416 

in  arith.  progression, 


425 


-,  infinity  of,  413 


arith.  progressions,  85,  395, 
41.S-20,  436 

-,  large,     352^,     362, 


365,  386,  388 

-,  law  of  apparition  of. 


396,  398,  406 
repeti- 
tion of,  396-8 

-,  miscellaneous  results 


on,  436-9 

-,  number    of,    352-4, 


429-35,  450 

,  product  of,  126 

represented  by  quad- 
ratic forms,  417 


poly- 
nomials, 333,  414,  418, 
420-1 

-,  sum  of  two,   421-4, 


435 

Primes,  tables  of,  347,  381 
,  test  for,  35,  276,  302, 

305,  360-65,  370,  374,  376- 

8,  380,  396-404,  426-8,  445 

,  to  base  2,  22,  353-4 

Primitive  diN-isor  of  a^-b",  388 

X-root,  202 

non-deficient  number, 

31 

number,  327,  334 

root,  63,  65,  72,  103, 

117,  181-204,  222,  378-9 
,    imaginary-. 


235-252 


136 


of  unity,  133, 


Probability,  138,  302,  308, 
328,  330,  333,  335,  407,  438 

Product  of  consecutive  inte- 
gers, 79,  263-4,  269,  331 

differences,  269 

divisors,   58, 

332 

Pronic,  357 

Quadratic  forms,  109,  130, 
158,  207,  210,  219,  276, 
318,  330,  361-5,  369-70, 
400,  415-8,  420-1 

residues,  23,  25,  29, 

6&-8,  71,  76,  92,  109,  165, 


185,  189,  190,  196-8,  202, 

210,  213-4,  218,  221,  231, 

240,    245-6,    253-5,    275, 

277,    360,    363,   365,   373, 

382,  393,  395-6,  403 
Quasi-Mersenne  number,  390 
Quotient  (a*"')-l)/m,  102, 

10.5-112 
{{p-l)\  +  l}/p,  109, 

112 

Rank  (see  matrix) 
Recurring  series,  376-7,  393- 

411 
,  algebraic  the- 

or>'  of,  407 
Reducible  law  of  recurrence, 

409-10 
Redundantem,  3,  4 
Remainders  on  dividing  n  by 

1, . .  . ,  n,  290,  313,  327-31 
Roots  of  unity,  133,  136,  183- 

4,  245,  250,  256,  419 

Secondary  nrnnber,  327 

root,  191 

Series  of  composition,  332 
Lame.  41 1 

ano,  393  JohS^.h^.^^.     Date 

Sieve    of    Eratostb    ^    ^  ^ ^ime 

347-8,353-6,424,     ^^^^"""^^ 

Similar  modulo  k,  26l    Stab  by No.  Sect Sew  by. 

Simple  system  of  r 

455,  458 

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Symbols,  E(n),  281;  Er{n), 
296;  Fia,  N),84;  Fr,  375; 
Him),  Hm,  264;  Jkin),  147; 
Mg,  31;  M(n),  441;  0,  305; 
Pm,  33;  <^(n),  61,  113; 
<j)k{n),  140;  qu,  105,  109; 
sHn),  48;  s„,  95;  5„.  m,  96; 
a(n),  53,  279,  446;  akin), 
Tin),  Tin),  279;  Tkin),291; 
ein),429;  C/„,u„,393;  f(s), 
292;  [x],  115,  276;  fn,  42; 
*before  author,  not  avail- 
able. 

Symmetric  functions  mod. 
p,  70,  95,  106,  143 

number,     112,    455, 

463-4  • 

Tables,  10,  14,  16,  18,  21-2, 
25,  27,  30-2,  37-8,  45, 
48-9,  54-5,  110-2, 126, 135, 
137,  140,  156-7,  160-79, 
181,  183,  185,  187-203,  213, 
217,  219,  222,  244-5,  248- 
51,  254,  262,  296,  308,  318, 
331,  339-41,  347-58,  361^, 
366-7,  379,  381-4,  386,  388, 
390-1.  399.  417.  422.  A^9. 


Score Press Strip  Sect. 


Solution    of   alg.    ec 

407-8 
Sous-double,  33  .„ -  .  •         j  ■ 

SnnnrP9     ^9     ^4    984     anv  defects  appearing  in  either  will  be  made  good 
isquares,    O^,   t>%   ^»4        ;jj^     ^   additional   charge.     "Bound  to  wear." 

361,  366,  453-464 
Stencil,  349,  356,  35{ 
Substitutions,  75,  78-80,  82, 

85,  158,  232,  262 
Sum   of   divisors,  5,   18,  19, 

22,  42,  48,  52-8,  135,  139, 

279-325,  445,  450 

A:th  powers  of 


divisors,  38,  123,  151,  286- 

325,  450 
integers 

<n,  95,  106,  121, 123,  126, 

140,  332 

four  squares,  283 

two  squares. 


247,  286,  340,  360,  381-2, 

390,  402-3 
Superfluos,  3,  4 
Symbolic,  99,  119,  124, 141-2, 

144-5,  148,  248,  250,  278, 

296,  395,  399,  402,  449 


Ubervollstandig,  3 
Unvollkommen,  3 
Unvollstandig,  3 

Venvandte,  38,  47 
^'ollkommen,  3 
Vollstiindig,  3 

Wilson's  theorem,  59-91,  99, 

103,  275 
,  converse    of, 

63,  427-8 
,  generalization 

of,  65,  68-74,   77-84,  87, 

90-1  (see  Galois) 

Zeta  function,  121,  125-7, 
134, 139, 149, 292-3,  298-9, 
310,318,322,324,328,331, 
439,  448 


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