MATHEMATICAL MONOGRAPHS.
EDITED BY
MANSFIELD MERRIMAN and ROBERT S. WOODWARD.
No. 13.
THE THEORY
OF
NUMBERS
BY
ROBERT D. CARMICHAEL,
Associate Professor of Mathematics in Indiana University
FIRST EDITION
FIRST THOUSAND
NEW YORK
JOHN WILEY & SONS, Inc.
London: CHAPMAN & HALL: Limited
1914
Copyright, 19 14.
BY
ROBERT D. CARMICHAEL.
$0*0*13
THE SCIENTIFIC PRESS
ROBERT DRUMMOND AND COMPANY
BROOKLYN, N. Y.
EDITORS' PREFACE.
The volume called Higher Mathematics, the third edition
of which was published in 1900, contained eleven chapters by-
eleven authors, each chapter being independent of the others,
but all supposing the reader to have at least a mathematical
training equivalent to that given in classical and engineering
colleges. The publication of that volume was discontinued in
1906, and the chapters have since been issued in separate
Monographs, they being generally enlarged by additional
articles or appendices which either amplify the former pres-
entation or record recent advances. This plan of publication
was arranged in order to meet the demand of teachers and
the convenience of classes, and it was also thought that it
would prove advantageous to readers in special lines of mathe-
matical literature.
It is the intention of the publishers and editors to add other
monographs to the series from time to time, if the demand
seems to warrant it. Among the topics which are under con-
sideration are those of elliptic functions, the theory of quantics,
the group theory, the calculus of variations, and non-Euclidean
geometry; possibly also monographs on branches of astronomy,
mechanics, and mathematical physics may be included. It is
the hope of the editors that this Series of Monographs may
tend to promote mathematical study and research over a wider
field than that which the former volume has occupied.
302093
PREFACE
The purpose of this little book is to give the reader a con-
venient introduction to the theory of numbers, one of the most
extensive and most elegant disciplines in the whole body of
mathematics. The arrangement of the material is as follows :
The first five chapters are devoted to the development of those
elements which are essential to any study of the subject. The
sixth and last chapter is intended to give the reader some
indication of the direction of further study with a brief account
of the nature of the material in each of the topics suggested.
The treatment throughout is made as brief as is possible con-
sistent with clearness and is confined entirely to fundamental
matters. This is done because it is believed that in this way
the book may best be made to serve its purpose as an intro-
duction to the theory of numbers.
Numerous problems are supplied throughout the text.
These have been selected with great care so as to serve as excel-
lent exercises for the student's introductory training in the
methods of number theory and to afford at the same time a
further collection of useful results. The exercises marked with
a star are more difficult than the others; they will doubtless
appeal to the best students.
Finally, I should add that this book is made up from the
material used by me in lectures in Indiana University during
the past two years; and the selection of matter, especially of
exercises, has been based on the experience gained in this way.
R. D. Carmichael.
4
CONTENTS
CHAPTER I. ELEMENTARY PROPERTIES OF INTEGERS
PAGE
j i. Fundamental Notions and Laws 7
j 2. Definition of Divisibility. The Unit 8
\ 3. Prime Numbers. The Sieve of Eratosthenes 10
& 4. The Number of Primes is Infinite. 12
& 5. The Fundamental Theorem of Euclid 13
I 6. Divisibility by a Prime Number 13
§ 7. The Unique Factorization Theorem 14
I 8. The Divisors of an Integer 16
§ 9. The Greatest Common Factor of Two or More Integers 18
10. The Least Common Multiple of Two or More Integers 20
11. Scales of Notation 22
12. Highest Power of a Prime p Contained in n\ 24
13. Remarks Concerning Prime Numbers. . '. 28
CHAPTER II. ON THE INDICATOR OF AN INTEGER
14. Definition. Indicator of a Prime Power 30
15. The Indicator of a Product 30
16. The Indicator of any Positive Integer 32
17. Sum of the Indicators of the Divisors of a Number 35
CHAPTER III. ELEMENTARY PROPERTIES OF CONGRUENCES
18. Congruences Modulo m 37
19. Solutions of Congruences by Triai 39
20. Properties of Congruences Relative to Division 40
21. Congruences with a Prime Modulus 41
22. Linear Congruences 43
CHAPTER IV. THE THEOREMS OF FERMAT AND WILSON
23. Fermat's General Theorem 47
24. Euler's Proof of the Simple Fermat Theorem 48
: 25. Wilson's Theorem 49
5
D CONTENTS
PAGE
§ 26. The Converse of Wilson's Theorem 50
§ 27. Impossibility of 1 -2 -3 • ... •« — i-f i=m*, «>s 51
§ 28. Extension of Fermat's Theorem : 52
§ 29. On the Converse of Fermat's Simple Theorem 55
§ 30. Application of Previous Results to Linear Congruences 56
§31. Application of the Preceding Results to the Theory of Quad-
ratic Residues 57
CHAPTER V. PRIMITIVE ROOTS MODULO m
§ 32. Exponent of an Integer Modulo m 61
§ i2>- Another Proof of Fermat's General Theorem 63
§ 34. Definition of Primitive Roots 65
§ 35. Primitive Roots Modulo p 66
§ 36. Primitive Roots Modulo pa, p an Odd Prime '. 68
§ 37. Primitive Roots Modulo 2pa, p an Odd Prime 70
§ 38. Recapitulation 71
§ 39. Primitive X-roots 71
CHAPTER VI. OTHER TOPICS
§ 40. Introduction 76
§ 41. Theory of Quadratic Residues 77
§ 42. Galois Imaginaries 80
§ 43. Arithmetic Forms 81
§ 44. Analytical Theory of Numbers 83
§ 45. Diophantine Equations 84
§ 46. Pythagorean Triangles 85
§ 47. The Equation xn+yn = in 9i
THE THEORY OF NUMBERS
CHAPTER I
ELEMENTARY PROPERTIES OF INTEGERS
§ i. Fundamental Notions and Laws
In the present chapter we are concerned primarily with
certain elementary properties of the positive integers i, 2, 3,
4, '...•. . It will sometimes be convenient, when no confusion
can arise, to employ the word integer or the word number in
the sense of positive integer.
We shall suppose that the integers are already defined,
either by the process of counting or otherwise. We assume
further that the meaning of the terms greater, less, equal, sum,
difference, product is known.
From the ideas and definitions thus assumed to be known
follow immediately the theorems :
I. The sum of any two integers is an integer.
II. The difference of any two integers is an integer.
III. The product of any two integers is an integer.
Other fundamental theorems, which we take without proof,
are embodied in the following formulas:
IV. a+b = b+a.
V. aXb = bXa.
VI. (a+b)+c = a+(b+c).
VII. (aXb)Xc = aX(bXc).
VIII. aX(b+c) =aXb+aXc.
Here a, b} c denote any positive integers.
8 ♦/• * 2/2 •••{ ; •: XHIJO^y Of NUMBERS
These formulas are equivalent in order to the following
five theorems: addition is commutative; multiplication is
commutative; addition is associative; multiplication is asso-
ciative; multiplication is distributive with respect to addition.
EXERCISES
Prove the following relations:
(1+2+ . . . +n)K
of each of the following series: \\
i3+33 + 53+ • • • +(2w-i)3.
5. Discover and establish the law suggested by the equations i2 = 0+1, 22 = i-f-3,
32 = 3+6, 42 = 6+io, . . .; by the equations i = i3, 3+5 — 23, 7+9+11=3?,
i3 + i5 + i74-i9 = 43,
§ 2. Definition of Divisibility. The Unit
Definitions. An integer a is said to be divisible by an
integer b if there exists an integer c such that a — be. It is clear
from this definition that a is also divisible by c. The integers
b and c are said to be divisors or factors of a; and a is said to
be a multiple of b or of c. The process of finding two integers
b and c such that be is equal to a given integer a is called the
process of resolving a into factors or of factoring a; and a is
said to be resolved into factors or to be factored.
We have the following fundamental theorems:
I. If b is a divisor of a and c is a divisor of b, then c is a
divisor of a.
Since & is a divisor of a there exists an integer /3 such that
a = &j8. Since c is a divisor of b there exists an integer 7 such
that b = cy. Substituting this value of b in the equation a = 5/3
we have a = cyfi. But from theorem III of § i it follows that
7jS is an integer; hence, c is a divisor of a, as was to be proved.
ELEMENTARY PROPERTIES OF INTEGERS 9
II. If c is a divisor of both a and b, then c is a divisor of the
sum of a and b.
From the hypothesis of the theorem it follows that integers
a and /3 exist such that
a = ca, b = c(3.
Adding, we have
a + b = ca+cP = c(a-\-p)=c8,
where 5 is an integer. Hence, c is a divisor oi0j-b.
III. If c is a divisor of both a and b, then c is a divisor of the
difference of a and b.
The proof is analogous to that of the preceding theorem.
DEFINITIONS. If a and b are both divisiblsJby c, then c
is said to be a common divisor or a common fa»r«: a and b.
Every two integers have the common factor ifHrTne greatest
integer which divides both a ariU b is called the greatest common
divisor of a and b. More generally, we define in a similar way
a common divisor and the greatest common divisor of n integers
0i, a2, . . . , On.
Definitions. If an integer a is a multiple -of each of two
or more integers it is called a common multiple of these integers.
The product of any set of integers is a common multiple of the
set. The least integer which is a multiple of each of two or
more integers is called their least common multiple.
It is evident that the integer i is a divisor of every integer
and that it is the only integer which has this property. It is
called the unit.
Definition. Two or more integers which have no common
factor except i are said to be prime to each other or to be rela-
tively prime.
Definition. If a set of integers is such that no two of
them have a common divisor besides i they are said to be prime
each to each.
EXERCISES
i. Prove that n3 — n is divisible by 6 for every positive integer n.
2. If the product of four consecutive integers is increased by i the result
is a square number.
,S. Show that 24n+2+i has a factor different from itself and i when n is a
positive integer.
10 THEORY OF NUMBERS
§ 3. Prime Numbers. The Sieve of Eratosthenes
Definition. If an integer p is different from 1 and has
no divisor except itself and 1 it is said to be a prime number
or to be a prime.
Definition. An integer which has at least one divisor
other than itself and 1 is said to be a composite number or to
be composite.
All integers are thus divided into three classes:
1) The unity
2) Prime numbers; ^j
3) Composite numbers.
We have seen that the first class contains only a single
number. The third class evidently contains an infinitude of
numbers; for, it contains all the numbers 2?, 23, 24, . . . .
In the next section we shall show that the second class also
contains an infinitude of numbers. We shall now show that
every number of the third class contains one of the second
class as a factor, by proving the following theorem:
I. Every integer greater than 1 has a prime factor.
Let m be any integer which is greater than 1. We have
to show that it has a prime factor. If m is prime there is the
prime factor m itself. If m is not prime we have
w = mim2,
where m\ and ni2 are positive integers both of which are less than
m. If either ni\ or ni2 is prime we have thus obtained a prime
factor of m. If neither of these numbers is prime, then write
rn\=mrimf2, m\>i, w/2>i.
Both ih\ and mr2 are factors of m and each of them is less than
mi. Either we have now found in m\ or m'2 a prime factor
of m or the process can be continued by separating one of these
numbers into factors. Since for any given m there is evideix. I
only a finite number of such steps possible, it is clear that w
ELEMENTARY PROPERTIES OF INTEGERS 11
must finally arrive at a prime factor of m. From this conclu-
sion the theorem follows immediately.
Eratosthenes has given a useful means of finding the prime
numbers which are less than any given integer m. It may be
described as follows:
Every prime except 2 is odd. Hence if we write down every
odd number from 3 up to m we shall have in the list every prime
less than m except 2. Now 3^s_a_prime. Leave it in the list;
but beginning to count from 3 strike out every third number
in the list. Thus every number divisible by 3, except 3 itself,
is cancelled. Then begin from 5 and cancel every fifth num-
ber. Then begin from the next uncancelled number, namely
7, and strike out every seventh number. Then begin from
the next uncancelled number, namely 11, and strike out every
eleventh number. Proceed in this way up to m. The uncan-
celled numbers remaining will be the odd primes not greater
than m.
It is obvious that this process of cancellation need not be
carried altogether so far as indicated; for if p is a prime greater
than Vw, the cancellation of any pth number from p will be
merely a repetition of cancellations effected by means of another
factor smaller than p, as one may see by use of the following
theorem.
II. An integer m is prime if it has no prime factor equal to
or less than I, wJiere I is the greatest integer whose square is
equal to or less than m.
Since m has no prime factor less than /, it follows from
theorem I that it has no factor but unity less than /. Hence,
if m is not prime it must be the product of two numbers each
greater than /; and hence it must be equal to or greater than
(7-f i)2. This contradicts the hypothesis on /; and hence
we conclude that m is prime.
EXERCISE
By means of the method of Eratosthenes determine the primes less than
200.
12 THEORY OF NUMBERS
§4. The Number of Primes is Infinite
I. The number of primes is infinite.
We shall prove this theorem by supposing that the number
of primes is not infinite and showing that this leads to a con-
tradiction. If the number of primes is not infinite there is a
greatest prime number, which we shall denote by p. Then
form the number
iV = i-2-3-. . .-/>+!.
Now by theorem I of § 3 N has a prime divisor q. But every
non-unit divisor of iV, is obviously greater than p. Hence q
is greater than />, in contradiction to the conclusion that p is
the greatest prime. Thus the proof of the theorem is complete-
In a similar way we may prove the following theorem:
II. Among the integers of the arithmetic progression 5, II,
17, 23, ... , there is an infinite number of primes.
If the number of primes in this sequence is not infinite
there is a greatest prime number in the sequence; supposing
that this greatest prime number exists we' shall denote it by p.
Then the number N,
iV^i-2-3-. . -#-i,
is not divisible by any number less than or equal to p. This
number N, which is of the form 6» — 1, has a prime factor.
If this factor is of the form 6& — 1 we have already reached a
contradiction, and our theorem is proved. If the prime is of
the form 6^i-+-i the complementary factor is of the form 6^2 — 1»
Every prime factor of 6&2--1 is greater than p. Hence we
may treat 6^2 — 1 as we did 6n — 1 , and with a like result. Hence
we must ultimately reach a prime factor of the form 6^3 — 1;
for, otherwise, we should have 6w — 1 expressed as a product
of prime factors all of the form 6/+1 — a result which is clearly
impossible. Hence we must in any case reach a contradiction
of the hypothesis. Thus the theorem is proved.
The preceding results are special cases of the following more
general theorem :
ELEMENTARY PROPERTIES OF INTEGERS 13
III. Among the integers of the arithmetic progression a, a-\-d,
a-\-2d, a-\-$d3 . . . , there is an infinite number of primes, pro-
vided that a and b are relatively prime.
For the special case given in theorem II we have an elemen-
tary proof; but for the general theorem the proof is difficult.
We shall not give it here.
EXERCISES
i. Prove that there is an infinite number of primes of the form 4» — i. ,
2. Show that an odd prime number can be represented as the difference of
two squares in one and in only one way.
3. The expression mp — np, in which m and n are integers and p is a prime,
is either prime to p or is divisible by p2.
4. Prove that any prime number except 2 and 3 is of one of the forms 6w+i,
6n— 1.
§ 5. The Fundamental Theorem of Euclid
// a and b are any two positive integers there exist integers
q and r, q>o, o=r<b, such that
a = qb-\-r..
If a is a multiple of b the theorem is at once verified, r being
in this case o. If a is not a multiple of b it must lie between
two consecutive multiples of b; that is, there exists a q such
that
qb<a<(q+i)b.
Hence there is an integer r, o<r<b, such that a = qb-\-r. In
case b is greater than a it is evident that q = o and r = a. Thus
the proof of the theorem is complete.
§ 6. Divisibility by a Prime Number
I. If p is a prime number and m is any integer, then m either
is divisible by p or is prime to p.
This theorem follows at once from the fact that the only
divisors of p are i and p.
II. The product of two integers each less than a given prime
number p is not divisible by p.
14 THEORY OF NUMBERS
Let a be a number which is less than p and suppose that b
is a number less than p such that ab is divisible by p, and let
b be the least number for which ab is so divisible. Evidently
there exists an integer m such that
mb<p<(m + i)b.
Then p—mb<b. Since ab is divisible by p it is clear that mab
is divisible by p; so is ap also; and hence their differei^B
ap — mab, =a(p — mb), is divisible by p. That is, the prodiH
of a by an integer less than b is divisible by p, contrary to tH
assumption that & is the least integer such that ab is divisibH
by p. The assumption that the theorem is not true has thiH
led to a contradiction; and thus the theorem is proved.
III. If neither of two integers is divisible by a given primm
number p their product is not divisible by p.
Let a and b be two integers neither of which is divisible
by the prime p. According to the fundamental theorem off
Euclid there exist integers m, n, a, /3 such that
a = mp-\-a, o<a<p}
b = np+(3, o<(3<p.
Then ab = (mp+a)(np+p) = (mnp+<*+p)p+aP.
If now we suppose ab to be divisible by p we have a/3 divisible
by p. This contradicts II, since a and /3 are less than p. Hence
ab is not divisible by p.
By an application of this theorem to the continued product,
of several factors, the following result is readily obtained:
IV. If no one of several integers is divisible by a given prime !
p their product is not divisible by p.
§ 7. The Unique Factorization Theorem
I. Every integer greater than unity can be represented in one
and in only one way as a product of prime num1 ers.
In the first place we shall show that it "s a T ways possible
to resolve a given integer m greater than v lty into prime
ELEMENTARY PROPERTIES OF INTEGERS 15
factors by a finite number of operations. In the proof of the-
orem I, § 3, we showed how to find a prime factor pi of m by
a finite number of operations. Let us write
m = p\m\.
If mi is not unity we may now find a prime factor p2 of nil.
Then we may write
m = pinii = pip2ffi2-
If ni2 is not unity we may apply to it the same process as that
applied to nil and thus obtain a third prime factor of m. Since
mi>m2>ni3> ... it is clear that after a finite number of
operations we shall arrive at a decomposition of m into prime
factors. Thus we shall have
m = pip2 • . . pr
where pi, p2, . . . , pr are prime numbers. We have thus
proved the first part of our theorem, which says that the decom-
position of an integer (greater than unity) into prime factors
is always possible.
Let us now suppose that we have also a decomposition of
m into prime factors as follows:
m = qiq2 . . . qs.
Then we have
pip2 • • . pr = qiq2 . . . q*
Now pi divides the first member of this equation. Hence it
also divides the second member of the equation. But pi is
prime; and therefore by theorem IV of the preceding section
we see that pi divides some one of the factors q\ we suppose
that pi is a factor of qi. It must then be equal to qi. Hence
we have
p2p3 • • - pr = q2q3 . . . qa.
By the same argument we prove that p2 is equal to some q,
say qo. Then we have
p3p± • • • pr = qzq± . . . q„
16 THEORY OF NUMBERS
Evidently the process may be continued until one side of the
equation is reduced to i. The other side must also be reduced
to i at the same time. Hence it follows that the two decom-
positions of m are in fact identical.
This completes the proof of the theorem.
The result which we have thus demonstrated is easily the
most important theorem in the theory of integers. It can
also be stated in a different form more convenient for some
purposes:
II. Every non-unit positive integer m can be represented in
one and in only one way in the form
m = p1aip2a2 . . . pt
an
where pi, p2, . . . , pn are different primes and «i, qe, . • • ,
un are positive integers.
This comes immediately from the preceding representation
of m in the form m = pip2 . . . pr by combining into a power
of pi all the primes which are equal to pi.
COROLLARY i. If a and b are relatively prime integers
and c is divisible by both a and b, then c is divisible by ab.
COROLLARY 2. If a and b are each prime to c then ab is
prime to c.
Corollary 3. If ' a is prime to c and ab is divisible by c,
then b is divisible by c.
§ 8. The Divisors of an Integer
The following theorem is an immediate corollary of the
results in the preceding section :
I. All the divisors of m,
are of the form
pi(ilp2fi2 . . . pnPn, o<ft^a*;
and every such number is a divisor of m.
/
ELEMENTARY PROPERTIES OF INTEGERS 17
From this it is clear that every divisor of m is included once
and only once among the terms of the product
(l+^l+/>l2+ • • • +^ia0(l+/>2+^22+ • • . +^2a>) • • •
(l+pn + pn2+ ... +pnan),
when this product is expanded by multiplication. It is obvious
that the number of terms in the expansion is (ai + i)(«2 + i) • • •
(a»+i). Hence we have the theorem:
II . The number of divisors of m is (a i + 1 ) («2 + 1 ) • • . («» + 1 ) •
Again we have
n(i+pi+pi2+ . . . +y)«n^.1>"1.
» » pi — i
Hence,
III. The sum of the divisors of m is
piai+l-I p2a*+\-I
«n + l _
pi -I p2 — l pn—I
In a similar manner we may prove the following theorem:
IV. The sum of the hth powers of the divisors of m is
Pih-i ' ' ' : ' r pnh-i '
EXERCISES
i. Find numbers x such that the sum of the divisors of x is a perfect square.
2. Show that the sum of the divisors of each of the following integers is twice
the integer itself: 6, 28, 496, 8128, 33550336. Find other integers x such that
the sum of the divisors of x is a multiple of x.
3. Prove that the sum of two odd squares cannot be a square."
^4. Prove that the cube of any integer is the difference of the squares of two
integers.
""> kg. In order that a number shall be the sum of consecutive integers, it is neces-
sary and sufficient that it shall not be a power of 2.
6. Show that there exist no integers x and y (zero excluded) such that y2 = 2x2.
Hence, show that there does not exist a rational fraction whose square is 2.
7. The number m = pia^pta- . . . pnan, where the p's are different primes and
the a's are positive integers, may be separated into relatively prime factors in
2s-1 different ways.
8. The product of the divisors of m is \/mc where v is the number of divisors
of m.
18 THEORY OF NUMBERS
§ 9. The Greatest Common Factor of Two or More
Integers
Let m and n be two positive integers such that m is greater
than n. Then, according to the fundamental theorem of
Euclid, we can form the set of equations
m = qn-{-ni, o<n\<n,
n = qini-\-n2, o<n2<ni,
ni=q2n2+m, o<n-s<n2,
nk-2 = qk-ink-i+nk, o<nk<nk-i,
nk-i=qknk. + 0
If m is a multiple of n we write n = n0, k = o, in the above equa-
tions.
DEFINITION. The process of reckoning involved in
determining the above set of equations is called the Euclidian
Algorithm.
I. The number nk to which the Euclidian algorithm leads is
the greatest common divisor of m and n.
In order to prove this theorem we have to show two things :
1) That nk is a divisor of both m and n;
2) That the greatest common divisor d of m and n is a
divisor of nk.
To prove the first statement we examine the above set of
equations, working from the last to the first. From the last
equation we see that nk is a divisor of nk-\. Using this result
we see that the second member of next to the last equation is
divisible by nk. Hence its first member nk-2 must be divisible
by nk. Proceeding in this way step by step we show that
U2 and ni, and finally that n and m, are divisible by nk.
For the second part of the proof we employ the same set of
equations and work from the first one to the last one. Let
d be any common divisor of m and n. From the first equation
we see that d is a divisor of ni. Then from the second equation
it follows that J is a divisor of n2. Proceeding in this way we
ELEMENTARY PROPERTIES OF INTEGERS 19
show finally that d is a divisor of fit. Hence any common
divisor, and in particular the greatest common divisor, of m
and n is a factor of nk.
This completes the proof of the theorem.
Corollary. Every common divisor of m and n is a factor
of their greatest common divisor.
II. Any number n% in the above set of equations is the differ-
ence of multiples of m and n.
From the first equation we have
n\=m — qn
so that the theorem is true for i = i. We shall suppose that
the theorem is true for every subscript up to i—i and prove
it true for the subscript i. Thus by hypothesis we have *
nt-2 = ±(a*-2W — j8<-2«),
ni-i = zF(ai-im — (3i-in).
Substituting in the equation
we have a result of the form
Hi = zb (at m — fan) .
From this we conclude at once to the truth of the theorem.
Since nt is the greatest common divisor of m and n, we have
as a corollary the following important theorem:
III. // d is the greatest common divisor, of the positive integers
m and n, then there exist positive integers ajindJLsuch that
am—ph=±d.
If we consider the particular case in which m and n are rela-
tively prime, so that d^h we see that there exist positive
integers a and 0 such that am — f}n=zLi. Obviously, if m and
n have a common divisor d, greater than i, there do not exist
* If i = 2 we must replace w^_2 by n.
20
THEORY OF NUMBERS
integers a and /3 satisfying this relation; for, if so, d would be
a divisor of the first member of the equation and not of the
second. Thus we have the following theorem:
IV. A necessary and sufficient condition that m and n are
relatively prime is that there exist integers a and /3 such that
am—(3n = zki.
The theory of the greatest common divisor of three or more
numbers is based directly on that of the greatest common
divisor of two numbers; consequently it does not require to
be developed in detail.
EXERCISES
i. If d is the greatest common divisor of m and n, then m/d and n/d are rela-
tively prime.
2. If <i is the greatest common divisor of m and n and k is prime to n, then
d is the greatest common divisor of km and n.
3. The number of multiplies of b in the sequence a, 2a, 3a, . . . , ba is equal
to the greatest common divisor of a and b.
~p?- 4. If the sum or the difference of two irreducible fractions is an integer, the
denominators of the fractions are equal.
5. The algebraic sum of any number of irreducible fractions, whose denomi-
nators are prime each to each, cannot be an integer.
6*. The number of divisions to be effected in finding the greatest common
divisor of two numbers by the Euclidian algorithm does not exceed five times
the number of digits in the smaller number (when this number is written in the
usual scale of 10).
§ 10. The Least Common Multiple of Two or More
Integers
I. The common multiples of two or more numbers are the
multiples of their least common multiple.
This may be readily proved by means of the unique factori-
zation theorem. The method is obvious. We shall, however,
give a proof independent of this theorem.
Consider first the case of two numbers; denote them by
m and n and their greatest common divisor by d. Then we
have
m = dfx, n = dv,
where fi and v are relatively prime integers. The common
multiples sought are multiples of m and are all comprised in the
ELEMENTARY PROPERTIES OF INTEGERS . 21
numbers am, = adfx, where a is any integer whatever. In order
that these numbers shall be multiples of n it is necessary and
sufficient that ad/x shall be a multiple of dv; that is, that a/x
shall be a multiple of v; that is, that a shall be a multiple of
v, since /* and v are relatively prime. Writing a = bv we have
as the multiples in question the set bdjiv where 5 is an arbitrary
integer. This proves the theorem for the case of two numbers;
for dfiv is evidently the least common multiple of m and n.
We shall now extend the proposition to any number of
integers m, n, p, q, . . . . The multiples in question must
be common multiples of m and n and hence of their least common
multiple p. Then the multiples must be multiples of ju and p
and hence of their least common multiple pi. But m is evi-
dently the least common multiple of m, n, p. Continuing in a
similar manner we may show that every multiple in question
is a multiple of /i, the least common multiple of m, n, p, q, . . . .
And evidently every such number is a multiple of each of the
numbers m, n, p, q, . . . .
Thus the proof of the theorem is complete.
When the two integers m and n are relatively prime their
greatest common divisor is i and their least common multiple
is their product. Again if p is prime to both m and n it is prime
to their product mn\ and hence the least common multiple
of m} n, p is in this case mnp. Continuing in a similar manner
we have the theorem :
II. The least common multiple of several integers, prime
each to each, is equal to their product.
EXERCISES
i. In order that a common multiple of n numbers shall be the least, it is neces-
sary and sufficient that the quotients obtained by dividing it successively by the
numbers shall be relatively prime.
2. The product of n numbers is equal to the product of their least common
multiple by the greatest common divisor of their products n — i at a time.
3. The least common multiple of n numbers is equal to any common mul-
tiple M divided by the greatest common divisor of the quotients obtained on
dividing this common multiple by each of the numbers.
4. The product of n numbers is equal to the product of their greatest common
divisor by the least common multiple of the products of the numbers taken n— 1
at a time.
22 THEORY OF NUMBERS
§ ii. Scales of Notation
I. If m and n are positive integers and n>i, then m can be
represented in terms of n in one and in only one way in the form
ni = a0nh+a1n?>~1-\- . . . +aA_1w+aftj
where
ao^o, o^ai<n, i = o, i, 2, . . . , h.
That such a representation of m exists is readily proved by
means of the fundamental theorem of Euclid. For we have
. ni = non+ah, o^ah<n,
n0 = nin+ah-u o^ah-i<n,
ni = n2n-\-ah-2, o^ah-2<n}
nhs =nh-2n+a2, o^a2<n,
nh-2 =nh-in+ai, o^ci<n,
nh-i=ao, o<ao<n.
If the value of nh-i given in the last of these equations is sub-
stituted in the second last we have
nh-2 = aon+ai.
This with the preceding gives
nh-3 = aon2+ain+a2.
Substituting from this in the preceding and continuing the
process we have finally
m = aon7l+ainn~l-\- . . . +ah_1n+ah,
a representation of m in the form specified in the theorem.
To prove that this representation is unique, we shall suppose
that m has the representation
ni = b0nk+b1nk~1 + . . . -^b^-^n-^h,
where
frop^o, o<bi<n, i = o, 1, 2, . . . , k,
ELEMENTARY PROPERTIES OF INTEGERS 23
and show that the two representations are identical. We
■have
a0nh + . . . +ah-in+ah = b0nk-+ . . . +h-in+bk.
Then
aonh+ . . . +ah-in-(b0nk+ . . . +bk-iti)=bk — ah.
The first member is divisible by n. Hence the second is also.
But the second member is less than n in absolute value; and
hence, in order to be divisible by n, it must be zero. That is,
bt = ah. Dividing the equation through by n and transposing
we have
a0nh~1+ . . . -\-ah-2n — (b0nk-l-{- . . . +bk-2n)=bk-i—ah-i
It may now be seen that bk~i=afl-i. It is evident that this
process may be continued until either the a's are all eliminated
from the equation or the b's are all eliminated. But it is
obvious that when one of these sets is eliminated the other is
also. Hence, h = k. Also, every a equals the b which multi-
plies the same power of n as the corresponding a. That is,
the two representations of m are identical. Hence the repre-
sentation in the theorem is unique.
From this theorem it follows as a special case that any posi-
tive integer can be represented in one and in only one way in
the scale of 10; that is, in the familiar Hindoo notation. It
can also be represented in one aild in only one way in any other
scale. Thus
120759 = i. 76+o.75 + i.74 + 2.73+o.72+3.71 + 2.
Or, using a subscript to denote the scale of notation, this may
be written
(120759)10 = (1012032)7.
For the case in which n (of theorem I) is equal to 2, the
only possible values for the a's are o and 1. Hence we have
at once the following theorem:
II. Any positive integer can be represented in one and in only
one way as a sum of different powers of 2.
24
THEORY OF NUMBERS
EXERCISES
i. Any positive integer can be represented as an aggregate of different powers
of 3, the terms in the aggregate being combined by the signs + and — appropri-
ately chosen.
2. Let m and n be two positive integers of which n is the smaller and suppose
that 2 Sn<2 + . By means of the representation of m and n in the scale of
2 prove that the number of divisions to be effected in finding the greatest common
divisor of m and » by the Euclidian algorithm does not exceed 2k.
§ 12. Highest Power of a Prime p Contained in n\.
Let n be any positive integer and p any prime number not
greater than n. We inquire as to what is the highest power
p^oi the prime p contained in n\.
In solving this problem we shall find it convenient to employ
the notation
[7]
to denote the greatest integer a such that as^r. With this
notation it is evident that we have
and more generally
M
L p1
(1)
If now we use H { x } to denote the index of the highest power
of p contained in an integer x, it is clear that we have
(0
since only multiples of p contain the factor p. Hence
sl.„-[l]+s{..,...g]}.
ELEMENTARY PROPERTIES OF INTEGERS 25
Applying the .same process to the H-i unction in the second
member and remembering relation (i) it is easy to see that
we have
Continuing the process we have finally
Bi«H3+K+r?]+ • • • ■
the series on the right containing evidently only a finite num-
ber of terms different from zero. Thus we have the theorem:
I. The index of the highest power of a prime p contained
in nl is
RMfM?]
+ . . .
The theorem just obtained may be written in a different
form, more convenient for certain of its applications. Let
n be expressed in the scale of p in the form
n = a0ph+a1ph~l + . . . -\-ah-!p-\-ah,
where
aoT^o, o^at<p, i = o, i, 2, . . . , h.
Then evidently
[^]=ao/>*-1.+ai/>*-2+ • • • +a*-2p+ah-1,
26 THEORY OF NUMBERS
Adding these equations member by member and combining
the second members in columns as written, we have
m?m
+ .
i-o p—i
^aoph+aip?t~1+ . . . +ah-(ao+ai+ . . . +ah)
p-i
= n — (ao+ai + . . . +flft)
p-i
Comparing this result with theorem I we have the following
theorem :
II. If n is represented in the scale of p in the form
n = a0ph+aiph-1+ . . . +ah,
where p is prime and
a05*o, o^al<p, i = o, i, 2, . . ., h,
then the index of the highest power of p contained in n ! is
n — (ao+ai-\- . . . +ah)
p-i
Note the simple form of the theorem for the case p = 2;
in this case the denominator p — i is unity.
We shall make a single application of these theorems by
proving the following theorem:
III. If n, a, (3, ... j \ are any positive integers such that
n=a+fi+ . . . +X, then
^TTTTT! {A)
is an integer.
Let p be any prime factor of the denominator of the frac-
tion (A). To prove the theorem it is sufficient to show that
the index of the highest power of p contained in the numerator
is at least as great as the index of the highest power of p con-
ELEMENTARY PROPERTIES OF INTEGERS
27
tained in the denominator. This index for the denominator
is the sum of the expressions
[f]+[?l+[?
mm?
+ . . .
+ .
The corresponding index for the numerator is
(B)
m?\
+
+
(O
But, since n=a+fi+ . . . + \, it is evident that
Prom this and the expressions in (B) and (C) it follows that
the index of the highest power of any prime p in the numerator
of (A) is equal to or greater than the index of the highest power
of p co'ntained in its denominator. The theorem now follows
at once.
Corollary. The product of n consecutive integers is divisible
by n\.
EXERCISES
i. Show that the highest power of 2 contained in 1000! is 2994; in 1900! is 21893.
Show that the highest power of 7 contained in 10000! is 71665.
2. Find the highest power of 72 contained in 1000!
3. Show that 1000! ends with 240 zeros.
4. Show that there is no number n such that 3' is the highest power of 3 con-
tained in n\.
5. Find the smallest number n such that the highest power of 5 contained
in n\ is 531. What other numbers have the same property?
6. If n = rs, r and s being positive integers, show that n\ is divisible by (r!)'9;
by <>!)r; by the least common multiple of (r!)5 and (s\)T.
7. If n^a+p+pq+rs, where a, /3, p, q, r, s, are positive integers, then n\ is
divisible by
28 THEORY OF NUMBERS
S. When m and n are two relatively prime positive integers the quotient
(m+n — i)!
mini
as an integer.
9*. If m and n are positive integers, then each of the quotients
(mn)l (2w)!(2»)!
w!(w!)n' mlnl(m-\-n)l!
is an integer. Generalize to & integers w, n, p, . . . .
10*. If »=a+/3+^g+rs where a, /3, p, q, r, s are positive integers, then n\
is divisible by
ali3lrlpl(ql)p(sl)r.
ii*. Show that
(rst)l /
is an integer (r, s, t being positive integers). Generalize to the case of n integers
r, s, t,u, . . . .
§ 13. Remarks Concerning Prime Numbers
We have seen that the number of primes is infinite. But
the integers which have actually been identified as prime are
finite in number. Moreover, the question as to whether a large
number, as for instance 2257 — 1, is prime is in general very
difficult to answer. Among the large primes actually identified
as such are the following:
,61
I, 275-5 + I, 289-I, 2127-I,
No analytical expression for the representation of prime num-
bers has yet been discovered. Fermat believed, though he con-
fessed that he was unable to prove, that he had found such an
analytical expression in
22 +1.
Euler showed the error of this opinion by finding that 641 is a
factor of this number for the case when ^ = 5.
The subject of prime numbers is in general one of exceeding
difficulty. In fact it is an easy matter to propose problems
ELEMENTARY PROPERTIES OF INTEGERS 29
about prime numbers which no one has been able to solve.
Some of the simplest of these are the following:
i. Is there an infinite number of pairs of primes differing
by 2?
2. Is every even number (other than 2) the sum of two
primes or the sum of a prime and the unit?
3. Is every even number the difference of two primes or
the difference of 1 and a prime number?
4. To find a prime number greater than a given prime.
5. To find the prime number which follows a given prime.
jo\/To find the number of primes not greater than a given
number.
7. To compute directly the nth prime number, when n is
given.
CHAPTER II
ON THE INDICATOR OF AN INTEGER
§ 14. Definition. Indicator of a Prime Power
Definition. If m is any given positive integer the num-
ber of positive integers not greater than m and prime to it is
called the indicator of m. It is usually denoted by <j>(m), and
is sometimes called Euler's ^-function of m. More rarely,
it has been given the name of totient of m.
As examples we have
4>(i) = 1, 0(2) = 1, 0(3) = 2, 0(4) =2.
If p is a prime number it is obvious that
for each of the integers 1,2,3, . . . , p — 1 is prime to p.
Instead of taking m = p let us assume that m = pa, where
a is a positive integer, and seek the value of 4>{pa). Obviously,
every number of the set 1, 2, 3, ... , ^either is divisible
by p or is prime to p\ The number of integers in the set
divisible by p is p"'1. Hence pa — pa~1 of them are prime
to p. Hence 4>{pa) = pa-pa~1. Therefore
If p is any prime number and a is any positive integer, then
*<r)=r(i-|-}
§ 15. The Indicator of a Product
I. If (i and v are any two relatively prime positive integers
then
<f>(fj.v) =<£(m)0(V).
•80
ON THE INDICATOR OF AN INTEGER 31
In order to prove this theorem let us write all the integers
up to iiv in a rectangular array as follows:
i 2 3 . . . h . . . m
jliH-I jU+2 ^+3 . . . M-f^ . . . 2/x
2/i + I 2jU + 2 2/Z+3 . . . 2fi+k . . . 3M
(y-l)jti+I (y-l)/* + 2 (i>-i)m+3 • • • (^-i)m+^ . .
V]±
(A)
If a number /z in the first line of this array has a factor in
common with /x then every number in the same column with
h has a factor in common with /*. On the other hand if h is
prime to n so is every number in the column with h at the top.
But the number of integers in the first row prime to /x is <£(/*).
Hence the number of columns containing integers prime to fi
is 0(/x) and every integer in these columns is prime to p.
Let us now consider what numbers in one of these columns
are prime to v\ for instance, the column with h at the top.
We wish to determine how many integers of the set
h, n+h, 2>i+h, . . . , (p—i)n+h
are prime to v. Write
sn+h = qsv-\-r5
where 5 ranges over the numbers s = o, i, 2, . . . , v — i and
o£r,0. Clearly sn+h is or is not prime to v according as
r8 is or is not prime to v. Our problem is then reduced to that
of determining how many of the quantities rs are prime to v.
First let us notice that all the numbers rs are different;
for, if rs = r% then from
sn+h = q8v+rs, tii+h = qtv+rt,
we have by subtraction that (s—t)n is divisible by v. But
fx is prime to v and s and / are each less than v. Hence (s — t)fj,
can be a multiple of v only by being zero; that is, s must equal t.
Hence no two of the remainders rs can be equal.
Now the remainders rs are v in number, are all zero or posi-
tive, eat:h is less than v, and they are all distinct. Hence they
32 THEORY OF NUMBERS
are in some order the numbers o, i, 2, . . . , v — 1. The num-
ber of integers in this set prime to v is evidently 4>{v).
Hence it follows that in any column of the array (A) in which
the numbers are prime to fi there are just </>(*>) numbers which
are prime to v. That is, in this column there are just 4>(v)
numbers which are prime to iiv. But there are <£(/x) such
columns. Hence the number of integers in the array (A)
prime to fxv is 0(m) 4>{v) .
But from the definition of the ^-function it follows that
the number of integers in the array (A) prime to ixv is 4>(jiv).
Hence,
which is the theorem to be proved.
Corollary. In the series of n consecutive terms of an
arithmetical progression the common difference of which is prime
to n, the number of terms prime to n is <f>(n).
From theorem I we have readily the following more general
result :
II. If mi, W2, . . . , mk are k positive integers which are
prime each to each, then
4>(mim2 . . . mk) = <t>(mi) <j>(m2) . . . <i>(mk)..
§ 16. The, Indicator of any Positive Integer
From the results of §§14 and 15 we have an immediate
proof of the following fundamental theorem:
If m = p1aip2a2 . . . pnan where pi, p2, . . . , pn are differ-
ent primes and a\, a2, . . • , a„ are positive integers, then
m=m(i-£)(t-±^ . . . (x-i).
For,
4>{m) = 4>{p^)<t>(p2a*) . . . 4>{pnan)
ON THE INDICATOR OF AN INTEGER 33
On account of the great importance of this theorem we shall
give a second demonstration of it.
It is clear that the number of integers less than m and
divisible by p\ is
m
The number of integers less than m and divisible by p2 is
m
}~2
The number of integers less than m and divisible by pip2 is
m
p\p2
Hence the number of integers less than m and divisible by
either p\ or p2 is
m
Pl p2 plp2
Hence the number of integers less than m and prime to p\p2 is
m m . m I i \ / i
m — | =m[ i ) i
pl p2 Plp2 \ pl/ \ p2
We shall now show that if the number of integers less than
m and prime to p\p2 . . . pi, where i is less than n, is
X pin P2) ' '
i---
then» the number of integers less than m and prime to pip',
. . • pipt+i is
-) • • • («-—)•
p2l \ Pi+l/
From this our theorem will follow at once by induction.
i ii
Pil\ P<<
m—m\ i
34 THEORY OF NUMBERS
From our hypothesis it follows that the number of integers
less than m and divisible by at least one of the primes
pi, P2, . . • , pi is
or
P\ plp2 plp2pS
where the summation in each case runs over all numbers of
the type indicated, the subscripts of the fs being equal to or
less than i.
Let us consider the integers less than m and having the
factor pi+i but not having any of the factors pi, p2, . . . , pi.
Their number is
Pi+l pi+1 ( P\ Plp2 Plp2p3
where the summation signs have the same significance as before.
For the number in question is evidently tn/pt+i minus the
number of integers not greater than m/pt+i and divisible by
at least one of the primes pi, p2, . . . , pu
If we add (A) and (B) we have the number of integers less
than m and divisible by one at least of the numbers pi, p2,
. . . , pi+i> Hence the number of integers less than m and
prime to pi, p2, . . . , pi+i is
m—2j — \-2j 2j
'i pip2 pip2pz
where now in the summations the subscripts run from i to
i+i. This number is clearly equal to
w I
Pi)\ P2) ' V Pi+l)'
From this result, as we have seen above, our theorem follows
at once by induction.
ON THE INDICATOR OF AN INTEGER 35
§ 17. Sum of the Indicators of the Divisors of a Number
We shall first prove the following lemma:
Lemma. If d is any divisor of m and m = nd, the number
of integers not greater than m which have with m the greatest com-
mon divisor d is <t>(n).
Every integer not greater than m and having the divisor
d is contained in the set d, 2d, 3d, . . . , nd. The number of
these integers which have with m the greatest common divisor
d is evidently the same as the number of integers of the set
1, 2, . . . , n which are prime to m/d, or n; for ad and m have
or have not the greatest common divisor d according as a is
or is not prime to m/d, =n. Hence the number in question
is cj>(n).
From this lemma follows readily the proof of the following
theorem :
If di, d2, . . . , dr are the different divisors of m, then
<t>(d1) + <f>(d2)+ . . . +<f>(dr)=m.
Let us define integers mi, m2, . . . , mr by the relations
m = dimi=d2m2= . . . =dTmr.
Now consider the set of m positive integers not greater than
m, and classify them as follows into r classes. Place in the
first class those integers of the set which have with m the great-
est common divisor mi; -their number is <t>(di), as may be seen
from the lemma. Place in the second class those integers
of the set which have with m the greatest common divisor m2\
their number is 4>(d2)- Proceeding in this way throughout,
we place finally in the last class those integers of the set which
have with m the greatest common divisor mr', their number
is 4>{dr). It is evident that every integer in the set falls into
one and into just one of these r classes. Hence the total num-
ber m of integers in the set is 4>{d\) -\- 4>{d2) -{- . . . -\-<j>(dr).
From this the theorem follows immediately.
36 THEORY OF NUMBERS
EXERCISES
1. Show that the indicator of any integer greater than 2 is even.
2. Prove that the number of irreducible fractions not greater than 1 and with
denominator equal to n is <f>(n).
3. Prove that the number of irreducible fractions not greater than 1 and
with denominators not greater than n is
*(i)+0(2)+*(3)+ • - . +*(»).
4. Show that the sum of the integers less than n and prime, to n is %n<f>{n)
if «>i.
5. Find ten values of x such that <f>(x) = 24.
6. Find seventeen values of * such that </>(#) = 72.
7. Find three values of n for which there is no x satisfying the equation
<f>(x) = 271.
8. Show that if the equation
<f)(x) = n
has one solution it always has a second solution, n being given and x being the
unknown.
9. Prove that all the solutions of the equation
<f>(x) = 4n— 2, n>i,
are of the form pa and 2pa, where p is a prime of the form 45—1.
10. How many integers prime to n are there in the set
a) 1-2, 2-3, 3-4, . . . , »(»+i)?
V) 1-2-3, 2-3-4, 3-4-5, . . . , «(»+i)(»+2)?
\ Hi LI Hi »(»+i)?
4)
222 2
1-2-3 2-3-4 3'4-5 »(»+i)(»+2).
6. 6 6 6
it*. Find a method for determining all the solutions of the equation
<j>(x) = n,
where n is given and x is to be sought.
12*. A number theory function <f>(n) is denned for every positive integer n;
and for every such number n it satisfies the relation
<t>{di)+<t>{d2)+ . . . +<f>(dr)=n,
where di, (k, . . . , dr are the divisors of n. From this property alone show
that
«"-(-i)(-s)-- (-£>■'
where Pi, P2, • • • , />* are the different prime factors of n.
CHAPTER III
ELEMENTARY PROPERTIES OF CONGRUENCES
§ 1 8. Congruences Modulo m
Definitions. If a and .b are any two integers, positive
or zero or negative, whose difference is divisible by m, a and b
are said to be congruent modulo m, or congruent for the modulus
m, or congruent according to the modulus m. Each of the
numbers a and b is said to be a residue of the other.
To express the relation thus defined we may write
a = b-\-cm,
where c is an integer (positive or zero or negative). It is more
convenient, however, to use a special notation due to Gauss,
and to write
a=b mod m, ,
an expression which is read a is congruent to b modulo m, or
a is congruent to b for the modulus m, or a is congruent to b
according to the modulus m. This notation has the advantage
that it involves only the quantities which are essential to the
idea involved, whereas in the preceding expression we' had the
irrelevant integer c. The Gaussian notation is of great value
and convenience in the study of the theory of divisibility.
In the present chapter we develop some of the fundamental
elementary properties of congruences. It will be seen that
many theorems concerning equations are likewise true of con-
gruences with fixed modulus; and ft is this analogy with equa-
tions which gives congruences (as such) one of their chief claims
to attention. *
37
38 THEORY OF NUMBERS
As immediate consequences of our definitions we have the
following fundamental theorems:
I. If fl=cmodm, Ncmodw,
then a=bmodm\
that is, for a given modulus, numbers congruent to the same num-
ber are congruent to each other.
For, by hypothesis, a — c=Cim, b—c = C2m, where c\ and
C2 are integers. Then by subtraction we have a — b = (ci—C2)m\
whence a = b mod m.
II. // a = bmodm, a=j8 mod w,
then azka=b±(3 mod m;
that is, congruences with the same modulus may be added or sub-
tracted member by member.
For, by hypothesis, a — b = c\m, a — ^ = c2m; whence
(ad=a) — (bdtP) = (ci±c2)m. Hence a±a=^±/3modw.
III. // a=b mod ot,
then ca=cb mod m,
c being any integer whatever.
The proof is obvious and need not be stated.
IV. If a = bmodm, a=(3modm,
then aa m b(3 mod m ;
that is, two congruences with the same modulus may be multiplied
member by member.
For, we have a = b+c\m, a = (3+c2m. Multiplying these equa-
tions member by member we have aa = bp-\-m(bc2+ficiJrciC-2m).
Hence aa=b$ mod m.
A repeated use of this theorem gives the following result:
V. If a=b mod m,
then an = bn mod m
whtre n is any positive integer.
ELEMENTARY PROPERTIES OF CONGRUENCES 39
As a corollary of theorems II, III and V we have the follow-
ing more general result :
* VI. If f(x) denotes any polynomial in x with coefficients
which are integers (positive or zero or negative) and if further
a=b mod m, then
f(a) =f(b) mod m.
§ 19. Solutions of Congruences by Trial
Let f(x) be any polynomial in x with coefficients which
are integers (positive or negative or zero). Then if x and c
are any two integers it follows from the last theorem of the
preceding section that
f(x) =f(x-\-cm) mod m. (i)
Hence if a is any value of x for which the congruence
/(#)=omodra (2)
is satisfied, then the congruence is also satisfied for x=a-\-cm,
where c-is any integer whatever. The numbers* a -\-cni are
said to form a solution (or to be a root) of the congruence, c
being a variable integer. Any one of the integers a-\-cm may
be taken as the representative of the solution. We shall often
speak of one of these numbers as the solution itself.
Among the integers in a solution of the congruence (2)
there is evidently one which is positive and not greater than
m. Hence all solutions of a congruence of the type (2) may
be found by trial, a substitution of each' of the numbers 1, 2,
. . . , m being made for x. It is clear also that m is the maxi-
mum number of solutions which (2) can have whatever be
the function f(x) . By means of an example it is easy to show
that this maximum number of solutions is not always possessed
by a congruence*; in fact, it is not even necessary that the
congruence bave a solution at all.
This is illustrated by the example
x2 — 3=0 mod 5.
40 THEORY OF NUMBERS
In order to show that no solution is possible it is necessary to
make trial only of the values i, 2, 3, 4, 5 for x. A direct sub-
stitution verifies the conclusion that none of them satisfies
the congruence; and hence that the congruence has no solution
at all.
On the other hand the congruence
x5 — x=o mod 5
has the solutions x=i, 2, 3, 4, 5 as one readily verifies; that
is, this congruence has five solutions — the maximum number
possible in accordance with the results obtained above.
EXERCISES
1. Show that
(a+b)v = av+bp.modp
where a and b are any integers and p is any prime.
2. From the preceding result prove that
a? =a mod p
for every integer a.
3. Find all the solutions of each of the congruences xn=x mod 11,
x10=i mod n, xb = i mod n. *
§ 20. Properties of Congruences Relative to Division
The properties of congruences relative to . addition, sub-
traction and multiplication are entirely analogous to the prop-
erties of algebraic equations. But the properties relative to
division are essentially different. These we shall now give.
I. If two numbers are congruent modulo m they are con-
gruent modulo d, where d is any divisor of m.
For, from o=5 mod m, we have a = b-\-cm — b-\-cfd. Hence
a=b mod d.
II. If two numbers are congruent for different moduli they
are congruent for a modulus which is the least common multiple
of the given moduli.
ELEMENTARY PROPERTIES OF CONGRUENCES 41
The proof is obvious, since the difference of the given num-
bers is divisible by each of the moduli.
III. When the two members of a congruence are multiples of
an integer c prime to the modulus, each member of the congruence
may be divided by c.
For, if ca=cb mod m then ca — cb is divisible by m. Since
c is prime to m it follows that a — b is divisible by m. Hence
a =b mod m.
IV. // the two members of a congruence are divisible by an
integer c, having with the modulus the greatest common divisor 5,
one obtains a congruence equivalent to the given congruence by
dividing the two members by c and the modulus by 5.
By hypothesis
ac=bc mod m, c=8ci, m—hm\.
Hence c(a — b) is divisible by m. A necessary and sufficient
condition for this is evidently that c\{a — b) is divisible by mi.
This leads at once to the desired result.
§21. Congruences with a Prime Modulus
The congruence *
aoxn+aixn~l-\- . . . +an=omod p, ao^omod p,
where p is a prime number and the a's are any integers, has not
more than n solutions.
Denote the first member of this congruence by f(x) so that
the congruence may be written
f(x) =o mod p. (i)
Suppose that a is a root of the congruence, so that
f(a) =o mod p.
Then we have
f(x) ^f(x)-f(a) mod p.
*
* The sign ^ is read is not congruent to.
42 THEORY OF NUMBERS
But, from algebra. f(x)—f(a) is divisible by x — a. Let (x — a)a
be the highest power of x — a contained in f(x) —f(a) . Then
we may write
f(x)-f(a) = (x-a)«f1(x), (2)
where /i(x) is evidently a polynomial with integral coefficients.
Hence we have
}{x) = (x-a) a/i 0) mod p. (3)
We shall say that a occurs a times as a solution of (1); or that
the congruence has a solutions each equal to a.
Now suppose that congruence (1) has a root b such that
b=£a mod p. Then from (3) we have
f(b)m(b^a)afi(b)modp.
But f(b)=omodp, (b-a)a^omodp.
Hence, since p is a prime number, we must have
fi(b) =0 mod p.
By an argument similar to that just used above, we may
show that /1 (x) — /1 (b) may be written in the form
where /5 is some positive integer. Then we have
f(x) = (x-a)a(x-byf2{x) mod p.
Now this process can be continued until either all the
solutions of (1) are exhausted or the second member is a prod-
uct of linear factors multiplied by the integer a0. In the for-
mer case there will be fewer than n solutions of (1), so that
our theorem is true for this case. In the other case we have
f(x)^a0(x-a)a(x-by . . . (>-/)xmod/>.
We have now n solutions of (1) : a counted a times, b counted
0 times, . . . , I counted X times; «+/?+ . . . +X = w.
ELEMENTARY PROPERTIES OF CONGRUENCES 43
Now let 77 be any solution of (i). Then
f(rl)=ao(ri-a)a(r,-b)f' . . . (v-l)x=omodp.
Since p is prime it follows now that some one of the factors
7} — a, r] — b, . . . , t)—1 is divisible by p. Hence 77 coincides
with one of the solutions a, b, c, . . . , I. That is, (1) can
have only the n solutions already found.
This completes the proof of the theorem.
EXERCISES
1. Construct a congruence of the form
aoxn-\-alxn~1-\-. . . -\-an=omodm, a0^omodf»,
having more than n solutions and thus show that the limitation to a prime mod-
ulus in the theorem of this section is essential.
2. Prove that
x*-i=(x-i)(x-2)(x-3)(x-4)(x-s)(x-6) mod 7
for every integer x.
3. How many solutions has the congruence x5=i mod 11? the congruence
£5=2 mod n?
§ 22. Linear Congruences
From the theorem of the preceding section it follows that
the congruence
ax =c mod p, a^o mod p,
where p is a prime number, has not more than one solution.
In this section we shall prove that it always has a- solution.
More generally, we shall consider the congruence
ax=c mod m
where m is any integer. The discussion will be broken up
into parts for convenience in the proofs.
I. The congruence
ax=\ mod m, (1)
44 THEORY OF NUMBERS
in which a and m are relatively prime, has one and only one solu-
tion.
The question as to the existence and number of the solu-
tions of (i) is equivalent to the question as to the existence
and number of integer pairs x, y satisfying the equation,
ax— my = i, (2)
the integers x being incongruent modulo m. Since a and m
are relatively prime it follows from theorem IV of § 9 that
there exists a solution of equation (2). Let x=a and y = p
be a particular solution of (2) and let x = a and y = ]3 be any
solution of (2). Then we have
aa — mP = i,
aa—mfi=i;
whence
a(a — a)—m(p — ~(3) =0.
Hence a — a is divisible by m, since a and m are relatively prime.
That is, a = amodm. Hence a and a are representatives of
the same solution of (1). Hence (1) has one and only one
solution, as was to be proved.
II. The solution x=a of the congruence w=imodw, in
which a and m are relatively prime, is prime to m.
For, if aa — 1 is divisible by m, a is divisible by no factor
of m except 1.
III. The congruence
flx=cmodm (3)
in which a and m and also c and m are relatively prime, has one
and only one solution.
Let x = y be the unique solution of the congruence
a=i mod w. Then we have ayx=cy = imodm. Now* by
I we see that there is one and only one solution of the con-
gruence ayx = imo<im\ and from this the theorem follows at
once.
Suppose now that a is prime to m but that c and m have
the greatest common divisor 8 which is different from 1 . Then
it is easy to see that any solution x of the congruence ax=c mod m
ELEMENTARY PROPERTIES OF CONGRUENCES 45
must be divisible by <5. The question of the existence of solu-
tions of the congruence ax=c mod m is then equivalent to the
question of the existence of solutions of the congruence
x c t m
a — sb - mod — ,
8 8 8
where x/8 is the unknown integer. From III it follows that
this congruence has a unique solution x/8=a. Hence the
congruence ax^cmodm has^the unique solution x=8a. Thus
we have the following theorem:
IV. The congruence ax=cmodm, in which a and m are
relatively prime, has one and only one solution.
Corollary. The congruence ax=c mod p, a=|=o mod p,
where p is a prime number, has one and only one solution.
It remains to examine the case of the congruence ax=(?mod m
in which a and m have the greatest common divisor d. It is
evident that there is no solution unless c also contains this
divisor d.l Then let us suppose that a = ad, c = yd, m = txd.
Then for every x such that ax=c mod m we have ax =7 mod ft;
and conversely every x satisfying the latter congruence also
satisfies the former. Now ax=yraodn has only one solu-
tion. Let j3 be a non-negative number less than jjl which satis-
fies the congruence ax =7 mod /*. All integers which satisfy
this congruence are then of the form fi+ixv, where v is an integer.
Hence all integers satisfying the congruence ax=cmodm are
of the form P+jjiv; and every such integer is a representative
of a solution of this congruence. It is clear that the numbers
0, 0+M, P + 2H, . . . , fi + (d-l)ii (A)
are incongruent modulo m while every integer of the form
fi+nvis congruent modulo m to a number of]the set (-4). Hence
the congruence ax=c mod m has the d solutions (^4).
This leads us to an important theorem which includes all
the other theorems of this section as special cases. It may be
stated as follows:
V. Let
flx=cmodw
46 THEORY OF NUMBERS
be any linear congruence and let a and m have the greatest common
divisor d (d^-i). Then a necessary and sufficient condition for
the existence of solutions of the congruence is that c be divisible
by d. If this condition is satisfied the congruence has just d solu-
tions^ and all the solutions are congruent modulo m/d.
EXERCISES
J $
i. Find the remainder when 240 is divided by 31; when 2 43 is divided by 31.
2. Show that 228+i has the factor 641.
3. Prove that a number is a multiple of 9 if and only if the sum of its digits
is a multiple of 9.
4. Prove that a number is a multiple of n if and only if the sum of the digits
in the odd numbered places diminished by the sum of the digits in the even
numbered places is a multiple of 11.
CHAPTER IV
THE THEOREMS OF FERMAT AND WILSON
§ 23. Fermat's General Theorem
Let m be any positive integer and let
ai, fl2, • • • , a*(m) (A)
be the set of <f>(m) positive integers not greater than m and
prime to m. Let a be any integer prime to m and form the set
of integers
aai, aa2, . . . , aa^m). (B)
No number aat of the set (B) is congruent to a number aa$
unlessy = z; for, from
aai^= aaj mod m
we have Oi= a j mod. m\ whence a% = ah since both a\ and aj
are positive and not greater than m. Theref ore j — i . Further-
more, every number of the set (B) is congruent to some number
of the set (A). Hence we have congruences of the form
aai^a^ mod m,
aa2 = au mod my
00*<m)=a«0(w)mod*»
No j^oriu|jfrers in the second members are equal, since aa^aaj
un[^^t=j. Hence the numbers atl, aiv . . , , at (m) are
the numbers #i, #2, ... , a0(m) in some order. Therefore,
if we multiply the above system of congruences together mem-
47
48 THEORY OF NUMBERS
ber by member and divide each member of the resulting con-
gruence by ai-d2 . . . a^m) (which is prime to m), we have
This result is known as Fermat's general theorem. It may
be stated as follows :
If m is any positive integer and a is any integer prime to m,
then
Corollary i. If a is any integer not divisible by a prime
number p, then
ap~1 = i mod p.
Corollary 2. If p is any prime number and a is any
integer, then
ap=a mod p.
§ 24. Euler's Proof of the Simple Fermat Theorem\
The theorem of Cor. 1, § 23, is often spoken of as the simple
Fermat theorem. It was first announced by Fermat in 1679,
but without proof. The first proof of it was given by Euler
in 1736. This proof may be stated as follows:
From the Binomial Theorem it follows readily that
(a+i)p=ap+imodp
since
p\
r\(p-r)\
, o<r<p,
is obviously divisible by p. Subtracting a-\-i %>m each side
of the foregoing congruence, we have
(a + i)p-(> + i)=ap-arnod£.
THE THEOREMS OF FERMAT AND WILSON 49
Hence if av — a is divisible by p, so is (a + i)p — (a+i). But
ip — i is divisible by p. Hence 2P — 2 is divisible by p\ and
then 3P — 3; and so on. Therefore, in general, we have
ap=a mod p.
If a is prime to p this gives ap~x = 1 mod ^, as was to be proved.
If instead of the Binomial Theorem one employs the Poly-
nomial Theorem, an even simpler proof is obtained. For,
from the latter theorem, we have readily
'(ai+a2+ • . . +aa)p=aip+a2p+ . . . +aap mod p.
Putting ai=a2= . . . =aa=i we have
ap=a mod p,
from which the theorem follows as before.
§ 25. Wilson's Theorem
From the simple Fermat theorem it follows that the con-
gruence
xp~1 = i mod^
has the p — i solutions 1, 2, 3, . . . , p — i. Hence from the
discussion in § 21 it follows that
xp — i==(x — i)(x — 2) . . . (x — p — i) mod p,
this relation being satisfied for every value of x. Putting x=o
we have
(_I) = (_I)p-i.Iv2.3 , , . p-imodp.
If p is an odd prime this leads to the congruence
i- 2-3 . . . p — i-\-i=omod p.
Now for p = 2 this congruence is evidently satisfied. Hence
we have the Wilson theorem :
Every prime number p satisfies the relation
i* 2*3 . . . p — i + i=omod^.
50 THEORY OF NUMBERS
An interesting proof of this theorem on wholly different
principles may be given. Let p points be distributed at equal
intervals on the circumference of a circle. The whole number
of ^-gons which can be formed by joining up these p points
in every possible order is evidently
i
„ 2p
p(p-i)(p-2) . . . 3-2-i;
for the first vertex can be chosen in p ways, the second in p — i
ways, . . . , the (p — i)th in two ways, and the last in one
way; and in counting up thus we have evidently counted each
polygon 2p times, once for each vertex and for each direction
from the verfcqx around the polygon. Of the total number
of polygons \{p — i) are regular (convex or stellated) so that
a revolution through 360°/^ brings each of these into coin-
cidence with Its former position. The number of remaining
p-gons must be divisible by p\ for with each such ^>-gon we may
associate the p — i ^-gons which can be obtained from it by
rotating it through successive angles of 360°/^. That is,
-^-p(p-i)(p-2) . . . 3-2-i--0-i)=omod/>.
2p 2
Hence
(p — i)(p — 2) . . . 3-2-1 — />+i=o mod p;
and from this it follows that
1-2 .. . ^—1 + 1=0 mod p,
as was to be proved.
§ 26. The Converse of Wilson's Theorem
Wilson's theorem is noteworthy in that its converse is also
true. The converse may be stated as follows:
Every integer n such that the congruence
1-2-3 . . . n— 1 + 1=0 mod n
is satisfied is a prime number.
THE THEOREMS OF FERMAT AND WILSON 51
For, if n is not prime, there is some divisor d of n different
from i and less than n. For such a d we have 1-2-3 • • •
n— 1=0 mod d; so that 1-2 .. . w-i + i^omodi; and
hence 1-2 . . . n— 1 + 1=^0 mod n. Since this, contradicts our
hypothesis the truth of the theorem follows.
Wilson's theorem and its converse may be combined into
the following elegant theorem:
A necessary and sufficient condition that an integer n is prime
is that
1-2-3 • • • w-i + i=omodw.
Theoretically this furnishes a complete and elegant test
as to whether a given number is prime. But, practically,
the labor of applying it is so great that it is useless for verifying
large primes.
§27. Impossibility of 1-2-3 • • • n~ i + i=w* for n>$.
In this section we shall prove the following theorem:
There exists no integer k for which the equation
i- 2
3 . . . n— i-\-i=n*
is true when n is greater than 5.
If n contains a divisor d different from 1 and n, the equa-
tion is obviously false; for the second member is divisible
by d while the first is not. Hence we need to prove the theorem
only for primes n.
Transposing 1 to the second member and dividing by n — 1
we have
1-2-3 • • • n — 2=nt~1+nt~2+ . . . +n+i.
If w>5 the product on the left contains both the factor 2 and
the factor \{n — 1); that is, the first member contains the fac-
tor n— 1. But the second member does not contain this fac-
tor, since for n = i the expression w*_1+ . . . -\-n-\-i is equal
to kj^o. Hence the theorem follows at once.
52 theory of numbers
§ 28. Extension of Fermat's Theorem
The object of this section is to extend Fermat's general
theorem and incidentally to give a new proof of it. We shall
base this proof on the simple Fermat theorem, of which we
have already given a simple independent proof. This theorem
asserts that for every prime p and integer a not divisible by p,
we have the congruence
ap~1 = i mod p.
Then let us write
a*-1 = i+hp. (1)
Raising each member of this equation to the pth power we
may write the result in the form
aviV-D =I+hlp2 (2)
where hi is an integer. Hence
^-"simod^,
By raising each member of (2) to the p power we can readily
show that
/^^imod^3.
It is now easy to see that we shall have in general
wUere a is a positive integer; that is,
a*(pa) = 1 m0(i p«m
For the special case when p is 2 this result can be extended.
For this case (1) becomes
a = 1 + 2/2.
Squaring we have
<z2 = i+4/*0+i).
Hence,
a2 = i+8Ai, (3)
THE THEOREMS OF FERMAT AND WILSON 53
where h\ is an integer. Therefore
a2 = i mod 23.
Squaring (3) we have
a22 = i + 2*h2;
a22 = 1 mod 24.
It is now easy to see that we shall have in general
ifa>2. That is,
a2" = 1 mod 2a
a^2a) = i mod 2a if «>2.
Now in terms of the ^-function let us define a new function
X(m) as follows:
\(2a) = <£(2aX if a = o, 1, 2;
X(2«)=^(2a) if a>2-
\(pa) = <i>(pa) if p is an odd prime;
\{2ap^p2^ . . . pnan) =M,
where M is the least common multiple of
x(2«), x(/>!«0, x(/>2««), . . . , X(/>.««),
2, ^1, p2, - . . , pn being different primes.
Denote by m the number
rn = 2ap1aip2a* . . . pnan.
Let a be any number prime to m. From our preceding results
we have
#(*■>«! mod '?«."
bWsi mod#i%
ax(p2«2) = x moc] ^2«s
aX(P„«») = lmod^an>
54 THEORY OF NUMBERS
Now any one of these congruences remains true if both of
its members are raised to the same positive integral power,
whatever that power may be. Then let us raise both members
of the first congruence to the power \(m)/\(2a); both members
of the second congruence to the power X(w) /Mpi"1) 5 • • • J
both members of the last congruence to the power \(m)/\(pnan).
Then we have
a\(m) _ x mo(J 2a}
From these congruences we have immediately
a\(m) m j mocJ Mf
We may state this result in full in the following theorem :
// a and m are any two relatively prime positive integers, the
congruence
aHm) s j m0(j m
is satisfied.
As an excellent example to show the possible difference
between the exponent \(m) in this theorem and the exponent
4>(m) in Fermat's general theorem, let us take
w = 26-33-5-7-i3-i7-i9'37-73.
Here
X(»=24-32, 4>0) = 221-310.
In a later chapter we shall show that there is no exponent
v less than \(m) for which the congruence
a" = i mod m
is verified for every integer a prime to m.
From our theorem, as stated above, Fermat's general the-
orem follows as a corollary, since X(m) is obviously a factor
of 4>(m),
<Km) = <K2«)*(/>i«0 • ■ • 4>(£»an).
THE THEOREMS OF FERMAT AND WILSON 55
EXERCISES
i. Show that a16=i mod 16320, for every a which is prime to 16320.
2. Show that a12 = i mod 65520, for every a which is prime to 65520.
3*. Find one or more composite numbers P such that
ap~l = i modP
for every a prime to P. (Compare this problem with the next section.)
§ 29. On the Converse of Fermat's Simple Theorem
The fact that the converse of Wilson's theorem is a true
proposition leads one naturally to inquire whether the con-
verse of Fermat's simple theorem is true. Thus, we may ask the
question: Does the existence of the congruence 2n_1 = i mod n
require that n be a prime number? The Chinese answered
this question in the affirmative and the answer passed unchal-
lenged among them for many years. An example is sufficient
to show that the theorem is not true. We shall show that
234o = j mocj 241
although 341, = 11 -31, is not a prime number. Now 210— i
= 3-11-31. Hence 210=imod34i. Hence 2340=i mod 341.
From this it follows that the direct converse of Fermat's the-
orem is not true. The following theorem, however, which is
a converse with an extended hypothesis, is readily proved.
If there exists an integer a such that
an~l = i mod n
and if further there does not exist an integer v less than n—i such
that
av=i mod n,
then the integer n is a prime number.
For, if n is not prime, <f>(n)<n—i. Then for v = <f>(n) we
have a" = 1 mod n, contrary to the hypothesis of the theorem.
56 THEORY OF NUMBERS
§30. Application of Previous Results to Linear Con-
gruences
The theorems of the present chapter afford us a ready means
of writing down a solution of the congruence
ax=c mod m.
We shall consider only the case in which a and m are relatively
prime, since the general case is easily reducible to this one,
as we saw in the preceding chapter.
Since a and m are relatively prime we have the congruences
Hence either of the numbers x,
is a representative of the solution of (1). Hence the following
theorem :
If ax=c mod m
is any linear congruence in which a and m are relatively prime,
then either of the numbers x,
x = cax{m)-1, x = ca^m)-\
is a representative of the solution of the congruence.
The former representative of the solution is the more con-
venient of the two, since the power of a is in general much less
in this case than in the other.
EXERCISE
Find a solution of ^x = ^ mod 26-3«5-i7. Note the greater facility in apply-
ing the first of the above representatives of the solution rather than the second.
the theorems of fermat and wilson 57
§ 31. Application of the Preceding Results to the Theory
of Quadratic Residues
In this section we shall apply the preceding results of this
chapter to the problem of rinding the solutions of congruences
of the form
az2+/3z+Y=o mod /z
where a, /3, 7, fi are integers. These are called quadratic con-
gruences.
The problem of the solution of the quadratic congruence
(1) can be reduced to that of the solution of a simpler form of
congruence as follows: Congruence (1) is evidently equivalent
to the congruence
4«222 -f 4^2+40:7 = 0 mod 4.afi. (1)
But this may be written in the form
(2az+(3)2 = (32— 40:7 mod 4<xfjL.
Now if we put
2a2+/3=xmod 4a/x (2)
and
|82 — 4«7 = a, 4ctfjL = m,
we have
x2 = amodm. (3)
We have thus reduced the problem of solving the general con-
gruence (1) to that of solving the binomial congruence (3)
and the linear congruence (2). The solution of the latter may
be effected by means of the results of § 30. We shall there-
fore confine ourselves now to a study of congruence (3). We
shall make a further limitation by assuming that a and m
are relatively prime, since it is obvious that the more general
case is readily reducible to this one.
The example
#2 = 3 mod 5
58 THEORY OF NUMBERS
shows at once that the congruence (3) does not always have a
solution. First of all, then, it is necessary to find out in what
cases (3) has a solution. Before taking up the question it will
be convenient to introduce some definitions.
Definitions. An integer a is said to be a quadratic
residue modulo m or a quadratic non-residue modulo m accord-
ing as the congruence
#2 = amodw
has or has not a solution. We shall confine our attention to
the case when m > 2 .
We shall now prove the following theorem :
I. If a and m are relatively prime integers, a necessary con-
dition that a is a quadratic residue modulo m is that
ahMm) _ j mo(J m
Suppose that the congruence x2 = a mod m has the solu-
tion x=a. Then a2 = a mod m. Hence
aX<m) m aJX(m) mod m
Since a is prime to m it is clear from a2 = amod m that a is prime
to m. Hence ax(w) m 1 mod m. Therefore we have
I=a§M«) modra.
That is, this is a necessary condition in order that a shall be
a quadratic residue modulo m.
In a similar way one may prove the following theorem:
II. If a and m are relatively prime integers, a necessary con-
dition that a is a quadratic residue modulo m is that
ai4im) = i mod m.
When m is a prime number p each of the above results
takes the following form: If a is prime to p and is a quadratic
residue modulo p, then
THE THEOREMS OF FERMAT AND WILSON 59
We shall now prove the following more complete theorem,
without the use of I or II.
III. If p is an odd prime number and a is an integer not
divisible by p, then a is a quadratic residue or a quadratic non-
residue modulo p according as
aKp-D = +I or ^-i)=_imod^
This is called Euler's criterion.
Given a number a, not divisible by p, we have to determine
whether or not the congruence
x2 = a mod p
has a solution. Let r be any number of the set
i, 2, 3, ... , p-i (A)
and consider the congruence
rx = amod p.
This has always one and just one solution x equal to a number
5 of the set (.4). Two cases can arise :T either for every r of the
set (A) the corresponding s is different from r orjfor some r
of the set (.4) the corresponding 5 is equal to r. The former
is the case when a is a- quadratic non-residue modulo p\ the
latter is the case when a is a quadratic residue modulo p. We
consider the two cases separately.
In the first case the numbers of the set (.4) go in pairs such
that the product of the numbers in the pair is congruent to a
modulo p. Hence, taking the product of all \{p — i) pairs,
we have
1-2-3 . . . p — i = -]-a^p~1) mod p.
But
1-2-3 . . . p — i = — i mod p.
Hence
al(p-u = __j mod p^
whence the truth of one part of the theorem.
60 THEORY OF NUMBERS
In the other case, namely that in which some r and corre-
sponding s are equal, we have for this r
r2 = a mod p
and
(p—r)2 = a mod p.
Since x2=amod p has at most two solutions it follows that
all the integers in the set (A) except r and p—r fall in pairs
such that the product of the numbers in each pair is congruent
to a modulo p. Hence, taking the product of all these pairs,
which are ^(p—i) — i in number, and multiplying by r{p—r)
we have
1-2-3 . . . p — i = (p— r)ra^p~l^~l mod p
= -r2ai(p-i)-imodp
= -a-ah{lp-l)-lmodp
= _ai(p-Dmo(i^.
Since 1-2-3 • • • P "~ 1 m r~ 1 m°d P we have
a^p-v = +imodp;
whence the truth of another part of the theorem.
Thus the proof of the entire theorem is complete.
CHAPTER V.
PRIMITIVE ROOTS MODULO m-
§ 32. Exponent of an Integer Modulo m
Let
01, 02, ... , 0£(m) (^)
be the set of <j>(m) positive integers not greater than m and
prime to m; and let a denote any integer of the set (A). Now
any positive integral power of a is prime to m and hence is
congruent modulo w to a number of the set (A). Hence,
among all the powers of 0 there must be two, say an and av,
n>v, which are congruent to the same integer of the set (A).
These two powers are then congruent to each other; that is,
0n=a" mod m.
Since 0" is prime to m the members of this congruence may be
divided by 0". Thus we have
an~"=i mod m.
That is, among the powers of a there is one at least which is
congruent to 1 modulo m.
Now, in the set of all powers of a which are congruent to
1 modulo m there is one in which the exponent is less than in
any other of the set. Let the exponent of this power be d,
so that ad is the lowest power of a such that
ad= 1 mod m.
61
62 THEORY OF NUMBERS
We shall now show that if aa= i mod m, then a is a multiple
of d. Let us write
a = d8+p, o^(3<d.
Then
aa=imodm, (2)
ad8=i mod m, (3)
the last congruence being obtained by raising (1) to the power
5. From (3) we have
ads+p=ap mod m;
or
ap=i mod m.
Hence /3 = o, for otherwise d is not the exponent of the lowest
power of a which is congruent to 1 modulo m. Hence d is a
divisor of a.
These results may be stated as follows:
I. If m is any integer and a is any integer prime to m, then
there exists an integer d such that
ad = 1 mod m
while there is no integer /3 less than d for which
a&=i mod m.
Further, a necessary and sufficient condition that
a"=i mod m
is that v is a multiple of d.
DEFINITION. The integer d which is thus uniquely deter-
mined when the two relatively prime integers a and m are given
is called the exponent of a modulo m. Also, d is said to be
the exponent to which a belongs modulo m.
Now, in every case we have
PRIMITIVE ROOTS MODULO M 63
if a and m are relatively prime. Hence from the preceding
theorem we have at once the following:
II. The exponent d to which a belongs modulo m is a divisor
of both <f>(m) and \(m).
§ 33. Another Proof of Fermat's General Theorem
In this section we shall give an independent proof of the
theorem that the exponent d of a modulo m is a divisor of <j>(m) ;
from this result we have obviously a new proof of Fermat's
theorem itself.
We retain the notation of the preceding section. We shall
first prove the following theorem :
I. The numbers
1, a, a2, ... , a*-1 (A)
are incongruent each to each modulo m.
For, if fl^^modw, where o<a<d and o<0<d, a>ft,
we have aa~^=i mod m, so that d is not the exponent to which
a belongs modulo m, contrary to hypothesis.
Now any number of the set (A) is congruent to some number
of the set
Hi, 42, ... , a*(TO). (B)
Let us undertake to separate the numbers (B) into classes
after the following manner: Let the first class consist of the
numbers
(I) ao, ai, a2, . . . , «„_!,
where at is the number of the set (B) to which ai is congruent
modulo m.
If the class (I) does not contain all the numbers of the set
(B), let at be any number of the set (B) not contained in (I)
and form the following set of numbers:
(IIr) aoat, aiat, a2#«, . . . , ad_!a<.
64 THEORY OF NUMBERS
We shall now show that no number of this set is congruent to
a number of class (I) . For, if so, we should have a congruence
of the form
ataj^aic mod m\
hence
(Ha* = ak mod m,
so that
aiad = ak+d~3 mod m;
or <h = dk+d~j mod my
so that at would belong to the set (I) contrary to hypothesis.
Now the numbers of the set (II') are all congruent to num-
bers of the set (B) ; and no two are congruent to the same num-
ber of this set. For, if so, we should have two numbers of
(II') congruent; that is, akdi=ajdi mod m, or ak=aj mod m;
and this we have seen to be impossible.
Now let the numbers of the set (B) to which the numbers
of the set (II') are congruent be in order the following:
(II) 00, 01, 02, ... , 0d-!-
These numbers constitute our class (II).
If classes (I) and (II) do not contain all the numbers of the
set (B) , let dj be a number of the set (B) not contained in either
of the classes (I) and (II) : and form the set of numbers
(III') aodj, aidj, 4x2a j, . . . , ad_1dJ.
Just as in the preceding case it may be shown that no number
of this set is congruent to a number of class (I) and that the
numbers of (III') are incongruent each to each. We shall
also show that no number of (III') is congruent to a number
of class (II). For, if so, we should have dkdj=(3i mod m. Hence
akdj=dldi mod m\ or dj=dl+d~kdi mod m, from which it
follows that dj is of class (II), contrary to hypothesis.
Now let the numbers of the set (B) to which the numbers
of the set (III') are congruent be in order the following:
(HI) TO, 71, 72, • • • , 7d-i-
These numbers form our class (III).
PRIMITIVE ROOTS MODULO M 65
It is now evident that the process may be continued until
all the numbers of the set (B) have been separated into classes,
each class containing d integers, thus:
(i)
(ii)
(in)
( ) Xo, Xi, X2,
OtO,
ai,
CL2, ■
• • j ad-l>
Po,
01,
02, .
■ • > Pd-1>
7o,
71,
72, •
• • , 7d-l,
^-1-
The set (5), which consists of 4>{m) integers, has thus been
separated into classes, each class containing d integers. Hence
we conclude that d is a divisor of <j>(m). Thus we have a second
proof of the theorem:
II. If a and m are any two relatively prime integers and d
is the exponent to which a belongs modulo m, then d is a divisor
of <t>(m).
In our classification of the numbers (B) into the rectangular
array above we have proved much more than theorem II;
in fact, theorem II is to be regarded as one only of the con-
sequences of the more general result contained in the array.
If we raise each member of the congruence
ad=i mod m
to the (integral) power <f>(m)/d, the preceding theorem leads
immediately to an independent proof of Fermat's general
theorem.
§ 34. Definition of Primitive Roots
Definition. Let a and m be two relatively prime integers.
If the exponent to which a belongs modulo m is <f>(m), a is said
to be a primitive root modulo m (or a primitive root of m).
In a previous chapter we saw that the congruence
aHm) = j mod m
66 THEORY OF NUMBERS
is verified by every pair of relatively prime integers a and m.
Hence, primitive roots can exist only for such a modulus m as
satisfies the equation
<t>(m)=\(m). (i)
We shall show later that this is also sufficient for the existence
of primitive roots.
From the relation which exists in general between the
^-function and the X-function in virtue of the definition of the
latter, it follows that (i) can be satisfied only when m is a prime
power or is twice an odd prime power.
Suppose first that m is a power of 2, say m = 2a. Then (1) is
satisfied only if a = o, 1, 2. For a = oor 1, 1 itself is a primitive
root. For a = 2, 3 is a primitive root. We have therefore
left to examine only the cases
m = pa, m = 2pa
where p is an odd prime number. The detailed study of these
cases follows in the next sections.
§35. Primitive Roots Modulo p.
We have seen that if p is a prime number and d is the
exponent to which a belongs modulo p, then J is a divisor of
0(^)=^-i. Now, let
di, d2, d3, . . . , dr
be all the divisors of p — 1 and let $(dl) denote the number of
integers of the set
1, 2, 3, ... , p-i
which belong to the exponent d%. If there is no integer of the
set belonging to this exponent, then \p(di) =0.
PRIMITIVE ROOTS MODULO m 67
Evidently every integer of the set belongs to some one and
only one of the exponents d\, d2, . . . , dr. Hence we have
the relation
*(<*l) + *(*0 + • • • ++(dT)=p-I. (l)
But
*(<Zl)+*(<fe)+ • • • +<t>(dr)=p-I. (2)
If then we can show that
*(<*.)£*(<*.) (3)
for i = i, 2, . . . , r, it will follow from a comparison of (i)
and (2) that
${di) = <t>{dt).
Accordingly, we shall examine into the truth of (3).
Now the congruence
xd* = imodp (4)
has not more than di roots. If no root of this congruence
belongs to the exponent di, then \J/(di)=o and therefore in this
case we have \p(di) <4>(di). On the other hand if a is a root
of (4) belonging to the exponent di, then
a, a\ a*, . . . , ad* (5)
are a set of dt incongruent roots of (4) ; and hence they are the
complete set of roots of (4).
But it is easy to see that a* does or does not belong to the
exponent di according as k is or is not prime to di, for, if a*
belongs to the exponent /, then / is the least integer such that
kt is a multiple of di. Consequently the number of roots in
the set (5) belonging to the exponent di is <f>(di). That is,
in this case \j/(di) = 4>(di) . Hence in general ^(dt) £^(J«).
Therefore from (1) and (2) we conclude that
t(di) = 4>(di), i=i, 2, . . . , r.
68 THEORY OF NUMBERS
The result thus obtained may be stated in the form of the
following theorem:
I. If p is a prime number and d is any divisor of p — i , then
the number of integers belonging to the exponent d modulo p
is <f)(d).
In particular:
II. There exist primitive roots modulo p and their number
is 4>{p-i).
§36. Primitive Roots Modulo pa, p an Odd Prime
In proving that there exist primitive roots modulo pa, where
p is an odd prime and a> 1, we shall need the following theorem:
I. There always exists a primitive root y modulo p for which
yp~l — 1 is not divisible by p2.
Let g be any primitive root modulo p. If gp~1 — 1 is not
divisible by p2 our theorem is verified. Then suppose that
gv~x — 1 is divisible by p2, so that we have
where k is an integer. Then put
7 = g+Xp
where x is an integer. Then 7=g mod p, and hence
yh = gh mod p;
whence we conclude that 7 is a primitive root modulo p. But
I ! 2 I
= P(kp+tllr*x+(P-*)(p-*)f-Zx2p+ . .
Hence
7p-1-i=^(-^"2x) mod^2.
PRIMITIVE ROOTS MODULO M t)9
Therefore it is evident that x can be so chosen that yp~1 — i
is not divisible by p2. Hence there exists a primitive root 7
modulo p such that 7P_1 — 1 is not divisible by p2. Q. E. D.
We shall now prove that this integer 7 is a primitive root
modulo pa, where a is any positive integer.
If
7*= 1 mod p,
then k is a multiple of p — i, since 7 is a primitive root modulo
p. Hence, if
7l=imodf,
then k is a multiple of p — 1 .
Now, write
yt-^j+kp.
Since yp~1 — 1 is not divisible by ^>2, it follows that h is prime
to ^>. If we raise each member of this equation to the power
(3pa~2, a>2J we have
where / is an integer. Then if
y>p«-2(P-i)^imod^
|8 must be divisible by p. Therefore the exponent of the lowest
power of 7 which is congruent to 1 modulo pa is divisible by
pa~l. But we have seen that this exponent is also divisible
by p — 1. Hence the exponent of 7 modulo pa is pa~1(p—i)
since (j>(pa)=pa~1(p — i). That is, 7 is a* primitive root mod-
ulo pa.
It is easy to see that no two numbers of the set
7, 72, T3, . • • , 7pa"1(p~1) (A)
are congruent modulo pa; for, if so, 7 would belong modulo pa
to an exponent less than pa~1(p~i) and would therefore not
be a primitive root modulo pa. Now every number in the set
70 THEORY OF NUMBERS
(-4) is prime to pa; their number is <f>(pa) = pa~1(p — i). Hence
the numbers of the set (A) are congruent in some order to the
numbers of the set (B) :
0i, 02, as, ... , 0po-i(P_i), (B)
where the integers (B) are the positive integers less than pa
and prime to pa.
But any number of the set (B) is a solution of the congruence
/"^-"simodf. (i)
Further, every solution of this congruence is prime to pa. Hence
the integers (B) are a complete set of solutions of (i). Therefore
the integers (A) are a complete set of solutions of (i). But
it is easy to see that an integer 7* of the set (A) is or is not a
primitive root modulo pa according as k is or is not prime to
pa~1(p—i). Hence the number of primitive roots modulo
The results thus obtained may be stated as follows:
II. If p is any odd prime number and a is any positive integer,
then there exist primitive roots modulo pa and their number is
§ 37. Primitive Roots Modulo 2pa, p an Odd Prime
In this section we shall prove the following theorem :
// p is any odd prime number and a is any positive integer,
then there exist primitive roots modulo 2pa and their number is
*U(2f)}.
Since 2pa is even it follows that every primitive root modulo
2pa is an odd number. Any odd primitive root modulo pa is
obviously a primitive root modulo 2pa. Again, if 7 is an even
primitive root modulo pa then y-\-pa is a primitive root modulo
2pa. It is evident that these two classes contain (without
repetition) all the primitive roots modulo 2pa. Hence the
theorem follows as stated above.
primitive roots modulo yyl 71
§ 38. Recapitulation
The results which we have obtained in §§ 34-37 inclusive
may be gathered into the following theorem:
In order that there shall exist primitive roots modulo m, it is
necessary and sufficient that m shall have one of the values
m=i, 2, 4, pa, 2pa
where p is an odd prime and a is a positive integer.
If m has one of these values then the number of primitive roots
modulo m is <t>{4>{m) } .
§39. Primitive X-roots
In the preceding sections of this chapter we have developed
the theory of primitive roots in the way in which it is usually
presented. But if one approaches the subject from a more
general point of view the results which may be obtained are
more general and at the same time more elegant. It is our
purpose in this section to develop the more general theory.
We have seen that if a and m are any two relatively prime
positive integers, then
aHm) = j m0(J ^
Consequently there is no integer belonging modulo m to an
exponent greater than \(m). It is natural to enquire if there
are any integers a which belong to the exponent \(m). It turns
out that the question is to be answered in the affirmative, as
we shall show. Accordingly, we introduce the following defini-
tion:
Definition. If aHm) is the lowest power of a which is
congruent to 1 modulo m, a is said to be a primitive X-root
modulo m. We shall also say that it is a primitive X-root of
the congruence £x(m) = 1 mod m. To distinguish we may speak
of the usual primitive root as a primitive 0-root modulo m.
72 THEORY OF NUMBERS
From the theory of primitive 0-roots already developed
it follows that primitive X-roots always exist when m is a power
of any odd prime, and also when m= i, 2, 4; for, for such values
of m we have \(m) = <f>(m).
We shall next show that primitive X-roots exist when m = 2a,
a>2, .by showing that 5 is such a root. It is necessary and suf-
ficient to prove that 5 belongs modulo 2a to the exponent
2a_2 = X(2a). Let d be the exponent to which 5 belongs modulo
2a. Then from theorem II of § 32 it follows that d is a divisor
of 2a-2 = X(2a). Hence if d is different from 2a~2 it is 2a"3
or is a divisor, of 2a~3. Hence if we can show that fa"3 is not
congruent to 1 modulo 2a we will have proved that 5 belongs
to the exponent 2a~2. But, clearly,
52«-3 = (I+22)2«-3=I+2«-l+/.2aj
where / is an integer. Hence
52a_ ^1 mod 2a.
Hence 5 belongs modulo 2a to the exponent X(2a).
By means of these special results we are now in position to
prove readily the following general ,theorem which includes
them as special cases:
I. For every congruence of the form
xx(m) = i mod m
a solution g exists which is a primitive \-root, and for any such
solution g there are 4>{\(m)\ primitive roots congruent to powers
ofg- ^ .
If any primitive X-root g exists, gv is or is not a primitive
X-root according as v is or is not prime to \{m) ; and therefore
the number of primitive X-roots which are congruent to powers
of any such root g is <f>{\(m) J .
The existence of a primitive X-root in every case may easily
be shown by induction. In case wis a power of a prime the
theorem has already been established. We will suppose that
it is true when m is the product of powers of r different primes
PRIMITIVE ROOTS MODULO M 73
and show that it is true when m is the product of powers of
r-f-i different primes; from this will follow the theorem in
general.
Put m = p1a'p2a2 • • • prarparT^, n = piaip2a2 . . . prar,
and let h be a primitive X-root of
xHn) = i mod n. (i)
Then
h-\-ny
is a form of the same root if y is an integer.
Likewise, if c is any primitive X-root of
^r+V^imod^1 (2)
a form of this root is
where z is any integer.
Now, if y and z can be chosen so that
h+ny = c+parr+11z
the number in either member of this equation will be a common
primitive X-root of congruences (i) and (2); that is, a com-
mon primitive X-root of the two congruences may always be
obtained provided that the equation
Pfi . . . pSry-p
+ 1
has always a solution in which y and z are integers. That this
equation has such a solution follows readily from theorem
III of § 9; for, if c — h is replaced by 1, the new equation has a
solution y, z; and therefore for y and z we may take y = y(c — h),
z=~z(c — h).
Now let g be a common primitive X-root of congruences
(1) and (2) and write
gv=i mod m,
74 THEORY OF NUMBERS
where v is to be the smallest exponent for which the congruence
is true. Since g is a primitive X-root of (i) v is a multiple of
Mpiai . • -'.prar). Since g is a primitive X-root of (2) v is a
multiple of X (^^j1) . Hence it is a multiple of X(w). But
gx(m) = imodm; therefore v = \(m). That is, g is a primitive
X-root modulo m.
The theorem as stated now follows at once by induction.
There is nothing in the preceding argument to indicate
that the primitive X-roots modulo m are all in a single set
obtained by taking powers of some root g\ in fact it is not in
general true when m contains more than one prime factor.
By taking powers of a primitive X-root g modulo m one
obtains 0{X(m)J different primitive X-roots modulo m. It is
evident that if 7 is any one of these primitive X-roots, then the
same set is obtained again by taking the powers of 7. We may
say then that the set thus obtained is the set belonging to g.
II. 7/ X(w)>2 the product of the <f>{\(m)} primitive \-roots
in the set belonging to any primitive \-root g is congruent to 1
modulo m.
These primitive X-roots are
8> 8 ) 8 > • • • 5 8
where
I, Ci, C2) • * • , Cft
are the integers less than \(m) and prime to X(w). If any one
of these is c another is \(m)—c, since \(m)>2. Hence
1+C1+C2+ • • • +cM = o mod X(ra).
Therefore
gi+Cl+C2+... +cME=I mod w.
From this the theorem follows.
Corollary. The product of all the primitive \-roots modulo
m is congruent to 1 modulo m when \{m) > 2.
PRIMITIVE ROOTS MODULO M 75
EXERCISES
i. If *i is the largest value of x satisfying the equation \(x)=a, where a is
a given integer, then any solution x2 of the equation is a factor of X\.
2*. Obtain an effective rule for solving the equation \{x)—a.
3*. Obtain an effective rule for solving the equation <f>(x)=a.
4. A necessary and sufficient condition that ap_1 = i mod P for every in-
teger a prime to P is that P = i mod X(P).
5. If ap~l = i mod P for every a prime to P, then (1) P does not contain a
square factor other than 1, (2) P either is prime or contains at least three dif-
ferent prime factors.
6. Let p be a prime number. If a is a root of the congruence xd= 1 mod p and
a is a root of the congruence xs = i mod p, then aa is a root of the congruence
xds = i mod p. If a is a primitive root of the first congruence and a of the second
and if d and 5 are relatively prime, then aa is a primitive root of the congruence
xdb = i mod p.
CHAPTER VI
OTHER TOPICS
§ 40. Introduction
The theory of numbers is a vast discipline and no single
volume can adequately treat of it in all of its phases. A short
book can serve only as an introduction; but where the field
is so vast such an introduction is much needed. That is the
end which the present volume is intended to serve; and it
will best accomplish this end if, in addition to the detailed theory
already developed, some account is given of the various direc-
tions in which the matter might be carried further.
To do even this properly it is necessary to limit the number
of subjects considered. Consequently we shall at once lay
aside many topics of interest which would find a place in an
exhaustive treatise. We shall say nothing, for instance, about
the vast domain of algebraic numbers, even though this is one
of the most fascinating subjects in the whole field of mathe-
matics. Consequently, we shall not refer to any of the exten-
sive theory connected with the division of the circle into equal,
parts. Again, we shall leave unmentioned many topics con-
nected with the theory of positive integers; such, for instance,
is the frequency of prime numbers in the ordered system of
integers — a subject which contains in itself an extensive and
elegant theory.
In §§ 41-44 we shall speak briefly of each of the following
topics: theory of quadratic residues, Galois imaginaries, arith-
metic forms, analytical theory of numbers. Each of these alone
would require a considerable volume for its proper development.
All that we can do is to indicate the nature of the problem in
each case and in some cases to give a few of the fundamental
results.
76
OTHER TOPICS 77
In the remaining three sections we shall give a brief intro-
duction to the theory of Diophantine equations, developing
some of the more elementary properties of certain special
cases. We shall carry this far enough to indicate the nature
of the problem connected with the now famous Last Theorem
of Fermat. The earlier sections of this chapter are not required
as a preliminary to reading this latter part.
§41. Theory of Quadratic Residues
Let a and m he any two relatively prime integers. In § 31
we agreed to say that a is a quadratic residue modulo m or a
quadratic non-residue modulo m according as the congruence
x2 = a mod m
has or has not a solution. We saw that if m is chosen equal
to an odd prime number p, then a is a quadratic residue modulo
p or a quadratic non-residue modulo p according as
aKp-i) = ! or fli(P-D=_imo(J|,,
This is known as Euler's criterion.
It is convenient to employ the Legendre symbol
-;)
to denote the quadratic character of a with respect to p. This
symbol is to have the value +1 or" the value — 1 according
as a is a quadratic residue modulo p or a quadratic non-residue
modulo p. We shall now derive some of the fundamental prop-
erties of this symbol, understanding always that the numbers
in the numerator and the denominator are relatively prime.
From the definition of quadratic residues and non-residues
it is obvious that
($)-(
if a = b mod p. (1)
Pi
78 THEORY OF NUMBERS
It is easy to prove in general that
(?)(!)-(t)
(2)
This comes readily from Euler's criterion. We have *to con-
sider the three cases
(I)-- ©-+« ©-+■. ©-<
f)— • (|)—-
The method will be sufficiently illustrated by the treatment
of the last case. Here we have
aiiP-D^^j modp, ^Kp-D=_t mod/>.
Multiplying these two congruences together member by member
we have
O^-^imOd^, '.
whence
(f)--m-
as was to be proved.
If m is any number prime to p and we write m as the product
of factors
w = e.2*V<z'Y" • • •
where q', q", q/;;, ... are odd primes, a is zero or a positive
integer and e is + 1 or — 1 according as m is positive or negative,
we have
"Hmmm • • • ■ «
as one shows easily by repeated application of relation (2).
Obviously,
©-
OTHER TOPICS 79
Hence, it follows from (3) that we can readily determine the
quadratic character of m with respect to the odd prime p, that
is, the value of
■ •• ©:,. •
provided that we know the value of each of the expressions
where q is an odd prime.
The first of these can be evaluated at once by means of
Euler's criterion; for, we have
(f)-<
and hence
' =(_I)KP-D.
(f)-<-
Thus we have the following result: The number — 1 is a quad-
ratic residue of every prime number of the form 4& + 1 and
a quadratic non-residue of every prime number of the form
4^+3-
The value of the second symbol in (4) is given by the formula
(?)
= (_I)i(P-i).
The theorem contained in this equation may be stated in the
following words: The number 2 is a quadratic residue of every
prime number of either of the forms 8& + 1, 8^ + 7; it is a quad-
ratic non-residue of every prime number of either of the forms
8£+3, 8£ + 5.
The proof of this result is not so immediate as that of the
preceding one. To evaluate the third expression in (4) is still
more difficult. We shall omit the demonstration in both of
these cases. For the latter we have the very elegant relation
©©=<->
i(P-i)(a-i)
80 THEORY OF NUMBERS
This equation states the law which connects the quadratic
character of q with respect to p with the quadratic character
of p with respect to q. It is known as the Law of Quadratic
Reciprocity. About fifty proofs of it have been given. Its
history has been a very interesting one; see Bachmann's
Niedere Zahlentheorie, Teil I, pp. 180-318, especially pp.
200-206.
For a further account of this beautiful and interesting
subject we refer the reader to Bachmann, loc. cit., and to the
memoirs to which this author gives reference.
§ 42. Galois Imaginaries
If one is working in the domain of real numbers the equation
#2+i=o
has no solution; for there is no real number whose square is
— 1. If, however, one enlarges the " number system" so as
to include not only all real numbers but all complex numbers
as well, then it is true that every algebraic equation has a root.
It is on account of the existence of this theorem for the enlarged
domain that much of the general theory of algebra takes the
elegant form in which we know it.
The question naturally arises as to whether we can make a
similar extension in the case of congruences. The congruence
#2 = 3 mod 5
has no solution, if we employ the term solution in the sense in
which we have so far used it. But we may if we choose intro-
duce an imaginary quantity, or mark, j such that
jf2 = 3 mod 5,
just as in connection with trie equation #2+i=o we would
introduce the symbol i having the property expressed by the
equation
;2 = _ T
OTHER TOPICS 81
It is found to be possible to introduce in this way a general
set of imaginaries satisfying congruences with prime moduli;
and the new quantities or marks have the property of combining
according to the laws of algebra.
The quantities so introduced are called Galois imaginaries.
We cannot go into a development of the important theory
which is introduced in this way. We shall be content with
indicating two directions in which it leads.
In the first place there is the general Galois field theory
which is of fundamental importance in the study of certain
finite groups. It may be developed from the point of view
indicated here. An excellent exposition, along somewhat
different lines, is to be found in Dickson's Linear Groups with
an Exposition of the Galois Field Theory.
Again, the whole matter may be looked upon from the geo-
metric point of view. In this way we are led to the general
theory of finite geometries, that is, geometries in which there
is only a finite number of points. For a development of the
ideas which arise here see Veblen and Young's Projective
Geometry and the memoir by Veblen and Bussey in the Trans-
actions of the American Mathematical Society, vol. 7, pp.
241-259.
§ 43. Arithmetic Forms
The simplest arithmetic form is ax+b where a and b are
fixed integers different from zero and x is a variable integer.
By varying x in this case we have the terms of an arithmetic
progression. We have already referred to Dirichlet's cele-
brated theorem which asserts that the form ax+b has an infinite
number of prime values if only a and b are relatively prime.
This is an illustration of one type of theorem connected with
arithmetic forms in general, namely, those in which it is asserted
that numbers of a given form have in addition a given property.
Another type of theorem is illustrated by a result stated
in §41, provided that we look at that result in the proper
way. We saw that the number 2 is a quadratic residue of
82 THEORY OF NUMBERS
every prime of either of the forms Sk + 1 and &k + j and a quad-
ratic non-residue of every prime of either of the forms 8^+3
and 8^ + 5. We may state that result as follows: A given
prime number of either of the forms Sk-\-i and Sk + 7 is a divisor
of some number of the form x2 — 2, where x is an integer; no
prime number of either of the forms 8^+3 and 8^ + 5 is a
divisor of a number of the form x2 — 2, where x is an integer.
The result just stated is a theorem in a discipline of vast
extent, namely, the theory of quadratic forms. Here a large
number of questions arise among which are the following:
What numbers can be represented in a given form? What is
the character of the divisors of a given form? As a special
case of the first we have the question as to what numbers can
be represented as the sum of three squares. To this category
belong also the following two theorems: Every positive integer
is the sum of four squares of integers; every prime number of
the form 4^+1 may be represented (and in only one way) as
the sum of two squares.
For an extended development of the theory of quadratic
forms we refer the reader to Bachmann's Arithmetik der Quad-
ratischen Formen of which the first part has appeared in a
volume of nearly seven hundred pages.
It is clear that one may further extend the theory of arith-
metic forms by investigating the properties of those of the third
and higher degrees. Naturally the development of this subject
has not been carried so far as that of quadratic forms; but
there is a considerable number of memoirs devoted to various
parts of this extensive field, and especially to the consideration
of various special forms.
Probably the most interesting of these special forms are the
following :
n nn
niQti ** P n-l_l_ »-2/d_i_ 1 pn-1
a +j8 , ^=« +<* P + • • • +P >
a — p
where a and j8 are relatively prime integers, or, more generally,
where a and 0 are the roots of the quadratic equation
x2— ux+v — o where u and v are relatively prime integers. A
.. &<ftxA+* \ „ t4m^^^^. 83
development of the theory of these forms has been given by
the present author in a memoir published in 19 13 in the Annals
of Mathematics, vol. 13, pp. 30-70.
§ 44. Analytical Theory of Numbers
Let us consider the function
P(x)=-
I
\x\
Sp<l.
00 >
n (i-*2t)
fc=0
It
is clear that
we have
p(*)=
00
■ n
1
) I— X2*
= n (i+x2t
k=0
= 2 G(s)xs,
8=0
+X2
■2t+x3-
where G(o) = 1 and G(s) (for 5 greater than o) is the number of
ways in which the positive integer s may be separated into like
or distinct summands each of which is a power of 2.
We have readily
(i-xy2G(s)xs = (i-x)P(x)=P(x2)= I G(s)x2S;
8=0 s=0
whence
G(2s+i) =G(2s) =G(2s-i)+G(s), (A)
as one readily verifies by equating coefficients of like powers
of x. From this we have in particular
G(o) = i, G(i) = i, G(2) = 2, G(3) = 2,
G(4)=4, G(5)=4, G(6)=6, G(7)=6.
Thus in (-4) we have recurrence relations by means of which
we may readily reckon out the values of the number theoretic
function G(s). Thus we may determine the number of ways in
which a given positive integer 5 may be represented as a sum
of powers of 2.
84 THEORY OF NUMBERS
We have given this example as an elementary illustration
of the analytical theory of numbers, that is, of that part of the
theory of numbers in which one employs (as above) the theory
of a continuous variable or some analogous theory in order to
derive properties of sets of integers. This general subject
has been developed in several directions. For a systematic
account of it the reader is referred to Bachmann's Analytische
Zahlentheorie.
§ 45. Diophantine Equations
If f(x, y,z, . . . ) is a polynomial in the variables x, y, z, . . .
with integral coefficients, then the equation
f(x, y, z, . . . )=o
is called a Diophantine equation when we look at it from the
point of view of determining the integers (or the positive in-
tegers) x, y, z, . . . which satisfy it. Similarly, if we have
several such functions /*(#, y, z, . . . ), in number less than
the number of variables x, y, z, . . . , then the set of equations
ft(x, y,z, . . .)=o, i = i, 2, . . . ,
is said to be a Diophantine system of equations. Any set of
integers x, y, z, . . . which satisfies the equation [system]
is said to be a solution of the equation [system].
We may likewise define Diophantine inequalities by replac-
ing the sign of equality above by the sign of inequality. But
little has been done toward developing a theory of Diophantine
inequalities. Even for Diophantine equations the theory is
in a rather fragmentary state.
In the next two sections we shall illustrate the nature of
the ideas and the methods of the theory of Dipohantine equa-
tions by developing some, of the results for two important
special cases.
OTHER TOPICS 85
§ 46. Pythagorean Triangles
Definitions. If three positive integers x, y, z satisfy
the relation
x2+y2 = z2 (1)
they are said to form a Pythagorean triangle or a numerical
right triangle; z is called the hypotenuse of the triangle and x
and y are called its legs. The area of the triangle is said to be
\xy.
We shall determine the general form of the integers x, y, z,
such that equation (i) may be satisfied. Let us denote by v
the greatest common divisor of x and y in a particular solution
of (1). Then v is a divisor of z and we may write
x = vu, y = w, z = vw.
Substituting these values in (i) and reducing we have
u2+v2=w2, (2)
where u, v, w are obviously prime each to each, since u and v
have the greatest common divisor i.
Now an odd square is of the form 4& + 1. Hence the sum
of two odd squares is divisible by 2 but not by 4; and therefore
the sum of two odd squares cannot be a square. Hence one
of the numbers u, v is even. Suppose that u is even and write
equation (2) in the form
u2 = (w— v)(w+v). (3)
Every common divisor oi w—v and w+v is a divisor of their
difference 2V. Therefore, since w and v are relatively prime,
it follows that 2 is the greatest common divisor oi w—v and
w+v. Then from (3) we see that each of these numbers is
twice a square, so that we may write
w— V = 2b2, w+v = 2a2
86 THEORY OF NUMBERS
where a and b are relatively prime integers. From these two
equations and equation (3) we have
w = a2+b2, v = a2 — b2, u = 2ab. (4)
Since u and v are relatively prime it is evident that one of the
numbers a, b is even and the other odd.
The forms of u, v, w given in (4) are necessary in order that
(2) may be satisfied. A direct substitution in (2) shows that
this equation is indeed satisfied by these values. Hence we
have in (4) the general solution of (2) where u is restricted to
be even. A similar solution would be obtained if v were re-
stricted to be even. Therefore the general solution of (1) is
x = 2vab, y = v(a2 — b2), z = v(a2+b2)
and
x = v(a2 — b2), y = 2vab, z = v(a2+b2)
where a, b, v are arbitrary integers except that a and b are rela-
tively prime and one of them is even and the other odd.
By means of this general solution of (1) we shall now prove
the following theorem:
I. There do not exist integers m, n, p, q, all different from
zero, such that
q2-\-n2 = m2, m2-\-n2=p2. (5)
It is obvious that an equivalent theorem is the following:
II. There do not exist integers m, n, p} q, all different from
zero such that
p2+q2 = 2m2, p2—q2 — 2n2. (6)
Obviously, we may without loss of generality take m, n,
p, q to be positive; and this we do.
The method of proof is to assume the existence of integers
satisfying equations (5) and (6) and to show that we are thus
led to a contradiction. The argument we give is an illustra-
tion of Fermat's famous method of " infinite descent.' '
If any two of the numbers p, q, m, n have a common prime
factor /, it follows at once from (5) and (6) that all four of them
OTHER TOPICS 87
have this factor. For, consider an equation in (5) or in (6)
in which these two numbers occur; this equation contains a
third number, and it is readily seen that this third number is
divisible by /. Then from one of the equations containing the
fourth number it follows that this fourth number is divisible
by /. Now let us divide each equation of system (6) through
by t2; the resulting system is of the same form as (6). If
any two numbers in this resulting system have a common prime
factor h, we may divide through by h2; and so on. Hence if
a pair of simultaneous equations (6) exists then there exists a
pair of equations of the same form in which no two of the num-
bers m, n, p, q have a common factor other than unity. Let
this system of equations be
pi2+qi2 = 2Mi2, pi2-qi2 = 2tii2. (7)
From the first equation in (7) it follows that pi and q\ are
both even or both odd; and, since they are relatively prime,
it follows that they are both odd. Evidently pi>qi. Then
we may write
pi=qi + 2<x,
where a is a positive integer. If we substitute this value of
pi in the first equation of (7), the result may readily be put in
the form
(qi+a)2+a2 = mi2. (8)
Since qi and mi have no common prime factor it is easy to see
from this equation that a is prime to both qi and mi, and hence
that no two of the numbers qi -\-a, a, mi have a common factor.
Now we have seen that if a, b, c are positive integers no two
of which have a common prime factor, while
a2+b2 = c2,
then there exist relatively prime integers r and s} r>s, such
that
c = r2-\-s2, a = 2rs, b = r2—s2
or
c = r2+s2, a = r2 — s2, b = 2rs.
88 THEORY OF NUMBERS
Hence from (8) we see that we may write
qi-\-a = 2rs, a=r2—s2 (9)
or
qi+a = r2 — s2, a = 2TS. (10)
In either case we have
pi2-qi2 = (pi-qi)(pi+qi) = 2a-2(q1+a)=Srs(r2-s2).
If we substitute in the second equation of (7) and divide by 2
we have
4rs(r2— s2)=n\2.
From this equation and the fact that r and 5 are relatively
prime it follows at once that r, s, r2 — s2 are all square numbers;
say,
r = u2, s = v2, r2—s2 = w2.
Now r—s and r+s can have no common factor other than
1 or 2 ; hence from
w2 = (y2 _ S2^ — (y _ 5) (y +s^ = (u2 _ v2^ (u2 _^_v2^
we see that either
U2 ~\-V2 = 2Wi2 , U2 — V2 = 2W22 (11)
or
U2~\-V2=Wi2, U2—V2 = W22.
And if it is the latter case which arises, then
Wi2 -\-W22 = 2U2 , Wi2— W22 = 2V2. (12)
Hence, assuming equations of the form (6) we are led either to
equations (n) or to equations (12); that is, we are led to new
equations of the form with which we started. Let us write
the equations thus:
p22+q22 = 2tn22, p22-q22 = 2n22; (13)
that is, system (13) is identical with that one of systems (11),
(12) which actually arises.
OTHER TOPICS 89
Now from (9) and (10) and the relations pi=qi + 2a, r>s,
we see that
pi = 2rs+r2-s2>2s2+r2-s2 = r2+s2 = u*+v*.
Hence u<p\. Also,
wi2 ^w2 = r+s <r2+s2.
Hence wi<pi. Since u and w\ are both less than pi it follows
that p2 is less than pi. Hence, obviously, p2<p- Moreover,
it is clear that all the numbers ^2, £2, W2, n^ are different from
zero.
' From these results we have the following conclusion : If
we assume a system of the form (6) we are led to a new system
(13) of the same form; and in the new system p2 is less than p.
Now if we start with (13) and carry out a similar argument
we shall be led to a new system
P32 +qs2 = 2 w32, ps2 - qz2 = 2fl32,
with the relation pz<p2\ starting from this last system we shall
be led to a new one of the same form, with a similar relation of
inequality; and so on ad infinitum. But, .since there is only
a finite number of positive integers less than the given positive
integer p this is impossible. We are thus led to a contradic-
tion; whence we conclude at once to the truth of II and like-
wise of I.
By means of theorems I and II we may readily prove the
following theorem:
III. The area of a numerical right triangle is never a square
number.
Let the sides and hypotenuse of a numerical right triangle
be u, v, w, respectively. The area of this triangle is \ui). If
we assume this to be a square number t2 we shall have the
following simultaneous Diophantine equations
U2+V2=W2, UV = 2t2. (14)
90 THEORY OF NUMBERS
We shall prove our theorem by showing that the assumption
of such a system leads to a contradiction.
If any two of the numbers u, v, w have a common prime
factor p then the remaining one also has this factor, as one
sees readily from the first equation in (14). From the second
equation in (14) it follows that t also has the same factor. Then
if we put u = pui, v — pvij w = pwi, t = ph, we have
Ui2 +V\2 = Wi2, U1V1 = 2ti2,
a system of the same form as (14). It is clear that we may
start with this new system and proceed in the same manner as
before, and so on, until we arrive at a system
U2 + V2=W2, UV = 2~t2, (15)
where u, v, w are prime each to each.
Now the general solution of the first equation (15) may be
written in one of the forms
u = 2ab, v = a2 — b2, w = a2-\-b2,
u = a2b2, v = 2ab, w = a2+b2.
Then from the second equation in (15) we have
}2 = ab(a2-b2)=ab(a-b)(a+b).
It is easy to see that no two of the numbers a, b, a — b, a-\-b
in the last member of this equation have a common factor; for,
if so, u and v would have a common factor, contrary to hypoth-
esis. Hence each of these four numbers is a square. That is,
we have equations of the form
a = tn2, b = n2, a+b = p2, a — b=q2\
whence m2—n2 = q2, m2-\-n2= p2.
But, according to theorem I, no such system of equations can
exist. That is, the assumption of equations (14) leads to a
contradiction. Hence the theorem follows as stated above.
OTHER TOPICS 91
§47. The Equation xn+yn = zn.
The following theorem, which is commonly known as Fer-
mat's Last Theorem, was stated without proof by Fermat
in the seventeenth century:
// n is an integer greater than 2 there do not exist integers
x, y, z, all different from zero, such that
xn+yn = z\ (1)
No general proof of this theorem has yet been given. For
various special values of n the proof has been found; in par-
ticular, for every value of n not greater than 100.
In the study of equation (1) it is convenient to make some
preliminary reductions. If there exists any particular solution
of (1) there exists also a solution in which x, y, z are prime
each to each, as one may show readily by the method employed
in the first part of § 46. Hence in proving the impossibility
of equation (1) it is sufficient to treat only the case in which
x, y, z are prime each to each.
Again, since n is greater than 2 it must contain the factor
4 or an odd prime factor p. If n contains the factor p we write
n = mp, whence we have
(xmy+(ym)p=(zmy.
If n contains the factor 4 we write n = 4m, whence we have
(xmy+(ymy={zmy.
From this we see that in order to prove the impossibility of
(1) in general it is sufficient to prove it for the special cases when
n is 4 and when n is an odd prime p. For the latter case the
proof has not been found. For the former case we give a
proof below. The theorem may be stated as follows:
I. There are no integers x, y, z, all different from zero, such
that
92 THEORY OF NUMBERS
This is obviously a special case of the more general theorem :
II. There are no integers p, q, a, all different from zero, such
that
P4-q4=a2. (2)
The latter theorem is readily proved by means of theorem
III of §46. For, if we assume an equation of the form (2),
we have
(p*-q*)p2q2 = PY<*2- (3)
But, obviously,
(2^Y)2+(^4-g4)2 = (^+g4)2. (4)
Now, from (3) we see that the numerical right triangle deter-
mined by (4) has its area p2q2(p4: — q4) equal to the square num-
ber p2q2a2. But this is impossible. Hence no equation of the
form (2) exists.
EXERCISES
1. Show that the equation a4+4/34 = 72 is impossible in integers a, /3, 7, all
of which are different from zero.
2. Show that the system p2—q2 = km2, p2-\-q2 = kn2 is impossible in integers
p, q, k, m, n, all of which are different from zero.
3*. Show that neither of the equations m4— 4tii=dbt2 is possible in integers
m, n, t, all of which are different from zero.
4*. Prove that the area of a numerical right triangle is not twice a square
number.
5*. Prove that the equation mi-\-n*=a2 is not possible in integers m, n, a all
of which are different from zero.
6*. In the numerical right triangle a2+b2=c2, not more than one of the num-
bers a, b, c is a square.
7. Prove that the equation x -\-y =z2 implies an equation of the form
m +n=2* t .
8. Find the general solution in integers of the equation x2+2y2=t2.
9. Find the general solution in integers of the equation x2+y*=zA.
10. Obtain solutions of each of the following Diophantine equations:
xz-\-y3+zz = 2t3,
x*+y4+4z* = t\
X*+V4+Z4=2t*.
INDEX
Algebraic numbers, 76.
Analytical theory of numbers, 83-84.
Arithmetic forms, 81-83.
Arithmetic progression, 13.
Bachmann, 80, 82, 84.
Bussey, 81.
Carmichael, 83.
Circle, Division of, 76.
Common divisors, 9, 18-20, 21.
multiples, 9, 20-21.
Composite numbers, 10. fc
Congruences, 37-46.
Linear, 43-46, 56.
Solution by trial, 39-40.
with prime modulus, 41-
43-
Descent, Infinite, 86.
Dickson, 81.
Diophantine equations, 84.
Dirichlet, 81.
Divisibility, 8.
Divisors of a numbers, 16, 17.
Equations, Diophantine, 84.
Equation xn+yn*=zn, 91-92.
Eratosthenes, n.
Euclid, Theorem of, 13.
Euclidian algorithm, 18.
Euler, 28, 48.
Euler's criterion, 59, 77.
<£-function, 30.
Exponent of an integer, 61-63.
Factorization theorem, 14.
Factors, 14, 16, 17, 18.
Fermat, 28, 48, 86.
Fermat's general theorem, 47, 63.
last theorem, 91.
simple theorem, 48, 55.
theorem extended, 52-54.
Forms, 81-83.
Fundamental notions, 7.
Galois imaginaries, 80.
Gauss, 37.
Greatest common factor, 18-20, 21.
Highest power of p in n !, 24-28.
Imaginaries of Galois, 80.
Indicator, 30-36.
of any integer, 32-34.
of a prime power, 30.
of a product, 30-32'.
Infinite descent, 86.
Mm), 53.
Law of quadratic reciprocity, 80.
Least common multiple, 20-21.
Legendre symbol, 77.
0O), 30-
Prime each to each, 9.
Prime numbers, 10, 12, 13, 28-29, 51)
"76, 81, 82.
Primitive roots, 61-75.
X-roots, 71-74.
<]>-roots, 71.
Pythagorean triangles, 85-90.
93
94
INDEX
Quadratic forms, 82.
Quadratic reciprocity, 80.
Quadratic residues, 57-60, 77-80.
Relatively prime, 10.
Residue, 37, 58.
Scales of notation, 22-24.
Sieve of Eratosthenes, 10.
Totient, 30.
Triangles, Numerical, 85.
Unit, 8.
Veblen, 81.
Wilson's theorem, 49-81.
Young, 81.
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