ftoiT
& 8& \%1
n. i nu eim i rnc
- [] JY}!/ '--r
8| 10 j*
ftos \ Cms
iQ v -r t?"
b'll Ifr
NAME
This book is due for return on or before the
last date shown above,
voM 'if
NUMBER THEORY AND ITS HISTORY
Abacist vs. Atgorismisl
From Gregor Rds<;h: Margarita Philosophic*
Strasbourg 1504
NUMBER THEORY
AND ITS HISTORY
By Oystein Ore
STEMMING PHOFKSSOtt tit M.VTHKM ATtCS
YALE UNIVERSITY
1NKW TOKK T OttO IN TO LOMHJiN
Mt G RAW- IT ILL BOOK COMPANY, TNC
! 9 4 8
NUMBER THEORY AND TTS HISTORY
Copyright, 1948, by the McCravv-1 1 ill Book Company, Tnn. Printed in the
United States of Ameriaa. All rights reserved. This book, or parts thereof,
may not be reproduced in any form without permission of the publishers.
VI
PREFACE
This book is based upon a course dealing with the theory of
numbers and its history which has been given at. Yale for several
years, Although the course has been attended primarily by college
students in their junior and senior years it has been open to all
interested. The lectures were intended to give the principal ideas
and methods of number theory as well as their historical back-
ground and development through the centuries. Most texts on
number theory contain inserted historical notes but in this course
I have attempted to obtain a presentation of the results of the
theory integrated more fully in the historical and cultural frame-
work. Number theory seems particularly suited to this form of
exposition, and in my experience it lias Contributed much to making
the subject more informative as well as more palatable to the
students.
Obviously, only some of the main problems of number theory
could be included in this book. In making a selection, topics of
systematic and historical importance capable of a simple presenta-
tion have been preferred. While many standard aspects of number
theory had to be discussed , the treatment is often new, and much
material has been added that lias not heretofore made its appear-
ance in texts. Also, in several instances I have found it desirable
to introduce and define modern algebraic concepts whose useful-
ness is readily explained by the context.
The questions of number theory are of importance not only to
mathematicians. Now, as in earlier days, these problems seem to
possess a particular attraction for many laymen, and number
theory is notable as one of the few fields of mathematics where the
suggestions and conjectures of amateurs or nonprofessional mathe-
maticians have exerted an appreciable influence. It may be men-
tioned incidentally that there have been few college classes that
I can recall in which there were not to be found some students
VI PREFACE
who had already played with the strange properties of numbers.
To make the theory available to readers whose mathematical
knowledge may be limited, every effort has been made to reduce
to a minimum the technical complications and mathematical
requirements of the presentation. Thus, the book is of a more
elementary character than many previous texts, and for the under-
standing of a greater part of the subject matter a knowledge of
the simplest algebraic rules should be sufficient. Only in some of
the later chapters has a more extended familiarity with mathe-
matical manipulations been presupposed.
T am indebted to Prof. Otto Neugcbauer for valuable comments
on the historical material and to Paul T. Bateman for numerous
suggestions for mathematical improvements that have been em-
bodied in the text. In reading the proofs I was assisted by M.
Gerstenhabcr and E. V. Schenkman, who have also checked the
numerical computations.
Oybtein Ore
NEW Havkx, Con's 7 .
August, 1948
CONTENTS
Preface
Chapter 1. Counting and Recording of Numbers
Number and counting , 1
Basic number groups , . 1
The number systems 2
Large numbers , 4
5. Finger numbers .,...,.,.. 5
fi. Recordings of numbers 6
7. Writing of numbers , , 8
8. Calculations 14
Positional numeral systems 16
if. Hindu-Arabic numerals 1!)
Chapter 8, Properties of Numbers. Division
1. Number theory and numerology . 25
2. Multiples and divisors 2S
3. Division and remainders 30
4. Number systems 34
5. Binary number systems 37
Chapter S. Euclid's Algorism
[ 1, Greatest common divisor. Euclid's algorism .......... 41
. 2. The division lemma . 44
3. Least common multiple . 45
I 4. Greatest common divisor and least common multiple for several
1 numbers 47
Chapter / h Prime Numbers
1. Prime numbers and the prime factorization theorem 50
2. Determination of prime factors 52
yuj CONTENTS
3. Factor tables * ' ' 53
4. For mat' a factorization method - • ■ ■ «4
5. Euler's factorization method . ™
$. The sieve of Eratosthenes
69
7. Mersenne arid Format primes -
8. The distribution of primes **>
Chapter B, fh$ Aliquot Parts
1. The divisors of a number
2. Perfect numbers . . . .
3. Amicable numhei
Sfi
. yi
06
4. Greatest common .divisor and least common multiple 100
5. Euler's function 10S)
Chapter 0. Indeterminate Problems
1. Problems and puzzles
2. Indeterminate problems .- . , .
8. Problems with two unknowns * 124
4. Problems with several unknowns , 131
Chapter 7. Theory of Linear Indeterminate Problems
1. Theory of linear indeterminate equations wil.li two unknowns ... H2
2. Linear indeterminate equations in several unknowns 153
3. Classification of systems of numbers , . . . , - - . 158
Chapter 8. Diopkantine Problem*
1. The Pythagorean triangle • li86
2. The Plimpton Library tablet l«J
3. Diophantos of Alexandria . , 1 71)
4. Al-Karkhi and Leonardo Pisano l8->
5. From Diophantos to Format ***
6. The method of infinite descent w?
7. Fennat's last theorem , ^
CONTENTS ix
Chapter y. Cm§rmm&&
J. The Disquisitiones arithmetieae . ,..«.,, 209
2. Trie propeHios of fiongrueiices 211
3. Residue systems 213
4. Operations with congruences ,,,,,....,.,..... 216
5. Casting out nines 225
Chapter 10. Analysis of Congruences
1. Algebraic congruences . 234
2. Linear congruences . , - 8f0
3. Simultaneous congruences ami the Chinese remainder theorem . 24J1
4. Further study of algebraic congruences ,..,.,.,...,. 249
Chapter 11. Wilson's Theorem ami Its Consequences
1. Wilson's theorem 250
2. Gauss's generalization of Wilson's theorem 263
3. Representations of numbers m the sum of two squares 267
Chapter IS- Ruler's Theorem and. lis Consequence*
L Euler's theorem 272
2. Format's theorem 277
3. Exponents of numbers 279
4. Primitive roots for primes 2S4
5. Primitive roots for powers of primes .,.....,,.,... 2H5
6. Universal exponents 290
7. Indices 291
S. -Number theory and the splicing of telephone cables 302
Chapter 13. Theor-/ of Decimal tizpan-sions
1. Decimal fractions ,,,,,..,..,.... 311
2. The properties of decimal fractious 315
Chapter 14. The Converse of Fer mat's Tficorem
1. The converse of Fermat'a theorem 326
2. Numbers with the Format property 331
x CONTENTS
Chapter tS. TM Clawiral Crm,strudim Problems
1. The classical r;on^trud.ion problems ™
2. The construction of regular polygons 3-16
3. Examples of cnnstruntibJo polygons 3o2
Supplement, .......,.-■ ■"<'
Bibliography. . , - ' iml
General Name Index * ■ - - "' (,i
Subject Index 3(io
CHAPTER 1
COUNTING AND RECORDING OF NUMBERS
1-1. Numbers and counting. All the various forms of human
culture and human society, even the most rudimentary types,
seem to require some concept of number and some process for
counting. According to the anthropologists, every people has
some terminology for the first numbers, although in the most
primitive tribes this may not extend beyond two or three. In a
general way one can say that the process of counting consists in
matching the objects to be counted with some familiar set of
objects like fingers, toes, pebbles, sticks, notches, or the number
words. It may be observed that the counting process often goes
considerably beyond the existing terms for numerals in the
language.
1-2. Basic number groups. Almost all people seem to have
used their fingers as /the most convenient and natural counters.
In many languages this is easily recognized in the number termi-
nology. In English we still use the term digits for the numerals.
For numbers exceeding 10 the toes have quite commonly been
used as further counters.
Very early in the cultural development it became necessary to
perform more extensive counts to determine the number of cattle,
of friends and foes, of days and years, and so on. To handle
larger figures the counting process must be systematized. The
first step in this direction consists in arranging the numbers into
convenient groups. The choice of such basic groups depends
naturally on the matching process used in counting.
The great preponderance of people use a basic decimal or decadic
group of 10 objects, as one should expect from counting on the
1
2 NUMBER THEORY AND ITS HISTORY
fingers. The word for 10 often signifies one man. Quinary systems
based on groups of 5 or one hand also occur, but the vigesimal
systems based on a 20 group are much more common, corresponding
of course to a complete count of fingers and toes. Among
the American Indian peoples the vigesimal system was in wide-
spread use; best known is the well-developed Mayan system.
One finds traces of a 20 system in many other languages. We
still count in scores. The French quatre-vingt for 80 is a remnant
of a previously more extensive 20 count. In Danish the 20
system is still used systematically for the names of numbers less
than 100.
The largest known basic number, 60, is found in the Babylonian
sexagesimal system. It is difficult to explain the reasons for such
a large unit group. It has been suggested by several authors that
it is the result of a merger of two different number systems. We
still use this system when measuring time and angles in minutes
and seconds. "Other basic numbers than those mentioned here
are quite rare. We may detect a trace of a 12 or duodecimal
system in our counts in dozens and gross. Certain African tribes
use basic groups of 3 and 4. The binary or dyadic system, in which
2 or a pair is the basic concept, has been used in a rudimentary form
by Australian indigenes. The dyadic system is, however, a system
whose simple properties often have a special mathematical use-
fulness.
1-3. The number systems. When the basic counting group is
fixed, the numbers exceeding the first group would be obtained by
counting afresh in a new group, then another, and so on. For
instance, in a quinary sj^stem where the basic five group might be
called one h(and), one would count one h. and one, one h. and two,
2h. (10), 3h. and 2 (17), and so on. After one had reached five
hands (25), one might say hand of hands (h.h.) and begin over
again. So as an example, one would denote 66 by 2hh and 3h and
1, that is, 2 X 25 + 3 X 5 + 1. Clearly this process can be
extended indefinitely by introducing higher groups
hhh = 125 = 5 3 , hhhh = 625 = 5 4
COUNTING AND RECORDING OF NUMBERS 3
In this manner one arrives at a representation of any number as
an expression
a n • 5 n + an _x • 5 n_1 + • • • + a 2 • 5 2 + ai • 5 + a (1-1)
where each coefficient a,{ is one of the numbers 0, 1, 2, 3, 4.
To be quite correct, one should observe that this particular
example historically is fictitious, since no people is known to have
developed and used a completely general system (1-1) with the
base 5. But this systematic procedure for the construction of a
number system was certainly the guiding principle in the evolution
of our decadic number system and of many other systems. To
confirm this assertion further one can turn to the philological
analysis of our number terms. Through the laws of comparative
linguistics one can trace a word like eleven to one left over, and
similarly twelve to two over. There is some indication that our
fundamental word ten may be derived from an Indo-European root
meaning two hands. The word hundred comes from an original
term ten times (ten). It is further interesting to note that the
names for thousand are unrelated in the various main branches of
the Indo-European languages; hence it is probably a rather late
construction. The word itself seems to be derived from a Proto-
Germanic term signifying great hundred.
In our decadic system all numbers are put in a form analogous
to (1-1)
a n • 10 w H + a 2 • 10 2 + a x • 10 + a (1-2)
where the coefficients take values from to 9. In general, in the
subsequent chapters, we shall understand by a number system with
the base b a system in which we represent the numbers in the form
a n • b n + • • • + a 2 • b 2 + ax • b + a (1-3)
where the coefficients a* are numbers from to b — 1.
It should be mentioned that relatively few peoples developed
their number systems to this perfection. Also, in many languages
one finds other methods for the construction of numbers. As an
example of irregular construction let us mention that in Welsh the
4 NUMBER THEORY AND ITS HISTORY
number words from 15 to 19 indicate 15, 15 + 1, 15 + 2, 2 X 9,
15 + 4. Subtraction occurs often as a method; for instance, in
Latin, un-de-viginti = 20 — 1 = 19, duo-de-sexaginta = 60 — 2 =
58. Similar forms exist in Greek, Hindu, Mayan, and other
languages.
The Mayan number system was developed to unusually high
levels, but the system has one peculiar irregularity. The basic
group is 20, but the group of second order is not 20 X 20 = 400
as one should expect, but 20 X 18 = 360. This appears to be
connected with the division of the Mayan year into 18 months
each consisting of 20 days, supplemented with 5 extra days. The
higher groups in the system are
360 X 20, 360 X 20 2 , • • •
1-4. Large numbers. As one looks at the development of
number systems in retrospect it seems fairly simple to construct
arbitrarily large numbers. However, in most systems the span of
numbers actually used is very limited. Everyday life does not
require very large numbers, and in many languages the number
names do not go beyond thousands or even hundreds. We
mentioned above that the term one thousand seems to have made a
relatively late appearance in the Indo-European languages. The
Greeks usually stopped at a myriad or ten thousand. For a long
period the Romans did not have names or symbols for groups
above 100,000. There exists in Rome an inscription on the
Columna Rostrata commemorating the victory over Carthage at
Mylae in the year 260 b.c. in which 31 symbols for 100,000 were
repeated to signify 3,100,000. The Hindus had a peculiar attrac-
tion to large numbers, and immense figures occur commonly in
their mythological tales and also in many of their algebraic
problems. As a consequence, there existed particular names for
the higher decadic groups to very great powers of 10. For instance,
in a myth from the life of Buddha one finds the denominations up
to 10 153 .
Even our own number system has not been developed system-
atically to this extent. The word for one million is a fairly recent
COUNTING AND RECORDING OF NUMBERS 5
construction, which seems to have originated in Italy around a.d.
1400. The concept one billion has not found its final niche in our
system. In American and sometimes in French terminology this
means one thousand millions (10 9 ) while in most other countries
of the world one billion is one million millions (10 12 ), while
one thousand millions is called a milliard. It is probably only
through the expenditures of the world wars that numbers of this
size have reached such common use that confusion is likely to
occur. When a billion is defined to be a thousand millions a
trillion becomes one thousand billions (10 12 ), a quadrillion one
thousand trillions, and so on. On the other hand when a billion
is one million millions, one million billions is a trillion (10 18 ), one
million trillions is a quadrillion (10 24 ), and so on. While' this
discrepancy is not apt to cause any serious misunderstandings in
everyday life, some universal agreement on usage and nomen-
clature would, nevertheless, be desirable.
The intellectual effort that lies behind a systematic extension of
the number system is well illustrated by the fact that Archimedes
(278-212 b.c), the most advanced Greek mathematician, deems
it worth while to devote a whole treatise, The Sand Reckoning, to
this purpose. This work is addressed to his relative, King Gelon
of Syracuse, and begins as follows:
There are some, King Gelon, who think that the number of grains of
sand is infinite in multitude; and I mean by the sand not only that
which exists about Syracuse and the rest of Sicily, but also that which is
found in every region, whether inhabited or uninhabited. Again there
are some, who, without regarding it as infinite, nevertheless think that
no number has been named which is great enough to exceed its size.
Under this guise of aiming at finding a number exceeding the
totality of grains of sand in the universe, as then known, Archi-
medes proceeds to construct a systematic enumeration method for
arbitrarily high numbers.
1-5. Finger numbers. For the communication of numbers
from one individual to another it is often desirable to have some
other representation than the vocal expressions of the number
6 NUMBER THEORY AND ITS HISTORY
names in the language. We now mainly use written numbers, a
representation which we shall study subsequently. Before the
advent of a fairly general writing ability the finger numbers were
widely used as a universal numerical language. The numbers
were indicated by means of different positions of fingers and hands.
In a rudimentary way we still occasionally express numbers by
our fingers. The finger numbers were in use in Europe both in
the classical period and in the Middle Ages; they were used by the
Greeks, Romans, Arabs, Hindus, and many other people. The
human figures in ancient drawings and statues often show peculiar
finger positions which denote numbers. For instance, Pliny states
that the statue of Janus on the Forum in Rome represented the
number 365, the days in the year, on its fingers.
In the Orient the finger numbers are still in common use. They
enable buyers and sellers in the bazaars to bargain about prices
independent of language differences. When the bargainers cover
their hands with a piece of cloth, the finger numbers have the
added advantage that the negotiations are secret to other parties.
Our best information about finger numbers in early times is due
to the works of the Venerable Bede (a.d. 673-735), an English
Benedictine monk from the cloisters in Wearmouth and Jarrow.
His treatise De temporum ratione deals with the rules for calculating
the date of Easter, and as an introduction it contains a description
of the use of finger numbers (Fig. 1-1) . The finger numbers were
probablv only in actual use for fairly moderate figures. Bede's
numbers have a natural limit of 10,000, but he enlarges the method
rather artificially so that it becomes possible to express numbers
up to 1,000,000. To some limited extent it was possible to cal-
culate with finger numbers. In Europe they seem to have dis-
appeared gradually with the ascendency of the Hindu-Arabic
number system.
1-6. Recordings of numbers. Neither the spoken numbers nor
the finger numbers have any permanency. To preserve numbers
for the purpose of records it is necessary to have other representa-
tions. Furthermore, without some memory aids the performance
of calculations is extremely difficult.
COUNTING AND RECORDING OF NUMBERS
Fig. 1-1. Finger numbers. {From. Luca di Burgo Pacioli, Summa de arith-
metica geometria, second edition, Venice, 1523. Courtesy of D. E. Smith Collection,
Columbia University.)
8 NUMBER THEORY AND ITS HISTORY
Many procedures have been devised to record numbers. The
method of representing numbers by means of knots tied on strings
has been used quite widely, in ancient China and on some of the
South Sea Islands, and the quipus of the Incas in Peru are well
known. In some localities split bamboo sticks have served as
number records.
The most natural method for such records seems to con-
sist in letting the counting process proceed by indicating each
individual item through a mark on some suitable permanent
material; for instance, dots or lines drawn in clay or on stones,
s:ratches, notches, or scores on wooden sticks, chalk marks on
slate or boards, and, of course, our present method of check marks
on writing material.
The use of wooden tallies for recordings of numbers has been
common in most European countries and in isolated districts it
still occurs for special purposes. The English words score and
count, from computare (putare, to cut), point to such methods. An
important function of the tallies was to serve as contracts. In
this case the tallies were ordinarily made in duplicate, one for each
party, obtained by splitting a single piece of wood in two. Fraud-
ulent changes were prevented quite effectively by cutting the
number figures simultaneously over both parts. This system
reached its highest level in the well-known Exchequer tallies, which
formed an essential part of the British official accounting system
from the twelfth century on (Fig. 1-2). On the Exchequer tallies
the two pieces were unequal; the main piece, called the stock,
served as a receipt while the separated, thinner leaf or foil was the
record of payment. This tally system remained legally valid in
England until the year 1826, and it had its official funeral pyre in
1834 when the burning of the accumulated tallies resulted in the
fire that destroyed the old Parliament buildings.
1-7. Writing of numbers. The use; of marks or notches to
denote numbers is clearly a primitive form of writing, and it is
likely to have been one of the first attempts in this direction. One
can still see traces of this original procedure in many systems of
number writing, for instance, quite plainly in the Roman numerals
COUNTING AND RECORDING OF NUMBERS 9
; s ■'?'">,.
a wggggafrjg
Fig. 1-2. Exchequer Tallies. (From H. Jenkimon, Exchequer Tallies, Archae-
logica, London, 1011. Courtesy of Society of Antiquaries, London.)
10 NUMBER THEORY AND ITS HISTORY
I, II, III, IIII. To facilitate the reading of the tally marks when
they became numerous, each basic group would naturally be indi-
cated in some special way, for instance, by a cross-notch, which
we still use. Through this simple procedure one has already
arrived at the essential principle of some of the most important
systems of numerals. For the purpose of classification they may
be called simple grouping systems. Let us illustrate by a few
examples:
EARLY EGYPTIAN NUMERALS (3400 B.C.)
(Hieroglyphic)
10 100
ii ::: n
i 1 1
E X o mp ,e ; fniWnS'.V =13,545
Quite familiar are the symbols of the Roman system:
ROMAN NUMERALS
1
2
5
10 50 100
500
1,000
I
II
V
X L C
D
M
Example: MDCCCXXVII = 1,827.
The Roman symbols corresponding to 50 and 500 form inter-
mediate groupings within the basic decimal system, and they serve
to clarify and simplify the writing of numbers. The subtraction
principle in Roman numerals whereby a smaller unit preceding a
higher one indicates subtraction (for instance, IX = 9, IV = 4),
also shortens the representation. It may be mentioned that this
COUNTING AND RECORDING OF NUMBERS 11
use of subtraction in a systematic manner is an innovation of the
last few centuries; in the classical period or even in medieval times
it was used only rarely. Similar simplifications through subtrac-
tive notations occur in other numeral systems.
The Herodianic Greek numerals belong in principle to the simple
grouping type:
ATTIC OR HERODIANIC GREEK NUMERALS
I 5 10 100 1000 10,000
I r Z\ H X M
<-+■■ xph h h z\ ^r 1 1 ••»
These symbols are derived from the initials of the Greek numbers:
nENTE (5), AEKA (10), HKATON (100)
XIAI02 (1,000), MTPIOS (10,000)
The simple grouping system in several instances developed into
( a type of numeration that may be called a multiplicative grouping
system. In such systems one has special ciphers for the numbers
in the basic group, e.g., 1, 2, . . . , 9, and a second class of symbols
for the higher groups, e.g.,
10 = t, 100 = h, 1,000 = th,-->
The ciphers would then be used multiplicati very to show how many
of the higher groups should be indicated. This would lead to
representations of the type of the example
3,297 = 3th 2h 9t 7
The traditional Chinese- Japanese numeral system is a multipli-
12 NUMBER THEORY AND ITS HISTORY
cative grouping as shown in the illustration. It should be noted
that the writing is vertical instead of horizontal.
6 yr
7-t
CHINESE-JAPANESE NUMERALS
Example: 3468
1 — 10 ")"
2 ^~ 100 ^~
3 ^- 1000 -f"
4 23
<ZS
CHINESE MERCANTILE SYSTEM
I -10
•) ") x tf 1 ± i k +
Fig. 1-3.
A third method of number writing may be called a ciphered
numeral system. In the case of a decadic system one would denote
the numbers from 1 to 9 by special symbols; similarly the multiples
of 10 up to 90, the hundreds up to 900, and so on, would have their
COUNTING AND RECORDING OF NUMBERS 13
individual signs. All numbers can then be represented as a com-
bination of such symbols in a very compact form. The Egyptian
hieroglyphic number writing later developed into the hieratic and
demotic systems, which most nearly can be classified as being
ciphered. Other examples are afforded by Coptic and Hindu
Brahmi numerals.
HIERATIC NUMERALS
1-9 I II in — «j in ^ -=- ((«
10-90 A A A — *A J}) >1 ^ ^
100-900
y V ") -) •") m) V ?» (y
The usual Greek numerals are of a type that may be called
alphabetic. The Greeks ciphered by means of the letters of the
alphabet supplemented by a few symbols borrowed from the
Semitic.
ALPHABETIC GREEK NUMERALS
1-9 <xj876£6 4 7
10-90 i k A /a v j o 7r 9
100-900 p <r t v <p X y/ o> 7?
Example '.y>ju/&= 742
The higher units were obtained by special marks on the lower ones;
for instance,
,a = 1,000, ,(3 = 2,000
Alphabet numerals were used also by the Hebrews and the
Syrians and in early Arabic and Gothic writing.
14
NUMBER THEORY AND ITS HISTORY
1-8. Calculations. Most of the numeral systems that we have
mentioned in the preceding are not well suited for calculations,
such as addition, subtraction, multiplication, and division. The
peculiar difficulties that one encounters in performing these oper-
Fig. 1-4. Roman abacus. (From Marcus Welser, Opera historica et philologica,
Niirnberg, 1682).
ations can easily be ascertained in the; familiar system of Roman
numerals. In most cultures the ability to handle computations
has been considered an advanced and complicated art. On the
other hand such knowledge is essential for the functioning of
society when it reaches a certain stage of development; compu-
tations are essential for trade and commerce, for bookkeeping and
accounting, and for many other purposes.
COUNTING AND RECORDING OF NUMBERS 15
One finds as a consequence that devices to facilitate computa-
tions are in widespread use in the world. Best known is the
abacus or reckoning board. The reckoning board was particularly
an instrument of the merchants and tradesmen, and it could be
applied universally regardless of differences in languages and
numbers. This explains the close resemblance between Roman
abaci, the Chinese swanpan, the Japanese soroban, the Russian
tschotu. In the main they consist of balls in movable rows or on
beads, not essentially different from the frames of balls used in our
kindergartens to teach the rudiments of counting and calculation.
The only preserved Greek abacus was found on the island of
Salamis. It is of a different type, a marble slab with engraved
lines and Attic number symbols. There exist several Greek and
Roman illustrations of persons using such abaci. The numbers
were marked by means of small stones, in Latin calculi, whence the
origin of our terms calculus and to calculate. In medieval Europe
simplified abaci consisting only of lines, one each for units, tens,
hundreds, and so on, were in common use. The abacus pattern
could be drawn afresh on paper or parchment each time calcula-
tions were to be performed. The patterns could be carved perma-
nently on a comptoir board or table, and they were often sewn on
tablecloths, hence our word bureau derived from the Latin burra
or woolen cloth. These checkered tabulating boards or abaci also
originated such well-known terms as exchequer and checks. The
process of calculating on these boards was called casting (on the
lines), a term that is still preserved in various connections. We
shall encounter it later in the ancient checking method for calcu-
lations known as casting out nines. The numbers were indicated
on the board by means of special markers or counters, in French
jetons (casters), or also by means of special reckoning coins stamped
for this purpose. The abacus gradually lost ground as the knowl-
edge of calculation with Hindu- Arabic numerals was spread. In
modern times mechanized calculation has again gained the upper
hand in any extended computations through the use of the calcu-
lating machine.
16 NUMBER THEORY AND ITS HISTORY
1-9. Positional numeral systems. We shall now turn to the
history of our own numeral system in which we express every
number by means of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. It belongs
to a type of numeral systems which are usually called positional.
Such systems are based upon the principle of local value, so that a
symbol designates a value or class which depends on the place it-
takes in the numeral representation. For instance, in the three
numbers 352, 325, and 235, the digit 2 signifies respectively 2,
2 X 10, and 2 X 100.
Clearly the positional systems are closely related to the mul-
tiplicative grouping systems, and one obtains a positional system
from a multiplicative grouping system simply by omitting the
special symbols designating the higher class groups. As an
example one may consider the Chinese-Japanese numerals. It
may be observed, however, that this need not be the historical
process through which a positional system originated.
The only complication which the positional notation involves
lies in the necessity of introducing a zero symbol to express a void
or missing class; for instance, 204 is different from 24. The
essential discovery in the positional system may be considered to
lie in the invention of this symbol. The many advantages of the
positional system are not difficult to perceive. First, the numeral
notation is very compact and easily readable. Next, it is possible
to express arbitrarily large numbers only by the digits in the basic
group. Finally, and not least important, in comparison with
other systems the execution of calculations in the positional
system becomes extremely simple.
The positional system is interesting culturally because it affords
an illustration of an invention made independently in several
civilizations. The earliest known numeral system to embody the
principle of position is the sexagesimal Babylonian system, which
we have mentioned previously. This system evolved from an
earlier Sumerian system (about 3000 B.C.), which was also sexa-
gesimal, but whose numeral representation was a simple grouping
system. There exists an overwhelming material iof Babylonian
COUNTING AND RECORDING OF NUMBERS
17
cuneiform tablets dated from 2000 B.C. to 200 b.c. that throws
light upon the customs and institutions of this region. It is
;v i% v Sp~ 2
Fig. 1-5. Chinese-Japanese numerals (brush form). The first column gives
the numbers 1-10, the second represents 100, 1,000, 10,000, 100,000,000. The
three remaining columns give the examples 3,468, 15,702, and 860,531.
surprising that a considerable number of these tablets have been
found to be mathematical texts and tables of a rather advanced
nature. The .numeral representations in the cuneiform texts use
18 NUMBER THEORY AND ITS HISTORY
the symbols J and <^ to denote 1 and 10 respectively. Within
the basic 60 group the numbers are written by means of a simple
grouping system, for instance,
«CF
35
To simplify the writing a subtractive symbol f*~ , lal or minus, is
applied, as in the example
«7f-
= 20-1-19
The numbers exceeding 60 were written according to the positional
principle. To illustrate,
M'W ^JT^ = * x 6 ° 3 + 28 x 6 ° 2
<TT -^Sll N\ + 52 X 60 + 20 = 319,940
During a considerable part of the time in which they were in use,
the Babylonian numerals were deficient because no sign for zero
existed. As a consequence, the numeral representation was
ambiguous. Often the true value of a number can be decided
upon only through the context, although at times the spacing of
the symbols may be of assistance. A zero sign does not come into
regular use until after 300 b.c. Even so the numeral representa-
tion does not become unique since the zero is introduced only
within the numeral and not at the end, so that, for instance, W
may mean 3 or 3 X 60 or even 3 X 60 2 .
The Mayans also achieved the distinction of having created a
complete positional numeral system. Their number system, as
we have already mentioned, was a vigesimal system with a devia-
tion from the normal scheme in that the second number group was
360 = 20 X 18 instead of 400 = 20 2 . In the Mayan numeral
system the first four numbers were denoted by dots, for instance,
• • • = 3. By crossing four dots one obtained the line ,
representing 5. The numbers in the basic 20 group were obtained
from these two symbols by simple grouping, so that
COUNTING AND RECORDING OF NUMBERS 19
A symbol <22> for zero was used systematically as we do. Since
the representation of the numerals was vertical, an example would
appear as follows:
= 7x (I8x2O 2 )+I3x(I8x2O)+0x20 + 10 = 55090
1-10. Hindu-Arabic numerals. Our numerals as we now use
them are commonly known as the Hindu-Arabic numerals. Most
historical evidence points to India as the country of their origin.
To the Arabs who were instrumental in their transmission to
Europe, they were known as the "Hindu numbers. " Considerable
material on early Hindu numerals is available from manuscripts
and inscriptions. Although there is some difference of opinion
among the scholars, it seems plausible that the number symbols
from which our present digits have developed belonged "to the
Brahmi branch of numerals. This was originally a ciphered
numeral system with the following first nine symbols:
BRAHMI SYMBOLS ( 100 B.C.)
1 2345 6789
- = = v p ( p? c 7?
The use of a positional system with a zero seems to have made its
appearance in India in the period a.d. 600-800.
Around a.d. 800 the system was known among the Arabs
in Bagdad and it gradually superseded the older type Arabic
numerals. One of the greatest Arab mathematicians of this time
was Mohammed ibn Musa al-Khowarizmi, whose work, Al-Jabr
wal-Muqabalah, contributed much to the spread of calculations
with the new system, first in the Arab world and later in Europe.
20 NUMBER THEORY AND ITS HISTORY
This treatise is of interest also because it is believed that its title
Al-Jabr has given rise to the term algebra of modern mathematics.
Through the Arabs the Hindu numerals were introduced in
Europe An interesting early form, the Gobar numerals, appeared
in Spain. The name Gobar, or dust, numerals is derived from the
Indian custom of calculating on the ground or on a board covered
with sand The earliest preserved manuscript using Gobar
numerals dates from a.d. 976. The Gobar numerals can also be
found on the apices or jetons introduced by Gerbert, later Pope
Sylvester II (died a.d. 1003), for calculations on the abacus.
GOBAR OR WESTERN ARABIC NUMERALS (1000 A.D.)
123 A 567890
The works of al-Khowarizmi were translated into Latin, and
through a perversion of his name the art of computing with Hindu-
Arabic numerals became known as algorism. This term took on
various other forms; in Chaucer it appears as augrime. The
word is still preserved in mathematics, where a repeated calculating
process is called an algorism. Other terms have been taken over
from the Arabs. The Hindus early denoted the zero by a dot or
a circle and used the term sunya, or the void, for it. Translated
into Arabic this became as-sifi, which is the common root of the
words zero and cipher.
During the eleventh and twelfth centuries a number of European
scholars went to Spain to study Arab learning. Among them one
should mention the Englishmen Robert of Chester and Athelard
of Bath, both of whom made translations of al-Khowarizmi's
works. Still more important for the spread of the new numerals
was the Liber abaci (a.d. 1202), a compendium of arithmetic,
algebra, and number theory by Leonardo Fibonacci or Pisano,
the only outstanding European mathematician of the Middle
Ages. He expresses himself strongly in favor of calculations
COUNTING AND RECORDING OF NUMBERS 21
"modi Indorum," which he learned as a boy from Arab teachers in
North Africa before returning to his native town of Pisa. Another
text which was widely studied was the Algorismus of John of
Halifax or Sacrobosco (about a.d. 1250).
Through the works of these and other scholars, but probably
even more through merchants and trade, the knowledge of the
Hindu-Arabic numerals was disseminated. The numerals took
a great variety of shapes, some quite different from those now in
use, but through the introduction of printing the forms became
standardized and have since remained almost unchanged.
The transition to the new numerals was a long-drawn-out
process. For several centuries there was considerable ill feeling
between the algorismists, the users of the new numerals, and the
abacists, who adhered to the abacus and the Roman numerals.
Tradition long preserved Roman numerals in bookkeeping, coinage,
and inscriptions. Not until the sixteenth century had the new
numerals won a complete victory in schools and trade. Even as
late as the famous work of Nikolaus Copernicus (died a.d. 1543)
on the solar system, De revolutionibus orbium coelestium, one finds
a strange mixture of Roman and Hindu- Arabic numerals and even
numbers written out fully in words. The abacus or counter
method of calculation remained in use much longer. To illustrate,
let us quote from The Ground of Artes (1540) by Robert Recorde,
one of the Englishmen who had most influence on arithmetic:
Both names are corruptly written: Arsemetrick for Arithmetic, as
the Greeks call it, and Augrime for Algorisme as the Arabians found it;
which both betoken the Science of Numbring, for Arithmos in Greek
is called number: and of it comes Arithmetick, the Art of Numbring.
So that Arithmetick is a Science or Art teaching the Manner and Use of
Numbring: This Art may be wrought diversely, with Pen or with
Counters. But I will first show you the working with the Pen, and
then the other in order. [See Frontispiece.]
To complete this brief sketch of the development of our number
system, it should be mentioned that the first satisfactory expo-
sition of the use of decimal fractions was given by the Flemish
22
NUMBER THEORY AND ITS HISTORY
LIBRimARITHMETlCE PRACT1CAE
comuniaignores)tibi fiKin medium ftatuera ferretquatenus his con
ten cus ad altiorcs fecretiffimafqj naeToqt pofthac fpeculationes trafce
das. DI S.Fidelirer ropcifK pre eeptor colendiflime:fed tame adhuc
unum ex promiffis reftat declarandu.-quodToTuran obi abduxj't ob/
liuio. MAG.Q^uodefttakC DIS.Inur.fecudothetoricefpopS/
deras prxter haec que difta funrquedam alia:& perpulchras arris cal
culatoriae rradere regulas -.quodCnifi tibi moleftu foreOteopere ad/
implere:&hj's finemarithmerice tueimponere velim. MAG. Fa/
ciam ut peris.lao equidem q? nihil ex polhciris obliuioe autnegligen
ria tranGs.Primo auce Algorithm!! vulgi plineas:& denarios pro/
te<fbles pandarmutno modo perfiguras numeroru(quas .cifrasvo
canOverueoa perdenarios proieftiles quecunq? numeru reprgfenta/
re:addere:fubtraheremulriphcareautdiuidereinfuper& numeriiaii
que igrtotiig notos reperire pofljs. Q_uod quanru 8f im'Iiraris 8cio/
cunditaris tibi allaturum Gain fequenribus pacebit, DI S. Eyaer/
gpfermortem ad hacc quantocyus verras*
Libriquatti Algbnthmuscumdenarfjs
proi«fh'libus:J(eucalcuIaris Tra&atus
quintus. Capuulumprimu. MAG.
De Numeratione
Dreprefemationc numeri cu denarrjs ,piee"tiIibusCquibu»
Z ip cifn's urimuOneceflarie funt line? cifrarii repfenratcs loc*
candemqj cum ipfis fignificatione habena'a ut infra*
Spacia vero fuB lineis contenta refpe&ulinee .pxi'mc fuprapofite me
dietatcreprf fentat ut f In linea ifta denarius pofirus/ dece Ggnifi/
cat:fed is qui in fpacio cofiftit/rm quincp reprefenrat. Si igirur numc
rum aliqueo. denarios proieetiles reprefentare vblueris : tot denarios
fcSm bneam Scfpadoifc exigentia ad numerum ,ppofirum reprefen/
randum ponas: 8f cum. quinq? denarios in linea aliquahabucris:pro
ipfis/Ieuan's fi placer in fpacio prbximefuperiori unit ponas. Si ve/
ro in fpacio aliquo duos denarios reperies/ipfis leuaris/unu adlinei
immediate fuperioreponas.urhic § Nectefugiatgdigiriapplica/
rionemlinearuGgnificaaone ftaugeri & nunuipoifcAd quacunqj
Fig. l-6a. Instructions for computing on the lines in the form of a dialogue
between magister and disciple. (From Gregor Reisch, Margarita Philosophica,
Strassbourg, 1504.)
COUNTING AND RECORDING OF NUMBERS
23
&
TRAC..V.ALGOR1THMI CALCVLARIS
cnim lineam digitus applicaturtcadem unu ranram fifgnificabit: ncc
tuncaliqinferioifc donee digirii depofueris aliqdrcpffntar.ut
-ooo— j.reprefentanaq depofito digiro triginta important,Q_uod
__ urfingula fkdlim acmodo breuiorifianrin mulriplicarione
Kdiuifione huiufcemodi digitj application* pfepe urendu eft,
De Additione Capitulum feomdum. D1S.
DdirionumCm fallarex dj(?bs>o modo fieri arbirror: utfdli
a cetnumerus cui debet fieriaddirio per denarios ad b'neas co
perenres ponarur;& fimili modo numerus addendus eiappli
cetur. MAG. Oprirnefentis ffiridem exemplo probare porueris.
DI S.Sit caufa exempli numerus cuj deber fieri addirio .2 r». ira lo
«atus Z5 & numerus addendus. n c.in huncmodumadditus.
ff I o
De Subtra<ftione, Capitulum. iff- MAG*
Ecaliter fubrra<fh'one fades c£ ur cum a quo debet fieri fubtra
n c"h'o p lineas competenres ponas :8c ab code numerii fubtra/
hendum fubleuando ro!las/in inferiori inchoado.Si veto nu
Weru aliquem a fuperiori proximo fublcua re no poffis:a linea proxi
mefibifuprapofira unit auferas/SCilludin lineam qua fubtra Aone?
perficere non poruiftiin decern refoluens fubrractjoncm .ppofira per
ficics.Exempinfia £\ .^i.auferre voIucro:unumquidcalineainfe
rioriaccipio 8c* quattuor in bnea fecuda quero neciuenio. ob hoc unu
de linea tertia accipiens/ipfum in lineainferiorcin dree refoluo cinq;
feiheer ponendo ad eandem lineam : unum vero ad fpacium fur/urn
ut fie it Mo fa Ao dc linea fecunda tollanrur rfimiurcr in fimilibus
faciendum eft.
De IMultiplicatione Capi'mlum-iir)/ I^AG.
I vero aliquem numerum mulriplicare plaeuerir:ipfum ad Ii
f neas comperenres ponasrnumcrum aurc muiriplicatem mc/
teretineas:&45quoJiberdcnueromulriplkadofuHato/muI/
riplicanre inregrum refpecftu line? in qua opera ris in loco dexrro aut
finjftrodeponas:&idipfumufq5 0esmu!riplicadimrriplicad fueonr
irerandii eftin fugioreiniriii fumedo.ut fi caufa exepli to ,iC,pcr.t\
mulriplicarevolueris.2tf.utpmiflumeftad lineas ponant: arcfinli/
nea feciida inclioanres :unu demat :8c ^5 eodc in pane dexrra ad linea
eande.^.apponant.ut fie cf> 8c fimili modo dcfmmdo facicndfi eft.
uthitc f Er in eade linea digiro rerenro/eu quimfpa^ioinfi-oon pro
ximo reperit accipieres:ar,p eodcCqma no inregerfed dimidiuseft)
Ru4ntegra,^,fed riKdiecace ejus fcj.j ,in bnea eandc ponam 9 . ut X
Fig. l-6b. Examples of computation on the line.
ZS
A
y
24 NUMBER THEORY AND ITS HISTORY
mathematician Simon Stevin in his work La Disme (a.d. 1585).
John Napier seems to have been one of the first to use the comma
or point to separate decimals from the integers as we still do.
Bibliography
Cajori, F. : A History of Mathematical Notations, The Open Court Publishing
Company, Chicago, 1928.
: A History of Elementary Mathematics, The Macmillan Company, New
York, 1924.
Hill, G. E. : The Development of Arabic Numerals in Europe Exhibited in
Sixty-four Tables, Clarendon Press, Oxford, 1915.
Rouse-Ball, W. W. : A Short Account of the History of Mathematics, Macmillan
& Co., Ltd., London, 1940.
Smith, D. E. : History of Mathematics, Ginn & Company, Boston, 1923.
and L. C. Kabpinski: The Hindu-Arabic Numerals, Ginn & Company,
Boston, 1911.
CHAPTER 2
PROPERTIES OF NUMBERS. DIVISION
2-1. Number theory and numerology. The properties of the
series of natural numbers, one of the basic and most essential con-
cepts of mathematics, are the object of the theory of numbers.
One finds that there exist many simple rules regarding numbers
that are quite easy to discover and not too difficult to prove.
However, number theory also includes an abundance of problems
whose content can be comprehended and expressed in simple terms,
yet whose solution has for centuries defied all mathematical
investigation. Other problems whose solutions have been success-
fully obtained have yielded only to attacks by some of the most
ingenious and advanced methods of modern mathematics. The
simplicity in form of its problems and the great variation in the
methods and tools for their solution explain the attraction that
number theory has had for mathematicians and laymen. The
innumerable individual contributions, calculations, speculations,
and conjectures bear witness to the continued interest in this field
of mathematics throughout the centuries.
The origins of the study of number properties go back probably
almost as far as counting and the arithmetic operations. It does
not take long before it is discovered that some numbers behave
differently from others; for instance, some numbers can be divided
into smaller equal parts and others not. The operations with
fractions lead immediately to the study of divisibility of numbers,
the least common multiple, and the greatest common divisor.
Other approaches have led to early number-theory questions.
The solution of puzzles and amusement problems is one of them.
To us who are accustomed to have an easy access to modern
25
26 NUMBER THEORY AND ITS HISTORY
diversions it may be difficult to realize the entertainment value
that mathematical brain-teasers possessed for earlier generations.
A brief revival of this interest may be seen in the great number of
problems that circulated in the military camps during the war.
This interest was attested by the numerous urgent requests for the
correct answers, some from men earnestly interested (others
undoubtedly put to settle bets). Many such problems are very
old and appear in the earliest sources of mathematical information.
The entertainment value of mathematical questions was developed
especially by the Hindus. The Hindu mathematician Brahma-
gupta (a.d. 588-660) states in one of his works: "These problems
are stated merely for pleasure. The wise man can devise a
thousand others or he can solve the problems of others by the rules
given here. As the sun obscures the stars, so does the man of
knowledge eclipse the glory of other mathematicians in an assembly
of people by proposing algebraic problems and still more by solving
them." According to tradition Bhaskara (about a.d. 1140) wrote
his famous Lilavati ("the beautiful"), a collection of problems in
poetic form, to comfort one of his daughters. The presentation cf
mathematical problems in verse was facilitated by the Hindu
custom of using metaphors in the pronunciation of numbers.
A symbolic correspondence between numbers and objects or
philosophical concepts and ideas was a trait common to many of
the ancient cultures. One finds traces of such symbolism in most
mythologies, and it is even preserved in some of our popular
superstitions in regard to numbers. From this association, it is
not a long step to speculations about properties of numbers and
their implied relation to the corresponding concepts. Such numer-
ological studies permeate the writings of the classical and medieval
philosophers. The Pythagorean school (about 500 b.c.) was
particularly devoted to symbolic number speculations in philosophy
and nature. Their influence was considerable ; even Plato touches
upon numerology in several instances in his Republic, while
Aristotle warns against arguments based upon such foundations.
The later Neo-Pythagoreans tend to ascribe much of their mystical
lore to the early school, but it seems certain that they have been
PROPERTIES OF NUMBERS. DIVISION 27
under the influence also of other and sometimes even older sources.
At present it is extremely difficult to ascribe any rational content
to many of these numerological diatribes. To illustrate such
passages let us make a few excerpts from a long eulogy on the
number 7 from the work, On the Creation of the World, by the
prominent Jewish philosopher Philo Judaeus:
And such great sanctity is there in the number seven, that it has a
preeminent rank beyond all the other numbers in the first decade. For
the other numbers, some produce without being produced, others are
produced but have no productive power themselves; others again both
produce and are produced. But the number seven alone is contemplated
in no part. And this proposition we must confirm by demonstration.
Now the number one produces all the other numbers in order, being
itself produced absolutely by no other; and the number eight is pro-
duced by twice four, but itself produces no other number in the decade.
Again, four has the rank of both, that is, of parents and offspring, for
it produces eight when doubled, and it is produced by twice two. But
seven alone, as I said before, neither produces nor is produced, on which
account other philosophers liken this number to Victory, who has no
mother, and to the virgin goddess, whom the fable asserts to have
sprung from the head of Jupiter: and the Pythagoreans compare it to
the Ruler of all things. ...
Among the things then which are perceptible only by intellect, the
number seven is proved to be the only thing free from motion and
accident; but among things perceptible by the external senses, it dis-
plays a great and comprehensive power, contributing to the improvement
of all terrestrial things and affecting even the periodical changes of the
moon. And in what manner it does this, we must consider. The num-
ber seven when compounded of numbers beginning with the unit, makes
eight-and-twenty, a perfect number, and one equalized in its parts.
Numerology has unquestionably stimulated investigations in
number theory and bequeathed to us some most difficult problems.
Let us mention the perfect numbers, which are equal to the sum of
their aliquot parts (divisors). The discovery of a pair of amicable
numbers symbolizing friendship, like 220 and 284, one the sum of
the parts of the other, would not be possible without intimate
study of divisibility properties of numbers.
28 NUMBER THEORY AND ITS HISTORY
A subject closely related to numerology is gematria (gematry),
a name perhaps obtained as a corruption of the word geometry.
By assigning number values to the letters in the alphabet in some
order, each name and object received a number value. This letter
weighting or gematry served to predict relations between persons
or future events. Together with astrology it was one of the most
popular ancient branches of superstitious learning, and both have
persisted to our present days. The origin of gematry was directly
connected with the form of the Hebrew and Greek numeral systems.
Such alphabetic systems automatically assigned a number to each
name and person. The names of the Bible have been a favorite
field for gematry. Most famous is the Number of the Beast,
given in the Revelation of St. John (13:18): "Here is wisdom.
Let him that hath understanding count the number of the beast;
for it is the number of a man and his number is six hundred three
score and six." In spite of the innumerable researches on this
question through the centuries it seems impossible to arrive at
any definite solution. Clearly many names will have the same
number. In the violent theological feuds of the Reformation it
was a vicious stroke to write the opponent's name in such a way
that his number became the fatal 666 of the beast. Let us
mention also another number replacement that occurs in early
theological writings. They often conclude with the number 99,
in Greek Q& , and this is a gematry substitute for
Amen = aurjv = 1 + 40 + 8 + 50 = 99
as one easily verifies by the list of Greek numerals.
2-2. Multiples and divisors. Number theory, as we have
already stated, is primarily concerned with the properties of the
natural numbers. However, it is convenient for most purposes to
enlarge the system under consideration and investigate the whole
set of integers
0, ±1, ±2, ± • • • (2-1)
The two numbers ±a are sometimes said to be associated. They
are characterized by the fact that they have the same absolute
value \a\.
PROPERTIES OF NUMBERS. DIVISION 2{\
Now let o be an arbitrary integer. The multiples of o are all
numbers
0, ±o, ±2a, =fc • • • (2-2)
z.e., all numbers of the form ka where k is integral. One sees that
if ka and /ia are two multiples of a, then their sum, difference, and
product
ka±ha = (k ± A)a, ka • ha = kah • a
are also multiples of a.
A simple example is the multiples 2n of 2, that is, the even
numbers.
When a relation
c = ab (2-3)
holds between the integers a, b, and c ^ 0, one says that a is a
divisor or factor of c and that c is divisible by a. We also call (2-3)
a decomposition or factorization of c. Clearly b is also a divisor
of c and uniquely determined by a. This leads to an observation
that is useful in certain problems, namely, that the divisors of a
number occur in pairs (a, b). The divisors in such a pair can
only be equal (a, a) when c = a 2 is a square number.
From (2-3) one obtains a new factorization
c= (-a)(-6)
where the divisors are associated with a and 6. Each number has
the obvious decomposition
c =l. c= (-1)(- C )
and ±1 together with ±c are called trivial divisors. Other
remarks about divisors are the following: If c x and c 2 are two
numbers such that C\ divides c 2 and conversely, then the two
numbers are associated ci = ±c 2 . If c\ = ab\ and c 2 = ab 2
are two numbers divisible by a, then their sum and difference are
divisible by a.
ci ± c 2 = a(&i ± b 2 )
When c = ab is divisible by a, and d = cb\ is divisible by c, then
d = abbi is divisible by a.
30 NUMBER THEORY AND ITS HISTORY
In most questions regarding divisors we shall assume tacitly that
the number c is positive and that one only considers decompositions
(2-3) with positive divisors a and b. Clearly all other factori-
zations can be written down as soon as the positive factorizations
of positive integers have been obtained. In certain problems one
is interested only in the proper divisors, consisting of all positive
divisors including 1 that are actually less than c; that is, the
number c is excluded. This is the point of view in the classical
Greek problems.
In a decomposition (2-3) the factors a and b cannot both be
greater than Vc. One can suppose, therefore, that in a pair of
divisors (a, b) one has a ^ Vc and b ^ Vc. This limits the
possible numbers that one has to try out in determining the
factorizations of a number to divisors that do not exceed Vc. For
instance, when c = 60, one has Vc < 8, and one finds the six
pairs of divisors
1, 60 3, 20 5, 12
2, 30 4, 15 6, 10
Problems.
1. Find the divisors of the numbers 96 and 220.
2. Prove that a number is a square only when the number of (positive)
divisors is odd.
2-3. Division and remainders. Let M be an arbitrary
integer. Every other integer a will either be a multiple of b or fall
between two consecutive multiples q • 6 and (q + 1)6 of 6. Thus
one can write
a = qb + r (2-4)
where r is one of the numbers
0, 1, 2, ...,|6| - 1 (2-5)
In (2-4) r is called the least positive remainder or simply the
remainder of a by division with b, while q is the incomplete quotient
or simply the quotient. As an example, let us divide 321 by 74
321 - 4 • 74 + 25
PROPERTIES OF NUMBERS. DIVISION 31
Similarly, if 46 is divided by -17,
46= (-2)(-17) + 12
It should be noted that when a and b in (2-4) are given, q and r
are uniquely determined so that each integer a can be written in
one way in the form qb + r, where r is one of the b numbers (2-5).
For instance all numbers are even or odd, i.e., belong to one of the
two forms 2q or 2q + 1. When these numbers are squared, one
finds respectively
V, 4 9 2 + Aq + 1
so that we have:
Theorem 2-1. The square of a number is either divisible by 4
or leaves the remainder 1 when divided by 4.
One can write the division (2-4) in the ordinary fractional form
a r
where r/b is zero or a positive fraction less than 1 and q is the
greatest integer that is less than or equal to a/b. Such quotients
occur so often in number theory that it is convenient to introduce
a special notation for them,
~a~
called the greatest integer contained in a/b.
Examples.
This notation may be extended to arbitrary real numbers. If for
a real number
<* = q + P, 0^p<l
then we write q = [a] for the integer q.
Examples.
W = 3, [e] = 2, [^J = 4
32 NUMBER THEORY AND ITS HISTORY
Sometimes there is an advantage in performing the division in
a slightly different manner from (2-4) : We select a multiple kb as
near as possible to a on the number axis and obtain
a = kb + s (2-6)
where s now is a number between - 6/2 and 6/2. Such a represen-
tation as (2-6) we call a division with the least absolute remainder.
Again k and s in (2-6) are uniquely determined except when 6 is
even and the remainder is s = ±6/2, when one can write the
division in two ways
a = kb + ^ = (fc + 1)6 - -
If it is desirable always to have a unique remainder one can agree
to use s = 6/2 in this case.
Example.
As examples of division with smallest absolute remainder, let us divide 35
by 9 and 46 by -17
35 = 4-9-1, 46 = (-3)(-17) -5
It is often convenient to apply the smallest absolute remainder in
representing numbers. For instance, every number is representa-
ble in one of the three forms
3/e, 3/e ± 1
or one of the five forms
5k, 5fc ± 1, 5k ± 2
or in the four forms
4k, 4fc + 2, 4k ±1
In the last classification the odd numbers must belong to the forms
m = 4fc ± 1. As a consequence
m 2 = 16k 2 ± 8/c + 1 = 8k(2k ± 1) + 1
so that we can say:
PROPERTIES OF NUMBERS. DIVISION 33
Theorem 2-2. The square of an odd number is of the form
8g + 1.
Examples.
5 2 = 3 • 8 + 1, 7 2 = 6 • 8 + 1
Some similar results are given among the problems.
Any number can be written in the form
n = 10a + b, ^ b ^ 9
where b is the last digit in the decadic representation of the number.
By squaring one obtains
n 2 = 100a 2 + 20a6 + b 2
so that n 2 has the same last digit as b 2 . But when one considers
the squares of the numbers from to 9, one finds that they end in
one of the six digits 0, 1, 4, 5, 6, 9, so that one can say:
Theorem 2-3. The last digit in the square of a number must
be one of the numbers 0, 1, 4, 5, 6, 9.
For certain problems in number theory, for instance, with some
factorization methods, it is of importance to be able to decide
quickly whether a number can be a perfect square. By applying
the same method as above to the last two digits of a number and
looking up the squares of the numbers from to 99, one finds that
for a square the last two digits are limited to the following 22
possibilities:
Table of the last two digits in a square number
00 21 41 64 89
01 24 44 69 96
04 25 49 76
09 29 56 81
16 36 61 84
Problems.
1. Divide the following pairs of numbers with respect to both least positive
and least absolute remainders:
(a) 125 and 23 (b) 87 and 13
(c) -111 and -17 (d) 81 and 18
34 NUMBER THEORY AND ITS HISTORY
2. Prove that the square of a number not divisible by 2 or 3 is of the form
I2n + 1.
3. Prove that the fourth power of a number not divisible by 5 is of the form
5n + 1.
4. Consider an analogue of theorem 2-3 for third and fourth powers.
5. Prove that in the decadic number system the fifth power of any number
has the same last digit as the number itself.
6. Show that n(rt 2 — 1) is divisible by 24 when n is an odd number.
2-4. Number systems. As we mentioned previously, a variety
of different number systems have been in use. We observed in
this connection that when the basic counting group contained b
elements, the systematic extension of the counting process would
lead naturally to a representation of the natural numbers in the
form
a = a n • b n + On-i • & rt_1 + • • • + a 2 • b 2 4- a t • 6 + a (2-7)
where the numbers a* take the values 0, 1, 2, • • • , 6 — 1.
In analogy to our numerals in the decimal system we can
indicate the number (2-7) by the abbreviation
The question arises immediately how one can find the form
of a number in a system with a given base number, or more
generally how one can pass from one system to another. In (2-7)
clearly the last number a indicating the units is the least positive
remainder of a by division with b,
a = qi - b + a
where
Qi = a n • 6 n_1 + • • • + a 2 * b + ai
To determine ai one divides qi by b
qi = 92 • b + a x
where
Q2 = a n - b n ~ 2 + \- a 2
When q 2 is divided by b one finds the remainder a 2 , and through
the repetition of this procedure all a/s can be determined.
PROPERTIES OF NUMBERS. DIVISION 35
Examples.
1. To represent the number 1,749 in a system with the base 7, one performs
the divisions
1,749 = 249-7 + 6
249 = 35-7 + 4
35 = 5-7 +
so that one finds
1,749 = (5, 0, 4, 6)7
2. Similarly, to represent the number 19,151 to the base 12:
19,151 = 1,595 12 + 11
1,595 = 132-12 + 11
132 = 11-12+0
so that
19,151 = (11, 0, 11, ll)i2
In the preceding method for finding a number expressed to a
base b, the digits a , a x , . . . are determined from the lowest upward.
One can also proceed in a manner that yields the digits in the
reverse order a n , a n _ x , .... For this purpose, one determines the
highest power of b such that b n is less than a while the next power
b n+l exceeds a. Then from (2-7) it follows that the division of a
by b n must have the form
a = a n • b n + r n _!
where
r n _i = a n _i • V*- 1 -\ \- a
From the remainder r n _ x one determines a n _ x in the same
manner, and so on. This method is facilitated by a table of the
various powers of the base number b.
Example.
Represent 1,832 to the base 7. One calculates
7 2 = 49, 7 3 = 343, 7 4 = 2,401
36 NUMBER THEORY AND ITS HISTORY
and from the divisions
1,832 = 343 • 5 + 117
117 = 49 • 2 + 19
19 = 7 • 2 + 5
one concludes
1,832 = (5, 2, 2, 5) 7
From time to time it has been suggested that our venerable
decimal system be discarded in favor of some other system. Most
often the numbers 6, 8, or 12 are proposed as the new bases. The
arguments for such a change are of various kinds. In the case of
the bases 6 and 12, it is pointed out that division by 3 becomes
simple; in decimals one has the infinite expansion
J = 0.333 • • •
while with the bases 6 or 12
i = (0, 2) 6 = (0, 4)ia
On the other hand, fractions with denominator 5 would become
complicated in these systems; for instance,
i = (0, 1, 1, • • • )e
If one should wish to have simple expansions for all fractions with
denominators 2, 3, 4, 5, one would be led to the Babylonian sexa-
gesimal system. Large bases will give short representations of
numbers, but they have the drawback that the size of the multi-
plication tables to be memorized is considerably increased. A
12 X 12 multiplication table instead of the usual 10 X 10 table
may be admissible, but a 60 X 60 table is clearly out of the
question. Small bases lead to long number representations but
very simple multiplication tables. On the whole there is little
evidence that a change of bases will materially reduce the time
consumed by numerical computations. The reformers usually
pass lightly over the resulting complications and the necessity of
changing records, tables, and machines. For one thing, in order
to avoid a state of utter confusion in the transition period it would
PROPERTIES OF NUMBERS. DIVISION
37
be necessary to invent and use a completely new system of ciphers,
because otherwise no one would know whether 23 should mean 23
or 19 (if the base were 8) or 27 (if the base 12 had been decided
upon).
Problems.
1. Write the two numbers 1,947 and 21,648 to the four bases 3, 5, 7, and 23.
2. Write the number of seconds in 24 hours in the sexagesimal system.
3. Write the number of seconds of arc in 360° in the sexagesimal system.
2-5. Binary number systems. Number systems with other
bases than 10 have applications in several branches of mathematics;
particularly, the use of low base numbers 2 and 3 is helpful in many
types of problems. In the triadic or ternary system each number
is represented by means of the digits 0, 1, and 2 while in the dyadic
or binary system each number appears as a series of marks or 1.
As an example, let us expand the number 87 to the base 2. One
finds as before
87 = 43 • 2 + 1
43 = 21 • 2 + 1
21 = 10 • 2 + 1
10 = 5-2 +
5 = 2-2 + 1
2 = 1-2 +
1 = 0-2 + 1
Hence
87 = (1, 0, 1, 0, 1, 1, l) a = 2 6 + 2 4 + 2 2 + 2 1 + 2° (2-8)
Let us consider the series of numbers
87, 43, 21, 10, 5, 2, 1 (2-9)
occurring in the divisions. Each number is half the preceding with
the remainder thrown away. One obtains the digits in (2-8), in
38 NUMBER THEORY AND ITS HISTORY
reverse order, by writing or 1 for each number in (2-9) depend-
ing on whether it is even or odd. This schematic method not
only simplifies the determination of the representation in the
dyadic system, but leads also to a peculiar multiplication procedure.
To illustrate let us multiply 87 by 59. We form two chains of
numbers, the first obtained by successively taking half the pre-
ceding number as above, the second proceeding by doubling.
87
59
43
118
21
23(3
10
J&2T
5
944
2
OtSSS"
1
3,776
5,133
In the second column one strikes out the numbers corresponding
to even numbers in the first, and the sum of the remaining terms
gives the desired product. The proof lies in the dyadic represen-
tation of 87
87 • 59 = (1 + 2 + 2 2 + 2 4 + 2 6 )59
This method for performing multiplication reduces the opera-
tions to addition, together with doubling or duplication, and halv-
ing or mediation. In medieval treatises on computation, these two
processes of duplication and mediation were considered to be
separate arithmetic operations besides the four usual ones. The
principle of reducing multiplication to duplication is very old; it
was used by the early Egyptians, and it may well have been the
first approach to a systematic multiplication procedure. The
method given above is sometimes called Russian multiplication
because of its use among Russian peasants. Its great advantage
to the inexperienced calculator lies in the fact that it makes
unnecessary the memorizing of the multiplication table.
There are a great number of games and puzzles whose solutions
depend on the use of the dyadic number system. One is the fairly
PROPERTIES OF NUMBERS. DIVISION
39
well-known Chinese game of Nim, which is discussed at some length
in two of the books cited below (Hardy and Wright, and Uspensky
and Heaslet). Another puzzle for children consists of a set of
cards, each with a certain group of numbers on it. One is asked to
think of a number, and to indicate on which cards it can be found.
It is then possible immediately to pronounce the number in
question. The cards contain the numbers up to a certain limit
arranged in such a way that the first card contains all numbers
whose lowest digit in the dyadic system is 1, that is, the odd
numbers; the second contains all numbers whose second digit is 1,
beginning with 2; the third all whose third digit is 1, beginning
with 4, and so on. When it is known on which cards a given
number occurs, its dyadic expansion is known. The number itself
is the sum of the first numbers on the cards where it appears. As
a simple example let us take four cards containing all numbers
less than 2 4 = 16. One finds that they must have the forms
1 9
2
10
4
12
8
12
3 ii.
3
11
5
13
9
13
5 13
6
14
6
14
10
14
7 15
7
15
7
15
11
15
Problems.
1. Construct such cards for all numbers up to 31.
2. Expand 365 to the bases 2 and 3.
3. Multiply 178 and 147 by Russian multiplication.
Bibliography
Albert, A. A. : College Algebra, McGraw-Hill Book Company, Inc , New
York, 1946.
Bell, E. T.: Numerology, The Williams & Wilkins Company, Baltimore, 1933.
: The Magic of Numbers, McGraw-Hill Book Company, Inc., New
York, 1946.
Birkhopf, G., and S. MacLane: A Survey of Modern Algebra, The Macmillan
Company, New York, 1941.
40 NUMBER THEORY AND ITS HISTORY
Hardy, G. H., and E. M. Wright: An Introduction to the Theory of Numbers,
Clarendon Press, Oxford, 1938.
Hopper, Vincent F. : Medieval Number Symbolism, Columbia University-
Press, New York, 1938.
MacDuffee, C. C: An Introduction to Abstract Algebra, John Wiley & Sons,
Inc., New York, 1940.
Uspensky, J. V., and M. A. Heaslet: Elementary Number Theory, McGraw-
Hill Book Company, Inc., New York. 1939.
CHAPTER 3
EUCLID'S ALGORISM
3-1. Greatest common divisor. Euclid's algorism. Let a and
b be two integers. If a cumber c divides a and b simultaneously,
we shall call it a common divisor of a and b. Among the common
divisors of two numbers there must exist a greatest one, which we
shall call the greatest common divisor (g.c.d.) of a and b. It is
usually denoted by the symbol (a, b). Since every number has
the divisor 1 it follows that (a, 6) is a positive number. If (a, b) =
1 we say that the two numbers are relatively prime. In this case
±1 are the only common divisors.
Examples.
When the divisors of the two numbers 24 and 5G are determined one finds
that their g.c.d. is 8. The numbers 15 and 22 are relatively prime.
We shall now prove:
Theorem 3-1. Any common divisor of two numbers divides
their greatest common divisor.
To establish this theorem we shall introduce a procedure known
as Euclid's algorism, one of the basic methods of elementary
number theory. It occurs in the seventh book of Euclid's Elements
(about 300 b.c); however it is certainly of earlier origin. Let a
and b be the two given numbers whose g.c.d. is to be studied.
Since there is only question of divisibility, there is no limitation in
assuming that a and b are positive and a ^ b. We divide a by o
with respect to the least positive remainder
a = Qib + r u ^ r x < b
Next we divide 6 by r t
b = q 2 n + r 2 , ^ r 2 < n
42
42
NUMBER THEORY AND ITS HISTORY
and continue this process on r± and r 2 , and so on. Since the
remainders r ly r 2 , ... form a decreasing sequence of positive
integers, one must finally arrive at a division for which r n+i =
a = qib + n
b = q 2 n + r 2
n = q$r 2 + r 3
(3-1)
fn—2 — Qn r n—1 i r r,
Example.
Let us perform Euclid's algorism on the two numbers 76,084 and 63,020.
76,084 = 63,020
63,020 = 13,064
13,064 - 10,764
10,764 = 2,300
2,300 = 1,564
1,564 = 736
736 = 92
1 + 13,064
4 + 10,764
1 + 2,300
4 + 1,564
1 + 736
2 + 92
We shall now show that in Euclid's algorism (3-1) the last
nonvanishing remainder r n is the g.c.d. of a and b. The first step
is to show that r n divides a and b. It follows from the last division
in (3-1) that r n divides r n _ 1 . The next to the last division shows
that r n divides r n _ 2 since it divides both terms on the right.
Similarly from
r n— 3 = q n —l r n—2 ~T" 7"n— 1
one concludes that r n divides r n _ 3 , and successively one sees that
r n divides all r/s and finally a and b.
The second step consists in showing that every divisor c of a
and b divides r n \ this clearly implies that r n is the g.c.d. of a and b
EUCLID'S ALGORISM 43
and has the property required by theorem 3-1. But from the
first division in (3-1 ) one sees that any common divisor c of a and
b divides r 1} since r x = a - q x b; from the second, in the same way,
c divides r 2 ; and by continuing this process one establishes that
all rt's, and hence r n , are divisible by c.
Example.
From the previous algorism on the two numbers 76,084 and 63,020 it follows
that their g.c.d. is 92.
Euclid's algorism gives a very simple and efficient method for
the determination of the g.c.d. of two numbers. The French
mathematician Lame (1795-1870) has shown that the number of
divisions in the algorism is at most five times the number of digits
in the smaller number. Another observation of importance is
that all arguments used above remain valid for any chain of
relations (3-1) without any limitations on the numbers n; there-
fore, the conclusions are the same, except that r n may possibly be
the negative value of the g.c.d. One could, for instance, have
used the least absolute remainders in the divisions (3-1). It has
been shown by the German mathematician Kronecker (1823-
1891), one of the leading contributors to number theory in the
last century, that no Euclid algorism can be shorter than the one
obtained by least absolute remainders. (For a more detailed
study of the algorism, see the book by Uspensky and Heaslet,
cited in the bibliography of Chap. 2.)
Example.
Let us perform the algorism for 76,084 and 63,020 by least absolute
remainders .
76,084 = 63,020 • 1 + 13,064
63,020 = 13,064 ■ 5 - 2,300
13,064 = 2,300 • 6 - 736
2,300 = 736 • 3 + 92
736 = 92-8
44 NUMBER THEORY AND ITS HISTORY
Problems.
Find Euclid's algorism for least positive and least absolute remainders and
determine the g.c.d. for the pairs of numbers:
1. 139 and 49 2. 1,124 and 1,472 3. 17,296 and 18,416
3-2. The division lemma. From the algorism of Euclid one
can derive various other properties of the g.c.d. An important
consequence is the division lemma:
Theorem 3-2. When a product ac is divisible by a number b
that is relatively prime to a, the factor c must be divisible by b.
Proof: Since a and b are relatively prime, the last remainder r n
in the algorism must be 1 so that it has the form
a = qib + ri
Tn_ 2 = q n r n -i + 1
We multiply each of these equations by c and obtain
ac = q\bc + r±c
-2C = q n r n -ic + c
Since ac is divisible by b according to our assumption, the first
relation shows that r x c is divisible by b. From the second relation
one finds that r 2 c is divisible by b, and successively one finds that
all r^ and finally c are divisible by b, as we set out to show.
Theorem 3-2 leads to the further result :
Theorem 3-3. When a number is relatively prime to each of
several numbers, it is relatively prime to their product.
Proof: Let a be relatively prime to b and to c. If a has a com-
mon divisor d with 6c, the product is divisible by d. But (d,b) =
1 since d divides a; thus d must divide c, according to theorem
3-2, contrary to the fact that also (d, c) = 1. The extension of
theorem 3-3 to several factors is immediate.
Another consequence of Euclid's algorism is:
EUCLID'S ALGORISM 45
Theorem 3-4. For the greatest common divisor of two
products ma and nib, one has the rule
(mo, mb) = m(a, b) (3-2)
Proof: In Euclid's algorism (3-1) for the numbers a and b let us
multiply each equation by m.
am = q^bm + r^m
rn_ 2 m = q n r n _ 1 m + r n m
r n _im = q n+1 r n m
Clearly this is the algorism for am and bm so that their g.c.d. is
r n m = m(a, b) as the theorem requires.
A useful observation is the following:
Theorem 3-5. Let d = (a, b) be the greatest common divisor
of two numbers a and b so that
a = aid, b = bid (3-3)
Then the two numbers a± and b t are relatively prime.
Proof: It follows from the rule in (3-2) that
d = (a, b) = d(ai, bi)
or («!, bi) = 1.
This result applies in elementary arithmetic in the reduction of
fractions. Any fraction
a _ ai
b~h
can be represented in reduced form with numerator and denominator
that are relatively prime.
3-3. Least common multiple. A number m is said to be a
common multiple of the numbers a and 6 when it is divisible by
both of them. The product ab is a common multiple. Since there
is only question of divisibility properties, there is no limitation in
considering only the positive multiples. Among the common
multiples of a and b there is a smallest one, which we shall denote
by [a, b] and call the least common multiple (l.c.m.) of a and b. The
46 NUMBER THEORY AND ITS HISTORY
l.c.m. and the greatest common divisor have properties that in
many ways are quite analogous. Corresponding to theorem 3-1
one has:
Theorem 3-6. Any common multiple of a and b is divisible by
the least common multiple.
Proof: Let m be a common multiple of a and b. We divide m
by [a, b]
m = q[a, b] + r, f£ r < [a, b]
Since m and [a, b] are both divisible by a and b, it follows that the
remainder r has the same property. Since [a, b] is the smallest
common multiple, this implies that r = and [a, b] divides m.
To determine [a, b] we write a and b in the form of (3-3)
where d = (a, b). Any multiple of a has the form ha = haid.
If this number is to be divisible by b = bid, the factor ha must be
divisible by &i. Because a x and b\ are relatively prime, this is
possible only when h is divisible by b\ so that h = kb\. Thus any
common multiple of a and b has the form
m = kb\a = ka\b\d = ka\b = k —
d
For k = 1 one obtains the l.c.m. so that one has the result:
Theorem 3-7. When a and b are two numbers with the greatest
common divisor d = (a, b), the least common multiple is
ab
[a, b] = - (3-4)
The formula (3-4) can be written symmetrically in regard to
the l.c.m. and the g.c.d.
[a, b](a, b) = ab
Example.
Find the l.c.m. of the two numbers 76,084 and 63,020. We have already
found their g.c.d. to be 92 so that
76,084 • 63,020
[76,084, 63,020] = — — — = 52,117,540
An immediate consequence of theorem 3-7 is:
EUCLID'S ALGORISM 47
Theorem 3-8. The least common multiple of two numbers
a and b is equal to their product ab if, and only if, they are relatively
prime.
Corresponding to theorem 3-4 for the g.c.d., one has the analo-
gous formula for the l.c.m.
Theorem 3-9.
[ma, mb] = m[a, b] (3-5)
Proof: According to (3-4) and (3-2) one finds
r 7 ., ma • mb a • b
[ma, mb] = — = m - — — • = m[a, b]
(ma, mb) (a, 6)
Problem.
Determine the l.c.m. for the pairs of numbers for which the g.c.d. was found
in Sec. 3-1.
3-4. Greatest common divisor and least common multiple for
several numbers. So far the greatest common divisor and the
least common multiple have been denned only for two numbers,
but there is no difficulty in extending these concepts. Let us
consider first the case of three numbers a, b, and c. A common
divisor is any number dividing them all. Among these common
divisors there is a greatest common divisor, which shall be denoted
by
d = (a, b, c)
To calculate d, we observe that it is the largest number dividing c
and (a, b) simultaneously, so that
d = ((a,b), c) (3-6)
Example.
Let us determine the g.c.d. of the three numbers 76,084, 63,020, and 196.
In a previous example we have already found (76,084, 63,020) = 92; conse-
quently d = (92, 196) =4.
The formula (3-6) reduces the computation of the g.c.d. of
Lhree numbers to that of two numbers. Instead of beginning with
(a, b) one could have taken (b, c) first, so that
d= ((o, 6), c) = (a, (6, c)) (3-7)
48 NUMBER THEORY AND ITS HISTORY
This rule is called the associative law for the g.c.d. As in theorem
3-1, one sees that every common divisor of a, b, and c divides
(a, b, c). From theorem 3-4 one concludes by means of (3-6) that
{ma, mb, mc) = m(a, b, c) (3-8)
Corresponding to theorem 3-5, it follows that if one writes
a = aid, b = bid, c = cid
then
(«i, h, a) = 1
The g.c.d.
d n — {p>\, a2, • • ' , a n )
of an arbitrary set of numbers is defined analogously. Since d n
is the g.c.d. of a n and the numbers a\, • • • , a n _i, one concludes
that
d n = («n— 1, a n ), dn—i — {fli, ' ' • , ttn_i)
This leads to a stepwise calculation
d 2 = (oi, a 2 ), d 3 = (d 2 , a 3 ), • • •
All the rules just mentioned for the g.c.d. of three numbers hold
in the general case.
Let us mention briefly the corresponding concepts for the l.c.m.
A common multiple of three numbers a, b, and c is a number divisible
by all of them. Among these multiples there is a least common
multiple
m = [a, b, c\
Since the l.c.m. must be divisible by [a, b] and also by c, one
concludes that
m = [[a, b], c]
Example.
To find the l.c.m. of the three numbers 24, 18, and 52, one calculates
[24, 18] = 72 and m = [72, 52] = 936.
The l.c.m. divides all other multiples. It obeys the associative
law,
[[a, b],c] = [a, [b, c]]
FA JC LID'S ALGORISM 49
and from theorem 3-9 one derives the rule
[ma, mb, mc] = m[a, b, c]
It is not difficult to see that when one writes
m = a' a = b'b = cc
one must have
(a', &', c') = 1
To define and calculate the l.c.m.
m n = [a lt a 2 , • • • , a n ]
of a set of numbers, one can proceed stepwise as for the g.c.d.
w 2 = [ai, a 2 ], m 3 = [m 2 , a 3 ], ■ • •
All properties mentioned for three numbers readily extend to this
general case.
It may be recalled finally that the determination of the l.c.m.
occurs naturally in elementary arithmetic in bringing fractions to
their least common denominator to perform addition and sub-
traction.
Problems.
1. Find the g.c.d. and l.c.m. of the numbers
(a) 63, 24, 99 (6) 16, 24, 62, 120
2. Find the l.c.m. of the integers from 1 to 10.
CHAPTER 4
PRIME NUMBERS
4-1. Prime numbers and the prime factorization theorem. An
integer p > 1 is called a prime number or simply a prime when its
only divisors are the trivial ones, ±1 and ±p. The primes below
100 are
2
13
31
53
73
3
17
37
59
79
5
19
41
61
83
7
23
43
67
89
11
29
47
71
97
The number 2 is the only even prime. A number m > 1 that is
not a prime is called composite. The lowest composite numbers are
10
16
12
18
14
20
15
9
Analogously one introduces the negative prime numbers —2, —3,
— 5, ... , and the negative composite numbers — 4, — 6, ... . In
the following sections we shall, as usual, consider only the positive
factors in our study of the divisibility of numbers.
In regard to divisibility the primes have simple properties. We
mention first:
Lemma 4-1. A prime p is either relatively prime to a number n
or divides it.
Proof: This may be concluded from the fact that the greatest
common divisor of p and n is either 1 or p.
50
PRIME NUMBERS 5
Lemma 4-2. A product is divisible by a prime p only when p
divides one of the factors.
Proof: When ab is divisible by p and a is not divisible by this
prime, p is relatively prime to a and according to the division
lemma must divide 6. The same argument can be extended to a
product of several factors.
Lemma 4-3. A product q t . . . q r of prime factors q { is divisible
by a prime p only when p is equal to one of the g/s.
Proof: We have just seen in lemma 4-2 that p must divide some
prime q i} and since p > 1 one must have p = q { .
Lemma 4-4. Every number n > 1 is divisible by some prime.
Proof: When n is a prime, this is evident. When n is composite,
it can be factored n = ab where a > 1. The smallest possible
one of these divisors a must be a prime.
We are now ready to prove the main theorem about factori-
zations.
Theorem 4-1. Every composite number can be factored
uniquely into prime factors.
Proof: The first step is to show that every composite number n
is the product of prime factors. According to lemma 4-4 there
exists a prime p 1 such that n = p x ni. If m is composite, one
can draw out a further prime factor m - p 2 n 2 , and this process
can be continued with the decreasing numbers m, n 2 , ... until
some n k becomes a prime.
After the existence of a prime factorization thus has been
established, the second step consists in proving that it can only
be done in one way. Let us suppose that there exist two different
prime factorizations
n = p x p 2 -"p k = qiq 2 " -qi (4-1)
Since each p t divides the product of the q'&, it follows from lemma
4-3 that pi is equal to some q h and conversely that each q is equal
to some p. This shows that both sides of (4-1) contain the
same primes. The only difference might be that a prime p could
occur a greater number of times on one side than on the other.
However, by canceling p a sufficient number of times one would
52 NUMBER THEORY AND ITS HISTORY
obtain an equation with p on one side but not on the other, and
this contradicts lemma 4-3.
The idea of the prime-factorization theorem, as well as the
lemmas used in proving it, can be found in Euclid's Elements in
Books VII and IX.
4-2. Determination of prime factors. The actual determination
of the factorization of a number into prime factors is a problem of
great importance in number theory. Unfortunately, for large
numbers it often involves overwhelming computations.
The procedure nearest at hand consists in trying out all the
lowest primes as possible divisors of the given number n. When
a prime factor p has been found, one can write n = pm and
determine the factorization of the smaller number m. The work
is limited by the previous remark that if a number is composite it
must have a factor not exceeding Vn, so that only primes p ^ \/n
need be divided into n. Another useful observation is that when
the smallest prime factor p of n is found to be greater than Vn,
the other factor m in n = pm must be a prime. Thus if m = ab
were composite, both a and b would exceed 'vn, and one would
obtain the contradiction
n = pab > yfn \^n 'v / n = n
Example.
1. Find the prime factorization of n = 893. Since Vn < 30 only the
primes below 30 need be examined. One finds 893 = 19 • 47.
2. Find the prime factorization of the number n = 999,999. One finds
successively
n = 3 2 • 111,111 = 3 3 • 37,037 = 3 3 • 7 • 5,291
= 3 3 • 7 • 11 • 481 = 3 3 • 7 • 11 • 13 • 37
3. Find the prime factorization of n = 377,161. There are no obvious
factors, and since V n < 614 a considerable number of primes may have to be
divided into n. One finds that the smallest prime factor is p = 137 and
n = 137 • 2,753. Here the second factor is a prime since \n < 73.
This method of trial and error is quite satisfactory for relatively
small numbers, perhaps not exceeding four digits; for larger
PRIME NUMBERS 53
numbers the work involved is prohibitive, as one soon realizes
A great number of methods and devices have been invented to
facilitate the determination of a factor. There exist criteria that
under special circumstances make it possible to decide rather easily
whether a number is a prime or not. Some of these will be
mentioned later on.
Problems.
1. Find the prime factorization of the numbers: (a) 365, (b) 2 468 (c)
262,144. ' ' '
2. Find the prime factorization of the two numbers: (a) 99,999, (6)100,001
3. Mersenne determined the factorization of the number 51,001 180160
Find the prime factors of this number. '
4-3. Factor tables. The simplest way to obtain the factori-
zation of a number that is not too large is through the use of a
factor table. There exist various types of these tables. The most
detailed ones contain the complete factorization of every number
up to some limit, but such tables are unwieldy and can give space
only for relatively few numbers. To increase the capacity, most
factor tables indicate only the least prime dividing each entry.
Since it is quite simple to determine whether a number is divisible
by the lowest primes 2, 3, 5, and 7, the numbers divisible bv them
are often excluded from the tables.
One of the first fair-sized factor tables was published by Rahn
or Rohnius (Zurich, 1659) in an appendix to a book on algebra; the
table contained the numbers up to 24,000 excluding those divisible
by 2 and 5. In a translation of this work by Brancker (London,
1668) the table was extended to 100,000 by John Pell (1610-1685),'
an English mathematician particularly interested in number
theory. For a considerable period these tables were the only ones
available and they were reprinted several times in other works.
The great interest in number theory in the eighteenth century
created a demand for factor tables to higher limits. The strong
appeals from the German scientist J. H. Lambert (1728-1777)
made him a center of correspondence regarding factor tables, and
several calculations were initiated. Only one of these tables was
54 NUMBER THEORY AND ITS HISTORY
published, and even this one had an inglorious fate. It was com-
puted by Felkel, a schoolmaster in Vienna. The first volume,
which appeared in 1776 and extended to 408,000, was planned to
be a part of a more ambitious program reaching several millions,
most of it ready in manuscript. The tables were published at
the expense of the Austrian imperial treasury, but since there was
a disappointing number of subscribers, the treasury confiscated
the whole edition except a couple of copies, and the paper was used
in cartridges in a war against the Turks.
In the nineteenth century several large factor tables were com-
puted by Chernac, Burckhardt, Crelle, Glaisher, and the German
lightning calculator, Dase. By their combined efforts all numbers
up to 10,000,000 were covered, published in individual volumes
for each million. The most remarkable effort in this field was,
however, the table calculated by J. P. Kulik (1773-1863), a
professor of mathematics at the University of Prague. It repre-
sents the results of a twenty-year hobby and gives the factorization
of the numbers up to 100,000,000. The manuscript was deposited
in the library of the Vienna Academy and has not been published.
The best factor table now available is the one-volume table
extending to 10,000,000 prepared by D. N. Lehmer. There exist,
furthermore, various special tables and punch-card devices due to
D. N. Lehmer and D. H. Lehmer that greatly facilitate the deter-
mination of factors of numbers beyond the reach of tables.
4-4. Fermat's factorization method. We shall present a couple
of simple methods that are sometimes very helpful in finding the
factorization of a given number. The first method is due to the
French mathematician and lawyer Pierre de Fermat (1601-1665),
whose name we shall encounter repeatedly in the following.
Fermat must be awarded the honor of being the founding father
of number theory as a systematic science. His life was quiet and
uneventful and entirely centered around the town of Toulouse,
where he first studied jurisprudence, practiced law, and later
became prominent as councilor of the local parliament. His
leisure time was devoted to scholarly pursuits and to a voluminous
correspondence with contemporary mathematicians, many of
PRIME NUMBERS 55
whom, like himself, were gentlemen-scholars, the ferment of
intellectual life in the seventeenth and eighteenth centuries.
Fermat possessed a broad knowledge of the classics, enjoyed
literary studies, and wrote verse, but mathematics was his real
love. He published practically nothing personally, so that his
works have been gleaned from notes that were preserved after his
death by his family, and from letters and treatises that he sent to
his correspondents. In spite of his modesty, Fermat gained an
outstanding reputation for his mathematical achievements. He
made considerable contributions to the foundation of the theory of
probability in his correspondence with Pascal and introduced
coordinates independent of Descartes. The French, when too
exasperated over the eternal priority squabble between the
followers of Newton and Leibniz, often interject the name of
Fermat as a cofounder of the calculus. There is considerable
justification for this point of view. Fermat did not reduce his
procedures to rule-of-thumb methods, but he did perform a great
number of differentiations by tangent determinations and inte-
grations by computations of numerous areas, and he actually gave
methods for finding maxima and minima corresponding to those
at present used in the differential calculus.
In spite of all these achievements, Fermat's real passion in
mathematics was undoubtedly number theory. He returned to
such problems in almost all his missives; he delighted to propose
new and difficult problems, and to give solutions in large figures
that require elaborate computations; and most important of all,
he announced new principles and methods that have inspired all
work in number theory after him.
Fermat's factorization method, which is the point interesting
us particularly for the moment, is found in an undated letter of
about 1643, probably addressed to Mersenne (1588-1648).
Mersenne was a Franciscan friar and spent most of his lifetime
in cloisters in Paris. He was an aggressive theologian and phi-
losopher, a schoolmate and close friend of Descartes. He wrote
some mathematical works, but a greater part of his importance
in the history of mathematics rests on the fact that he was a
56 NUMBER THEORY AND ITS HISTORY
favorite intermediary in the correspondence between the most
prominent mathematicians of the times.
Fermat's method is based upon the following facts. If a
number n can be written as the difference between two square
numbers, one has the obvious factorization
n = x 2 - y 2 = (x - y)(x + y) (4-2)
On the other hand when
n = ab, b ^ a
is composite, one can obtain a representation (4-2) of n as the
difference of two squares by putting
x — y = a, x + y = b
so that
b + a b - a
x = -~y~ ' V = ~y~ ( 4_3 )
Since we deal with the question of factoring n, we can assume that
n is odd; hence a and b are odd and the values of x and y are
integral.
Corresponding to each factorization of n there exists, therefore,
a representation (4-2). To determine the possible x and y in
(4-2), we write
x 2 = n + y 2
Since x 2 ^ n one has x ^ Vn. The procedure consists in sub-
stituting successively for x the values above Vn and examining
whether the corresponding
A(x) = x 2 — n
is a square y 2 . Let us illustrate by a simple example.
Example.
The number n = 13,837 is to be factored. One sees that Vn lies between
117 and 118. In the first step we obtain
A (118) = 118 2 - 13,837 = 87
PRIME NUMBERS 57
which is not a square. In the next step one has
A (119) = 119 2 - 13,837 = 324 = 18 2
so that we have found the factorization
13,837 = (119 - 18) (119 + 18) = 101 • 137
This example is too simple to illustrate the short cuts that serve
to facilitate the work with larger numbers. One important
observation is that one need not calculate each A(x) separately
Since
(x + l) 2 - n = x 2 - n + 2x + 1
one has
A (a: + 1) = A(x) + 2x + 1
and by applying this rule repeatedly one finds
&(x + 2) = A(x + 1) +2z + 3
A(x + 3) = A(x + 2) + 2x + 5
This makes it possible to compute the successive A(x)'s by simple
additions.
Example.
We shall take the formidable number n = 2,027,651,281, on which Fermat
applied his method. The first integer above Vn is 45,030 and the calcula-
tions proceed as follows:
X
= 45,030
x 2 — n =
49,619
31
2x + 1 =
90,061
139,680
90,063
32
229,743
90,085
33
319,808
90,067
34
409,875
90,069
45,035 499,944
58 NUMBER THEORY AND ITS HISTORY
45,035 499,944
90,071
36 590,015
90,073
37 680,088
90,075
38 770,163
90,077
39 860,240
90,079
45,040 950,319
90,081
x = 45,041 1,040,400 = 1,020 2 = y 2
This shows that we have the factorization
n = (45,041 + 1,020) (45,041 - 1,020)
= 46,061 • 44,021
where each factor can be shown to be a prime. In this chain of computations,
each of the various numbers 49,619, 139,680, . . . should be looked up in a
table of squares to determine whether it is actually a perfect square. However,
in most cases this step may be eliminated since the last two digits will already
show that the number is not a square. The small table of 22 entries that we
computed on page 33 giving the possible two last digits of a square number is
most convenient for this purpose. Of all the numbers in the preceding chain
it is only necessary to look up the numbers 499,944 and 1,040,400, since 44
and 00 may be the last two digits in a square.
Fermat's method is particularly helpful when the number n has
two factors whose difference
2y = b — a
is relatively small, because a suitable y will then quickly appear.
In the choice of the example discussed above it is clear that Fermat
had this in mind. By means of certain other improvements that
can be introduced in the procedure, it becomes one of the most
effective factorization methods available.
Problem.
Factor the following numbers by means of Fermat's method: (a) 8,927,
(6) 57,479, (c) 14,327,581.
PRIME NUMBERS 59
4-5. Euler's factorization method. Frenicle tie Bessy (1605-
1675) was an official at the French mint and was well known for
his unusual facility in numerical computations. He was also a
mathematician of no mean ability and was in frequent corre-
Fig. 4-1. Leonhard Euler (1707-1783).
spondcnce with Fermat. In a letter of August 2, 1641, he pro-
poses the following problem : Fse the fact that
221 = 10 2 + ll 2 = 5 2 + 14 2 (4-^1)
to find the factors of this number. The same idea, that two
different representations of a number as a sum of two squares may
serve to factor it, was mentioned by Mcrsenne. However, Euler,
for whom the method is usually named, seems to have been the
first to put it to extensive use.
Leonhard Euler (1707-1783) was a remarkable scientist whose
contributions have left their imprint on almost all branches of
mathematics. His papers were rewarded ten times by prizes of
the French Academy. His productivity was immense; it has
been estimated that his collected works, which are still in the
60 NUMBER THEORY AND ITS HISTORY
process of being published, will fill upward of 100 large volumes.
Euler was born in Switzerland, but he was early called to the
Academy in St. Petersburg, later to the Academy in Berlin at the
request of Frederic II, and back again to St. Petersburg on still
more flattering terms. As a young man he lost the sight of one
eye and later in life he became totally blind, but even this calamity
did not halt his scientific work. One of his best known texts,
Complete Introduction to Algebra (1770), which contains much
material on elementary number theory, was dictated to a servant,
a former tailor, to prepare him to serve as his mathematical
secretary.
Euler carried on an extensive correspondence with contemporary
mathematicians, and the factorization by means of representation
of a number as the sum of two squares is mentioned first in a letter
of February 16, 1745, to Christian Goldbach (1690-1764). Gold-
bach was a German mathematician, onetime teacher of Peter II
and secretary for the Academy in St. Petersburg, who left
scientific work to embark upon a distinguished career in the
Russian civil service.
Euler's factorization method applies only to numbers which in
some way can be represented as a sum of two squares
N = a 2 + b 2 (4-5)
as, for instance,
41 = 5 2 + 4 2 , 269 = 10 2 + 13 2
Since we may assume that the number N to be factored is odd,
one of the numbers in (4-5), say a, is odd and the other, b, is even.
We have observed that the square of an odd number a 2 is of the
form 4n + 1, and since b 2 is divisible by 4, the number JV itself
must be of the form 4m + 1.
We shall assume now that there exists another representation of
N as the sum of two squares
N = c 2 + d 2 (4-6)
u 3, for instance, in the example (4-4) given by Frenicle. The nota-
PRIME NUMBERS 61
tion is again such that c is odd and d even. To show that the two
representations lead to a factorization of N, we proceed as follows.
From (4-5) and (4-6) we have
so that
or
a 2 + b 2 = c 2 + d 2
a 2 -c 2 = d 2 - b 2
(a - c)(a + c) = (d - b)(d + b) (4-7)
Let k be the greatest common factor of a — c and d — b so that
a - c = kl, d -b = km, (I, m) = 1 (4-8)
By our choice of notations a — c and d — b are even, hence k is
even. When (4-8) is substituted into (4-7) and k is canceled,
one obtains
I (a + c) = mid + b) (4-9)
Since I and m are relatively prime, a + c must be divisible by m
a + c = mn (4-10)
When this is applied in (4-9), finally
d + b = In (4-11)
The two expressions (4-10) and (4-11) also show that n is the
g.c.d. of a + c and d + b; thus n is even.
The desired factorization of N which results from (4-5) and
(4-6) is now
N = [© 2 + ©I (m2 + 12) (4 " 12)
To prove that this equation is correct, we multiply out the
expression on the right-hand side and find that it is equal to
{[(km) 2 + (kl) 2 + (nm) 2 + (nl) 2 ]
62 NUMBER THEORY AND ITS HISTORY
Here we substitute the values from (4-8), (4-10), and (4-11) so
that the new expression becomes
J[(d - b) 2 + (a - c) 2 + (a + c) 2 + {d + b) 2 ]
= J(2a 2 + 2b 2 + 2c 2 + 2d 2 ) = i(2N + 2N) = N
as we required.
Example.
For the number N = 221 the two representations (4-4) yield
a = 11, a — c = 6, k = 2
& = 10, a + c = 16, Z = 3
c=5, d — 6=4, m = 2
<2 = 14, d +b = 24, n = 8
The decomposition (4-12) is therefore
221 = (1 + 4 2 ) (2 2 + 3 2 ) =17-13
Clearly the decomposition (4-12) is never trivial in the sense
that any of the factors is equal to 1. To apply Euler's method
one has to determine two representations of a number as a sum of
two squares. This may be done by means of tables of squares,
as in the case of Fermat's method. Often the number is given
in such a form that one representation is immediate. To find
any representation of a number N as the sum of two squares, one
forms the differences N — x 2 for various x's and examines whether
they can be squares y 2 . Many cases are immediately excluded
by inspection of the last two digits of N — x 2 . As before, one
can reduce the calculations to additions by observing that one
obtains N — (x — l) 2 from N — x 2 simply by adding 2x — 1.
Examples.
1. Let us factor iV = 2,501. Since N = 50 2 + 1 we need only another
such representation. One finds for x = 50,
2,501 - x 2 = 1
2x - 1 = 99
2,501 - 49 2 = 100 = 10 2
PRIME NUMBERS 63
Thus one has
a = 1, a - c = -48, k = 8
b = 50, a +c = 50, Z = -6
c = 49, d -b = -40, to = -5
d = 10, d + & = 60, n = 10
so that the decomposition (4-12) is
2,501 = ( 4 2 -H 5 2 )(5 2 + 6 2 ) =41-61
2. Euler applied his method to decompose
N = 1,000,009 = 1,000 2 + 3 2
He finds a second representation
N = 972 2 + 235 2
and this leads to the factorization
N = 293 • 3,413
It is possible to show that if a number can be represented as
the sum of two squares one can find all factorizations by Euler's
method. Euler succeeded also in obtaining a. proof for the follow-
ing theorem due to Fermat: Every prime of the form 4n + 1 can
be represented as the sum of two squares. From our preceding
results we conclude that such a representation can be made in
only one way, since otherwise the number would be factorable.
The proof of the theorem of Fermat will be given in Chap. 11.
Let us illustrate the theorem on the primes of the form 4n + 1
below 100:
5 = 2 2 +l 2 , 13 = 3 2 +2 2 , 17 = 4 2 +1 2 , 29 = 5 2 + 2 2
37 = 6 2 +l 2 , 41=5 2 + 4 2 , 53 = 7 2 + 2 2 , 61 = 6 2 + 5 2
73 = 8 2 +3 2 , 89 = 8 2 +5 2 , 97 = 9 2 + 4 2
Euler's factorization method is capable of wide extensions. It
leads to the theory of representations of numbers by means of
quadratic forms, i.e.,
N = ax 2 + bxy + cy 2
64 NUMBER THEORY AND ITS HISTORY
Such representations can under certain conditions be used for
factoring in the same manner as the special form
N = x 2 + y 2
It would carry us too far to discuss the great number of other
aids and methods for factoring, some of them very ingenious. We
shall make only a final remark about the last digits of factors.
If, for instance, N has the last digit 1, one finds by checking all
possibilities that the two eventual factors must both end in 1,
or both in 9, or one in 3 and the other in 7. When other last
digits in N are examined, one finds the following table:
Last Digit Last Digit
in Number in Factors
1 (1, 1), (9, 9), (3, 7)
3 (1, 3), (7, 9)
7 (1, 7), (3, 9)
9 (1, 9), (3, 3), (7, 7)
The remaining digits 0, 2, 4, 5, 6, 8 are of no interest since there is
an obvious factor 2 or 5 in N. This method may be extended in
various ways, for instance, to several digits or to representations
of the number in number systems with a basis different from 10.
Problems.
1. Factor the numbers (a) 19,109, (6) 10,001 by Euler's method.
2. Express all primes of the form 4n + 1 between 100 and 200 as the sum
of two squares.
4-6. The sieve of Eratosthenes. The factorization theorem
states that every number can be represented uniquely as the
product of prime factors. Thus the prime numbers, as their
name already indicated in the Greek terminology, are the first
building stones from which all other numbers may be created
multiplicatively. As a consequence considerable efforts have been
concentrated on the study of primes.
PRIME NUMBERS 65
The first result that we shall mention has been derived in
Euclid's Elements (Proposition 20, Book IX).
Theorem 4-2. There is an infinitude of primes.
Euclid's proof runs as follows: Let a, b, c, . . . , k be any family
of prime numbers. Take their product P = ab • • • k and add 1.
Then P + 1 is either a prime or not a prime. If it is, we have
added another prime to those given. If it is not, it must be
divisible by some prime p. But p cannot be identical with any of
the given prime numbers a,b, . . .,k because then it would divide
P and also P + 1 ; hence it would divide their difference, which
is 1, and this is impossible. Therefore a new prime can always be
found to any given (finite) set of primes.
We may illustrate the construction of primes by Euclid's method
by the following examples:
2-3+1 = 7 = prime
2-3-5+1 = 31 = prime
2-3-5. 7 + 1 = 211 = prime
2- 3-5-7- 11 + 1 = 2,311 = prime
2 • 3 • 5 • 7 • 11 - 13 + 1 = 30,031 = 59 • 509
2 • 3 • 5 • 7 • 11 • 13 • 17 + 1 = 510,511 = 19 • 97 • 277
2 • 3 • 5 • 7 • 11 - 13 • 17 • 19 + 1 = 9,699,691 = 347 • 27,953
In Euclid's proof one could just as well have used the number
P - 1. When applied to the first primes, this leads to the factori-
zations
2-3-1 = 5 = prime 30,029 = prime
2-3-5-1= 29 = prime 510,509 = 61 - 8,369
209 = 11 • 19 9,699,689 = 53 • 197 • 929
2,309 = prime
There are many other numbers which could have served in a
66 NUMBER THEORY AND ITS HISTORY
similar manner to obtain arbitrarily large prime factors, for
instance, n! ± 1, where as usual
n! = 1 -2-3- --n
is n factorial. The reader may try to factor some of these
numbers.
Extensive tables of primes have been computed. Clearly every
factor table gives information about the primes within its range,
but it is desirable also to have separate lists of primes. Generally
available and unusually free from errors are the tables of primes up
to 10,000,000 prepared by D. N. Lehmer (1867-1938).
There exists an ancient method of finding the primes known as
the sieve of Eratosthenes. Eratosthenes (276-194 B.C.) was a
Greek scholar, chief librarian of the famous library in Alexandria.
He is noted for his chronology of ancient history and for his
measurement of the meridian between Assuan and Alexandria,
which made it possible to estimate the dimensions of the earth
with fairly great accuracy.
Eratosthenes' sieve method consists in writing down all numbers
up to some limit, say 100:
1 2 3 4 5 6 7 8 9 10 1112 13 1415 16
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
32 33 34 35 36 37 38 39 40 41 42 43 44 45 46
47 48 49 50 51~ 52 53 54 55 56 57 58 59 60 61
6263646566 67 6869 70 7172 73 747576
7778 79 808182 83 84 85 86 87 88 89 90 9T
92 93 94 95 96 97 98 99 100
From this series one first strikes out every second number
counting from 2, that is, the numbers 4, 6, 8, ... . In the example
above these numbers are marked by a bar. Counting from the
first remaining number, 3, every third number, that is, 6, 9, 12, . . .
is marked ; some of them will thus have a second bar. The next
remaining number is 5, which is a prime since it has not been
PRIME NUMBERS 67
struck out as divisible by 2 or 3; every fifth number 10, 15, 20, . . .
is eliminated. The first remaining number, 7, is a prime since it
is not divisible by 2, 3, or 5, and its multiples 14, 21, . . . are
marked. In this manner all primes may be determined suc-
cessively. Clearly the method is well adapted to mechanical
procedures. Since it is not necessary to write the numbers
explicitly, one can use stencils or punch cards. All larger factor
and prime tables have been constructed by means of such devices.
The following observation is essential in the application of
Eratosthenes' sieve. In the preceding example, when all multiples
of 7 have been marked in the fourth step, the remaining unmarked
numbers will now include all primes below 100, since no remaining
number N has any factor less than the next prime 11 > V~N.
This fact makes it possible to use the sieve of Eratosthenes to
calculate the number of primes up to prescribed limits. It is
customary to denote the number of primes not exceeding a number
a; by w{x) ; for instance, tt(12) = 5, tt(17) = 7. Let us return to our
example again. Here we had four primes below VlM, namely,
2, 3, 5, and 7. Let us perform the canceling in a slightly different
manner so that in the first step also the prime 2 is eliminated, in
the second step also 3 is canceled, and so on. What is left after the
four cancellations of multiples of 2, 3, 5, and 7 will be the number
1 and the primes between 10 and 100, hence altogether
t(100) - t(\/100) + 1
numbers. On the other hand in the first step one cancels 100/2
numbers out of 100. In the second step one cancels [100/3],
recalling that the bracket denotes the integral part of the quotient.
There is, however, some duplication since the [100/(2 • 3)] numbers
divisible by both 2 and 3 have been eliminated twice. After the
two steps there remain consequently
numbers. In the third application of the sieve one eliminates
68 NUMBER THEORY AND ITS HISTORY
[100/5] numbers, but there is duplication with respect to those
divisible by 2 • 5 and 3 • 5 so that the next further reduction is
R-1-KJ-KH
100
2-3-5
where the last term takes care of the fact that those numbers that
are divisible by 2 • 3 • 5 have been subtracted twice from [100/5].
Thus we conclude that out of the 100 original numbers there is
now left a total of
■ r i00 "| , poo] _ r loo "
|_2 • 5 J |_3 • 5 J |_2 • 3 • 5_
Alter the fourth step one verifies similarly that there remains
_f_io^ir^oo_~|_r_ioo_"|_r^po_-| , r_ioo_i
L2-3-5J L2-3-7J L2 - 5 • 7 J |_3 • 5 -7j + L2 -3 • 5 • 7 J
This makes it possible to calculate 7r(100), since ir(10) = 4 so
that
tt(100) - 3 = 100 - (50 + 33 + 20 + 14)
+ (16 + 10 + 7 + 6 + 4 + 2) - (3 + 2 + l 4- 0) + = 22
or tt(100) = 25 as one could have counted directly from the table
of primes.
It is clear that through the preceding considerations we have
been led to a general formula regarding the number of primes.
PRIME NUMBERS 69
LetN be the given number and p t , p 2 , ■ • . , P, the primes less than
VN. Then
,w - . ( viv) + 1 = * - nn _ m m
L-PlJ LP2J LPrJ
LP1P2J L.P1P3J
_pL_-|_... + ...
LP1P2P3J
It is not difficult to prove this result in general by means of
induction. Through the formula one can determine the numoer
of primes below N when the primes below VN are known. The
method is cumbersome; for instance, to find the number of primes
below 10,000 one must consider the primes less than 100. Some
simplification is derived from the fact that many of the terms must
vanish. However, as shown by Meissel (1870), the formula may
be considerably improved, and through various short cuts he
succeeded in finding
7T (100,000,000) = 5,761,455
These computations were continued by the Danish mathematician
Bertelsen, who applied them for the determination of errors in
prime tables. He announced the following result (1893):
t(1,000,000,000) = 50,847,478
which represents our most extended knowledge of the number of
primes.
Problem.
Determine the number of primes below 200 by the method given above
and check the result by actual count.
4-7. Mersenne and Fermat primes. Considerable effort hag
been centered on the factorization of numbers of particular types.
Some of them are numbers resulting from mathematical problems
70 NUMBER THEORY AND ITS HISTORY
of interest. Others have been selected because it is known for
theoretical reasons that the factors must have a special form.
Among the numbers that have been examined in great detail one
should mention the so-called binomial numbers
N = a n ±b n (4-13)
where a and b are integers. Certain factors can be obtained
immediately from their algebraic expression, since
a n - b n = (a - b) {a 71 ' 1 + a n ~ 2 b + h ab n ~ 2 + b 71 ' 1 ) (4-14)
as one verifies by performing the right-hand multiplication. By
putting —b for b in (4-14), one obtains for odd exponents n
a n + b n = (a + 6) {a 71 ' 1 - a n ~ 2 b + ab n ~ 2 + b n ~ l ) (4-15)
If one replaces a and b in (4-14) by a m and b m , it follows that
a nm -b nm = (a m -b m ) (a (n_1)m +a w(w ~ 2) 6 m H r-6 w(n-1) ) (4-16)
This expression may be used to factor a number (4-13) when the
exponent is composite. Thus, every number (4-13) has certain
algebraic factors that are fairly easily found, and the essential
difficulty lies in factoring these further or in establishing that they
are primes. Here one is aided by some knowledge of the type of
primes that can divide them.
Examples.
1. Factor
N = 10 9 - 3 9 = 999,980,317
One finds the algebraic factors
10 - 3 = 7 and 10 3 - 3 3 = 973 = 7 • 139
By using a factor table on the remaining factor, one finds the prime decom-
position
N = 7 • 19 • 139 • 54,091
2. The number
N = 10 9 + 3 9 = 1,000,019,683
PRIME NUMBERS 71
has the algebraic factors
10 + 3 = 13 and 10 3 + 3 3 = 1,027 = 13-79
and the final result is
N = 13 • 37 • 79 • 26,317
The prime factorization of the numbers
M n = 2 n - 1 (4-17)
has been the object of intensive studies. Their decomposition is
known and tabulated for a large number of exponents n. For
small exponents the reader can easily determine the factors; for
instance,
M 2 = 3 M 6 = 63 = 3 • 3 • 7
M 3 = 7 M 7 = 127
M 4 = 15 = 3 • 5 M 8 = 255 = 3 • 5 • 17
M 5 = 31
As an example of a more imposing factorization, let us give a prime
decomposition that the French mathematician Poulet worked on
as a pastime during the occupation in the Second World War.
2 135 - 1 = 7 • 31 • 73 • 151 • 271 • 631 • 23,311
• 262,657 • 348,031 • 49,971,617,830,801
The reason for the particular interest in the numbers (4-17) can
be found in the fact that they are directly associated with the
classical problem of the 'perfect" numbers, which we shall discuss in
the next chapter. Every number M n that is a prime gives rise to
a perfect number. These primes are known as Mersenne primes.
The historical justification for this nomenclature seems rather
weak, since several perfect numbers and their corresponding
primes have been known since antiquity and occur in almost
every medieval numerological speculation. Mersenne did, how-
ever, discuss the primes named after him in a couple of places
in his work Cogita physico-mathematica (Paris, 1644) and expressed
various conjectures in regard to their occurrence.
72 NUMBER THEORY AND ITS HISTORY
It is clear that a number of the type (4-17) cannot be a prime
when n ** rs is composite, because there would exist an algebraic
factorization of M n as in (4-16). Since in this case a — b = 2 —
Fig, 4-2. Marin Merseimc (1588-1648).
1 = 1, the factorization (4-14) is trivial. One concludes there-
fore that a Mersenne prime lias the form
M 9 = 2 P - 1
where the exponent p is itself a prime. As a consequence, these
numbers have been investigated for many primes p. For small
p one finds relatively many Mersenne primes, but for larger p they
seem to become more and more scarce. At present only 12
Mersenne primes are known; the first ones are
M 2 = 3 M 13 - 8,191
M 9 m 7 M l7 m 131,071
JSC* — 31 M 19 = 524,287
M 7 = 127
PRIME NUMBERS 73
The last two of these were determined by the early Italian mathe-
matician Cataldi (1552-1626) in his Trattato de numeri perfetti by
the direct procedure of dividing by all primes less than the square
root of the number. Cataldi was an enthusiastic protagonist for
mathematical studies. He founded the first mathematical
academy in his native town Bologna and distributed his works
free in Italian cities to create interest in the subject.
The next Mersenne prime M 31 was determined by Euler (1750) ;
another, M 61 , by Pervouchine (1883) and Seelhoff (1886) . Powers
(1911) found that M 8Q and, later (1914), M 107 are primes. The
largest and last of the known Mersenne primes is
M 127 = 170,141,183,460,469,231,731,687,303,715,884,105,727
The only reason for writing explicitly this huge number of 39
digits is that it is the largest number that has actually been verified
to be a prime. It was found by the French mathematician Lucas
in 1876. Lucas (1842-1891) discovered a new and very much
simpler method for testing the primality of the Mersenne numbers.
They have now been examined by means of Lucas's criterion for
the primes up to and including p = 257, and no new Mersenne
primes have been found. The examination of the last few re-
maining ones up to this limit has just been completed by H. S.
Uhler. (See Supplement.)
A family of numbers related to the Mersenne numbers are those
of the form
N n = 2 n + l (4-l 8 )
Fermat initiated the study of their factors and their primality.
Now, for a number of the type (4-18) to be a prime, it is clear that
the exponent n cannot have any odd factor. If, for example,
n = ab where 6 is odd, one would obtain an algebraic factorization
as in (4-15)
2 n + 1 = (2 a ) b -f 1
= (2° + 1) (2 a( ^ 1} - 2 a(b ~ 2) + 2 a(6_3) - |-l)
74 NUMBER THEORY AND ITS HISTORY
However, a number without odd factors must be a power of 2 so
that n = 2 l , and the numbers take the form
F t = 2 2 ' + 1 * (4-19)
These numbers are known as the Fermat numbers, and for the first
values of t they are seen to be primes
F = 3, Fi = 5, F 2 = 17, F 3 = 257, F 4 = 65,537
The next Fermat number is already so large that it is difficult to
factor, but on the basis of the few facts at hand Fermat made the
conjecture that they are all primes. He expresses this conjecture
repeatedly, in letters to Frenicle, Pascal, and others. In August,
1640, he states: 'Me n'en ai pas la demonstration exacte, mais j'ai
exclu si grande quantite de diviseurs par demonstrations infail-
libles, et j'ai de si grandes lumieres, qui etablissent ma pensee que
j'aurois peine a me dedire."
It was not until 100 years later (1739) that Euler exploded the
hypothesis by the simple expedient of showing that the next Fermat
number had a factor. Euler showed first theoretically that any
factor of a Fermat number must have the form
2 t+1 k + 1
For t = 5 one concludes, therefore, that the prime factors must
have the form p = 64A; + 1. From a prime table one finds that
the first primes of this kind are 193, 257, 449, 577, and finally 641,
which actually turns out to be a factor of F 5 .
Through this discovery the Fermat numbers lost much of their
attraction and actuality as a research object. However, through
one of the peculiar twists of the lines of mathematical investigation,
they reappeared with greater importance in an unsuspected and
quite surprising connection with a classical problem. In his
famous Disquisitiones arithmeticae the German mathematician C.
F. Gauss in 1801 among other things took up the ancient problem
of finding all regular polygons that can be constructed by means
of compass and ruler. We shall return to the Disquisitiones and
the problem of the regular polygons later on. It must suffice-
PRIME NUMBERS 75
to state here that after the investigations of Gauss, the problem
was reduced to the question of the existence of the Fermat
primes. As a consequence, they have been the object of numerous
studies, both theoretical and computational, and quite a few of
the larger Fermat numbers have been successfully factored. Of
Fermat's original conjecture there is no trace; no further primes
have been found. Students of the question now seem more
inclined to the opposite hypothesis that there are no further Fermat
primes than the first five already found. A survey of the present
state of the factorizations of Fermat and Mersenne numbers can
be found in a recent paper by D. H. Lehmer. 1
Problems.
1. Factor the numbers
10 8 ±3 8
2. Factor some of the first of the numbers
2"±1
beyond those given above.
4-8. The distribution of primes. By checking the entries in a
prime table one sees soon that aside from minor irregularities the
prime numbers gradually become more scarce. The sieve of
Eratosthenes shows that this must be the case since in the higher
intervals more and more numbers become effaced. For instance,
by actual count one finds that each hundred from 1 to 1,000 con-
tains respectively the following number of primes:
25, 21, 16, 16, 17, 14, 16, 14, 15, 14
while in the hundreds from 1,000,000 to 1,001,000, the corre-
sponding frequencies are
6, 10, 8, 8, 7, 7, 10, 5, 6, 8
and from 10,000,000 to 10,001,000,
2, 6, 6, 6, 5, 4, 7, 10, 9, 6
1 Lehmer, D. H., "On the Factors of 2" ± 1," Bulletin of the American
Mathematical Society, Vol. 53, 164-167 (1947).
76 NUMBER THEORY AND ITS HISTORY
A special computation by M. Kraitchik shows that for the
interval from 10 12 to 10 12 + 1,000 the corresponding figures are
4, 6, 2, 4, 2, 4, 3, 5, 1, 6
Except for the case p — 2 the primes are odd, so any two con-
secutive primes must have a distance that is at least equal to 2.
Pairs of primes with this shortest distance are called prime twins;
for instance,
(3, 5), (5, 7), (11, 13), (17, 19), (29, 31),
(10,006,427, 10,006,429)
In spite of the fact that these prime twins become quite rare in
the tables, it is still believed that their number is infinite. On
the other hand, one can also find consecutive primes whose distance
is as large as one may wish; in other words, there exist arbitrarily
long sequences of numbers that are all composite. To prove this
statement one need only observe that when
n! = 1 -2-3 n
the n — 1 numbers
n\ + 2, n\ + 3, • • • , n\ + n
are all composite.
These remarks show that there are great irregularities in the
occurrence of the primes. Nevertheless, when the large-scale
distribution of primes is considered, it appears in many ways quite
regular and obeys simple laws. The study of these laws in the
distribution of the primes falls in the field of analytic number
theory. This particular domain of number theory operates with
very advanced methods of the calculus and is considered to be
technically one of the most difficult fields of mathematics. Its
central problem is the study of the function ir(x), which indicates
the number of primes up to a certain number x. It was discovered
quite early by means of empirical counts in the prime tables that
PRIME NUMBERS
77
the function ir(x) behaves asymptotically like the function #/log x,
that is, for large values of x their quotient approaches 1
ir(x)
lim
oo z/log x
- = 1
(4-20)
(The logarithm here and in the following is the natural logarithm
to the base e.)
This does not, of course, mean that the difference t(x)
log x
becomes small, but only that this difference is small in comparison
with t(x). The result that is expressed in the formula (4-20) is
commonly known as the prime-number theorem.
For the purpose of approaching the prime function t{x), the
so-called integral logarithm is better than the function x/log x,
although for large values of x the two functions behave asymp-
totically alike. The integral logarithm is defined by means of an
integral
tv ^ r dt
«/2 log t
The following table indicates the accuracy of the approximation:
X
7r(x)
Li(a;)
1,000
168
178
10,000
1,229
1,246
100,000
9,592
9,630
1,000,000
78,498
78,628
10,000,000
664,579
664,918
100,000,000
5,761,455
5,762,209
1,000,000,000
50,847,478
50,849,235
Here the values of ir(x) up to 10,000,000 have been obtained
by actual count from tables of primes, while the two remaining
entries are the values of t(x) calculated by Meissel and Bertelsen,
which we have mentioned previously.
78 NUMBER THEORY AND ITS HISTORY
Already Euler had begun applying the methods of the calculus
to number-theory problems. However, the German mathe-
matician G. F. B. Riemann (1826-1866) is generally regarded as
the real founder of analytic number theory. His personal life
was modest and uneventful until his premature death from tuber-
culosis. According to the wish of his father he was originally
destined to become a minister, but his shyness and lack of ability
as a speaker made him abandon this plan in favor of mathematical
scholarship. He was unassuming to a fault, yet at present he is
recognized as one of the most penetrating and original mathe-
matical minds of the nineteenth century. In analytic number
theory, as well as in many other fields of mathematics, his ideas
still have a profound influence. His starting point was a function
now called Riemann' 's zeta function
This function he investigated in great detail and showed that its
properties are closely connected with the prime-number distri-
bution. He obtained various results and sketched the path of
future progress in a number of well-founded conjectures of which
all, except one that still remains undecided, have been shown to
be correct. On the basis of Riemann's ideas, the prime-number
theorem was proved independently in 1896 by the French mathe-
matician J. Hadamard (1865- ) and the Belgian C. J. de la
Vallee-Poussin (1866- ). Much progress has been made in
analytic number theory since this time, but it remains a peculiar
fact that the key to some of the most essential problems lies in
the so-called Riemann's hypothesis, the last of his conjectures
about the zeta function, which has not been demonstrated. It
states that the complex zeros of the function all have the real
component ^.
Let us present another important result regarding the distri-
bution of primes. As an example, the sequence of numbers
3, 7, 11, 15, 19, 23, 27, . . . (4-21)
PRIME NUMBERS 79
form an arithmetic series, i.e., consecutive terms in the sequence
have the same difference; in this case it is equal to 4. The general
term in the sequence (4-21) is
4:n—l, n = 1, 2, 3, • • •
The question arises whether this sequence contains an infinite
number of primes. To see that this is true, one can apply a
method that is a simple generalization of Euclid's idea for proving
that there is an infinite number of primes. The assumption that
there is only a finite number of primes
P\ = 3, P2, • ' • , Pk
in the sequence (4-21) leads to a contradiction, as we shall see.
One could then form the new number
N = 4pip 2 • ' • Pk — 1 = 4P - 1
which is not divisible by any p*. But any odd prime is of one of the
forms 4n + 1 or 4n — 1, and the product of two numbers of the
form An + 1 is again of this form, so at least one of the prime
factors p of N is of the form 4n — 1. But this prime cannot be
any of the p/s since they do not divide N; hence p is a new prime
in the sequence (4-21).
The same argument may be used to show that the arithmetic
series with the general term 6n — 1, that is,
5, 11, 17, 23, 29, 35, . . .
contains an infinite number of primes. In general, an arithmetic
series consists of terms
an + b, n = 1, 2, • • • (4-22)
where a and b are fixed numbers. If a and b have the common
divisor d, all numbers in the sequence are divisible by d. But
when one assumes that a and b are relatively prime, it can be
shown that the sequence contains an infinite number of primes.
This result is known as the theorem of Lejeune-Dirichlet (1805-
1859). It is another of the many theorems in number theory
that are simple to state and difficult to prove. Dirichlet's method
80 NUMBER THEORY AND ITS HISTORY
requires complicated mathematical tools and many results from
other fields. It is puzzling that many special cases can be ob-
tained very simply, as we illustrated above, yet the search for an
elementary proof has so far been unavailing.
Dirichlet's theorem states that the expressions (4-22) gives an
infinite number of primes when (a, b) = 1. Attempts to show
that other functions may have the same property have not suc-
ceeded. It has not even been possible to prove that an expression
as simple as n 2 + 1 gives an infinite number of prime values.
Related to these questions is the search for functions that will take
only prime values for n = 1, 2, . . . . We have already mentioned
Fermat's unsuccessful conjecture. The other results on this
problem are also all in the negative direction; one can show that
certain types of functions cannot have the desired property. 1
For instance, let us show that no polynomial with integral
coefficients
f(x) = a x r + a\X r ~ x + • • • + a r _ix + a r
can take only prime values for integral x. Let us assume that for
some x = n, the value f(n) = p is a prime. Then for any integral
t, the numbers f(n + tp) are divisible by p, since
f(n + tp) - f(n) = a [(n + tp) r - n r ] + a x [{n + tp)*- 1 - n*' 1 ]
+ • • • + a r -i[(n + tp) - n]
Also each difference
(n + tpY — n {
is divisible by p, as one sees by the binomial expansion. Since
every number f(n-\-tp) is divisible by p, these numbers are
composite unless
f(n + tp) = ±p or f(n + tp) = (4-23)
1 W. H. Mills (Bulletin American Mathematical Society, June, 1947) has re-
cently shown that there exists some real number A such that [.A 3 "] gives only
primes. (The bracket denotes greatest integer as before.)
PRIME NUMBERS 81
But a polynomial of degree r cannot take the same value more
than r times so that the cases in (4-23) cannot happen for more
than 3r values of t, at most, and for all other values /(n + tp) must
be composite.
Example.
When
f(x) = x 2 + 2x + 3
one finds /(2) = 11 and
/(2 + 1K) = 11(1 +6t + llt 2 )
is composite and divisible by 11 when t ?* 0.
In connection with the prime values that polynomials will take,
let us mention some peculiar examples of polynomials that take
prime values for a long series of consecutive values of the variable.
One is the polynomial
x 2 — x + 41
which produces a prime for the 41 values of x: 0, 1, 2, . , . , 40.
Similarly
x 2 - 79x+ 1,601
gives 80 consecutive prime values when x = 0, 1, • • • , 79. There
exist other examples of the same nature.
Let us conclude this review of facts and problems from the
prime-number theory by a few remarks regarding the additive
representation of numbers by means of primes. We have already
mentioned the extensive correspondence between Euler and Gold-
bach regarding mathematical questions, particularly number
theory. In some of these letters, dating from about 1742, Gold-
bach discusses the following two conjectures: Every even number
^6 is the sum of two odd primes. Every odd number ^9 is the
sum of three odd primes. Euler, whose mathematical intuition
was acute, states in reply that he also is convinced of the truth, of
these propositions, but he is unable to find any proof.
82
NUMBER THEORY AND ITS HISTORY
Factor Table
1
3
7
9
11
13
17
19
21
23
27
3
29
31
33
3
37
39
3
41
I 43
47
49
3
3
7
100
3
3
7
11
3
3
7
3
11
3
200
3
7
3
11
3
7
3
13
3
3
3
13
3
300
7
3
3
11
3
17
3
7
3
3
11
7
400
13
11
3
7
3
3
7
3
19
3
3
500
3
3
7
3
11
3
17
23
3
13
3
7
3
3
600
3
3
13
3
7
3
17
3
7
3
11
700
19
7
3
23
3
7
3
3
17
11
3
3
7
800
3
11
3
3
19
3
3
7
3
29
3
7
3
900
17
3
3
11
7
3
13
3
7
3
3
23
13
1,000
7
17
19
3
3
3
13
3
17
3
7
3
1,100
3
3
11
3
3
19
7
3
11
3
17
7
3
31
3
1,200
3
17
3
7
23
3
3
3
3
17
11
29
1,300
7
3
13
3
3
3
11
31
7
13
3
17
3
19
1,400
3
23
3
17
3
13
3
7
3
3
11
3
3
1,500
19
3
11
3
17
37
7
3
3
11
3
20
3
23
7
1,600
7
3
3
3
3
7
23
11
3
31
3
17
1,700
3
13
3
29
3
17
3
11
7
3
3
37
3
3
1,800
3
13
3
7
23
17
3
3
31
3
11
3
7
19
43
1,900
11
23
3
3
19
17
3
41
3
13
7
3
29
3
2,000
3
3
7
3
3
43
7
3
19
3
13
3
23
3
2,100
11
3
7
3
29
13
3
11
3
3
3
Ij
7
2,200
31
47
3
3
7
3
17
3
23
7
3
3
13
2,300
3
7
3
3
7
3
11
23
13
17
3
3
3
3
2,400
7
3
29
3
19
41
3
3
7
11
3
3
7
31
2,500
41
23
13
3
7
3
11
3
7
3
17
43
3
3
2,600
3
19
3
7
3
3
43
37
11
3
3
7
19
3
3
2,700
37
3
3
11
3
7
3
3
7
3
13
41
2,800
7
53
3
29
3
7
3
11
3
19
17
3
3
7
2,900
3
3
41
3
3
23
37
29
3
7
3
17
3
7
3
3,000
3
31
3
23
7
3
3
13
7
3
3
17
11
3,100
7
29
13
3
11
3
3
53
3
31
13
43
3
7
3
47
3,200
3
3
13
3
3
11
7
3
53
3
41
7
3
17
3
3,300
3
3
7
31
3
3
3
47
3
13
17
3,400
19
41
7
3
3
13
11
3
23
3
47
7
19
3
11
3
3,500
3
31
3
11
3
3
7
13
3
3
3
3
3,600
13
3
3
23
7
3
3
19
3
3
11
7
41
3,700
7
11
3
47
3
61
3
3
7
37
3
19
3
23
3,800
3
3
13
37
3
11
3
43
7
3
3
11
23
3
3
3,900
47
3
3
7
3
3
3
31
3
7
11
4,000
19
3
3
3
3
29
37
11
7
3
13
3
4,100
3
11
3
7
3
23
3
13
7
3
3
41
3
11
3
4,200
3
7
3
11
3
41
3
3
19
3
31
7
4,300
11
13
59
31
3
19
3
7
29
3
3
61
7
3
43
3
4,400
3
7
3
11
3
7
3
19
43
3
11
3
23
3
3
4,500
7
3
3
13
3
3
7
23
3
13
3
19
7
4,600
43
17
11
3
7
3
31
3
7
3
11
41
3
3
4,700
3
3
17
7
3
53
3
29
3
3
7
11
3
47
3
4,800
3
11
3
17
61
3
7
3
11
3
7
3
47
29
37
13
4,900
13
7
3
17
3
7
3
13
3
11
3
3
7
100
200
300
400
500
600
700
800
900
1,000
1,100
1,200
1,300
1,400
1,500
1,600
1,700
1,800
1,900
2,000
2,100
2,200
2,300
2,400
2,500
2,600
2,700
2,800
2,900
3,000
3,100
3,200
3,300
3,400
3,500
3,600
3,700
3,800
3,900
4,000
4,100
4,200
4,300
4,400
4,500
4,600
4,700
4,800
4,900
PRIME NUMBERS
Factor Table — (Continued)
83
51
59
61
67
37
29
17
3| 7
7 3
..23
3 13
..3
31 11
3 ..
11 3
69
71
11
3
11 | 13
3
73
77
79
29
3
131 | 19
3
3
83
87
11 19
7| 3
3
31 17
3 |23
3
7 '
3
19
3 13
3
23
3 I 7
7 1 3 131
17 . . 3
3 | 17 47
3 13
3
93 | 97
3
3
3
23 [41
3 | 13
3
7
31
7
3 |43
3
37
3
' 13
11 | 3
3 | 31
7
7 37
3 19
.. 3 I 11
31 | 3
3 11
23 19
3
17 3
..7
3 13
3 61
47 53
3
41
7
3
43
3
3
11
41
3
3
19
31
23
37
3
3
43
7
13
3 I 7
7 I 3
23
3
3 59
. . 3
. . 29
3 7
7 3
84 NUMBER THEORY AND ITS HISTORY
One verifies Goldbach's conjectures immediately for the smallest
numbers. For instance, for even numbers,
6=3 + 3 14 = 3 + 11
8=3+5 16 = 3 + 13
10 = 3 + 7 18 = 5 + 13
12 = 5 + 7 20 = 3 + 17
and for odd numbers,
9 = 3 + 3 + 3 17 = 3 + 7+ 7
11 =3 + 3 + 5 19 = 3 + 5+11
13 = 3 + 3 + 7 21 =3 + 5 + 13
15 = 3 + 5 + 7
The smallest integers 1, 2, 3, 5 must obviously be regarded as
exceptions. When the numbers become fairly large, there will
usually be numerous representations; for instance,
48 = 5 + 43 = 7 + 41 = 11 + 37 = 17 + 31 = 19 + 29
Goldbach's conjectures have been verified numerically up to
100,000 (N. Pipping). One should note also that the first con-
jecture implies the second. Take an odd number N and sub-
tract the odd prime p < N from it. Then N — p is even, and if
the even numbers could be expressed as the sum of two primes,
any odd number N would be the sum of three.
A problem for which Euler could find no attacking point could
be expected to be extremely difficult, and it was not until fairly
recently that essential progress was made. The Norwegian mathe-
matician V. Brun (1885- ) developed an extension of the sieve
method of Eratosthenes that enabled him to show that every suffi-
ciently large even number N can be written as a sum
N = JVi + N 2
where Ni and N 2 have at most nine prime factors. Later, others
improved the result to four prime factors, but it is still a far cry to
PRIME NUMBERS 85
Goldbach's conjecture, which requires a single prime factor in
each summand. Goldbach's second theorem, which we saw was
a weaker result that would follow from the first, is, however, much
nearer to its final solution. In 1937 the Russian mathematician
I. Vinogradoff succeeded in showing by analytic means that every
odd number that is sufficiently large is the sum of three odd primes.
How large the numbers have to be, however, he could not decide.
Bibliography
Cunningham, A. J. C. : Binomial Factorisations Giving Extensive Congruence-
Tables and Factorisation-Tables, Vols. 1-7, Francis Hodgson, London,
1923-25.
Kraitchik, M.: Theorie des nombres, Vols. I and II, Gauthier-Villars & Cie,
Paris, 1922, 1926.
Lehmeb, D.N.: "Factor table for the first ten millions containing the smallest
factor of every number not divisible by 2, 3, 5 and 7 between the limits and
10,017,000," Carnegie Institution of Washington Publication 105 (1909).
: "List of Prime Numbers from 1 to 10,006,721," Carnegie Institution
of Washington Publication 165 (1914).
CHAPTER 5
THE ALIQUOT PARTS
5-1. The divisors of a number. Several problems relating to
the divisors of a number can be solved by means of the main
theorem that every integer can be represented uniquely as the
product of prime factors. A number N shall be written
N = pi> 2 " 2 • • • V r ar (5-1)
where the p/s are the various different prime factors and on the
multiplicity, i.e., the number of times p { occurs in the prime
factorization. For any divisor d of N one has
N = ddi (5-2)
where d x is the divisor paired with d. When multiplied together,
the prime factorizations of d and d\ must give that of N so that
d = piV ' • • Vr Sr (5-3)
where the exponents 5; do not exceed the corresponding at in
(5-1). Since the second factor in (5-2) must contain the remain-
ing factors, it becomes
dx = pi" 1 - 51 ^" 2- * 2 ' ' • Pr ar ~ Sr
In the expression (5-3) for a divisor the exponent 5i can take the
«i + 1 values 0, 1, . . . , a\, similarly 8 2 the a 2 + 1 values 0, 1, ... ,
a 2 , and so on. Since each choice of 5i can be combined with any
choice of 8 2 , and so on, one concludes :
Theorem 5-1. The number of divisors of a number N in the
form (5-1) is
v(N) = («i + l)(a 2 + 1) ' ' ' («r + 1) (5-4)
THE ALIQUOT PARTS 87
Example.
The number
60 = 2 2 • 3 • 5
has
f(60) = (2 + 1)(1 + 1)(1 + 1) = 12
divisors. They are
1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 (5-5)
We shall now determine various expressions that may be formed
by means of the divisors of a number. We find first the product of
the divisors. In (5-2) let d run through all v(N) divisors of N.
The corresponding c?i will then also run through these divisors
in some order, so that the product of all d'a is the same as the
product of all di'a. This we write
Ud = Udt
where the symbol II, as usual, denotes the product. We form the
product of all i>(JV) equations (5-2) and obtain
N V(N) = (EfaXlfai) = (lie*) 2
This yields the desired result:
Theorem 5-2. The product of all divisors of a number N is
Ud = N^ N)
Example.
The product of all divisors of 60 is
Hd = 60 6 = 46,656,000,000
as one may check by multiplying together the divisors (5-5).
The result in theorem 5-2 can be expressed in a different manner
When
xi, x 2 , . . . ,x n (5-6)
is a set of n positive numbers, the geometric mean of the numbers
is defined to be
G = Vx\X2 • ' ' Xn
88 NUMBER THEORY AND ITS HISTORY
When applied to the product of the v(N) divisors of N, one sees:
Theorem 5-3. The geometric mean of the divisors of a number
Wis
G(N) - VN
The determination of the sum <r(N) of the divisors of a number
N is slightly more complicated. We shall first reduce the problem
to the case where N is a power of a single prime, a method that is
often applicable in similar problems. Let us write the given
number iVasa product of two relatively prime factors
N = ab
Since the prime factors of a and b are different, one concludes that
any divisor d of N must have the form
d = ajbi (5-7)
where a,- is a divisor of a and b { a divisor of 6. We denote the
divisors of a and b, respectively, by
1, a h a 2 , . . . , a, 1, b x , b 2 , . . . , b
so that their sums are
a (a) = 1 + a x + a 2 -\ f- a, <r(6) = 1 + b x + b 2 -\ \- b
In (5-7) let us take all divisors of N with the same a { . Their
sum is
a t -(l + bi + b 2 H h b) = a t -*(b)
Next, by taking this sum for all possible a* one obtains as the total
sum of all divisors of N
l(r (6) + ai(r(6) H 1- aa(b) = a(a)<r(b)
Thus we have derived the result that when a and b are relatively
prime
<r(N) = <r(ab) = a(a)a(b) (5-8)
We split a and b further into relatively prime factors and apply the
same rule (5-8) again. This may be continued until the factors
become the powers of the various primes dividing N. As a con-
THE ALIQUOT PARTS 89
sequence we conclude that when N has the prime factorization
(5-1)
°(N) = c(p^)a(p 2 ^) • . • a(7> r «0 (5-9)
For a prime power p a the divisors are
1, P,P 2 ,-.-, P a
so that
a{p a ) = l + p + p 2 H hp a
This is a geometric series in which the quotient of two consecutive
terms is p. It may be summed by the usual trick, multiplying the
series by p
V ' <?(p a ) = p + p 2 H \- p a + p a+1
and subtracting the original series
p • o{p a ) - a(p a ) = p« +1 - 1
so that
p a+1 — 1
0-(p a ) = — (5-10)
p — 1 v '
When this result is applied to each factor in (5-9), we have proved:
Theorem 5-4. The sum of divisors of a number N with the
prime factorization (5-1) is
Each prime power p a in the factorization (5-1) contributes a
factor (5-10) to the expression (5-11) for the sum of the divisors.
It is useful to observe the two following simple cases, which occur
commonly. When there is a single prime factor p, one has
When p = 2 a ,
*(P) = ~ T = V + 1
p - 1
*(2 a ) = 2 _ = 2« +1 - 1
90 NUMBER THEORY AND ITS HISTORY
Let us mention also that the average or arithmetic mean A(N)
of the divisors is obtained by dividing their sum a(N) by their
number v(N) so that
wlm (5 " 12)
Example.
The sum of the divisors of
60 = 2 2 • 3 • 5
is
<r(60) = (2 3 - 1)(3 + 1)(5 + 1) = 168
as one can verify by summing the divisors (5-5). Their average is
168
^(60) =— = 14
Since 14 > a/60 this illustrates the general fact that the arithmetic mean is
greater than the geometric mean.
The harmonic mean H of a set of numbers (5-6) is denned by
i i.(I + I + ... + I) (8 _ 13)
H n \Xi X2 Xn/
To determine the harmonic mean H(N) of the divisors of a number
N let us first find the sum of their inverse values. According to
(5-2) one has for any divisor d
1 _ a\
d~ N
where d x is the divisor paired with d. Here, as before, when d
runs through all divisors of N, so will d\. By summing all these
equations for the various d's, one obtains therefore
1 1 1 a(N)
where 21 is the usual summation symbol. According to (5-13),
THE ALIQUOT PARTS 91
we divide by the number v(N) of divisors and find by means of
(5-12)
H(N) v(N)^d v(N)N~ N
This gives the result:
Theorem 5-5. The product of the harmonic and arithmetic
mean of the divisors of a number is equal to the number itself
N = A(N) -H(N)
Since the arithmetic mean is greater than the geometric mean
VN of the divisors, one has the inequality
A(N) ^ VN ^ H(N)
in accordance with the general theory of means.
Example.
We have seen that the arithmetic mean of the divisors of 60 is 14. Therefore
#(60) = ff = 4f
Problems.
1. Find the number of divisors, their sum, and their means for (a) 220
(6) 365, (c) 6! = 1 • 2 • 3 • 4 • 5 • 6.
2. Find the sum of the squares of the divisors of a number.
3. Find the smallest numbers with 2, 3, 4, 5, or 6 divisors.
5-2. Perfect numbers. The perfect numbers are essential
elements in all numerological speculations. God created the
world in six days, a perfect number. The moon circles the earth
in 28 days, again a symbol of perfection in the best of all possible
worlds. In numerological terminology, the divisors are the parts of
which a number is created or reproduced. A perfect number is a
number that is the sum of its divisors, or, in more archaic language,
it is the sum of its aliquot parts. In this definition it must be
observed that Greek mathematics excluded the number itself as
a proper part. Therefore, to obtain the sum (r (N) of the aliquot
parts of a number N in the Greek sense, one must diminish the
92 NUMBER THEORY AND ITS HISTORY
sum <t(N) of all divisors we found in theorem 5-4 by the improper
divisor N so that
ffQ (N) = <r(N) - N (5-14)
The condition for a perfect number may then be expressed in the
formula
*o(N) = N (5-15)
or equivalently
<r(JV) = 2N (5-16)
By means of any of these conditions one can check whether a
number is perfect. For instance,
cr(6) = <r(2 • 3) = (2 + 1)(3 + 1) = 12
and
<r(28) = a(2 2 • 7) = (2 3 - 1)(7 + 1) = 56
so that both 6 and 28 are perfect numbers.
Only one general type of perfect numbers is known:
Theorem 5-6. A number of the form
p = 2^" 1 (2 P - 1) (5-17)
is perfect when
q = 2 P - 1
is a Mersenne prime.
This theorem represents the final proposition in the ninth book
of Euclid's Elements. The proof consists in computing
<r(P) = (2 p - l)(q + 1) = (2 p - 1)2 P = 2P
We have already mentioned that there are only 12 known
Mersenne primes, which one obtains for
p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127
From these one computes the 12 known perfect numbers. The
first four are
P 2 = 2 • (2 2 - 1) = 6
P 3 = 2 2 • (2 3 - 1) = 28
P 5 = 2 4 • (2 5 - 1) = 496
P 7 = 2 6 • (2 7 ~ 1) = 8,128
THE ALIQUOT PARTS 93
In Barlow's Number Theory (London, 1811) the author gives the
perfect numbers up to P 31 corresponding to the Mersenne prime
M Z1 obtained by Euler, at the time the greatest prime known.
This perfect number "is the greatest that will ever be discovered,
for, as they are merely curious without being useful it is not likely
that any person will attempt to find one beyond it." The great
efforts expended since that time in such computations show that
it is difficult to underestimate human curiosity. At present it
seems possible that further efforts along these lines will be made
by means of the tremendous calculators developed during the
Second World War, as soon as they are available for more peaceful
pursuits.
All of these perfect numbers are even; Euler succeeded in
proving the following theorem, which is the most general result
known for perfect numbers :
Theorem 5-7. Every even perfect number is of the type (5-17)
discussed by Euclid.
To prove this theorem we write the even perfect number P in
the form
P = 2*- 1 • q (5-18)
where q is some odd number. Since the two factors in (5-18) are
relatively prime, one finds as in (5-8)
a(P) = a(2^) • a(q) = (2*> - 1) • a(q)
The condition (5-16) for a perfect number states that one must
have
cr(P) = (2? - l)a(q) = 2P = 2*> ■ q
In this relation let us use the sum <r Q (q) of the proper divisors as
defined in (5-14) instead of a(q). One obtains
(2* - l)[cr (g) + q] = 7Pq
and this may be rewritten
q = (2 p - lkofe) (5-19)
This condition permits us to draw some strong conclusions. It
94 NUMBER THEORY AND ITS HISTORY
implies first that d = cr (q) is a proper divisor of q. On the other
hand <r (q) was the sum of all proper divisors of q, including d,
so that there cannot be any other proper divisors besides d. But
a number q with a single proper divisor d must be a prime and
d = 1. From (5-19) one concludes finally that
q = 2 P - 1
is a Mersenne prime. Thus the even perfect number (5-18) is of
the form (5-17) given by Euclid.
Do there exist any odd perfect numbers? This is one of the
celebrated unsolved problems in number theory. Extensive
numerical computations have failed to divulge any odd perfect
number less than 2,000,000. It has been possible to find various
conditions that such numbers must satisfy but they are insuffi-
cient to prove that odd perfect numbers cannot exist. (See
Supplement.)
For numbers that are not perfect there are the two possibilities:
<r (N) > N, <r (N) < N
Numbers of the first kind are called abundant, and those of the
second kind are deficient. This distinction is considered important
in numerology. For instance, Alcuin (735-804), the adviser and
teacher of Charlemagne, observes that the entire human race
descends from the 8 souls in Noah's ark. Since 8 is a deficient
number, he concludes that this second creation was imperfect in
comparison with the first, which was based on the principle of the
perfect number 6. The perfect numbers represent the happy
medium between abundance and deficiency.
The first few abundant numbers are
12, 18, 20, 24, 30, 36, . . .
There are only 21 abundant numbers up to 100, as the reader may
verify, and they are all even. The first odd abundant number is
945 = 3 3 • 5 • 7
for which
<r (945) = 975
THE ALIQUOT PARTS 95
There exists a table of the values of the sum <r(N) of the divisors of
the numbers up to 10,000, computed by J. W. L. Glaisher, so that
the character of numbers not exceeding this limit is easily de-
termined. There are some rules for abundant and deficient
numbers: for instance, a prime or a power of a prime is deficient-
any divisor of a perfect or deficient number is deficient; any
multiple of an abundant or perfect number is abundant.
We saw that the perfect numbers were defined by the condition
*q(N) = N
For certain abundant numbers the sum of the proper divisors may
turn out to be a multiple of the number itself. For example
<r (120) =2-120
as one easily verifies. A number of this kind is called multiply
perfect. When
<ro(N) = k-N
the integer k may be called the class of the multiply perfect number
so that 120 is of class 2 while the perfect numbers are of class 1 '
The problem of finding such multiply perfect numbers appears
to have been formulated first in 1631 by Mersenne in a letter to
Rene Descartes (1596-1650). Although Descartes's fame rests
mainly on his philosophical method and in mathematics on his
creation of analytical geometry and the invention of coordinate
systems, he was also greatly interested in number theory and made
various contributions to it. He must have speculated considerably
over the problem proposed by Mersenne, because about seven
years later he responded with a list of multiply perfect numbers
which he could not have discovered without great effort and
ingenuity. In the meanwhile, Fermat had also tackled the
problem and discovered a second multiply perfect number, namely,
672 = 2 5 • 3 • 7
for which
(T (672) = 2-672
96 NUMBER, THEORY AND ITS HISTORY
while Andre Jumeau, prior of Sainte-Croix, found a third
523,776 = 2 9 -3 • 11 ■ 31
also of class 2. Descartes in several letters to Mersenne gave
another multiply perfect number of class 2, namely, 1,476,304,896,
six others of class 3, and one of class 4. In addition he described
Fig. 5-1. Ren6 Descartes (1596-1650).
various general rules that permitted him to construct these
numbers. The subsequent letters exchanged between Mersenne,
Fermat, and Freniele contain several other multiply perfect
numbers. More recently many others have been discovered,
notably by E. Lucas, D. N. Lehmer, A. Cunningham, R. D.
Carraichael, and D. E. Mason. The most complete list to date is
due to P. Poulet (1929), and it contains 334 multiply perfect
numbers, some of class as high as 7. (See Supplement.)
5-3. Amicable numbers. Another type of numbers that are
prominent in the lore of number mysticism is the amicable numbers.
They are defined to be pairs of numbers such that each member is
THE ALIQUOT PARTS 97
composed of the parts of the other, thus symbolizing mutual
harmony, perfect friendship, and love. The existence of amicable
numbers seems to have been discovered somewhat later than the
perfect numbers, probably in the period of the flowering of the
Neo-Platonic mystical school in Greek philosophy. One of the
most influential of the Neo-Platonic philosophers, Iamblichus of
Chalcis (about a.d. 320), ascribes the knowledge of amicable
numbers to the earliest Pythagorean school (about 500 b.c).
This mythical tradition has, however, little credit with the his-
torians of the mathematical sciences.
In Arab mathematical writings the amicable numbers occur
repeatedly. They play a role in magic and astrology, in the
casting of horoscopes, in sorcery, in the concoction of love potions,
and in the making of talismans. As an illustration let us quote
from the Historical Prolegomenon of the Arab scholar Ibn Khaldun
(1332-1406) :
Let us mention that the practice of the art of talismans has also
made us recognize the marvelous virtues of amicable (or sympathetic)
numbers. These numbers are 220 and 284. One calls them amicable
because the aliquot parts of one when added give a sum equal to the
other. Persons who occupy themselves with talismans assure that
these numbers have a particular influence in establishing union and
friendship between two individuals. One prepares a horoscope theme
for each individual, the first under the sign of Venus while this planet is
in its house or in its exaltation and while it presents in regard to the
moon an aspect of love and benevolence. In the second theme the
ascendant should be in the seventh sign. On each one of these themes
one inscribes one of the numbers just indicated, but giving the strongest
number to the person whose friendship one wishes to gain, the beloved
person. I don't know if by the strongest number one wishes to designate
the greatest one or the one which has the greatest number of aliquot
parts. There results a bond so close between the two persons that
they cannot be separated. The author of the Ghai'a and other great
masters in this art declare that they have seen this confirmed by
experience.
Through the Arabs the knowledge of amicable numbers spread
to Western Europe. They are mentioned in the works of many
98 NUMBER THEORY AND ITS HISTORY
prominent mathematical writers around a.d. 1500, for instance
Nicolas Chuquet, Etienne de la Roche, known as Villefranche,
Michael Stiefel, Cardanus, and Tartaglia.
As we have already stated, a pair of numbers is said to be
amicable when the sum of the aliquot parts of one is equal to the
other, and conversely. In our previous terminology this can be
expressed that M and N are amicable when
«7 O (A0 = M, * (M) = N (5-20)
When one uses the sums of all divisors of the numbers, the con-
ditions (5-20) may be restated
<r(N) = <r(M) = N + M (5-21)
In ancient numerology there appears but a single set of amicable
numbers, namely, the pair
N = 220 - 2 2 • 5 • 11, M = 284 = 2 2 • 71
Even the discovery of the special relations between these two
fairly large numbers is evidence of considerable familiarity with
number properties. For this pair one has
M + AT = 504
and the formula for the sum of the divisors of a number yields
<r(N) = (2 3 - 1)(5 + 1)(11 + 1) = 504
= (2 3 - 1)(71 + 1)> v{M)
so that the condition (5-21) is fulfilled.
There is no indication of any other pair of amicable numbers
having been discovered before the work of Fermat. This is some-
what peculiar since Fermat found his new pair through the redis-
covery of a rule that actually had been formulated by the Arab
mathematician Abu-1-Hasan Thabit ben Korrah as early as the
ninth century. This rule we shall reformulate as follows:
For the various exponents n, write down in a table the numbers
Vn = 3 • 2 n - 1 (5-22)
THE ALIQUOT PARTS
99
n
1
2
3
4
5
95
6
7
Vn
5
11
23
47
191
383
As may be seen, each number is obtained by doubling the pre-
ceding and adding 1.
(5-23)
If, for some n, two successive terms Vn _ x and p n are both primes,
one examines the number
q n = 9 • 2 2 — 1 - 1 (5_ 24 )
If this number is also prime the pair
M = 2 n Pn _ lPn , N = 2 n q n (5-25)
is amicable. To illustrate the rule we observe that -p x = 5 and
p 2 = 11 are primes, and since q 2 = 71 is also prime we obtain the
classical pair 220 and 284 from (5-25).
To prove the rule of Thabit ben Korrah we compute bv means of
(5-22) and (5-24)
a{M) = (2 W+1 - lXp^ + l)(p n + 1) = Q • 2 2n ~ 1 (2 n+1 - 1)
*(N) = (2 W+1 - \){q n + 1) = 9 . 2 2n-i (2 n+i _ 1}
Since also
M + N = 2 n • (p n _ l7?n + ffw ) = 9 • 2 2n ~ 1 (2 n+1 - 1)
the pair is amicable.
The next pair of successive primes in the table (5-23) is
p 3 = 23 and p 4 = 47. In this case g 4 = 1,151 is also prime, and
we obtain the amicable pair announced by Fermat in 1636
17,296 = 2 4 • 23 • 47, 18,416 = 2 4 • 1,151
Descartes stated in letters to Mersenne in 1638 that he had been
led to the same rule and gave the third pair of amicable numbers
9,363,584 = 2 7 • 191 • 383, 9,437,056 = 2 7 • 73,727
corresponding to the primes p 6 = 191 and p 7 = 383 in the series
(5-23).
100 NUMBER THEORY AND ITS HISTORY
Euler took up the search for amicable numbers in a systematic
manner and developed several methods for finding them. In 1747
he gave a list of 30 pairs which he later expanded to more than 60.
Some of them are
2 • 5 • 7 • 19 • 107
J2 4
47-89
J2 3
19-41
2 • 5 • 47 • 359
[2 4
53-79
[2 5
199
2 4 • 23 • 479
J2 2
•5-251
J2 3
•17-79
2 4 • 89 • 127
[2 2
• 13 • 107
)2 3
•23-59
A rank amateur may occasionally make a contribution to number
theory, as was demonstrated again when the sixteen-year-old
Italian boy Nicolo Paganini in 1866 published the very small
pair of amicable numbers
1,184 = 2 5 • 37, 1,210 = 2 - 5 • ll 2
which had eluded all previous investigators. They were probably
found by trial and error. An extensive list of amicable numbers
is due to P. Poulet (1929). A complete survey of the existing
knowledge about amicable numbers has recently been published
by E. B. Escott. It contains a list of the 390 known amicable
pairs together with the names of their discoverers.
5-4. Greatest common divisor and least common multiple.
The algorism of Euclid enabled us to find the greatest common
divisor of two and more numbers and also their least common
multiple. When the prime-factor decompositions of the numbers
are known, the process becomes much simpler. Let a and b be
two given numbers, and
a = pi'V 2 • • • Pr ar , b = Pi^W ' ' • Pr* (5-26)
be their prime factorizations. It is convenient to write the two
decompositions formally as if the same primes occur in both.
This is possible since, for instance, if p\ should not divide b one
can take j3i = 0. Since one is interested in what happens in
regard to each prime pi, it often simplifies matters to use the
product symbol and write instead of (5-26)
a = Upi ai , b = lip/* (5-27)
THE ALIQUOT PARTS 101
When a number d is to divide both a and b, it cannot have any
prime factors different from those occurring in these numbers so
mat OTIfi ran wnfo oyj
that one can write
Vt t (5-28)
d = p x h p 2 h Sr
For each t the exponent 8 t in (5-28) cannot exceed any of the
corresponding «• and ft in (5-26). If, therefore, d is to be the
g.c d. of a and b, the exponent fc must be the smaller or minimal
write 6 eXp ° nents ai and ^ In mathematical shorthand we
«,• = min (a h ft), i = 1, 2, • • • , r
Similarly if
m = p^p 2 ^ --p*
is to be divisible both by a and b, none of the exponents * can be
less than ^ or ft. Therefore, if n is the l.c.m. of a and ^
«•■ must be equal to the greater or maximal of the two number,
a 4 and ft ; m symbols U(ffL
m = max {ai, ft), * = 1, 2, • • , r
Let us summarize these remarks:
Theorem 5-8. The greatest cummon divisor and the least
common multiplum of the two numbers a and 6 with the prime
decompositions (5-27) are respectively
(a, b) = u P r- «>*. a>, k 6] = IIp . max ( «, *>
It is evident that these rules (5-29) can be extended to three
or an arbitrary set of numbers.
Example.
For
a = 2 6 • 3 2 • 5, 6 = 2 5 • 3 3 • 7
one has the g.c.d. and l.c.m.
(a, 6) = 2 5 • 3 2 , [a, b] = 2 6 • 3 3 ■ 5 • 7
Let us show how some of the properties of the g.c.d and the
l.c.m. we derived previously (Chap. 3) follow quite simply also
102 NUMBER THEORY AND ITS HISTORY
from the formulas (5-29). If we multiply the two numbers
(a, b) and [a, b], the exponent to each p { becomes
min (ai, 8%) + max (a,-, &•)
But the sum of the smaller and the greater of two numbers is their
sum a{ + 6i. On the other hand a* + &• is the exponent of pi in
the product ab so that we have (3-4)
(a, 6) [a, b] = a& (5-30)
Let us next multiply the g.c.d. (a, 6) by a number c with the
prime decomposition
c = pi 71 p 2 72 • ' • Pr Jr
The exponent of pi in the product c • (a, b) is then
7,- + min (a,-, 8i)
But this is the same as the number
min (7,- + oti, 7i + 8i)
which is the exponent of pi in (ca, cb). This shows us that
c(a, b) = (ca, cb) (5-31)
as we obtained previously in (3-2). Similarly one sees that for
the l.c.m. (3-5)
c[a, b] = [ca, cb] (5-32)
holds, because one has the identity
ji + max (ai, 8i) = max (?,- + a { , y { + &■)
The laws (5-30), (5-31), and (5-32) were derived here by means
of the theorem of the unique factorization of a number into prime
factors. It is of interest to note that each of them is the expression
of some simple property of the process of forming maximum and
minimum of two numbers, namely,
min (a, 6) + max (at, 8) = a + B
7 + min (a, 8) = min (7 + a, 7 + 8)
7 + max (a, 8) = max (7 + a, 7 + 8)
THE ALIQUOT PARTS 103
A little later on we shall derive some properties of the g.c.d. and
the l.c.m. that depend on the same principle, but involve the
maximum and minimum of three numbers.
Let us discuss the two operations of forming the g.c.d. and the
l.c.m. from a somewhat different point of view. Each of them
d = (a, b), m = [a, b]
associates new elements d and m with the given ones a and b, much
in the way of ordinary addition and multiplication. These oper-
ations satisfy some very simple laws:
1. Idempotent law:
(a, a) = a, [a, a] = a
2. Commutative law:
(a, b) = (b, a), [ a , b] = [b, a]
3. Associative law:
((a, b), c) = (a, {b, c)), \[ a , b], c] = [a, [6, c]]
4. Absorption law:
[a, (a, b)] = a, (a, [ a , 6]) = a
Let us make a few comments on the four properties of the
operations. First, it should be noticed that the two operations
are dual, i.e., the conditions remain the same when the g.c.d. and
the l.c.m. are exchanged everywhere. The idempotent condition
states only that the g.c.d. and the l.c.m. of a number a with itself
is a. The commutative law and the associative law are exactly
the same as for addition and multiplication
a + b = b + a, ab = ba
a + (6 + c) = (a + b) + c, a(bc) = (ab)c
Since (a, b) is a divisor of a, the l.c.m. of a and (a, b) must be a
and the second part of the absorption law is equally trivial.
104 NUMBER THEORY AND ITS HISTORY
Here also, the four laws can be expressed as properties of the
operation of forming maximum and minimum of numbers. Let
p be some prime that divides a, b, and c to the powers p a , p®, and
p y , respectively. Then we leave it to the reader to verify that
the four laws are consequences of
2.
3.
min (a, a) = a, max (a, a) = a
min (a, 0) = min (p, a), max (a, 0) = max (0, a)
min {min (a, /3), 7} = min {a, min (0, 7)}
max {max (a, j3), 7} = max {a, max ((3, 7)}
4.
max {a, min (a, /3)} = a, min {a, max (a, 0)} = a
One reason for going into these simple rules for the g.c.d. and
the l.c.m. in some detail is that mathematicians quite recently have
come to realize that in many important mathematical systems
there exist operations with analogous properties and, furthermore,
that the mathematical theories of these systems are essentially
dependent on these laws of combination. It is far beyond the
scope of this book to discuss these theories; it must suffice to say
that they occur in the extension of number theory to other systems
than the ordinary integers; they appear in many theories of
algebra, in function theory and geometry, and even in logic. In
all cases, there are two operations corresponding in our special case
to g.c.d. and l.c.m. A special notation has been introduced for
such operations, namely,
d = a n b, m = a u b
while various names are in use, for instance, meet and join or union
THE ALIQUOT PARTS 105
and cross-cut. The two operations satisfy the same dual set of
axioms as those mentioned previously :
1.
a n a = a, a v a = a
2.
a nb = b n a, a v b = b u a
3.
a n (6 n c) = (a n b) n c, a u (6 u c) = (a u 6) u c
4.
a u (a n b) = a, a n (a u 6) = a
Systems that satisfy these axioms have been called lattices or
sometimes structures.
Besides the g.c.d. and the l.c.m. in number theory we shall
mention only a single other example of such systems. Let A and
Fig. 5-2.
B be two sets of points or, more general, of elements of some sort.
One can picture A and B as the shaded portions of the plane in the
illustration (see Fig. 5-2). Then the union or sum A u B of the
two sets consists of the elements that belong to either A or B,
while the cross-cut or intersection A n B contains the common
elements to A and B. In the figure A u B is the whole shaded
part of the plane while A n B is doubly shaded. The reader will
have no difficulty in verifying that the four pairs of axioms for a
lattice are satisfied.
106 NUMBER THEORY AND ITS HISTORY
In most such lattice systems there are further rules which the
two operations satisfy. This is, for instance, the case in the theory
we are particularly interested in here, namely, the g.c.d. and l.c.m.
of numbers. The two subsequent theorems can be interpreted as
giving laws of this kind.
Theorem 5-9. For any three integers a, b, and c one has the
rules
(a, [b, c]) = [(a, 6), (a, c)] (5-33)
and
[a, (b, c)] = ([a, b], [a, c]) (5-34)
connecting the greatest common divisor and the least common
multiple. One is the dual of the other.
One can express the law (5-33) in the form that the g.c.d. of a
number a with the l.c.m. of two numbers b and c may be found by
computing the g.c.d. of a with b and c separately and taking the
l.c.m. of the results. The second rule (5-34) can be stated
analogously. These laws are commonly called the distributive laws.
To prove the equality (5-33) it is probably simplest to use the
unique factorization theorem and verify for each prime that the
exponents to the various powers to which it is raised on both sides
are the same. We assume that some prime p divides the numbers
a, b, and c to powers with the exponents a, /3, and y, respectively.
Since b and c appear symmetrically in (5-33), there is no limitation
in arranging the notation such that j8 ^ y. Then the exponent
of the power of p contained in [b, c] is 0. Consequently, the left-
hand side of (5-33) contains p to a power with the exponent
min (a, /3) (5-35)
On the other hand, in (a, b) and (a, c) the prime p occurs with the
exponents
min (a, /3), min (a, y)
Since ^ y the first of these numbers is the larger, so that the
right-hand side of (5-33) also contains p to a power with the
exponent (5-35). This completes the proof of (5-33), and the
equality (5-34) may be derived quite analogously.
THE ALIQUOT PARTS 107
There exists another interesting identity, which we shall now
derive:
Theorem 5-10. For any three integers a, b, and c one has
([a, 6], [a, c], [b, c]) = [(a, b), (a, c), (b, c)] (5-36)
This formula is peculiar in that it states that a certain expression
involving the g.c.d. and l.c.m. of three numbers is self-dual, i.e.,
it remains the same when the two operations are interchanged.
As before, to prove (5-36) , let a, /3, and y be the exponents of the
powers to which some prime p divides a, b, and c. Since the
expression (5-36) is symmetrical in a, b, and c, we can arrange the
notation such that
a ^ ^ y
Then [a, 6], [a, c], and [b, c] contain p to powers with the exponents
a, a, and |8, respectively. In their g.c.d. the exponent therefore
is #. On the other hand, (a, b), (a, c), and (b, c) have the ex-
ponents /3, y, and y, respectively, for p so that their l.c.m. contains
p to a power with the exponent $. Thus for any prime p both
sides in (5-36) contain the same power and consequently they
are equal. One could also have derived (5-36) by using the rules
in theorem 5-9.
It should be noted that the rules (5-33) and (5-34) can be
considered to be equivalent to properties of maxima and minima
of three numbers a, 0, and y, namely :
min {a, max (j3, y)} = max (min (a, (3), min (a, y)}
max {a, min (|8, y)} = min {max (a, 0), max (a, y)}
Similarly the result in theorem 5-10 is a consequence of
min {max (a, j8), max (a, y), max (p, y)}
= max {min (a, j8), min (a, y), min (/3, 7)}
a law that remains invariant when maximum and minimum are
interchanged.
It is worth noting that the rules we have obtained in theorems
5-9 and 5-10 for the g.c.d. and l.c.m. have a much greater gener-
108
NUMBER THEORY AND ITS HISTORY
ality than may appear from their derivation in our special field.
These relations form a part of many other mathematical theories
of importance. We shall mention only one instance, namely, the
case of set operations, which we introduced above. Let us recall
the notations that when A and B are sets, A u B denotes the
union or sum set while A n B is
their intersection or common part.
Now let A, B, and C be three arbi-
Fig. 5-3.
Fig. 5-4.
trary sets. We shall see how the relations become evident on the
basis of simple illustrations.
In our notation the distributive law (5-33) takes the form
A n (B u C) = (A n B) u (A n C)
In Fig. 5-3 the shaded part is the set of points common to A
and the sum of B and C. Clearly this set may also be considered
the sum of two sets, namely, the common part of A and B and the
common part of A and C.
Corresponding to (5-34) one obtains
A u (B n C) = (A u B) n (.4 u C)
In Fig. 5-4 the sum of A and the common part of B and C has
been shaded. But it is evident that this set is also the common
part of the sum sets of A and B and of A and C, as the formula
requires.
THE ALIQUOT PARTS 109
The analogue of the relation (5-36) is
{A u B) n (A u C) n (B u C)
= (A n B) v (A nC) v (B nC)
In Fig. 5-5 the shaded part is found by inspection to consist of all
points that are common to the three sum sets
A u B, A v C, B v C
Fig. 5-5.
But it is also obvious that it is the sum of three sets, namely, the
intersection sets
An B, A n C, B n C
as we wanted to verify.
Problems.
1. Verify the relations in theorems 5-9 and 6-10 for the three numbers: 60,
72, 96.
2. The relations in the two theorems involve only the g.c.d. and the l.c.m.
There exist several other relations that also contain multiplication and division.
The reader may attempt to verify the relatively complicated identity
(o6, cd) = (a, c) (b, d) (^ , 0^))((~J ' (675))
5-5. Euler's function. When m is some integer, we shall con-
sider the problem of finding how many of the numbers
1, 2, 3, . . . , m - 1, m (5-37)
110 NUMBER THEORY AND ITS HISTORY
are relatively prime to to. This number is usually denoted by
<p (to), and it is known as Euler's <p-f unction of m because Euler
around 1760 for the first time proposed the question and gave its
solution. Other names, for instance, indicator or totient have
occasionally been used.
Example.
Among the positive integers less than 42 there are 12 that are relatively
prime to 42, namely:
1, 5, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41
so that ?>(42) = 12.
For the first few integers one finds
<p(2) = 1, *,(6) = 2, *>(10) = 4
<p(3) = 2, <p(7) = 6, <p(ll) = 10
*(4) = 2, *,(8) = 4, <p(12) = 4
*(5) = 4, *,(9) = 6,
The value <p(l) is in itself without meaning, but by special definition
one puts <p{\) = 1.
In some cases the determination of <p(m) is particularly simple.
When m = p is a prime, all numbers (5-37) except the last are
relatively prime to p ; consequently,
<p(p) = p - 1
When to = p a is a power of a prime, the only numbers in (5-37)
that have a common factor with m are the multiples of p,
p,2p, ... , p a ~ 1 p
Since there are p a ~ 1 of these multiples, one has
<p(p a ) = p a - p«~ l = p a ~ l (p - 1) = p a (l - i) (5-38)
We shall now tackle the general case. Let p be some prime
dividing w and let us first find the number <p p (m) of integers in
THE ALIQUOT PARTS 111
(5-37) that are not divisible by p. This is simple since those that
are divisible by p are the multiples
m
V, 2p, -..,-• V (5-39)
V
Since there are m/p of them, the remaining ones not divisible by
p are, in number,
m / 1\
<p p (m) = m = ra I 1 ) (5-40)
V \ vl
Next let q be some other prime dividing m. To find the number
«^pg( w ) of integers in (5-37) divisible neither by p nor q, one must
deduct from <p p {m) the m/q multiples of q
q,2q,...,-q (5-41)
Q
But this procedure involves a duplication since the multiples
(5-39) and (5-41) have some elements in common, namely, the
m/pq multiples of pq,
m
pq, 2pq, . . . , —pq
pq
Instead of deducting m/q, we must only subtract
m m
q pq
from (p p (m). Therefore the total of integers in (5-37) not divisible
by p or q is
-w-0-S-=0-;)-0-J)0-i)
(5-42)
We take the final step by mathematical induction. The results
in (5-40) and (5-42) show that for t = 1 and t = 2, if p\, p 2 , . . . ,Pt
are primes dividing m, there are
o» - (m) = m ( i -£)( i 4)-"( i 4) (M3)
g\ vl
112 NUMBER THEORY AND ITS HISTORY
integers in (5-37) that are divisible by none of them. We assume
that this formula (5-43) has been established for t primes and
wish to show that it is correct when there is one further prime
Pt+i- To determine the number of integers in (5-37) that are
not divisible by any of the p x , . . . , p t nor by pt+i, one must sub-
tract from (5-43) the number of multiples of p t+i
m
Pt+i, 2p t +i, • ■ • , Pt+i (5-44)
Pt+i
which have not already been stricken out. A multiple
m
ap f+1 , a = 1, 2, .
Pt+i
has, however, been taken into consideration previously if, and
only if, a is divisible by one of the primes p\, . . . , pt- But
according to our formula (5-43), there are
m
Pt+i
numbers in the series
o-a-o-s)
1,2,
m
Pt+i
not divisible by any p x , . . . , p t . One concludes that in (5-37)
there are
*■ '^ {m) = m ( 1 ~p[)'"( 1 ~i)
_J!L( 1 _!)...( 1 _I)
Pt+l\ Pi/ \ Pt/
\ Pi/ \ pt/\ Pt+l/
numbers not divisible by any p 1} p 2 , . . . , Pt+i- This establishes
our rule (5-43) in general.
To obtain Euler's (^-function let
m = pi ai • ■ • p r ar (5-45)
THE ALIQUOT PARTS 113
be the prime factorization of the given number. An integer in
(5-37) is relatively prime to m only when it is not divisible by any
of the primes p x , . . . , p r . The result that we just derived yields,
therefore :
Theorem 5-11. Let m be an integer whose various prime
factors are pi, . . . , p r . Then there are
„(„)-„(! -!)...(! -I) (5-46)
integers less than and relatively prime to m.
Example.
For m = 42 the prime factors are 2, 3, and 7, so that
*>(42) = 42 • (1 - i) • (1 - i) ■ (1 - i) = 12.
A table of the values of the ^-function for all numbers up to
10,000 has been computed by J. W. L. Glaisher.
The expression (5-46) for Euler's ^-function can be written in
slightly different forms. By means of the prime factorization
(5-45), one finds
<p(m) = p^ • • • p r (l - -^ • • • (l - -)
= Pi ai ~ 1 (Pl ~ \)---Pr ar - l {Pr- 1)
= (Pi" 1 ~ Vi ai ' 1 ) ■ ' • (Pr"' ~ Pr *" 1 )
When the last expression is compared with the formula (5-38) for
the ^-function of a prime power, one sees that
<p(m) = <p{ V r) • • • <p(Pr ar )
From this result one concludes further that when
m = a • 6
where a and b are relatively prime, one has
<p(m) = <p(a) ■ <p(b)
114 NUMBER THEORY AND ITS HISTORY
This is analogous to the property expressed in (5-8) for the sum
a (w) of the divisors of a number.
We shall deduce only one further fact about Euler's function :
Theorem 5-12. When d runs through all divisors of a number m,
the sum of all the corresponding ^-function values <p(d) is equal
to m
T.<p(d) = m (5-47)
Before we establish a proof of this theorem we shall illustrate
it.
Example.
The number
to = 42 = 2 • 3 • 7
has eight divisors, namely,
d = 1, 2, 3, 6, 7, 14, 21, 42
For these one finds
v (l) = 1, *(7) = 6
<p(2) = 1, <p(U) = 6
<p(S) = 2, *>(21) = 12
*(6) = 2, <p(42) = 12
and the sum of these values is to = 42.
In the special case where m = p a is a power of a prime, the
proof of the relation (5-47) is particularly simple. The divisors
of m are
1, P, V 2 , • ■ • i P a
and one has
*(1) + v (p) + *V) + *>(P 3 ) + • • • + <f(p a )
= 14-(p-l) + (p 2 -p) + (p 3 -p 2 )4-...4- ? )«-p^ 1 =^ (5-48)
since all terms except p a cancel out.
THE ALIQUOT PARTS 115
This result (5-48) facilitates the proof of the general theorem.
We assume that m has the prime factorization (5-45) and form
the product of the following expressions:
ya) + <p{ Vl ) + <p( Pl 2 ) + • • • + <p( Pl ai )]
X [<p(l) + <p(p 2 ) + <P(V2 2 ) + • • • + <p(p 2 a2 )]
X fe(l) + <p(p r ) + <p( Pr 2 ) + • • • + <p( V r ar )}
According to (5-48) these brackets are respectively equal to
p A p 2 a \ ..., p*
so that their product is m as in the right-hand side of (5-47).
When the product of the brackets is formed, one takes one term
from each of them and obtains all expressions
<P(P1 S1 MP2 S2 ) ' • • <f(Pr Sr ) = <p(Pl S W 2 ' ' * Pr Sr )
where each 5» may take some value 0, 1, . . . , a;. But this means
that the numbers
d = Pi Sl P2* 2 ■ ■ ■ Pr Sr
run through all divisors of m. The sum of all the terms resulting
from the multiplication of the brackets is therefore Y,<p(d), and
the formula (5-47) has been established.
Problems.
1. Determine <p(N) and verify theorem 5-12 for the numbers (a) N = 120,
(b) N = 365.
2. Prove the formula
<p(m 2 ) = m • <p(m)
3. Find all numbers m such that <p(m) divides m.
Bibliography
Escott, E. B.: "Amicable Numbers," Scripta Mathematica, Vol. 12, 61-72
(1946).
Glaisher, J. W. L. : Number-divisor Tables, Vol. 8, British Association Mathe-
matical Tables, Cambridge, 1940.
Poxjlet, P.: La Chasse aux nombres, 2 vols. Brussels, 1929, 1934.
CHAPTER 6
INDETERMINATE PROBLEMS
6-1. Problems and puzzles. Riddles, puzzles, and trick
questions constitute a part of the folklore over large parts of the
world. Curiously enough, the American Indians seem to have had
no feeling for this form of entertainment, since no trace of such
problem lore has been found among them by anthropologists.
However, in Europe, Africa, and Asia one finds a multitude of such
problems; they spread with the cultural interchanges, often with
a preservation of details that is remarkable. Many of the puzzles
have a germ of mathematical content. They can often be recog-
nized as interrelated by the faithfulness with which certain figures
and forms of questions are reproduced, even in localities separated
widely in time and place.
Let us illustrate these observations by an outstanding example.
One of our most important sources of ancient Egyptian mathe-
matical knowledge is the Papyrus Rhind, now in the British
museum. It is usually named for its previous owner, the Egyp-
tologist Henry Rhind, or sometimes for the scribe Ahmes who
copied it from earlier sources about 1800 b.c. The letters and
figures are in hieratic writing. The papyrus contains in some-
what systematic arrangement the solution of a variety of problems,
many of them practical everyday questions not very different from
those still encountered in our present-day school texts. But the
almost magical esteem of early mathematical learning may be
seen in the introductory statement to the manuscript wherein it
is promised to give "directions for obtaining knowledge of all
obscure things."
116
INDETERMINATE PROBLEMS 117
Most computations in the Papyrus Rhind have been relatively
easy to decipher. However towards the end one finds a curious
problem where the interpretation is not so certain. When tran-
scribed it consists of a column of terms (see Fig. 6-1).
Houses
7
Cats
49
Mice
343
Ears of wheat
2,401
Hekat measure
16,807
Total
19,607
It is preceded by a word that seems to mean estate, and in a
secondary column the same answer, 19,607, has been obtained in
a different manner. The figures one recognizes, of course, as the
five powers of the number 7. Some commentators assumed
directly that this would show that the Egyptian mathematicians
had mastered the concept of powers at this early time and that
the names houses, cats, and so on, were symbolic terms for powers
of the various orders.
However, scholars more familiar with the history of mathematics
have pointed out that problems with the same figures are known
from other sources. We have already mentioned the importance
of the Liber abaci (a.d. 1202) by Leonardo Pisano in the intro-
duction of Arabic numerals to Europe. Among the problems
included in this work is the following curious one: Seven old
women on the road to Rome, each woman has seven mules, each
mule carries seven sacks, each sack contains seven loaves, with
each loaf there are seven knives, and each knife is in seven sheaths.
How many objects are there, women, mules, sacks, loaves, knives,
and sheaths? Clearly Leonardo has borrowed this problem
from the medieval entertainment lore of his time, 3,000 years
after the compilation of the reckoning book of Ahmes. But one
need not go so far abroad. The old English children's rhyme
contains a jocular problem again based on the powers of the
sacred number 7
118 NUMBER THEORY AND ITS HISTORY
"As I was going to St. Ives,
I met a man with seven wives
Every wife had seven sacks
Every sack had seven cats
Every cat had seven kits
Kits, cats, sacks, and wives
How many were going to St. Ives?"
According to the first line none of them was going to St. Ives as
one would gleefully reveal to the victim after he had performed
the lengthy computations. It seems likely that Leonardo's problem
must have had the same twist of surprise in its original popular
version.
This brings us back to Ahmes's old problem, and it appears quite
likely that it may have been introduced, like Leonardo's, because
the figures were familiar from a trick question. As to a suitable
formulation the reader may use his own imagination; perhaps it
might have run somewhat as follows: "A man's estate included
seven houses, each house had seven cats, for each cat there were
seven mice, for each mouse there were seven ears of wheat,
and each ear would yield seven measures of grain. How many
things did he possess, houses, cats, mice, ears, and measures all?"
Perhaps the answer finally should be none, because the owner
was dead.
Numerous other examples of the preservation of the ideas of
mathematical puzzles may be given. Several medieval manu-
scripts containing collections of popular problems are still extant,
and many of these puzzles, with small variations, may be recog-
nized in our present-day magazines almost every week. Let us
mention only a few.
1. An old woman goes to market and a horse steps on her
basket and crushes the eggs. The rider offers to pay for the
damages and asks her how many eggs she had brought. She does
not remember the exact number, but when she had taken them
out two at a time, there was one egg left. The same happened
when she picked them out three, four, five, and six at a time, but
£ in •
c-_
mi
in
7
in
• "fin
i nnn
\?4
94
)l
„,™99 S^?0 D
no 9 i/ avw
343
'VII
103,2
111199991X1
III 9999 III
70S, 61
mi999inn
in 999 Sill
706,91
AWWV3
a»« a> n p
o o> g
in/-/ 7
t.db
□..
in '
t-Skh
s
}
drrfd
u ?
n-o - t'umt t'^u;
,9999 ¥¥ i
' 9999ii •
106,2 l
2 6,5
nn99|jj i
402,11
HI Will* If fe
706,91
dmdl
Fig. 6-1. Problem from Parvus Rhind.
Above, hieratic original; below, hieroglyphic transcription. Note that writing is
from right to left. {Courtesy of Mathematical Association of America.)
119
120 NUMBER THEORY AND ITS HISTORY
when she took them out seven at a time they came out even.
What is the smallest number of eggs she could have had?
2. A cistern can be filled by one pipe in one hour, by another
in two, by a third in three, and by a fourth in four. How long a
time will it take for all four pipes together to fill it?
3. Two men have a full eight-gallon jug of wine and also two
empty jugs taking five and three gallons. How can they divide
the wine evenly?
4. Three jealous husbands must ferry a river with their wives.
There is only one small skiff capable of taking two persons at a
time. How can one transport all six across so that no wife is ever
left with other men without her husband's being present?
5. Fifteen Christians and fifteen Turks were passengers on the
same ship when a terrible storm arose. To lighten the ship, the
captain orders that half the passengers should be thrown over-
board to save the others. All 30 are placed in a circle, and one
agrees to count out every ninth person and throw him over-
board. Providence intervenes, and it turns out that all the Turks
are thrown overboard and the Christians saved. How had the
passengers been arranged?
The importance of such problems as a matter of diversion has
of course decreased, and certainly they cannot compete with our
modern mechanized amusement industry; but as a part of our
folklore they are far from extinct. An interesting side light on
this fact came during the last war when long waiting was the
most nerve-racking and the most common activity of the soldiers.
At this time, as we already mentioned, teachers of mathematics
received a surprising number of requests from servicemen both for
the correct answers and for methods for solving puzzle problems.
Often they had arrived independently at the solution through the
most laborious guesses.
6-2. Indeterminate problems. There is a type of problem
that occurs quite commonly in puzzles and whose theory con-
stitutes a particularly significant part of number theory. These
problems may appropriately be called linear indeterminate problems,
for reasons that will become clear after some examples.
INDETERMINATE PROBLEMS 121
One of the earliest occurrences of such problems in Europe is to
be found in a manuscript containing mathematical problems
dating from about the tenth century. It is believed possible that
it may be a copy of a collection of puzzles which Alcuin prepared
for Charlemagne. The problem we are interested in runs as
follows :
1. When 100 bushels of grain are distributed among 100 persons
so that each man receives three bushels, each woman two bushels,
and each child half a bushel, how many men, women, and children
are there?
To formulate this problem mathematically let x, y, and z denote
the number of men, women, and children, respectively. The
conditions of the problem then give
x + y + s=100, 3x + 2y + |z = 100 (6-1)
As we shall see later there are several solutions but Alcuin gives
only the values
x = 11, y = 15, z = 74
From an Arabic manuscript copied about a.d. 1200, but un-
doubtedly composed earlier, we take this example :
2. One duck may be bought for 5 drachmas, one chicken for 1
drachma, and 20 starlings for 1 drachma. You are given 100
drachmas and ordered to buy 100 birds. How many will there be
of each kind?
When x, y, and z are the number of ducks, chickens, and starlings,
it follows that
x + y + z = 100, 5x + y + ^- = 100 (6-2)
One may observe that the same number occurs on the right-hand
side in both equations (6-1) and in (6-2). This particular pref-
erence in the choice of the figures in the questions is common in
Arabic, Chinese, and medieval European problems, and it
undoubtedly points to an interrelated or common background.
122 NUMBER THEORY AND ITS HISTORY
Even the use of the special number 100 shows a peculiar persist-
ence in problems from all these sources.
One finds similar questions in the many medieval collections of
problems. They occur in Leonardo's Liber abaci (a.d. 1202),
probably derived from Arabic sources, and in the following
centuries they became increasingly popular. To illustrate a fairly
common type of formulation we quote from a German reckoning
manual (Christoff Rudolff, 1526):
3. At an inn, a party of 20 persons pay a bill for 20 groschen.
The party consists of men (x), women (y), and maidens {z), each
man paying 3, each woman 2, and each maiden \ groschen. How
was the party composed?
Here the equations become
x + y + z = 20, Sx + 2y + | = 20 (6-3)
and the figures are so chosen that there is a unique solution x = 1,
y = 5, z = 14.
It is, of course, not certain that this type of problem originated
within a single cultural sphere, but if so, it seems likely that India
should be looked to for its source. As early as the arithmetic
of Aryabhata (around a.d. 500) one finds indeterminate problems.
Brahmagupta (born a.d. 598) in his mathematical and astronomical
manual Brahma- Sphuta-Siddhanta ("Brahma's correct system")
not only introduces them, but gives a perfected method for their
solution that is practically equivalent to our present procedures.
The method is called the cuttaca or pulverizer and is based upon
Euclid's algorism. Brahmagupta's examples are almost all of
astronomical character and refer to the comparisons between
periods of revolution of the heavenly bodies and determinations
of their relative positions.
We take the following problem from the Lilavati by Bhaskara,
a work we have already mentioned:
4. Say quickly, mathematician, what is the multiplier by which
two hundred and twenty-one being multiplied and sixty-five added
INDETERMINATE PROBLEMS 123
to the product the sum divided by one hundred and ninety-five
becomes exhausted?
Here one wishes to find some x satisfying the condition
221* + 65 = 195?/ (6-4)
In the Lilavati, as well as in other Hindu treatises on mathe-
matics, one finds many problems in the flowery style so customary
in Hindu writings. This problem is from the Bija-Ganita, literally
meaning seed-counting but denoting algebra, also composed by
Bhaskara :
5. The quantity of rubies without flaw, sapphires, and pearls
belonging to one person is five, eight, and seven respectively; the
number of like gems appertaining to another is seven, nine and
six; in addition, one has ninety-two coins, the other sixty-two
and they are equally rich. Tell me quickly then, intelligent
friend, who art conversant with algebra, the prices of each sort of
gem.
In Hindu mathematics colors were used to denote the various
unknowns, black, blue, yellow, red, and so on. If we prosaically
denote the prices of rubies, sapphires, and pearls by x, y, and z,
the condition becomes
5x + %y + 7z + 92 = 7x + 9y + 62 + 62 (6-5)
A further example may be taken from Mahaviracarya's work
Ganita-Sara-Sangraha, probably composed around a.d. 850:
6. Into the bright and refreshing outskirts of a forest which were
full of numerous trees with their branches bent down with the
weight of flowers and fruits, trees such as jambu trees, date-palms,
hintala trees, palmyras, punnaga, trees and mango trees — filled
with the many sounds of crowds of parrots and cuckoos found near
springs containing lotuses with bees roaming around them — a
number of travelers entered with joy.
There were 63 equal heaps of plantain fruits put together and
seven single fruits. These were divided evenly among 23 travel-
ers. Tell me now the number of fruits in each heap.
124 NUMBER THEORY AND ITS HISTORY
It is quite an anticlimax to state that if x is the number of fruits
in each heap, one must have
63z + 7 = 23?/ (6-6)
This beautiful Hindu forest contains a number of other problems,
but after all these ancient examples let us conclude with one with
a more modern touch. The following letter was one among several
similar ones received by the author during the recent war:
Dear Sir:
A group of bewildered GI's at Guadalcanal, most of whom have
been out of school for a good many years and have forgotten how to
solve algebraic problems, have been baffled by what appears to be a very
simple problem. Some of them affirm that it cannot be worked other
than through the trial and error method, but I maintain that it can be
worked systematically by means of some sort of formula or equation.
[7.] Here is the problem : A man has a theater with a seating capac-
ity of 100. He wishes to admit 100 people in such a proportion that will
enable him to take in $1.00 with prices as follows: men hi, women 2j£,
children 10 for one cent. How many of each must be admitted?
Can this problem be solved other than through the laborious trial
and error method? We shall greatly appreciate your assistance in
helping us to find the solution, thus relieving our weary brains.
Yours truly,
P.S. Through the trial and error method we found the answer to be
11 men, 19 women, and 70 children.
In terms of equations we have the conditions
x + y + z = 100, 5x + 2y + ^ = 100 (6-7)
Here again our familiar number 100 figures on the right. It is
also clear that the figures are not adapted to the present movie
prices and that we are confronted with an ancient problem which
has gained in actuality by being put in modern dress.
6-3. Problems with two unknowns. We have presented a
whole series of examples of linear indeterminate problems. As
INDETERMINATE PROBLEMS 125
we saw, they lead to one or more linear equations between the
unknown quantities. Furthermore, the number of unknowns is
greater than the number of equations so that if there were no
limitations on the kind of values the solutions could take, one
could give arbitrary values to some of the unknowns and find the
others in terms of them. For instance, in problem 4 one could
simply write
195y - 65
x =
221
and any value of y would give a corresponding value of x. How-
ever, by the terms of the problems, the choice of solutions is
limited to integral values and usually also to positive numbers.
But even with these restrictions, the solutions may be indetermi-
nate in the sense that there may be several, or even an infinite
number of them, as we shall see. On the other hand there may be
no solution at all.
Clearly many of our previous problems could be solved by
probing, by trial and error, and in medieval times this procedure
must have been commonly used. In several problems the possi-
bilities are rather limited so that not many attempts need to be
made. We have already mentioned that a method for solving
linear indeterminate problems was found quite aarly by the Hindu
school of mathematics. In Europe a corresponding method was
not discovered until a millenium later, and the date of rediscovery
can be fixed quite accurately. In 1612 there appeared in Lyons a
collection of ancient puzzles under the title: Problemes plaisans et
delectables, qui se font par les nombres. The author was Claude-
Gaspar Bachet, Sieur de Meziriac (1581-1638), a gentleman,
scholar, poet, and theologian, ardently devoted to classical learning.
His work proved popular and a second enlarged edition appeared
in 1624. Here one finds for the first time his rules for solving
indeterminate problems.
One of Bachet's problems runs about as follows :
8. A party of 41 persons, men, women, and children, take part
in a meal at an inn. The bill is for 40 sous and each man pays 4
126 NUMBER THEORY AND ITS HISTORY
sous, each woman 3, and every child \ sou. How many men,
women, and children were there?
In this case we have the equations
x + y + z = 41, 4x + Sy + | = 40 (6-8)
Bachet's procedure unfortunately is complicated by a lack of
algebraic symbolism. In the following sections we shall make a
more systematic study of the linear indeterminate problems. Here
we prefer to present a method of repeated reductions that is easy
to explain. It works quite well when the numbers involved are
not too large, as, for instance, in most of the examples we have
already mentioned. This method was used extensively by Euler
in his popular Algebra (1770), which devotes much space to inde-
terminate problems.
We shall deal first with a single linear equation
ax + by = c (6-9)
in two unknowns. As a preliminary example we take simply
x + 7y = Sl (6-10)
which may be written
x = 31 - 7y (6-11)
This shows that any integral value substituted for y in (6-11) will
give an integral value for x; for instance, y = Q,x = — 11; ory = 0,
x = 31. Thus there will be an infinite set of pairs of solutions.
But if one requires positive solutions, one must have both y >
and
x = 31 - 7y >
thus y < 4f-. This gives only four possibilities, y = 1, 2, 3, 4,
with the corresponding values, x = 24, 17, 10, 3, for the other
unknown.
This trivial example was introduced in order to show that when
one of the coefficients of x and y in (6-9) is unity, as in (6-10),
the solution is immediate. The guiding principle in the method
INDETERMINATE PROBLEMS 127
used below is to reduce the more general equations in successive
steps to this simple form. The first example given by Euler is:
Write the number 25 as the sum of two (positive) integers, one
divisible by 2 and the other by 3.
The two summands may be taken to be 2x and Sy so that
2x + Sy = 25 (6-12)
is the equation to be fulfilled. Since x has the smaller coefficient,
we solve for x and find by taking out the integral parts of the
fractional coefficients
x = ^H>. 12 . iy + l±i (6 _ 13)
Because x and y are integers, the quotient
t - i±» (6-14)
is integral. Conversely, any integral value t we may give to this
quotient (6-14) will make y integral
y = 2t - 1
and also x integral according to (6-13)
x = 12 - 2y + t = 14 - St
This shows that the general integral solution of (6-12) is
x = 14 - St, y = 2t - 1
and one can verify by substitution that they actually satisfy the
equation. Consequently, there is an infinite number of solutions,
one for each integral t. For instance, when t = 10, x = —16,
y= 19.
But if one is limited to positive values, one must have
x = 14 - St > 0, y = 2t - 1 >
hence
128 NUMBER THEORY AND ITS HISTORY
and there are only four permissible values, t = 1, 2, 3, 4. The
corresponding solutions are
x = 11, 8, 5, 2
V= 1, 3, 5, 7
This gives the decompositions
25 = 22 + 3 = 16 + 9 = 10 + 15 = 4 + 21
required in the original problem, as one could have verified without
much effort by probing.
This example requires only one reduction. In most cases two
or more steps are required. Let us illustrate this by another
example taken from Euler's Algebra.
A man buys horses and cows for a total amount of $1,770. One
horse costs $31 and one cow $21. How many horses and cows
did he buy?
When x is the number of horses and y the number of cows,
the condition
31x + 2ly = 1,770 (6-15)
must be fulfilled. Here y has the smaller coefficient so we solve
for y and find
1,770 - 31x _ 6 - lOz /n N
y = = 84 - x H (6-16)
"21 21 V
This requires that the quotient
6 - lOz
t
21
shall be integral. Our task is, therefore, to find integers x and t
such that
21* + lOx = 6 (6-17)
As can be seen, this is an equation of the same type as (6-15) but
INDETERMINATE PROBLEMS 129
with smaller numbers so that a first reduction has been performed.
Since x has the smaller coefficient, we derive from (6-17)
x = -2t + ^p (6-18)
We conclude that x can only be integral when
6 - t
is integral or
t = 6 - 10m
for some integer u. By substituting this value into (6-18) and
then x into (6-16), one finds
x = - 12 + 21m, y = 102 - 31m
Any integral value of u will give integers x and y satisfying the
equation (6-15) so that we have obtained the general solution.
The form of the problem requires, however, that x and y must be
positive. This leads to the conditions
-12 + 21m>0, 102 - 31m >
or
There are, therefore, three possible values u — 1, 2, 3, and the
corresponding solutions are
s = 9, 30, 51
y = 71, 40, 9
We could have made the solution of the problem unique, for
instance, by requiring in the formulation that the number of horses
would be greater than the number of cows.
As a last example of this type we shall take problem 4 in the
preceding section, stated by Bhaskara. It is of interest because
it permits us to mention some simplifications that often are avail-
able in the solution of indeterminate problems.
130 NUMBER THEORY AND ITS HISTORY
We observe first that in (6-4) the coefficients 221, 65, and 195
are all divisible by 13. This factor can, therefore, be canceled and
the equation becomes
17s + 5 = 15y (6-19)
Here, furthermore, both 5 and 15y are divisible by 5 so that 17s
must have this factor. But 17 is prime to 5 so that x must be
divisible by 5, and we can write
x = bx\
When this is substituted in (6-19), one can cancel by 5 and have
the still simpler equation
17si + 1 = 3y
By writing this
17xi + 1 _ 1 - xi
y = — — = fei + -3-
we see that
1 — Xi
= t
3
is integral. This gives #1 = 1 — 3* and
x = 5xi = 5 - 15*, y = Q - 17t (6-20)
as the general solution.
Let us ask for the positive solutions. One obtains, as previously,
the conditions
5 - 15* > 0, 6 - 17* >
or
^ 3' * ^ 17
This shows that all values * = 0, —1, —2, • • • will give positive
solutions in (6-20). To obtain positive values for this parameter
or auxiliary variable, it is convenient in (6-20) to write t — ~u
so that
x - 5 + 15u, y = 6 -f- 17u
INDETERMINATE PROBLEMS 131
becomes the general solution; all values u = 0, 1, 2, • • • give
positive answers, namely,
x = 5, 20, 35, 50, • • •
y = 6, 23, 40, 57, • • •
This example illustrates the fact that even when the solutions are
required to be positive there may be an infinite number of them.
Problems.
1. Divide 100 into two summands such that one is divisible by 7, the other
by 11. (Euler.)
2. Required,, such values of x and y in the indeterminate equation
7x + IQy = 1,921
that their sum x + y may be the least possible. (From Barlow, An Ele-
mentary Investigation of the Theory of Numbers, etc., London, 1811.)
3. In the forest 37 heaps of wood apples were seen by the travelers.
After 17 fruits were removed the remainder was divided evenly among 79
persons. What is the share obtained by each? (Mahaviracarya. )
4. Find two fractions having 5 and 7 for denominators whose sum is equal
35-
5. A party of men and women have paid a total of 1,000 groschen. Every
man has paid 19 groschen and every woman 13 groschen. What is the smallest
number of persons the party could consist of? (Modified from Euler. )
6. How many different ways may £1,000 be paid in crowns and guineas?
(Barlow.) [For non-English readers it may be recalled that one crown is
5 shillings, one pound 20 shillings, and one guinea 21 shillings.]
7. Solve problem 6 in Sec. 6-2.
8. Find a number that leaves the remainder 16 when divided by 39 and
the remainder 27 when divided by 56.
6-4. Problems with several unknowns. We turn now to those
indeterminate problems in which there are more than two un-
knowns. There exists then a certain number of linear conditions;
often the number of equations is just one less than the number of
unknowns. The procedure is to eliminate some of the unknowns
until one winds up with a single equation with two unknowns,
which is the case we have just discussed. Most common is the
case of two equations with three unknowns. In medieval times
132 NUMBER THEORY AND ITS HISTORY
problems of this kind were known as problema coed, a term of
unknown origin. The name probably refers to the fact that these
problems often appeared in the form that a check should be paid
by a certain number of people, as in the problem given by Bachet,
for instance. Sometimes they were also called problema potatorum,
referring to drinkers and the mixing of wine, or also problema
virginum, believed to have originated through certain problems
given in terms of Greek mythology from the so-called Palatine
Anthology. The same problems are also reproduced in Bachet' s
collection. In Euler's Algebra the regula coed is illustrated first
by the following example:
Thirty persons, men (x), women (y), and children (2), spend
50 thaler at an inn. Each man pays 3 thaler, each woman 2 thaler,
and each child 1 thaler. How many persons were there in each
category?
The equations are
x + y + z = 30, 3z + 2y + z = 50 (6-21 )
By subtracting the first from the second, one obtains an equation
with two unknowns
2x + y = 20
The positive solutions are obviously
x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
y = 20, 18, 16, 14, 12, 10, 8, 6, 4, 2,
and from the first of equation (6-21), one finds the corresponding
values
z = 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
Thus there are 11 solutions.
For a less trivial example let us take problem 8 of Sec. 6-3, from
Bachet. From the second of equation (6-8) it is clear that z must
be a number divisible by 3 so that one can write
z = Szi
When this is substituted in (6-8), the two equations become
x + y + 3zi = 41, 4x + Zy + z x = 40
INDETERMINATE PROBLEMS 133
When the last equation is multiplied by 3 and the first subtracted
from it, one finds the equation with two unknowns
lis + Sy = 79
To find its general solution we proceed as previously and write
79 - 11a; _ 3a; + 1
y = —j— = 10 - x - —£-
Therefore the quotient
3x + 1
must be integral so that
or
is an integer and
8
8* = 3x + 1
x = St ! —
3
t = 3m - 1
Substituting this in the expressions for x and y, one obtains t v e
general solution
x = 8w - 3, y = 14 - 11m
For positive solutions one must have
8m - 3 > 0, 14 - 11m >
or
| < u < 1-A-
This leaves as the only possibility u = 1 and x = 5, y = 3. From
either one of the equations (6-8), it follows that z = 33. Thus
there is a single solution to Bachet's problem.
Next let us consider the GI problem stated in (6-7). The second
of these shows that z must be divisible by 10, hence z = 10zi, and
the equations become
x + y + IO21 = 100, 5x + 2m + z x = 100 (6-22)
134 NUMBER THEORY AND ITS HISTORY
When the first equation is multiplied by 2 and subtracted from the
second, one finds
Zx - 19zi = - 100 (6-23)
This equation may be solved as before. One writes
19 *i ~ 100 a oo , *i ~ l
x = = 621 - 33 H —
Therefore
"^
is an integer, and one obtains z x = St + 1 and
2 = 30* + 10, x = 19t - 27
as the general solution of (6-23). When it is substituted in the
first equation (6-22), it follows that
y = 117 - 49*
To make all three numbers positive, one must have
19* - 27 > 0, 30* + 10 > 0, 117 - 49* >
or
t > 1-JQ-, t > —$, t K 2^9-
This is only possible for t = 2 so that one has the unique solution
x = 11, y = 19, and z = 70, as already indicated.
Let us discuss another type of problem with three unknowns,
which occurs later in the theory of congruences:
Find a number N that leaves the remainder 3 when divided by
11, the remainder 5 when divided by 19, and the remainder 10
when divided by 29.
The conditions are in this case
N = Ux 4- 3 = 19y + 5 = 292 4- 10 (6-24)
Combining the two last conditions one has
19y = 292 + 5
INDETERMINATE PROBLEMS 135
One finds
102 + 5
and this shows that
102 + 5 _ 5(22 + 1)
19 ~ 19
is integral. Since 5 is relatively prime to 19, it follows that
22 + 1
is integral. This gives in turn
t - 1
2 = 9^ + ^-
so that we write
t - 1
and find t = 2u + 1 and
2 = 19w + 9, y = 29m + 14 (6-25)
These values for y and z give numbers N leaving the remainder 5
when divided by 19 and the remainder 10 when divided by 29.
But one should also have the remainder 3 when the number is
divided by 11. When the two first conditions in (6-24) are com-
bined, one finds
Ux = 19t/ + 2
Since the form of y is given by (6-25), it follows that x and u must
be integers satisfying
llx = 551w + 268
This gives
u + 4
x = 50m + 24 +
so that
w + 4
11
136 NUMBER THEORY AND ITS HISTORY
is integral. We have, therefore, u = lly — 4 and
x = 55h> - 176
This in turn gives
N = 6,061V - 1,933
as the general form of the numbers with the desired residue proper-
ties. The positive values of v give positive N, and the smallest
solution is obtained f or v = 1 when N = 4,128.
Let us finally consider some problems in which the number of
unknowns is at least two greater than the number of equations.
In this case also, one can eliminate some of the unknowns and end
up with a single equation with several unknowns. For instance
there may be two equations and four unknowns and one of them
may be eliminated to obtain a single equation with three unknowns.
Problem 5 in Sec. 6-2, which we quoted from the Bija-Ganita
of Bhaskara, is formally of this type. But (6-5) may be written
simply as
y = z - 2x + 30
and the solution is trivial. One can choose any integral positive x
and z arbitrarily greater than 2x — 30 and find the corresponding
y. In the solution given by Bhaskara the proportions x : y : z of
the various prices are prescribed, and one obtains an ordinary
equation in a single variable.
A less trivial example from the same source, stated by Bhaskara
to be a problem from ancient authors, runs as follows:
Five doves are to be had for 3 drammas, seven cranes for 5, nine
geese for 7, and three peacocks for 9. Bring 100 of these birds
for 100 drammas, for the prince's gratification.
When x, y, z, and t denote the numbers of doves, cranes, geese,
and peacocks, the equations are
x + y + z + t = 100, %x + f y + f z + 3* = 100 (6-26)
When the first equation is multiplied by 3 and the second sub-
tracted from it, one finds after the fractions have been cleared
189x + 180^/ + 175^ = 15,750
INDETERMINATE PROBLEMS 137
Here one may apply the same reduction method as for equations
with two unknowns. Solving for z, which has the smallest co-
efficient, it follows that
14x + by
175
2 = 90 — x — y —
so that
14x + by
is an integer. Thus we have the reduced equation
175w = 14z + by
or
This shows that
x
y = 35w — Sx + -
5
x
is integral, and proceeding backward one finds
x = bv, y = 35w — 14u
z = 90 + 9v - 36w, t = u + 10
This represents the general solution of (6-26) in integers. The
variables or parameters u and v are arbitrary integers. It should
be noticed that in the case of one equation with two unknowns
we obtained a general solution with one parameter, while in the
case of three unknowns there will be two of them, as above.
Our problem requires that the solutions shall consist of positive
numbers. This in general leads to a set of inequalities, which at
times may be bothersome to analyze to find all possibilities. In
our special case we must have v > since x > 0, and consequently
also u > since y > 0. These conditions then insure that x >
and t > 0. From y > and z > one concludes
v > 4m — 10, v < f u
138
NUMBER THEORY AND ITS HISTORY
and therefore
or
Thus, there are the possibilities 1, 2, 3, 4, 5, 6, for u. Let us
choose one of them, for instance, u = 2. When substituted in the
general solution, this gives
x = 5v, y = 70 - Uv, z = 18 + 9v, t = 12
Here y > so that v is limited, namely v < 5. The four possi-
bilities v = 1, 2, 3, 4 correspond to the solution sets
X
y
z
t
V = 1
5
56
27
12
«; = 2
10
42
36
12
*; = 3
15
28
45
12
v = 4
20
14
54
12
The first three of these are those actually given by Bhaskara. All
the other possibilities for u and v may be investigated similarly.
We shall leave it to the reader to derive all sets of solutions; it
may only be stated that there are altogether 16 of them.
From Euler's Algebra we take our final example:
Someone buys 100 head of cattle for 100 thaler at the following
prices: a steer, 10 thaler; a cow, 5 thaler; a calf, 2 thaler; and a
sheep, | thaler. How many did he buy of each kind?
Here are the equations.
x + y + z + t = 100, 10z + by + 2z + }* = 100
We multiply the last equation by 2 and subtract the first from it,
giving
19x + 9y + 32 = 100
This in turn leads us to
x — 1
z = 33 - Qx - Sy
INDETERMINATE PROBLEMS 139
Hence, the only requirement is that
x - 1
shall be integral. One finds, therefore, the general integral solution
in the form
x = 3m + 1, z = 27 - 19m - Sy, t = 72 + 16m + 2y
while y is arbitrary. Next one must analyze the conditions for
positive solutions. Since
x = 3u + 1 >
one must have u ^ and naturally also y > 0. This is already
sufficient to make t > 0. The remaining condition z > leads to
19m + 3y < 27
Clearly m can take only the values m = and u = 1. For u = 0,
« can take the eight values y = 1, 2, • • • , 8. For m = 1 there
are only the possibilities y = l,y = 2, and the two sets of solutions
are
s = 4, y = 1, 2 = 5. < = 90
4, 2, 2, 92
As these examples indicate, the determination of all positive
solutions may often prove quite cumbersome. How complicated
the matter may have appeared in earlier periods is evident from
the following lament by an Arabic writer on the subject about
a.d. 900. The title of his work is The Book of Precious Things in
the Art of Reckoning, and the preface opens in this manner:
In the name of God, the compassionate and merciful. The writer
is Shod j a C. Aslam known by the name of Abu Kamil. I am familiar
with a special kind of problems which circulate among high and low,
among learned and among simple people, which they enjoy and which
they find new and beautiful. But when one asks about the solution,
one receives inaccurate and conjectural replies and they see in them
140 NUMBER THEORY AND ITS HISTORY
neither principle nor rule. Many men, some distinguished and some
humble, have asked me about problems in arithmetic and I replied to
them for each separate problem with the single answer when there were
no others. But often a problem had two, three, four and more answers,
and often there was no solution. Indeed it happened to me in one
problem which I solved that I found very many solutions. I considered
the matter more penetratingly and came upon 2676 correct solutions.
At this my surprize was great and I had the experience that when I told
of the discovery, I was met with astonishment or was considered incom-
petent or those who did not know me had a false suspicion of me.
Then I decided to write a book on the subject of such computations to
facilitate the study and bring understanding nearer. This I have now
begun and I shall declare the solutions for those problems which have
several solutions and for those which have only one and for those which
have none, all by means of an infallible method. Finally I shall treat
a problem, which as I stated, has 2676 solutions. The suspicions will
again disappear and my statement will be confirmed and the truth will
show itself. It would carry too far if I should add more about the
opinions which have been expressed to me in regard to the great number
of solutions of this and similar problems.
To satisfy the curiosity of the reader it may be stated that the
problem is of the same type as those considered in the last two
examples. It has five unknowns and it leads to the equations
x + y + z + u + v = 100
2x + §y + iz + lu + v= 100
The reader is welcome to verify Abu KaimTs result.
To conclude, let us mention only a famous indeterminate
problem known as the "cattle problem of Archimedes." Although
the problem is ancient, it appears doubtful whether it ever had
any connection with Archimedes. It is in poetic form and in it
one is requested to find the number of head of cattle of various
colors in the herds of the sun-god Helios as they graze the slopes of
Sicily. The problem leads to seven equations in eight unknowns.
These equations are quite simple and present no theoretical
difficulty, but the numbers appearing in the solutions are enormous.
INDETERMINATE PROBLEMS 141
A later addition requires the solution of second-degree indetermi-
nate equations.
Problems.
1. Solve problems 1, 2, and 3 in Sec. 6-2.
2. The following problems are quoted from the letters of the German
mathematician Regiomontanus (1436-1476). In all of them, find the positive
integral solutions.
(a) 97x + 56y + 3z = 16,047
(b) 17x + 15 = lBy + 11 = 10s + 3
(c) 23x + 12 = 17y + 7 = 10s + 3
3. Write ^ as a sum of three fractions with relatively prime denominators.
4. Find the number of solutions in positive integers of the equation
5x + lly + 133 = 2,000
and find the solution for which the sum x + y + z is as small as possible.
5. In how many ways can one give change for (a) 25 cents, (b) 50 cents,
(c) $1.
Bibliography
Bachet, C. G. : Prollemes plaisans et delectables qui se font par les nombres.
Lyons, 1612. Many later editions.
Chace, A. B., L. S. Bull, H. P. Manning, and R. C. Archibald: The Rhind
Mathematical Papyrus, 2 vols., Oberlin, Ohio, 1927, 1929.
Colebrooke, H. T. : Algebra with Arithmetic and Mensuration from the Sanscrit
of Brahmegupta and Bhaskara, London, 1817.
Euler, L. : Vollstandige Anleitung zur Algebra, St. Petersburg, 1770. Many
English and French translations.
Rangacarya, M. : The Ganita-Sara-Sangraha of Mahaviracarya, with English
translation and notes, Madras, 1912.
CHAPTER 7
THEORY OF LINEAR INDETERMINATE
PROBLEMS
7-1. Theory of linear indeterminate equations with two
unknowns. After the many examples of linear indeterminate
problems we have given in the preceding chapter, it is time to
consider some of the more systematic aspects of their theory. The
following result is essential in many applications of number theory:
Theorem 7-1. When a and b are relatively prime, it is possible
to find such other integers x and y that
ax -f- by — 1
(7-1)
This may be stated slightly differently by saying that unity is a
linear combination of a and b. The proof is an immediate appli-
cation of Euclid's algorism. We suppose that a > b. To make
the notation more systematic, we write a = r\ and b = r 2 in stating
the algorism:
n = Qir 2 + r z
r 2 = 22*3 + r 4
r„_ 3 = g w _ 3 • r w _ 2 + ?V-i
r„_ 2 = #n-2 • r n -\ + 1 ,
(7-2)
The last remainder is 1 since a and b are relatively prime. We
shall now obtain a representation (7-1) by a stepwise process
142
LINEAR INDETERMINATE PROBLEMS 143
derived from (7-2). We begin at the bottom and write 1 as a
linear combination of r„_2 and ?v_i
1 = *»_2 — gn_2^n-l
Here we substitute from the next to the last division
7n_i = r„_ 3 — q n -3 r n-2
and one obtains after rearrangement
1 = — g f n-2»*n-3 + (1 + g f n-2Q , n-3)»'n-2
so that we have represented 1 as a linear combination of r n _ 3 and
r„_ 2 . From the third last relation one introduces
Tn—2 ~ ^n— 4 ~~ Qn— 4^"n— 3
and in a similar manner one expresses 1 linearly by means of r n _4
and r n _ 3 . This process is continued until one arrives at a linear
combination of r x = a and r 2 = b equal to 1, as the theorem
requires.
Let us illustrate the procedure on the example a = 109 and
b = 89. The algorism is
109 = 89-1 +20
89 = 20-4 + 9
20 = 9-2 + 2
9 = 2-4 + 1
where the various remainders have been underscored to keep them
separate from other figures occurring in the reductions. We begin
by writing
1=9-4-2
and substitute
2 = 20 - 2-9
from the third division. This gives
1 = 9-9 - 4-20
144 NUMBER THEORY AND ITS HISTORY
Here we substitute
9 = 89 - 4-20
from the second relation and obtain
1 =9-89-40-20
In the last step we use the first division and write
20 = 109-89
so that we arrive at the desired representation
1 =49-89-40-109
It should be observed that when the algorism (7-2) was used
to derive a solution of the equation (7-1), it was immaterial what
the values of the remainders r. were, as long as one had a set of
relations of the type (7-2). This remark may be used to shorten
the algorism in some cases, for instance, by taking least absolute
remainders instead of least positive remainders. To illustrate we
shall take a = 249 and b = 181, for which one finds
249 = 181-1+68
181 = 68-3-23
68 = 23-3-1
Again, in
1 = 3.23 -68
we substitute the expression for 23 from the second relation and
derive
1 =8-68-3-181
When 68 is eliminated in the same manner by means of the first
relation, we obtain as a solution to our linear equation
1=8 -249 - 11 -181
The work involved in the computation of a solution of the linear
equation (7-1) may be reduced considerably by a systematic
LINEAR INDETERMINATE PROBLEMS
145
arrangement. Before we give the necessary proofs we shall
illustrate the method on the example
1,027a; + 712*/ = 1
Here the algorism is
1,027 = 712 • 1 + 315 ,
712 = 315 • 2 + 82,
315 = 82-3 + 69,
82 = 69-1 + 13,
69 = 13-5 + 4,
13 = 4-3 + 1,
= -165 • 1,027 + 238-712
73 • 712 - 165 • 315
-19 -315
16-82
-3 -69
1 -13
0-4
+ 73-82
- 19-69
+ 16-13
- 3-4
+ 1-1
In the second column we have performed the substitutions required
by our method, beginning at the bottom and proceeding succes-
sively to the solution on top. The lowest equation has been added
as a supplement for a reason that will be clear instantly. To derive
the rules of computation we shall establish, let us rewrite separately
the coefficients in the right-hand column of equations above.
-165
238
73
-165
-19
73
16
-19
-3
16
1
-3
1
The two columns are the same except for one element at each
end, so that we have the rule that the last coefficient in one equa-
tion becomes the first in the next. Furthermore the signs alternate.
Consequently, to obtain our solution x = —165 and y = 238,
it would suffice to compute the positive values of the numbers in
the last column, and then give them the signs plus and minus
146
NUMBER THEORY AND ITS HISTORY
alternatingly. This may be executed according to the following
scheme :
1
238 =
165
•1+73
2
165 =
73
•2 + 19
3
73 =
19
•3 + 16
1
19 =
16
1+ 3
5
16 =
3
5+ 1
3
3 =
1
3+0
In the first column we have written the quotients which occur in
the divisions on the Euclid algorism. In the second column are
the positive coefficients, each computed as indicated, by multiply-
ing the corresponding quotient by the preceding coefficient and
adding to the product the next preceding coefficient.
The proof of these rules is quite simple. Suppose that for
some i we have found the relation
1 = -^i+l^' + A.{Tir\-\
To eliminate the remainder r l+1 one must substitute
r i+\ = r i—\ — fiQi—1
from the algorism and one obtains
1 = A,r,-_i - (Afti-i + A i+l )ri
This shows that the coefficient of r t -_i is the same as that of r i+1
in the preceding and also that one must put
Ai_ x = - (A^i + Ai+i)
However, these are exactly the rules that were verified in the exam-
ple above.
To give a final illustration of the scheme let us take the equation
1,726a; + l,229y = 1
LINEAR INDETERMINATE PROBLEMS
147
The algorism is
1,726 =
1,229 -1 + 497
639
1,229 =
497 • 2 + 235
455
497 =
235 • 2 + 27
184
235 =
27-8+19
87
27 =
19-1 + 8
10
19 =
8-2 + 3
7
8 =
3-2 + 2
3
3 =
2-1 + 1
1
1
The last column contains the computation of the successive coeffi-
cients by means of the quotients in the algorism. The two top
numbers give the solution x = -455, y = 639, as is readily
verified.
In this example, as well as in the preceding, we have assumed
positive coefficients in the linear equation to be solved, and this
is usually convenient. In order to find the solution of equations
with negative coefficients, one need only observe that if x and
?/o is a solution of (7-1), then
— xq, y Q) x , ~y , —x , —y
respectively are solutions of the equations
-ax + by = 1, ax - by = 1, -ax - by = 1
We shall turn next to the general linear equation in two unknowns
ax + by = c (7-3)
On the basis of the preceding analysis it is not difficult to find
when such an equation can have integral solutions. Clearly, if
the coefficients a and b in (7-3) have the greatest common divisor d,
this factor must also divide c when there is to be an integral solu-
tion. Conversely, let d divide c. Then one can divide (7-3)
148 NUMBER THEORY AND ITS HISTORY
by d and obtain an equation in which a and b are relatively prime.
Let us suppose that this has been done. It follows from theorem
7-1 that one can find integers x and y such that
ax + by = 1
When this equation is multiplied by c, one finds
a(cx ) + b(cy ) = c
and therefore
x = cx , y = cy
is a solution of (7-3). To summarize we state:
Theorem 7-2. The necessary and sufficient condition for the
equation
ax + by = c
to have a solution in integers is that the greatest common divisor
of a and b divide c.
Examples.
1. The equation
114x + 312y = 28
has no solution in integers since the g.c.d. of 114 and 312 is 6 and this number
does not divide 28.
2. The equation
208a; + 136y = 120 (7-4)
has solutions, since (208, 136) = 8 and this common divisor divides 120.
After canceling by 8, we have the equation
26x + lly = 15 (7-5)
As indicated above, we first find a solution of the equation
26z + 17y = 1 (7-6)
The algorism is
26 = 17
1+9
3
17 = 9
1 +8
2
9=8
1 + 1
1
1
LINEAR INDETERMINATE PROBLEMS 149
The last column contains the computation of the successive coefficients so that
a solution of (7-6) is x = 2, y = —3. When these values are multiplied by
15, one obtains a solution x = 30, y = -45 of any of the two equivalent
equations, (7-4) or (7-5).
3. To find a solution of the equation
l,726x + 1,229?/ = 3 (7-7)
we recall that we have already shown previously that the equation
l,726x + l,229y = 1
has the solution
x = -445, y = 639
When these numbers are multiplied by 3, one arrives at the solution
x = -1,335, y = 1,917
for (7-7).
So far we have derived only one solution of the indeterminate
equations we have studied. However, on the basis of one solu-
tion it is not difficult to find the general solution. We have
already established that when there exists a solution of (7-3), the
greatest common factor of a and b also divides c so that it may
be canceled. We shall suppose in the following, therefore, that
this has been done and a and b are relatively prime. When x
and y form some particular solution of (7-3), one has
ax + by = c (7-8)
To find the general solution x and y, we subtract (7-8) from
(7-3) and obtain
a(x - xq) + b(y - y ) =
which we prefer to write
a(x - xo) = ~b(y - y ) (7-9)
This equation shows that the product of a and x — x is divisible
by &. Since a is relatively prime to b, we conclude that x — x is
divisible by b so that one can write
x — Xq = tb
150 NUMBER THEORY AND ITS HISTORY
where t is some integer. When this is substituted back in (7-9)
and b is canceled, one obtains
y - Vo = -ta
Thus we have shown that one must have
x = x + tb, y = y — ta (7-10)
It may be verified directly that these values for x and y satisfy
(7-3) regardless of the value of the integer t; hence in (7-10)
we have the general form for the solution of the indeterminate
equation.
By means of the general solution one can answer various ques-
tions about the existence of solutions with particular properties.
Let us again consider some illustrations.
Examples.
1. Find the smallest positive integer that leaves the remainder 1 when
divided by 1,000 and the remainder 8 when divided by 761.
The number must have the form
N = 1,000s + 1 = 761y + 8 (7-11)
so that
l,000z - 761y = 7 (7-12)
A solution of the equation
l,000x - 7Qly = 1
is found by the previous method to be
x = 121, y = 159
When multiplied by 7, it gives the solution to (7-12)
x = 847, y = 1,113
and the general solution becomes
x = 847 - 761i, y = 1,113 - 1,000< (7-13)
The smallest positive values for x and y are obtained when t = 1 and they are
x = 86, y = 113 (7-14)
The corresponding number asked for in the problem is, according to (7-11),
A = 86,001.
LINEAR INDETERMINATE PROBLEMS 151
In forming the general solution one could also have used the particular
solution (7-14) so that
x = 86 - 761*, y = 113 - 1,000*
would be the general form, instead of (7-13). However, when * runs through
all integers, the totality of solutions is, of course, the same in both forms. Let
us also mention that since — * runs through all integers when * does, one could
also present the general solution as
x = 86 + 761*, ?/ = 113 -h 1,000*
2. Find the number of positive solutions to the equation
lOx + 28?/ = 1,240
We first cancel the common factor 2 so that the equation reduces to
5x + Uy = 620
By inspection one sees that a particular solution of
5x + 14t/ = 1
is x = 3, y = — 1. When this is multiplied by 620, it gives the particular
solution x = 1,860,7/ = —620 for the previous equation. The general solution
is therefore
x = 1,860 - 14*, y = -620 + 5*
To obtain positive solutions one must have
1,860 6
* < ——- = 132 - , * > 124
14 7
or
132 ^ * ^ 125
so that there are 8 positive solutions. For * = 124 one finds y = 0.
One can also look at this theory of linear indeterminate equations
from a geometric point of view. In analytic geometry, an equation
ax + by = c
represents a straight line. The points (x, y) in the plane whose
coordinates x and y are integers are called lattice points. To solve
the linear equation in integers means, therefore, to determine
those lattice points that lie on the line. The general form of the
integral solutions, as we have found it, shows that if (x , y ) is
152
NUMBER THEORY AND ITS HISTORY
a solution, then there are lattice points on the line for all the
abscissas
x , xq dt a, x ± 2a,
This means that the lattice points that represent solutions lie at
^x
Fig. 7-1.
even intervals on the line with abscissas differing by a, and simi-
larly, with ordinates y differing by b.
The situation has been illustrated in the figures representing
the two lines
2x + Zy = 11, 3z - Ay = 1
These two illustrations clarify another fact. One can write
the equation for the straight line in the form
y
a c
l x + l
LINEAR INDETERMINATE PROBLEMS 153
Therefore the slope of the line is positive when a and b have
different signs, as in the second example. But when the slope is
positive, the line must have an infinitely long portion in the first
quadrant, in which both x and y are positive, so that when a and
b have different signs, there will be an infinite number of positive
integral solutions, provided of course that there are any at all.
On the other hand, when a and b have the same sign, for instance,
both positive as in the first example, the line, if it goes through the
first quadrant, can have only a finite portion in this part of the
plane. In this case there can at most be a finite number of
positive solutions.
Problems.
1. Find the general integral solution and also the positive solutions for
each of the following equations :
(a) 39x - 56y = 11
(b) 311a; + 712y = 1,300
(c) 7x + 13y = 71
(d) 39x - 11% = 49
(e) 170x - 445y = 625
2. Find the number of positive solutions of the equations
(a) 33x + 41y = 1,946
(6) 31x - ly = 2
(c) 3x + \\y = 1,000
7-2. Linear indeterminate equations in several unknowns. So
far we have discussed the theory of linear indeterminate equations
in two unknowns in considerable detail. We turn next to such
equations with several unknowns. These equations, as we saw
in the examples, will appear in problems in which the number of
unknowns is one or more greater than the number of equations.
The basic result for several unknowns is quite analogous to the
main result for two unknowns as we expressed it in theorem 7-1.
In a slightly different form this theorem may be stated: When d is
154 NUMBER THEORY AND ITS HISTORY
the greatest common divisor of two numbers a and b, one can find
two other integers x and y such that
ax -\- by = d (7-15)
For several unknowns we shall derive the corresponding theorem :
Theorem 7-3. Let
ci\, a 2 , . . . , a n
be a set of integers with the greatest common divisor
d = (ai, a 2 , - - • , a n )
Then one can find such integers
that
aixi + a 2 x 2 + • • • + a n x n = d (7-16)
To prove this theorem one can proceed in various ways. We
shall prefer to use the induction method, and to this end we need
to observe that the result is obviously true when there is only one
number a± (then d = a x and x\ = 1). Also we have just men-
tioned that the theorem is true when there are two numbers a\
and a 2 . The induction consists in supposing that the theorem is
true when there are n — 1 numbers ai and applying this to prove
it for n numbers. We shall denote by
dn— i = (fli, ' ' ' j tt,i_i)
the g.c.d. of the n — 1 first numbers. According to our assump-
tion, we can find n — 1 numbers
y 1} ... , yn-i
such that
a\V\ + • • • + an-i^/n-i = dn-i (7-17)
But let us recall from the properties of the g.c.d. that d, the g.c.d.
of all ai's is also the g.c.d. of d n _\ and a n
d = (d n -l, On)
LINEAR INDETERMINATE PROBLEMS 155
This means that one can find two integers t and x n so that
When we put in the value of d n _i from (7-17), we arrive at the
relation
aiVit + • • • + a-n-iyn-it + a n x n = d
and this is exactly of the form (7-16) when we write
xi = yit, x 2 = y 2 t, " • , x n _ x = y n -it
This proof has the advantage of providing a fairly simple
method of computing a solution to (7-16).
Example.
The three numbers
a x = 100, a 2 = 72, a 3 = 90
have the g.c.d. d = 2, and we shall find numbers x\, x% and x% such that
lOOxi + 72x 2 + 60x 3 = 2 (7-18)
The g.c.d. of 100 and 72 is d% = 4 so that we begin by solving the equation
lOOr/i + 72y 2 = 4
or
252/i + 182/ 2 = 1
By the usual method one finds a solution 2/1 = —5, 2/2 = 7. In the next step
we solve
4i + 90x 3 = 2
or
2t + 45x 3 = 1
By inspection one sees that X3 = 1, t = —22 satisfies this equation. By mul-
tiplying 2/1 and 2/2 by —22 one obtains as a solution of (7-18)
xi = 110, x 2 = -154, x 3 = 1 (7-19)
Analogously, as in the case of two unknowns, one can prove
further:
Theorem 7-4. The necessary and sufficient condition for the
equation
a\Xi + • • • + a n x n = c (7-20)
156 NUMBER THEORY AND ITS HISTORY
to be solvable in integers x\, . . . , x n is that c be divisible by the
greatest common divisor of all numbers a*.
The proof is simple. Since d, the g.c.d. of all numbers a*,
divides each of them, it must also divide c if there is to be an
integral solution. On the other hand, when c is divisible by d,
one can divide all terms in (7-20) by d and obtain a new equation
in which the coefficients of the unknown have 1 for their g.c.d.
Let us suppose that this reduction has already been carried out
in (7-20). Then according to theorem 7-3 one can find such
integers y 1} . . . , y n that
«i2/i + • • • + a n y n = 1
When this relation is multiplied by c, one has
ai(cyi) + • • • + a n (cy n ) = c
so that the multiples Xi — cyi give a solution of (7-20).
Examples.
1. Let us take first the equation
lOOxi + 72a; 2 + 90x3 = 11
Since the g.c.d. of 100, 72, and 90 is d = 2, and since this number does not
divide c = 11, the equation has no integral solution.
2. On the other hand, the equation
100xi + 72x 2 + C0x 3 = 6 (7-21)
does have solutions according to our criterion since d = 2 divides c = 6. To
find one of them we divide (7-21) by 2 and obtain
50xi + 36x 2 + 45x 3 = 3 (7-22)
and then solve the equation
50yi + 36i/ 2 + 45y 3 = 1
This, however, is the same as (7-18) divided by 2, so that (7-19) gives
a solution to it. When these numbers are multiplied by 3, one finds
xi = 330, x 2 = -462, x 3 = 3
to be a solution of (7-21) and (7-22).
When there were only two unknowns, it was fairly simple to
derive the general solution as soon as one knew a particular set of
LINEAR INDETERMINATE PROBLEMS 157
values satisfying the equation. For several unknowns, the situa-
tion is more complicated, as we have already seen in the examples
in Chap. 6. We indicated there how one could find the general
solution by a series of reductions, and this method is probably
the best available. Let us illustrate it once more by deriving
the general solution of (7-22). We begin by writing
3 - 50;zi - 45^3 3 - 14zi -
36 *> X3+ 36
- 9x 3
Therefore
3 — 142! — 9x 3
v —
36
is an integer. When this expression is solved for x 3 ,
it follows
that
,, o , 3 + 4*!
x 3 = At 2x 1 +
Consequently
3 +Ax x
u =
9
is integral and
u - 3
Xi = 2u -\
4
This gives finally that
u - 3
v =
4
is integral and u = 4w + 3. When this is substituted, one finds
as the general solution
X\ = 9v + 6
x 2 = 5t + 5v + 3
x 3 = -At - 14u - 9
where v and t are arbitrary integers.
158 NUMBER THEORY AND ITS HISTORY
Problems.
1. Find an integral solution to each of the equations:
(a) 31a; + 49?/ - 22s = 2
(6) 120x + My + 1442 = 22
2. Find the general solution and the number of positive solutions of each of
the equations:
(a) Six + 49?/ + 222 = 1,000
(b) 102x + Zlly + 2022 = 10,000
7-3. Classification of systems of numbers. In mathematics
one deals with many systems of numbers, characterized by various
properties. It has gradually become evident that certain systems
are of particular importance, namely, those that reproduce them-
selves, or, as one prefers to say, are closed, under some or all of
the four arithmetic operations, addition, subtraction, multiplica-
tion, and division. For such systems there has come into use
fairly recently a nomenclature we shall now explain. Although we
shall use these terms only incidentally in subsequent chapters,
they are of such importance and common occurrence even in
fairly elementary mathematical writings that the reader should
be familiar with them.
Let S be a set of numbers of any kind. We shall say that S is
closed under addition if, for any two numbers a and b in S, their
sum a + b is also a number in S. It follows immediately that
the sum of three, four, or any finite number of elements in S will
again belong to S. Let us illustrate this definition by some
examples.
Example 1. The set of all natural numbers 1, 2, 3, ... is closed
under addition.
Example 2. The even integers form a system closed under
addition. The odd integers are not closed under addition.
Example 3. All positive real numbers form a system closed
under addition.
Example 4. The sets of all integers, all rational numbers, all
real numbers, and all complex numbers are closed under addition.
LINEAR INDETERMINATE PROBLEMS 159
The closure with respect to subtraction is defined analogously.
We say that a set of numbers S is closed with respect to subtrac-
tion, when the difference a — b of any two of its numbers again
belongs to S. Such a set is called a modul.
Example 1. The integers 0, ±1, ±2, . . . form a modul, but
the natural numbers 1, 2, 3, ... do not.
Example 2. The even integers form a modul.
Example 3. The sets of all rational numbers, all real numbers,
and all complex numbers are moduls.
Example 4. All purely imaginary numbers ib form a modul.
We shall now show that a modul has the following properties:
Theorem 7-5. (a) A modul contains 0.
(6) When a modul contains a it contains —a.
(c) A modul is closed with respect to addition.
Proof: When a is an element in a modul S, the difference
a — a = is in S. Consequently — a = — a is in S, and there-
fore also
a + b = a — ( — 6)
when a and b are in S.
As a consequence a modul is sometimes defined as a system
closed under both addition and subtraction.
Since number theory deals primarily with the integers, we are
interested in finding all moduls consisting only of integers. Clearly
one type of such moduls may be obtained simply by taking all
multiples k - a of some integer a because the sum and difference
of two multiples is again a multiple.
k\a ± k 2 a = (ki ± k 2 )a
It is remarkable, however, that all moduls of integers are of this
kind. This is a consequence of the following theorem, which we
shall now prove:
Theorem 7-6. Any modul M containing only integers consists
of all multiples of the greatest common divisor of the numbers in M.
We remark first that if a is some integer in a system that is
closed under addition, every multiple k • a is also in the system
160 NUMBER THEORY AND ITS HISTORY
since it is the sum of a taken k times. Second, let a and b be
some integers contained in a modul M. Then k • b is in M for
any integral ft and therefore also any difference
r = a — kb
In particular, we conclude that when a is divided by b, the
remainder r is in the modul.
To prove theorem 7-6 we select the smallest positive integer
d in M. Such an integer must exist except in the trivial case
where M consists of the single number 0. All positive and nega-
tive multiples of d are in M, and they exhaust M. If, for example,
m is some integer in M, we divide m by d
m = kd + r, d > r ^
and r, as we remarked, also belongs to M. But since d was the
smallest positive integer in M, this is possible only when r = 0,
and m = kd is a multiple of d. Obviously d is the g.c.d. of the
numbers in M .
It is of interest to connect the properties of moduls with the
linear indeterminate equations. We shall use the result expressed
in theorem 7-6 to derive the basic theorem 7-3 for equations.
Here were given n numbers
a\, a,2, . . . , o>n
All numbers of the form
x = xidi + • • • + x n a n (7-23)
with integral x/s will form a modul M, since the sum and difference
of two such numbers will be of the same kind. M consists of
integers and it is not difficult to find their g.c.d. All numbers a t -
belong to M because one can write, for instance, \
ai = lai + 0a 2 + • • • + 0a n
But
d = (ai, • • • , a«)
divides all a/s and consequently all numbers (7-23) so that d is
the g.c.d. of the numbers in M. From theorem 7-6 we know
LINEAR INDETERMINATE PROBLEMS 161
that M consists of all multiples of d; in particular, d belongs to M
so that it is also of the form (7-23)
d = x x ai + ■ • • + x n a n
with suitable X{. This is the content of theorem 7-3.
A set of numbers S is closed under multiplication when the
product a • b of any two of its elements a and b is again in S.
One concludes that then any finite number of elements in S has
a product belonging to S. A set closed under multiplication is
sometimes called a ray. Among the examples let us mention:
Example 1. The natural numbers as well as the integers.
Example 2. The even integers and also the odd integers.
Example 3. The rational numbers, the real numbers, and the
complex numbers.
Example 4. The real numbers between and 1.
A system of numbers that is closed under addition, subtraction,
and multiplication is called a ring. From theorem 7-5 we see
that the specific mention of addition in this definition is superfluous,
nevertheless it is usually included in the statement. One can
also say that a ring is a modul that is closed under multiplication.
Among the many examples are:
Example 1. The integers form a ring.
Example 2. The even numbers form a ring.
Example 3. The rational, real, and complex numbers define
rings.
Example 4. All complex numbers a + ib, where a and b are
integral, form a ring.
Example 5. All numbers of the form a + bV2 where a and 6
are integers form a ring. To verify this, we observe that the
difference of two such numbers is of the same form, and furthermore
(a + bV2) (c + dV2) = ac + 2bd + (be + ad) V2
Example 6. A similar argument shows that for a fixed integer
D all numbers a + bx^D with integers a and b form a ring.
162 NUMBER THEORY AND ITS HISTORY
We have shown in theorem 7-6 that a modul consisting of
integers must consist of all multiples k • a of some number a. It is
of interest to note that such a modul is also a ring, since the prod-
uct of two such multiples
(fcia)(fc 2 a) = (k 1 k 2 a)a
is another multiple of a.
We turn finally to the sets of numbers that are closed under
division. Such a set contains the quotient a/b of any two of its
elements, provided b j 6 - 0.
Exam-pie 1. The set of positive real numbers is closed under
division.
Example 2. The set consisting of the single number may be
considered to be closed under division. This is somewhat
improper of course, since by the definition of such systems the
division by was excluded.
As a result of the analogy we have already mentioned, between
the laws for addition and those for multiplication, one can derive
a theorem analogous to theorem 7-5.
Theorem 7-7. A set of numbers S, consisting not only of 0,
that is closed under division, must have the three properties:
(a) S contains 1.
(6) When a ^ is in S, so is aT 1 .
(c) S is closed with respect to multiplication.
Proof: When a 9^ is in S, the quotient a/ a = 1, hence the
notien
b in S,
quotient aT 1 = - must also belong to S. Consequently, for any
ah = -37
a
is in S.
A system that does not include and is closed under division
is called a multiplicative group. A simple example is the set of
LINEAR INDETERMINATE PROBLEMS 163
all positive and negative powers
1, a, a 2 , ... , aT 1 , a~ 2 , . . .
of some number a^O.
We now come to the last definition of this kind: A field is a
set of numbers that is closed under all four arithmetic opera-
tions, addition, subtraction, multiplication, and division. From
theorems 7-5 and 7-7, we see that in this definition the inclusion
of addition and multiplication is superfluous. One could also
have defined a field as a system that is a modul and in which the
elements after exclusion of form a multiplicative group. Again
we illustrate by some examples:
Example 1. The most trivial case of a field would be the
number alone. This case is so exceptional that it is ordinarily
excluded and not counted as a field.
Example 2. The rational, the real, and the complex numbers
all form fields.
Example 3. All numbers of the form a -f- b^2 with rational
a and b form a field. To verify this, one observes that the differ-
ence of two such numbers belongs to the set. Furthermore, the
reduction
a + by/2 _ ac — 2bd be - ad _
c+dy/2 ~ c 2 - 2d 2 + c 2 - 2d 2 V2
shows that the quotient can be written in the same form.
Example 4. Clearly the preceding example can be extended.
Let D be some fixed, positive or negative integer that is not
a square, so that \/D is not rational. As before, one can show
that the numbers a + by/D with rational a and b form a field.
Such fields are called quadratic fields.
There are many other rings and fields, some of great importance
in number theory and algebra. For our purposes the examples
given above are quite sufficient. We shall conclude these remarks
164 NUMBER THEORY AND ITS HISTORY
with a theorem that shows that the rational field in a sense is the
smallest possible field:
Theorem 7-8. Every field contains the rational field.
Proof: From theorem 7-7 one concludes that any field con-
tains the number 1. Since the sum of any number of these l's is
in the field, all natural numbers are in the field, and so are all
integers according to theorem 7-6. Since every rational number
is the quotient of integers, our theorem is proved.
Problems.
1. Under which arithmetic operations are the following sets of numbers
closed :
(a) The real numbers ^ 1
(6) The numbers of the form l/« where n is integral
(c) The numbers of the form n/2 where n is integral
\d) The complex numbers a + ai where a is integral
2. Show that all fractions whose denominators are powers of 2 form a ring.
3. Prove that all numbers of the form
a + ° — —
with integral a and b form a ring.
Bibliography
Albert, A. A. : College Algebra, McGraw-Hill Book Company, Inc., New York,
1946.
Birkhoff, G., and S. MacLane: A Survey of Modern Algebra, The Macmillan
Company, New York, 1941.
MacDuffee, C. C. : An Introduction to Abstract Algebra, John Wiley & Sons,
Inc., New York, 1940.
CHAPTER 8
DIOPHANTINE PROBLEMS
8-1. The Pythagorean triangle. Among the many classical
Greek schools of mathematics and philosophy the Pythagorean
was the oldest and most venerable. Pythagoras was born around
570 b.c, according to the best estimates. Tradition has it that
he came from the island of Samos and traveled widely before he
established his school in Crotona in Southern Italy. In Egypt
and Babylonia he absorbed the lore of mysticism and also learned
the laws of numbers and geometry. Pythagoras must have had
considerable personal charm and conviction; his school became
a fashionable center and attracted large numbers of students-
some as auditors while the more qualified were eligible to be
initiated in an inner circle of advanced and mystical learning.
The school continued actively for at least a century after the
death of Pythagoras, and it preserved its esoteric character as
a society of fellows searching for the divine laws of knowledge.
The extent of Pythagoras's own creative contributions to the
science of mathematics is difficult to estimate, both because his
doctrines were propounded only in his lectures and transmitted
orally without permanent records, and also because his disciples
generally effaced their personal roles by ascribing their discoveries
to the founder of the school. The strands of mathematical history
are further snarled because later Greek writers almost traditionally
ascribed the early mathematical discoveries to the Pythagoreans
when their provenance was not otherwise known.
The Pythagorean theorem states that in a right triangle the
square constructed on the hypotenuse is equal to the sum of the
squares on the two legs. When c is the length of the hypotenuse
165
166 NUMBER THEORY AND ITS HISTORY
and a and b the lengths of the two other sides, the theorem becomes
a 2 + b 2 = c 2 (8-1)
This result was certainly known to the Pythagoreans and they
may have been the first to give a satisfactory proof.
One of the simplest cases of the theorem occurs when the sides are
a = 3, & = 4, c = 5
since
32 + 42 = 52 (8 _2)
The knowledge of this particular case has been widespread.
One finds it in the earliest Chinese and Hindu works, together with
other examples where the sides may be represented by integers,
for instance,
a = 5, b = 12, c = 13
a = 8, b = 15, c = 17
In view of the particular interest of the Pythagoreans in rela-
tions that could be expressed in whole numbers, it would appear
natural that they should have investigated the problem of finding
right triangles with integral sides. There exists, according to
Proclus, a much later writer, a formula for a certain type of solu-
tion to the equation (8-1), which he ascribes to Pythagoras. The
formula is f
a = 2n + 1, b = 2n 2 + 2n, c = 2n 2 + 2n + 1 (8-3)
where n is any integer. It may be verified by substitution that
the values (8-3) actually satisfy the relation (8-1). For
n = 1, 2, 3, one finds the following triplets of solutions:
(3, 4, 5), (5, 12, 13), (7, 24, 25)
and this may be continued to give infinitely many others. Pythag-
oras's solution, as one sees, has the special property that the
hypotenuse exceeds the larger leg by one. Another special solu-
tion is ascribed to Plato. The first general solution of the Pythag-
orean problem is found in the tenth book of Euclid's Elements,
DIOPHANTINE PROBLEMS 167
shrouded in geometric terms according to the custom of Greek
mathematics at the time.
Let us examine how one may arrive at the general solution of
the equation (8-1) in positive integers. We remark that the
restriction to integers is not essential; if any rational solution had
been found one could write the three numbers on a common
denominator
°i 7 h ci
a = —■> b = —> c = —
m m m
and it would follow that
0l 2 + h 2 = Cl 2
would be an integral solution.
It is sufficient to find the primitive integral solutions of the
equation, i.e., those solutions in which there is no factor common
to a, b, and c, because if such a factor did occur the equation could
be canceled by d 2 . But for a primitive solution any pair of two
of the numbers a, b, and c must be relatively prime. If for instance
a and b had a common factor e, the left-hand side in (8-1), and
hence also c 2 , would be divisible by e 2 . But then c is divisible by
e, contrary to the assumption that the solution was primitive.
The next step is to see that in a primitive solution a, b, and c,
the numbers a and b cannot both be odd. This is a consequence
of theorem 2-1. Since the square of an odd number leaves the
remainder 1 when divided by 4, it would follow that if a and b
were both odd, the left-hand side in (8-1), hence also c 2 , would
leave the remainder 2 when divided by 4, contrary to the theorem
just mentioned. We suppose now that the notation is taken
such that a is even; consequently b and c are odd since there are
no common factors. The equation (8-1) may be written
a 2 = c 2 - b 2 = (c + b)(c - b)
According to the preceding statements both sides are divisible by
4, and when this factor is divided out, one has
r a\ 2 c + b c — b
= -H 7T- (8-4)
168 NUMBER THEORY AND ITS HISTORY
Here the two integral factors on the right are relatively prime,
because any common factor d would divide both the sum and the
difference of them. But since
c + b c — b
c -\- b c — b
and b and c are relatively prime, one must have d = 1.
When the two numbers on the right in (8-4) are relatively
prime, their prime factors are different, and their product cannot
be a square unless each of them is a square. We can put, therefore,
c + b 9 c — b 9
2 ' 2
and from this we obtain by substitution in (8-4)
a = 2uv, b = u 2 — v 2 , c = u 2 + v 2 (8-5)
To ensure that this solution is actually primitive, we observe that
any common factor of b and c must divide the sum and difference
of these numbers. But since
c + b = 2u 2 , c - b = 2v 2
and since u and v are relatively prime, the only possible common
factor is 2. This factor is excluded when one of the numbers u
and v is odd and the other even.
From the general primitive solution (8-5) where u and v are
integers subject to the conditions just mentioned, one finds the
general integral solution of the Pythagorean equation (8-1) by
multiplying by an arbitrary integer. The general rational solu-
tion is obtained by multiplication of (8-5) by a rational number.
A little later on, however, in connection with a problem by
Diophantos, we shall need the general rational solution, and it is
convenient to have the formulas in a slightly different form. Let
DI0PHANT1NE PROBLEMS 169
us divide both sides of a, b, and c in (8-5) by v 2 so that
a „ u b /u\ 2 c /u\ 2
-2 = 2-» -a = (-) - 1, - = - + l
vr v v l \vj v 2 \vj
There exists, therefore, to the given solution (8-5), a proportional
rational solution
a x = 2t, &! = t 2 - 1, a = t 2 + 1
where we have put t = u/v. When these values are multiplied
by some rational number, one obtains the general solution
a = 2tr, b = (t 2 - l)r, c - (t 2 + l)r (8-6)
where r and t are arbitrary rationals.
Some of the primitive integral solutions in the smallest numbers
may be obtained from (8-5) .
a
b
c
u = 2,
» = 1
4
3
5
u = 3,
v = 2
12
5
13
u = 4,
v = 1
8
15
17
u = 4,
» = 3
24
7
25
Extensive tables of integral Pythagorean triangles have been
computed; one, for instance, by A. Martin 1 gives all primitive
triangles for which the hypotenuse does not exceed 3,000.
There are a great number of questions one may ask in regard to
the Pythagorean triangles, and through the centuries they have
been the source of many number-theory problems. A simple one
suggested by the special Pythagorean solution (8-3) is: When
does the hypotenuse differ from one of the legs by 1 ? One cannot
have
c-b = 1
1 Proceedings, Fifth International Mathematical Congress, Cambridge, 1912.
170 NUMBER THEORY AND ITS HISTORY
because when the values (8-5) are substituted, one finds the
impossible equation in integers
2v 2 = 1
The other possibility
c — a — 1
leads to
u 2 + v 2 — 2uv = (u — v) 2 = 1
so that u = v + 1. When this is substituted in (8-5), one obtains
a = 2v 2 + 2v, b = 2v + 1, c - 2v 2 + 2v + 1
This is, however, the Pythagorean solution (8-3) when the a
and b are interchanged in the notation.
Other problems have been discussed, for instance, the determina-
tion of all integral triangles in which the legs differ by 1, of triangles
with special properties of the perimeter or area, of the number of
right triangles with a given side, and so on.
Problems.
1. Find all integral Pythagorean triangles in which one leg differs from the
hypotenuse by 2 or 3.
2. Find all integral Pythagorean triangles with hypotenuse not exceeding 50.
3. Try to find the general solution in integers of the equations
(a) 2z 2 +y 2 = z 2
(b) 3x 2 + y 2 = 2 2
by the method used to solve the Pythagorean triangle.
8-2. The Plimpton Library tablet. Our brief sketch of the
early history of the Pythagorean problem would have covered the
main facts until quite recently. However, in the last decade or
two new light has been thrown on the whole beginning of mathe-
matics through a deeper understanding of the extent of Babylonian
mathematics. The existence of early mathematical results among
the Babylonians had long been known or suspected, partly through
statements in Greek sources, partly through scattered cuneiform
texts. It had also been known that the larger Babylonian collec-
DIOPHANTINE PROBLEMS 171
tions, particularly those at the British Museum and the Louvre
abroad, and at Yale and the University of Pennsylvania in this
country, possessed a considerable number of undeciphered cunei-
form tablets of unusual types. They often contained columns of
figures, and for that reason they had in some cases been summarily
classified as "commercial accounts." Recent investigations,
particularly those by Neugebauer and Thureau-Dangin, have
revealed that they are actually mathematical tables and texts.
With this fact as a key, the reading was not difficult.
Through this rich mine of source material we have gained a sur-
prisingly intimate view of Babylonian mathematics and its role in
society. The tablets cover a period from 2000 B.C. to 200 B.C.,
but even the oldest ones contain methods that are quite advanced
so that the origin of such methods may safely be placed at a con-
siderably earlier period. The cuneiform tablets give calculations
of areas and volumes, to a large extent as practical problems
arising in connection with surveying and construction, digging of
dikes, and building of walls. Other problems contain questions
regarding the computation of simple and compound interest or
division of estates according to rather involved laws and customs.
One also finds theoretical problems, some of them strikingly like
those given in elementary mathematics today. It is evident that
the Babylonians were familiar with problems that led to second-
degree equations, and the square roots that occurred in their
solution were determined much as they are today, namely, by
means of tables. As a whole, Babylonian mathematics made
systematic and extensive use of numerical tables. Numerous
multiplication tables, tables of inverses, squares, and square
roots, tables of powers of a number, tables for finding the circum-
ference of a circle, and several other types have been preserved
and may be found in the Babylonian collections.
Let us dwell for a moment on the tables of inverses, which
are particularly common among the Babylonian tablets. The
operation of division appears to have been a relatively difficult
one to master in the development of arithmetic in all countries.
In medieval Europe a man capable of performing long division
172
NUMBER THEORY AND ITS HISTORY
was probably more rare than a man with a Ph.D. at present. The
simple procedure of successive duplications and mediations, which
we described in Chap. 2, has been widely used. The Babylonians
used tables of inverses to reduce division to multiplication. To
find the value of the fraction a/b, one wrote it as a multiplication
a • 1/b where the value of 1/6 could be found as a sexagesimal
fraction in the tables.
In constructing these tables of sexagesimal inverses, one runs
into the same trouble as in the ordinary expansion in decimal
fractions, namely, that some expansions are infinite and do not
break off, as for instance
i = 0.333 • • • , \ = 0.142857 142857 • • •
In the most common tables this difficulty is circumvented by
including only numbers whose inversions have a finite, in fact,
a rather short, sexagesimal expansion. According to Neugebauer
the standard type of table of reciprocals usually contains the
following pairs :
a
a- 1
a
a' 1
a
a" 1
2
30
16
3, 45
45
1, 20
3
20
18
3, 20
48
1, 15
4
15
20
3
50
1,12
5
12
24
2, 30
54
1, 6, 40
6
10
25
2, 24
1
1
8
7, 30
27
2, 13, 20
1, 4
56, 15
9
6, 40
30
2
1, 12
50
10
6
32
1, 52, 30
1, 15
48
12
5
36
1, 40
1, 20
45
15
4
40
1,30
1,21
44, 26, 40
We have preserved the sexagesimal notation in the table. In
checking the figures, the reader should recall that the Babylonians
used no decimal sign to indicate where the units begin, so that,
for instance, 60 may denote not only this figure but also 1 or
60 2 . One observes that in the table, entries like 7, 11, 13, 14,
DIOPHA NTINE PROBLEMS
173
and so on, which would give infinite expansions, have been
omitted.
Although they are comparatively rare, there also exist tables
that within their limits give the reciprocals of all numbers without
exception. For numbers with an infinite sexagesimal expansion,
Fig. 8-1. Ttible of inverses. (Courtesy of Yalu Babul
CotlecHon.)
one obtains a satisfactory approximation by breaking it off after
a certain number of places as in our ordinary numerical tables.
The Yale Babylonian Collection, which is particularly rich in
mathematical source material, contains one tablet (YBC 10,529)
which in its preserved part gives the reciprocals of all numbers
between 58 and SO with great accuracy. (Fig. 8-1.)
It is not difficult to determine which numbers have a finite
sexagesimal expansion
b = Go + 60 +
174 NUMBER THEORY AND ITS HISTORY
Clearly this can occur only when the fraction can be written in
the form
an t
i - «r (8 " 7)
The fraction on the right may be reduced to its lowest terms by
cancellation of some factors. Since the base number 60 has the
factorization into prime factors
60 = 2 2 • 3 • 5
it follows that in the reduced fraction a/b in (8-7), the denomi-
nator b can have only the prime factors 2, 3, and 5. This means
that with suitable exponents a, /3, and y, we have
b = T • 3* • 5 7 (8-8)
Numbers of this type (8-8) may be called regular for the given
base number 60. One sees conversely that if b is a regular number,
the fraction (8-7) can be extended, and be written in the right-
hand form as a fraction whose denominator is a power of 60, and
so one finds a finite sexagesimal expansion.
The reader may verify that all entries in the table of inverses
given above are regular, and also consider the question of finding
the length of the expansion when 6 has a given prime fac-
torization (8-8).
These discoveries in Babylonian mathematics also throw light
on the history of early Greek science. The knowledge of Greek
mathematics before Euclid has always been somewhat nebulous,
and it has been difficult to understand the rapid rise from its
primary stages, represented by Thales of Miletus (about 600 b.c.)
and the Pythagoreans, to the beautiful system one finds developed
at the time of Euclid (300 b.c.) or probably even earlier. It
must now be assumed that the Greeks absorbed much more from
the Babylonian storehouse of mathematical facts and methods
than had hitherto been suspected. This, however, it should be
explicitly stated, does not detract from the distinction of the
Greeks for having created the concept of the systematic mathe-
matical theory as we still understand and use it today, based upon
D10PHANTINE PROBLEMS
175
axioms or fundamental assumptions and developed by logical
deductions in its proofs. This achievement has been one of the
most important in the history of human thought.
In the transition from Babylonia to Greece, mathematical
knowledge changed its form. Greek mathematics is dominated
Fig. 8-2.
Plimpton mathematical tablet 'XV2. {Columbia University Library.
Courtesy Professor J. Mendelsohn.)
by the geometric figure. This preference may in part be due to
their feeling for beauty in lines and patterns, as shown in their
decorative art and architecture, but a more compelling reason for
the adoption of the geometric system was the logical consequence.
The geometric lines were understandable and complete, while the
numbers led to the logically incomprehensible, the unutterable
concept of the irrational. Babylonian mathematics, on the other
hand, was arithmetic and algebraic in character and expressed
itself through numerical computations. Approximations were
resorted to quite freely, thus obviating the necessity for the
irrational perfectionism.
Judging from the advanced state of Babylonian mathematics
176
NUMBER THEORY AND ITS HISTORY
as revealed by the tablets, it seemed a reasonable conjecture that
the Babylonians were in possession of the Pythagorean theorem.
However, it was not until quite recently that a factual proof was
found. In a new publication of cuneiform texts by Neugebauer
and Sachs (1945), there is included a description of a clay tablet
from the Plimpton Library at Columbia University, which bids
fair to be one of the most crucial records in the history of mathe-
matics. The tablet, catalogued as Plimpton 322, is composed in
Old Babylonian script so that it must fall in the period from
1900 b.c. and 1600 B.C., at least a millenium before the Pythag-
oreans. Unfortunately, the tablet is broken and one section is
missing, but there remain three complete columns of figures and
part of u, fourth which may be reconstructed (see Fig. 8-2). The
reader may verify from the photographic reproduction that when
we preserve the sexagesimal notation, the numbers in the three
columns run as follows:
1, 59
2, 49
1
56, 7
1, 20, 25 [3, 12, 1]
2
1, 16, 41
1, 50, 49
3
3, 31, 49
5, 9, 1
4
1,5
1,37
5
5, 19
8, 1
6
38, 11
59, 1
7
13, 19
20, 49
8
8, 1 [9, 1]
12, 49
9
1, 22, 41
2, 16, 1
10
45
1, 15
11
27, 59
48, 49
12
2, 41 [7, 12, 1]
4, 49
13
29, 31
53, 49
14
58
1, 46 [53]
15
Clearly the last column only enumerates the lines. The first two
columns are much more interesting. It is not difficult to verify
that they form the hypotenuse and one leg of a Pythagorean
triangle. When one squares the numbers in the middle column
DI0PHANT1NE PROBLEMS
177
and subtracts from each of them the square of the corresponding
number in the first column, one obtains a square number. There
are, however, four exceptions to this rule, and in the preceding
table the corrected figures have been given rather than the actual
figures on the tablet, which have been put in brackets. The
exception in line 2 is difficult to explain, while the number 9
instead of 8 in the ninth line must be a mere slip of the stylus.
The number in line 13 is the square of the correct one and in line
15 half of the side occurred originally. It is of course some-
what unsatisfactory to be compelled to make four corrections
in a table with 15 entry lines, but as we shall see, the fourth column
gives a further check on the values in the other two columns, con-
firming again the corrected figures.
It is of interest to compute the missing column of the last side
of the triangle ana also to use our previous solution of the Py-
thagorean triangle given in (8-5) to determine the values of the
numbers u and v that correspond to the solutions on the tablet.
This information is given in the following table, which the reader
may check:
6
c
a
u
V
119
169
120
12
5
3,367
4,825
3,456
64
27
4,601
6,649
4,800
75
32
12,709
18,541
13,500
125
54
65
97
72
9
4
319
481
360
20
9
2,291
3,541
2,700
54
25
799
1,249
960
32
15
481
769
600
25
12
4,961
8,161
6,480
81
40
45
75
60
2
1
1,679
2,929
2,400
48
25
161
289
240
15
8
1,771
3,229
2,700
50
27
56
106
90
9
5
178 NUMBER THEORY AND ITS HISTORY
All solutions are primitive except in line 11, where there is
a common factor 15, and line 15, where there is a factor 2.
The question naturally arises whether the Babylonians were
in possession of a method for solving the Pythagorean triangle
corresponding to the general solution we have already established
in (8-5). The answer must undoubtedly be in the affirmative,
for many reasons. Of course one cannot hope to discover an
explicit formula since no algebraic terminology existed at this
time. In Babylonian mathematics, as in all early expositions,
the reader was expected to infer the general rule from the examples
given. Evidently, the large solutions of the Pythagorean problem
found in the Plimpton Library tablet have not been obtained by
guesswork ; there are many much simpler solutions one would run
across before these. The last leg of the triangle, computed in each
case from the two given on the tablet, provides the key to the
construction of the table. These numbers are all very simple in
the sexagesimal system, as the reader may verify by rewriting
them, and furthermore they are all regular sexagesimal numbers
as we have defined this term, since they have only the prime
divisors 2, 3, and 5. According to our solution (8-5), this side is
determined by the formula a = 2uv so that u and v are also regular
sexagesimal numbers, as one sees by inspection of the table above.
Thus it appears that the table on page 176 has been constructed
by making a choice of small regular numbers for the parameters
u and v.
This particular method had been used with a special idea in
mind. The numbers representing the side a are all regular and
occur in the tables of inverses; this fact points to their application
in a division process, and indeed, the last, somewhat maculated,
column on the tablet contains the value of the quotient c 2 /a 2 for
each triangle. If one denotes by a the angle in the right triangle
opposing the side a (see Fig. 8-3), one has
c
a 2 sin 2 a
DIOPHANTINE PROBLEMS 179
Another remarkable fact now becomes apparent. If one proceeds
to compute the values of the quotients
c 1
- = — = cosec a
a sin a
it is a consequence of the particular choice of the side a that this
trigonometric function must have finite sexagesimal expansions.
Furthermore, the values of cosec a. form a very regular sequence
with a decrease of almost exactly 1/60 from one line to another,
so that one would have a table of this trigonometric function
constructed by means of right triangles with integral sides. Cor-
respondingly, the angle decreases from 45° to 31°, and it seems
natural to believe that there existed companion tablets with
similar values for the angles from 0° to 15° and from 16° to 30°.
How the Babylonians succeeded in finding values for c and a
such that the quotient c/a decreases so evenly cannot be con-
sidered fully explained. It is evident, however, that at this early
date the Babylonians not only had completely mastered the Py-
thagorean problem, but also had used it as the basis for the con-
struction of trigonometric tables. One can only hope that future
discoveries will produce further material, which will throw light
upon this fascinating subject.
8-3. Diophantos of Alexandria. Greek mathematics at its
height was preeminently geometric in character. However during
the later Alexandrian period, when Greek science and philosophy
as a whole was on the decline, and with it mathematics, the alge-
braic methods came more into the foreground. It is possible
180 NUMBER THEORY AND ITS HISTORY
that this change may have been caused, at least to some degree,
by outside influences from Babylon and perhaps even India.
During this period, Diophantos (perhaps a.d. 250), the most
renowned proponent of Greek algebra, lived in Alexandria. Prac-
tically nothing is known about his life. There exists a collection
of Greek problems in poetic form, the Palatine Anthology, which
was compiled probably not over a century after Diophantos's
death. It contains certain simple problems that can be solved
by equations, some of them indeterminate, and among them one
finds the following, containing all known personal information
about Diophantos:
Here you see the tomb containing the remains of Diophantos, it is
remarkable: artfully it tells the measures of his life. The sixth part of
his life God granted him for his youth. After a twelfth more his cheeks
were bearded. After an additional seventh he kindled the light of
marriage, and in the fifth year he accepted a son. Elas, a dear but
unfortunate child, half of his father he was and this was also the span
a cruel fate granted it, and he consoled his grief in the remaining four
years of his life. By this device of numbers, tell us the extent of his life.
If x is the age of Diophantos and if one interprets the poetic
statement to mean that the son died at the age when he was half
the father's ultimate age, the equation becomes
x x , x , „ , x
-H h- + 5 + - + 4 = :r.
6 12 7 2
Thus x = 84 was his age.
The known titles of works of Diophantos are the Arithmetics in
13 books, the Porisms, and a treatise on polygonal numbers. All
of them dealt with the properties of rational or integral numbers.
Unfortunately, the Porisms have been lost and only a part of the
Polygonal Numbers exists. Six or seven of the books of the
Arithmetics have been preserved and there is some doubt whether
the whole cycle was ever completed. In regard to the title it
should be pointed out that Greek mathematicians used the term
arithmetic in the sense of number theory, i.e., the systematic
DIOPHANTINE PROBLEMS 181
investigation of the properties of numbers, while ordinary com-
putations were classified as logistics.
In his mathematical presentation Diophantos uses stenographic
abbreviations with special signs, often composed of the initial
letters of the names of the concepts he wishes to designate; the
unknown quantity, powers, and various operations therefore have
fixed notations. This form of mathematical writing has been
called syncopated algebra, and it must be considered an early step
towards algebraic formalization and the creation of mathematical
language.
The Arithmetics deal with topics on algebraic equations and
more particularly with the solution of certain problems in which
it is required to find rational numbers satisfying prescribed condi-
tions. More than 130 problems of this latter type, of considerable
variety, are discussed, and Diophantos shows great ingenuity in
devising elegant methods for their solution. He is particularly
adept at selecting the unknowns in such a manner that the alge-
braic conditions become easily manageable. We shall reproduce
a few of his problems to illustrate the kind of problems he tackles.
They should be prefaced by the general remark that negative or
zero solutions are always excluded.
Problem 1 in Book II requires: To find two numbers such that
their sum is in a given proportion to the sum of their squares.
In modern notation we would write
x 2 + y 2
— T — = V
x + y
where x and y are the numbers to be found and p the given pro-
portion. This may be written
x 2 — xp + y 2 — yp =
and when it is considered to be a second-degree equation in x, one
finds the solutions
2 ±^-y* + py (8-9)
182 NUMBER THEORY AND ITS HISTORY
Since the problem should be solved in rational numbers, the
number under the square root sign must be a square. A typical
device for expressing this condition is based upon the following
observation: It is possible to express every number in the form
by a suitable choice of the number t. The square number which
occurs under the root sign in (8-9) can therefore be written
^ - y 2 + PV = (l + tyj (8-10)
When one performs the reduction, the terms that do not involve
y drop out, and one factor y may be canceled. There remains
a simple equation of the first degree, which gives
For any rational value of t, the corresponding y in (8-11) makes
the expression (8-10) a square
p* [> (i + a - * 2 ) f
j-y + w - [a i + ? J
When this is substituted in (8-9), one finds for x two solutions
* -*>! + ?• X = V ~^TJ (8 " 12)
The general solution of the problem is therefore given by (8-11)
and (8-12) with rational t. Diophantos, of course, has no for-
mulas, but he illustrates the methods for p = 10. His solution
x = 12, y = 6 corresponds to t = f .
The majority of Diophantos's indeterminate problems require
that one shall find certain sets of square or cube numbers with
special properties, and the solution of the Pythagorean triangle
often comes into play in his procedures. For instance, in Problem.
22 in Book IV it is proposed : To find three numbers such that odp
DIOPHANTINE PROBLEMS 183
is the mean proportional between the two others, and such that
the difference between any two of them shall be a square number.
If x > y > z are the three rational numbers to be found, it is?
required that the three differences
x — y = a 2 , y — z = b 2 , x - z = c 2
shall be square numbers. To satisfy the first two conditions one
must have
x = y + a 2 , z = y - b 2 (8-13)
with arbitrary a and b, but when these values are substituted in
the third it reduces to
a 2 + b 2 = c 2
Therefore, the three numbers a, b, and c must form a rational
Pythagorean triangle so that according to (8-6) we have
a = 2tr, b= (t 2 - l)r, c = (t 2 + l)r (8-14)
with rational values t and r. It remains to fulfill the condition
that y be the mean proportional between x and z, that is,
y 2 = xz
According to (8-13) this may be written
V 2 = (V + a 2 )(y-b 2 )
and after reduction one finds
a 2 b 2
V= ^b 2
From (8-13) follows further
x = -5 rs> z =
a 2 _ b 2 a 2_ tf
To obtain the general solution we must substitute the values
(8-14) for a, b, and c. Since the solutions are to be positive,
one must choose the sides of the Pythagorean triangle such that
a > b. The example on which Diophantos illustrates the pro-
184 NUMBER THEORY AND ITS HISTORY
cedure corresponds to the familiar case b = 3, a = 4, c = 5, and
he finds
„ _ 25G 7 . _ 144 _ SJ,
X — 7 , y — 7 , * 7
We shall give a few more examples of problems from the
Arithmetics of Diophantos, on which the reader may try his skill.
Problem 29, Book II : Find two square numbers such that when one
forms their product and adds either of the numbers to it, the result is
a square.
Problem 7, Book III : Find three numbers such that their sum is a
square and the sum of any two of them is a square.
Problem 9, Book III: Find three numbers in arithmetic series such
that the sum of any two of them is a square.
Problem 15, Book III: Find three numbers such that the product of
two of them minus the third is always a square.
Problem 11, Book IV: Find two numbers such that their sum is
equal to the sum of their cubes.
Problem 18, Book VI: Find a Pythagorean triangle in which the
length of the bisector of one of the acute angles is rational.
Quite appropriately, as a tribute to Diophantos's early contri-
bution to the subject, algebraic problems in which one is required
to find rational solutions are called Diophantine problems. In
modern terminology this concept is usually narrowed somewhat
to refer mainly to problems with integral solutions. As a conse-
quence, even our previous linear indeterminate problems are
commonly called linear Diophantine problems in spite of the fact
that these problems were not discussed by Diophantos, probably
because he considered them trivial. For instance any linear
equation with integral coefficients may be solved rationally by
giving arbitrary rational values to all unknowns except one and
expressing the remaining one by the others.
It seems unlikely that the large collection of problems in the
Arithmetics should be the creation of a single author, and some of
them must have been gleaned from previous sources. However,
any statements about iae earlier history of Diophantine problems
are entirely conjectural. It is possible that Greek algebra was
DIOPHANTINE PROBLEMS 185
further developed than our present sources indicate, and it is
always an open guess that Babylonian mathematics embraced
problems of this type.
8-4. Al-Karkhi and Leonardo Pisano. To the subsequent
Greek mathematicians, all of them of very minor stature, and to
the Arabs, Diophantos remained an outstanding name, almost
synonymous with algebra itself. Among the Arab mathemati-
cians, al-Karkhi of Bagdad, who died around a.d. 1030, was
probably the most scholarly and original. Two of his works are
known. One is the Al-Kafi fit hisab or Essentials of Arithmetic
which is of an elementary character and gives the rules for com-
putations. It is peculiar in that it avoids the use of Hindu
numerals throughout, although they were at this time quite
common in Bagdad. Among certain orthodox groups among
the Arabs there seems to have been strong objection to the Hindu
numbers, in many ways reminiscent of the opposition of the
abacists in Europe to the same numbers a few centuries later.
Al-Karkhi's second work, the Al-Fakhri, is a much more
important document in the history of mathematics. It derives
its name from al-Karkhi's friend, the grand vizier in Bagdad at
the time, to whom the treatise was dedicated. Al-Karkhi, in
many ways, was the Arabic successor to Diophantos, even to the
extent that the Al-Fakhri contains long sections that have been
copied verbatim from the Arithmetics. The general plan of the
two works is the same. Both contain basic algebraic theory with
applications to equations and especially to problems that should
be s;lved in rational numbers. Although al-Karkhi repeats many
of Diophantos's problems, he develops the methods further and
also introduces problems of quite different types. However, in
terms of our present-day algebraic symbolism, most of them do
not present great difficulties.
For instance, in Problem 1 in Section 5 in the Al-Fakhri it is
requested to find such numbers that the sum of their cubes is
a square number. This means that the equation
x B + y 3 = z 2
186 NUMBER THEORY AND ITS HISTORY
shall be solved in rational numbers. One can put
y = mx, z = nx
and by substitution and cancellation of x 2 , one obtains
n 2
x =
1 +ra d
where m and n may be arbitrary rational numbers. Al-Karkhi
gives the special solution x = 1, y = 2, z = 3, a set of numbers
that probably led to the problem's being put. The same method
is clearly applicable to much more general rational problems,
for instance,
ax n + by n = cz n ~ l
and others for which al-Karkhi gives illustrations.
In several problems he asks for rational solutions to two simul-
taneous equations that may be included in the general type
x 3 + ax 2 = y 2 , x 3 — bx 2 = z 2
where a and b are known integers. Again al-Karkhi puts
y = mx, z = nx
and from the two equations he derives
x = m 2 — a, x = n 2 + 6
Since these two numbers must be the same, the condition
m 2 — n 2 = a + b
must be satisfied for m and n. Here one puts m = n + t and
obtains
2nt + t 2 = a + b
or
a + b - t 2
n =
2t
From this value one finds m and in turn the general solution for
x, y, and z in terms of the arbitrary rational number t. Al-Karkhi,
DIOPHANTINE PROBLEMS 187
of course, only gives examples, for instance
x 3 + 4x 2 = y 2 , x 3 - hx 2 = z 2
is found to have the solution x = 21.
In regard to mathematical knowledge the Middle Ages in
Europe was a vacuous period with a single, brilliant star, Leonardo
Pisano. Leonardo was a mathematician of great originality and
creative power but also a direct successor to the Arabic mathe-
matical school, much in the way al-Karkhi was heir to the knowl-
edge and inspiration of Diophantos. Leonardo never mentions
his sources, but he was educated in North Africa and traveled
widely in the Eastern Mediterranean, and there can be no question
that he was familiar with works of the leading Arabic mathematical
writers. In Leonardo's main work, the Liber abaci (1202), one
finds many problems that have been borrowed literally from the
Al-Fakhri, and therefore sometimes originally from Diophantos;
others have their source in al-Khowarizmi's Al-Jabr wal-
Muqabalah.
Leonardo's fame was widespread, and true to the customs of
the time, he was presented with challenge problems from near
and far. Some of these were indeterminate problems. For
instance, Master Theodorus, court philosopher to Emperor
Frederic II proposed the problem of finding numbers x, y, and z
such that all three expressions
x + y + z + x 2 , x + y + z + x 2 + y 2 ,
x + y + z + x 2 + y 2 + z 2
become squares. This was a problem truly in the tradition of
Diophantos, and Leonardo gives as one solution
r _ JJ8. 7/ _ .48 „ _ 144
The Emperor Frederic II was a sincere patron of learning and
actively promoted the diffusion of Arabic knowledge in Europe.
No wonder therefore that he took an interest in such an out-
standing scholar as Leonardo. Probably in the year 1224, he
188 NUMBER THEORY AND ITS HISTORY
was summoned to take part in a mathematical tournament,
which was to be held in the presence of the emperor. The prob-
lems were formulated and presented by John of Palermo, another
scholar belonging to the entourage of the emperor. Leonardo
easily carried off the laurels by solving all problems in the most
admirable manner.
One of the problems was the solution of a particular cubic
equation, and after having shown that there could be no rational
root to it, Leonardo proceeded to compute the real root in sexa-
gesimal fractions with an accuracy that corresponds to 11 decimal
places. A much simpler problem was the following, which we
mention only because it belongs to a type that at the time enjoyed
considerable popularity :
Three men own a share in a heap of coins; the first owns §,
the second §, and the third f of the total. The money is divided
by having each man take an amount arbitrarily. The first man
afterwards returns \ of the coins he has taken, the second \, and
the third \ . The money thus returned is divided into three equal
shares, which are given to each man, and it turns out that now
everyone has his proper part. How much money was there, and
how much money did each obtain the first time? We leave the
solution to the reader.
Here we are more interested in the following indeterminate
problem proposed in the tournament: Find such a square number
that when 5 is added or subtracted one also obtains squares. In
mathematical symbols, one wishes to find a number x such that
a* + 5 = y 2 } x 2 - 5 = z 2 (8-15)
Leonardo gives the solution x = 3 j^-.
Again one cannot exclude the possibility that Leonardo may
have been familiar with this kind of problem since it occurs in
earlier Arab writings. However, in a treatise Liber quadratorum
(1225) written shortly after the tournament, Leonardo returns to
the problem and here his methods are entirely different from
those used by Arab mathematicians.
Let us discuss the general problem of rinding a square number
DIOPHANTINE PROBLEMS 189
such that when a given number h is added to it or subtracted from
it one obtains other square numbers. This means that we must
find a number x such that simultaneously
x 2 + h = a 2 , x 2 - h = b 2 (8-16)
and determine for which h rational solutions x can exist. We shall
first determine the solutions in integers and this depends, as we
shall see again, on the Pythagorean triangle. When the second
equation (8-16) is subtracted from the first, one has
2h = a 2 - b 2 = (a - b)(a + b) (8-17)
Since the left-hand side is even, a and b must both be odd or both
even. Therefore, a — b is even
a - b = 2k
and k must be a divisor of h, according to (8-17). It follows that
and by adding and subtracting the last two equations, one finds
a = — - -\- k, b = — — k
2k ' 2k
When these two expressions are substituted in the original equa-
tions (8-16), there results
so that we have now only a single condition
2
+ k 2
\2k)
190 NUMBER THEORY AND ITS HISTORY
Therefore the three numbers
* i.
form a Pythagorean triangle, and according to the solution we
have obtained, we can write
h
x = t(m? + n 2 ), — = t(m 2 - n 2 ), k = 2mnt
where t is some integer and the expressions in m and n define
a primitive solution of the triangle. When we take the product
of the last two expressions, we obtain as the general solu-
tion to (8-16)
x = t{m 2 + n 2 ), h = 4mn(m 2 - n 2 )t 2 (8-18)
We shall make a slight reduction in this solution. Let us
suppose that we have a solution x of (8-16), where x has the
factor t and h at the same time the factor t 2
x = x\t, h = hit 2
From the two equations
xft 2 + h x t 2 = a 2 , x x H 2 - ht 2 = b 2 (8-19)
it follows that a and b have the factor t
a = a\t, b = b\t
After the factor t 2 has been canceled in (8-19), one has
%i 2 + hi = a 2 , x 2 — hi = bi 2
When no further such reduction is possible, we shall say that we
have a primitive solution. When this reduction is applied to
(8-18), the solution becomes
x = m 2 + n 2 , h = 4mn(m 2 — n 2 ) (8-20)
The numbers m and n produce a Pythagorean triangle where
sides have no common factor. The hypotenuse x is then relatively
prime to the sides
DIOPHANTINE PROBLEMS
191
hence x is relatively prime to h, and (8-20) must be a primitive
solution to the problem in the sense just denned.
When one takes small values for m and n, one finds the following
primitive solutions:
m
n
h
X
2
1
24
5
3
1
96
10
3
2
120
13
4
1
240
17
4
3
336
25
When the equations (8-16) in our problem are to have integral
solutions, the number h must have the form we have derived
in (8-18). However, to determine when a given number h can be
represented in this manner is in itself a problem that is not easily
settled in general. After Leonardo, many mathematicians returned
to the problem and the permissible numbers even received a
special name, a congruum. This nomenclature is now obsolete
and must not be confused with the congruent numbers we shall
study in the next chapter.
Leonardo established the following simple property: A congruum
is divisible by 24. In the examples given above in the table this
is immediately verified. To prove it in general, we recall that of
the numbers m and n that give a primitive solution to a Pythag-
orean triangle, one is odd and the other is even. The product mn
is therefore divisible by 2, and so h is divisible by 8 according to
(8-20). It remains to show that h is divisible by 3. This is
immediate when m or n is divisible by 3. W T hen neither of them
is divisible by 3, one can write
so that
m = 3mi ±1, n = Zni ± 1
m 2 - n 2 = 9wi 2 - 9n x 2 ± Qm 1 ± 6^
is divisible by 3.
192 NUMBER THEORY AND ITS HISTORY
It is now time to return to the original tournament problem
(8-15) solved by Leonardo. Since in this case h = 5 is not
divisible by 24, there can be no integral solutions. We therefore
write x, a, and b as fractions with a common denominator
x x
a x
fc _£
x — — '
a = — - >
d
d
d
By substitution into (8-15) and clearing the fractions, one
obtains
x * + 5d 2 = a, 2 , x x 2 - 5d 2 = h 2 (8-21)
If there is to be any solution to (8-15) in rational numbers, it
must be possible to find some integer d such that bd 2 is a congruum.
Now in the condition
5d 2 = 4mn(m 2 — n 2 )
it is a natural first attempt to make m = 5, and one must then
satisfy
d 2 = 4n(5 2 - n 2 )
By trying out the first few integers, one sees that n = 4 gives
a square
d 2= 4-4(5 2 - 4 2 ) = 144 = 12 2
The values n = 4, m = 5, according to (8-20), result in the
solution
Xl = 5 2 + 4 2 = 41
for (8-21). Consequently
xi 41 „ 5
x = — = — = 3 —
d 12 12
is a solution to Leonardo's problem, as he actually stated. To
check the solution we have
«v + 5 = n 2 , /«v - 5 - (^Y
12/ \12/ \12/ \12/
DIOPHANTINE PROBLEMS 193
There are no indications that John of Palermo was himself a
prominent mathematician. On the other hand, when we look
back upon the process that was required for the solution of his
last problem, it is evident that it was not proposed haphazardly.
It is difficult to escape the conclusion that it was a problem drawn
from previous sources, and in that case most naturally from
Arabic scholars whom he encountered on his native Sicily, which
under Frederic II was one of the centers in the exchange of Euro-
pean and Oriental scholarship.
We have followed the development of Diophantine analysis
through the Arabs to its transmittal to Europe through Leonardo
Pisano. There exists, however, another branch of this field of
number theory, which we must mention although we shall not
pursue it in detail. The Hindus early became acquainted with
the works of Diophantos, but their own number theory took an
independent direction. We have already mentioned (Sec. 6-2) the
method of the pulverizer, a variation of the algorism of Euclid,
which gave the Hindus the solution of their linear indeterminate
problems. But both in the Brahma- Sputa- Siddhanta by
Brahmagupta and the Bija-Ganita by Bhaskara, one finds con-
siderable space devoted to indeterminate problems of the type
ex 2 + 1 = y 2
and more generally
ex 2 + a = y 2
Not only are the rational solutions found; the integral solutions
are also discussed. Later it has turned out that this kind of
problem is of systematic importance for various mathematical
questions, for instance, for continued fractions and number theory
in quadratic fields, both subjects that are left out of this book
with regret.
Problems.
The reader may try to find the rational solutions to the following equations
or sets of equations, all taken from the Al-Fakhri:
1. x 2 + 5 = y 2 3. x 2 - 2x - 2 = y 2
2. x 2 - 10 = y 2 4. 10 - x 2 = y
194 NUMBER THEORY AND ITS HISTORY
5. lOx - 8 - x 2 - y 2 9. x 4 + y* = z z
6. x 2 + x = y 2 , x 2 + 1 = 3 2 10. x 2 - y 3 = 2*
7. x 3 + y S = 2 2 1L x 6 + 5y 2 = 3 2
8. xV = z 3
8-5. From Diophantos to Fennat. Fermat represents a focal
point in the history of number theory; in his work the radiating
branches of earlier periods were united and their content recreated
in a richer and more systematic form.
The path from Diophantos to Fermat, although long in time,
is quite direct. During the Renaissance, at the rebirth of classical
learning, numerous manuscripts of Greek mathematical works
reached Western Europe. The general level of mathematics in
Europe had been extremely low during the Middle Ages, so low
that the Greek knowledge was a revelation whose true content at
times was found to be intolerably hard to decipher. Among the
works were copies of the writing of Diophantos, whose very name
had until then been unknown, and they represented a severe
challenge to the mathematicians of the sixteenth century.
The first reference to Diophantos in the Occident seems to have
been made by Regiomontanus in 1462. He reported that he had
discovered a manuscript of a certain Diophantos in the Vatican
library and that he was interested in making a translation from
the Greek, a task he never seems to have tackled.
The first printed edition and translation of Diophantos into
Latin was published in Heidelberg in 1575, by the German pro-
fessor Holzman, a name which he changed to the Greek form
Xylander. To show the impact of Greek mathematics on the
European scholars, let us reproduce a part of Xylander's foreword
to his translation of the Arithmetics. He mentions that he had
heard earlier of the existence of a Diophantos manuscript, but
. . . since no one had edited it, I gradually silenced my eagerness to
know it, and buried myself in the mastery of the works of such arithme-
ticians as I could obtain, and in my own cogitations on the subject.
Truth however compels me to offer with complete frankness the testi-
DIOPHANTINE PROBLEMS 195
mony which follows, however much to my disgrace. As for Cossica
or Algebra, since, self-taught — except for the mute teachings of books,
I had not only acquired command of the subject, but also had advanced
to the point of adding, giving variety, and in places even of making
corrections to what such great and devoted teachers as Christifer
Rodolphus Silesius, Michaelus Stifelius, Cardanus, Nonius, and others
had written about it, I fell into that mood of complacency, which
Heraclitus called "The Holy malady"; — in short I came to believe that
in Arithmetic and Logistic "I was somebody". And in fact by not a
few, and among them some true scholars, I was adjudged an Arithme-
tician beyond the common order. But when I first came upon the
work of Diophantos, his method and reasoning so overwhelmed me
that I scarcely knew whether to think of my former self with pity or
with laughter. It has seemed w orth while in this place to proclaim my
former state of ignorance, and at the same time to give some hint of the
work of Diophantos, which swept away from my befogged eyes the
cloud of darkness which enveloped them. The treatment of surds I
had mastered so well that I had even ventured to add to the inventive-
ness of others some things not inconsiderable, and these contributions
in the field of arithmetic were accounted of no small importance in view
of the difficulties of the subject, which had driven many from the whole
subject of mathematics. But how much more brilliant a performance
was it, in problems which seemed scarcely capable of solution even with
the help of surds, and where surds bidden to till the soil of Arithmetic,
true to their name, turned a deaf ear and fahed, to carry the solution
of the subtlest kind of problems to a point where surds are not invoked,
and are not so much as even mentioned.
Xylander emphasizes with admiration how Diophantos is able
to avoid irrational square roots or surds in his solutions. He puns,
as one sees, on the surd or deaf numbers, a term we have taken
over as a direct translation from Arab authors. This is in direct
analogy with the "unspeakable" numbers of the Pythagoreans,
and it is fully as satisfactory as our "irrational" roots, translated
from the Greek a\oyos or without ratio. Another ancient term in
Xylander is the Cossica or Rule of Coss, which in early English
texts most often appears as the Cossick Art, synonymous with
algebra or equation theory. It refers to the common terminology
196 NUMBER THEORY AND ITS HISTORY
of the time, of Italian origin, in which the unknown to be found
in a problem is termed the thing or cosa.
Xylander's source manuscript was quite unsatisfactory and
this is reflected in his translation. Nevertheless, the book created
much interest in problems of Diophantos's type. In 1621 Bachet
de Meziriac, whose acquaintance we have already made in connec-
tion with the linear indeterminate problems, published a new
edition with notes and comments. In some of these he sharply
and somewhat ungratefully criticizes Xylander, whose earlier
edition clearly had been of assistance to him. However, Bachet's
edition represents a great improvement. Furthermore, it is very
probable that it has the unique distinction of being the work that
introduced Fermat to the problems of number theory.
Fermat possessed a well-worn copy of Bachet's Diophantos,
which he also used as a notebook. In the margin he jotted down
several of his most important results as they occurred to him in
connection with the related problems in Diophantos. After
Fermat's death the entire book, together with Fermat's notes, was
published by his son Samuel (1670).
We shall discuss the content of a few of the various results
indicated by Fermat in his marginal comments to Diophantos.
Here one finds the result we have already mentioned in connection
with the factorization of numbers, and which we prove in Chap. 11,
namely, that every prime of the form 4n + 1 can be represented
as a sum of two integral squares in a single manner. By means
of the identity
(a 2 + b 2 ) (c 2 + d 2 ) = (ac ± bd) 2 + (ad^F be) 2 (8-22)
which was known to Leonardo Pisano and was used implicitly by
Diophantos, one can represent the product of any two numbers that
are sums of two squares as the sum of two other squares, and even
in two different ways. We have, for instance,
13 = 3 2 + 2 2 , 37 = 6 2 + l 2
and find by using (8-22)
13 • 37 - 20 2 + 9 2 = 16 2 + 15 2
DIOPHANTINE PROBLEMS 197
In the special case
2 = l 2 + l 2
one derives from (8-22)
2(a 2 + b 2 ) = (a + 6) 2 + (6 - a) 2
One concludes, therefore, that any product whose factors are 2
and primes of the form An + 1, can be represented as the sum of
two squares. Moreover, if one multiplies a sum of two squares
by a square number
P(a 2 + 6 2 ) = {ka) 2 + (kb) 2
the result is a sum of two squares. This leads to the criterion:
If N is an integer and n 2 its largest square factor, so that
N = N n 2
then N is the sum of two squares if the prime factors of N are
2 and primes of the form 4n + 1. Conversely, it may be shown
that these are the only numbers that are the sum of two squares.
Fermat also gives a formula for the number of such representa-
tions. We shall return to these questions in Chap. 11.
Examples.
1. The two numbers
56 = 7 • 2 3 , 99 = 3 2 • 11
cannot be the sum of two squares since the prime factors 7 and 11 are not
of the form 4n + 1.
2. The number
1,105 = 5 • 13 • 17
can be represented as the sum of two squares. To find the representations we
observe that
5 = 2 2 + l 2 , 13 = 3 2 + 2 2 , 17 = 4 2 + l 2
By application of the identity (8-22) one obtains
5 • 13 = 8 2 + l 2 = 7 2 + 4 2
5 • 17 = 9 2 + 2 2 = 7 2 + 6 2
13 • 17 = 14 2 + 5 2 = ll 2 + 10 2
198 NUMBER THEORY AND ITS HISTORY
and by a repeated application one finds the four representations
1,105 = 33 2 + 4 2 = 32 2 + 9 2 = 31 2 + 12 2 = 24 2 + 23 2
One verifies that there are no others.
Problem.
Which of the numbers 101, 234, 365, 1,947 can be written as the sum of
two squares?
Not all numbers are the sum of two squares, as we just observed.
Some, but not all, of the others, can be written as the sum of
three squares. For instance, the prime 43 is not the sum of two
squares, but one has
43 = 5 2 + 3 2 + 3 2
Similarly, not all integers are the sum of three squares, for instance,
the prime 47 is not so representable, as one easily verifies, but it is
the sum of four squares, even in two ways
47 = 6 2 + 3 2 + l 2 + l 2 = 5 2 + 3 2 + 3 2 + 2 2
Bachet made the conjecture that every positive integer can be
written as the sum of at most four squares, and he verifies it for
all numbers up to 120. Fermat states in one of the Diophantos
notes that he has a proof for this theorem. In a letter to the
French mathematician Roberval he returns to the difficulties he
had to overcome to find a proof and explains that he had finally
succeeded through the use of his favorite method of infinite descent,
a procedure he also had used to derive the results regarding the
representation of numbers as the sum of two squares. He con-
tinues: "I confess openly that in the theory of numbers I have
found nothing which I have enjoyed more than the proof of this
theorem and I should be pleased if you would attempt to find it,
even if it were only to let me know whether I value my discovery
higher than it deserves."
There seems to be little reason to doubt that Fermat was in
possession of a proof according to the indications he has given.
That the problem was difficult can be judged from the fact that
even the resourceful Euler in vain pitted his ingenuity against it,
DlOPHANTINE PROBLEMS 199
and not until 1770 did the French mathematician J. L. Lagrange,
the successor of Euler at the Academy in Berlin and later a friend
of Napoleon, publish the first proof. As so often happens, the
completion of one problem gives birth to another.
In the same year Edward Waring (1734-1798), professor in
Cambridge and both scientifically and personally one of England's
most peculiar mathematicians, published his Meditationes
algebraicae. In this work one finds several announcements and
conjectures on the theory of numbers, among them the fact that
every number can be represented as the sum of a limited number
of cubes, fourth, or higher powers. This Waring's problem has
occupied the mathematicians intensely. That such representa-
tions exist was proved by the German mathematician D. Hilbert
in 1909. Essential information regarding the number of powers
that are required in each case has been given by various mathe-
maticians; among the most important results, one should mention
particularly those of the English mathematicians G. H. Hardy
and J. E. Littlewood, Vinogradoff (Russian), and L. E. Dickson
of the University of Chicago.
8-6. The method of infinite descent. Fermat's method of the
descente infinie is illustrated by his comments on Problem 26 in
Book VI in Diophantos's Arithmetics. These remarks are inter-
esting in several ways. He begins by stating: "The area of a
rational right triangle cannot be a square number. The proof of this
theorem I have reached only after elaborate and ardent study. I
reproduce the proof here, since this kind of demonstration will make
possible wonderful progress in number theory." Then follows a
fairly complex indication of the proof and it is remarkable that in
the long statement he uses no mathematical symbolism whatever,
giving all terms in longhand words. Towards the end he breaks off
with the statement: "The margin is insufficient to give all details
of the proof."
We shall give the proof in ordinary algebraic symbols, but we
first reduce the problem to integers by the following observations.
When the area of a rational triangle is a square number and each
side is multiplied or divided by a factor, the area is multiplied
200 NUMBER THEORY AND ITS HISTORY
or divided by the square of this factor and it remains a square
number. One can therefore clear the fractions in the rational
sides to make them integral, and if they now should have any
common divisor, it may be canceled. It follows that it is sufficient
to show that the area of an integral, primitive Pythagorean
triangle cannot be a square number.
The proof of this theorem is a bit long, but each step, as will be
seen, is quite simple. The sides of a primitive Pythagorean
triangle
a = 2mn, b = m 2 — n 2 , c = m 2 + n 2 (8-23)
we have already found. The area of the triangle is
A = §a& = mn(m 2 - n 2 ) (8-24)
Since this integer shall be a square, one must have
mn(m — ri)(m -\- n) = t 2 (8-25)
In a primitive triangle the numbers m and n are relatively prime,
one even and the other odd, and one concludes, therefore, that
among the four numbers
m, n, m — n, m + n
any two are relatively prime. Since their product is a square,
according to (8-25), each one of them is a square
m = u 2 , n = v 2 , u 2 — v 2 = p 2 , u 2 + v 2 = q 2 (8-26)
where all four numbers u, v, p, q, also must be relatively prime
in pairs.
By adding and subtracting the last two equations in (8-26),
one finds
2u 2 = p 2 + q 2 , 2v 2 = q 2 - p 2 = (q - p) (q + p) (8-27)
Since one of the numbers m and n is odd and the other even,
u and v must have the same property, so that according to (8-26)
p and q must both be odd. This shows further that q — p and
q + p are both even so that the second equality in (8-27) yields
that v is even
v = 2v\
DIOPHANTINE PROBLEMS 201
We put this into the second equation in (8-27) and find
2vi = — - — * — ~ —
2 2
(8-28)
q — p q -\- p
Here the two factors — - — and — — — on the right are relatively
z z
prime, because a common factor would divide their sum q and
their difference p, but p and q are relatively prime.
In (8-28) we have two alternatives, depending on which of
Q ~ V
the factors on the right is even. Let us suppose that — - — is
even. Then, besides the factor 2 it must have some factor in
it!
factor of wi 2 , so that we can write
common with v 2 , while — - — must be equal to the remaining
l^v = 2k2 <L±JP = l 2 (8 _ 29)
2 2
where
v x 2 = k 2 l 2 (8-30)
The other alternative is that — - — is even, and in this case one
Zi
obtains similarly
" "" n = I 2 , ^—^- = 2k 2 (8-31)
Q-P 12 ? + P_oi2
where (8-30) still holds.
From (8-29) one finds
p = l 2 - 2k 2 , q = l 2 + 2k 2
and, in the alternative case (8-31),
p = 2k 2 - I 2 , q = 2k 2 + I 2
When these values for p and q are substituted in the first equation
(8-27), one finds in both cases
w 2 = ( Z 2 )2 + Qk 2 ) 2 (8-32)
202 NUMBER THEORY AND ITS HISTORY
We have now completed the steps preparatory to the use of the
infinite-descent argument. Our starting point was the integral
Pythagorean triangle (8-23) with an area (8-24) which was
supposed to be an integral square number. From this we have,
according to (8-32), derived a new triangle of the same kind with
the sides
I 2 , 2k 2 , u
The area of the new triangle is found to be
A 1 = ^ -l 2 -2k 2 = l 2 k 2 = v, 2 = (^ = - 4
so that it is also an integral square number, which clearly is smaller
than the area of the original triangle. From this second triangle
one could derive a third, a fourth, and so on, with the same prop-
erties and steadily decreasing integral areas. This, however,
clearly involves a contradiction since the area is always an integer
St 1. Our initial assumption that there existed some Pythagorean
triangle with a square number for its area is therefore inaccept-
able, and the theorem is proved.
In general, the method of infinite descent may be stated in the
following form: it is assumed that a problem can be solved in
positive integers and one derives from this a new solution in
smaller numbers; since positive integers cannot be decreased
indefinitely, one arrives at a contradictory situation so that the
assumption that the problem had a solution is impossible.
There are various consequences of the result that the area of
a Pythagorean triangle cannot be square. Let us return for a
moment to the concept of a congruum, which was introduced
in connection with the problem of Leonardo. The expression
(8-20) for a congruum was
h = Amn{m 2 — n 2 )
and when one compares it with the expression (8-24) for the
area of a Pythagorean triangle, one sees that the congruum is four
times the area of the triangle defined by m and n. We can state
therefore: A congruum cannot be a square number. Leonardo
DIOPHANTINE PROBLEMS 203
was aware of this result in his Liber quadratorum, but he did not
possess any satisfactory proof for it.
Another consequence is: There are no fourth powers whose
difference is a square; i.e., the equation
x 4 _ ^4 = z 2 (8 _ 33)
has no solution in integers x, y, and z different from zero.
If one could find two such integers x and y fulfilling the equation
(8-33), the Pythagorean triangle defined by m = x 2 , n = y 2
would, according to (8-24), have the area
x 2 y 2 (x 4 — y 4 ) = x 2 y 2 z 2
which is a square.
8-7. Fermat's last theorem. We now come to the most famous
of Fermat's remarks in his copy of Diophantos. In Problem 8 in
Book II Diophantos propounds: To decompose a given square
number into the sum of two squares.
To use a general notation, let a 2 be the given square for which
one wants to find x and y such that
a 2 = x 2 + y 2 (8-34)
As usual, Diophantos asks for rational solutions. For a suitably
chosen number m, one can then write
y — mx — a
When this is substituted into (8-34), one can cancel a factor x
and find
2am
x =
+ 1
Here m may be any rational number. Diophantos must proceed
only by illustrating the method on an example. He chooses
a = 4 and takes a solution that corresponds to m = 2 in our
formula, giving
x — 5 > y 5
One verifies that
(¥) 2 + (W 2 = 4 2
204
NUMBER THEORY AND ITS HISTORY
This problem to us is quite straightforward, but it was not always
so. In the oldest preserved Diophantos manuscript, copied in
the thirteenth century, we find at this point the following heartfelt
remark by the writer: "Thy soul, Diophantos, to Satanas, for the
difficulty of thy problems and this one in particular."
Fermat's comments in connection with this problem are, as
one should expect, considerably more constructive and of much
Fig, 8-4. Pierre de Fermat (1608-1665).
greater consequence: "However, it is impossible to write a cube
as the sum of two cubes, a fourth power as the sum of two fourth
powers and in general any power beyond the second as the sum
of two similar powers. For this I have discovered a truly wonder-
ful proof, but the margin is too small to contain it."
This is the famous Fermat's theorem, sometimes called Fermat's
last theorem, on which the most prominent mathematicians have
tried their skill ever since its announcement three hundred years
ago. In algebraic language, it requires that it shall be shown that
the Diophantine equation
x n +y n = z n
(8-35)
DIOPHANTINE PROBLEMS 205
has no solution in integers x, y, and z, all different from zero,
when n ^ 3.
For one case of the theorem Fermat obviously had a proof.
It follows from our previous results that the equation
x ± + y* = S 4 (8 _ 36 )
cannot have any integral solutions different from zero. One can
write this equation in the form
2 4 _ y 4 = (a .2 )2
and since we have shown that the difference between two fourth
powers cannot be a square, (8-36) is also impossible in integers.
This result also goes a little further. If the exponent n in (8-35)
is divisible by 4, one can write n = 4ra, and Fermat's equation
takes the form
(x w ) 4 + {y m f = (z m ) 4
and this equation is impossible as we have just shown.
By a similar remark one can reduce the general case to the case
where the exponent in (8-35) is an odd prime. Let us suppose
that n = pm where p > 2 is a prime. Then Fermat's equation
may be written
(x m ) p + (y m ) p = (z m ) p
so that it is sufficient to prove the equation impossible for prime
exponents p > 2.
The question whether Fermat possessed a demonstration of his
last problem will in all likelihood forever remain an enigma.
Fermat undoubtedly had one of the most powerful minds ever
applied to investigate the laws of numbers, and from his indica-
tions there is every reason to believe that he was able to prove the
various other assertions that he included in the Diophantos notes.
The remark that the margin was too small may perhaps sound a bit
like an excuse, but it was an observation he had to make also in
other instances. On the other hand, he may have made a mistake,
as in another case, where the conjecture, which he repeated in
several letters, that all Fermat numbers were primes proved
incorrect. Mathematicians occasionally may argue the point; the
206 NUMBER THEORY AND ITS HISTORY
consensus seems to be that in view of the numerous investigations
of the problem for three centuries from every conceivable angle,
by first- and second-rate mathematicians, by amateurs and
dilettanti, it is very unlikely that there should exist a proof based
on any methods one can reasonably assume Fermat could have
mastered. Such methods would undoubtedly have great conse-
quences in other problems of number theory, but Fermat mentions
them nowhere. Like so many of the other mathematicians who
later worked on the problem, including Kummer whose results
were the most incisive of all, he may have fallen into one of the
many pitfalls of insufficient reasoning that have beset the investi-
gations on the problem.
Fermat's problem has remained remarkably active throughout
its history, and results and research on it still appear frequently
in the mathematical journals. It must be admitted frankly that
if the specific result implied in the theorem were obtained, it would
probably have little systematic significance for the general prog-
ress of mathematics. However, the theorem has been extremely
important as a goal and a constant source of new efforts. Some
of the new methods it has inspired have proved to be basic not
only for number theory but also for many other branches of
mathematics.
As we mentioned, Fermat gave a proof of his theorem when
n = 4. The case n = 3 he presented repeatedly as a challenge
problem to French and English mathematicians, and it seems
unlikely that he should propose a problem to which he could not
himself give an answer, if requested. The first proof for the cubic
case was published by Euler in a French translation of his Algebra.
The case n = 5 was proved independently about 1825 by the
German mathematician Lejeune-Dirichlet and the French
Legendre, and the case n = 7 in 1839 by Lame.
The most significant advance in the investigations of the prob-
lem was made by the German mathematician E. Kummer (1810-
1893). He extended the domain of number theory to include not
only the rational numbers but also the algebraic numbers, i.e.,
numbers satisfying algebraic equations with rational coefficients.
DIOPHANTINE PROBLEMS 207
In 1843 Kummer submitted to Lejeune-Dirichlet a manuscript
containing a purported proof of Fermat's theorem based on
algebraic numbers. Dirichlet, who had made similar attempts
himself, immediately picked out the error in the reasoning: in the
domain of algebraic numbers the fundamental theorem no longer
holds that every number is representable essentially in one way as
a product of prime factors. This failure caused Kummer to
attack the problem with redoubled vigor, and a few years later he
succeeded in finding a substitute for the theorem of the unique
factorization in the theory of ideals, a theory that later has gained
importance in almost all parts of mathematics.
By means of the ideals, Kummer was able to derive very general
conditions for the insolubility of Fermat's theorem. Practically
all important progress in this field in the last century has been
made along the lines suggested by the theory of Kummer. Numer-
ous criteria have been developed by means of which Fermat's
equation has been proved impossible for all exponents at least
up to n = 600.
A curious twist was added to the history of Fermat's problem in
1908 when the German mathematician P. Wolfskehl, who had
made a few contributions related to the subject, bequeathed
100,000 marks to the Academy of Science in Gottingen for a
prize to be awarded for the first complete proof of Fermat's last
theorem. The prize probably added little or nothing to the
interest of the mathematicians in the problem, but an immediate
consequence was a deluge of alleged proofs by laymen eager to
gain money and glory. This interest of the dilettanti in the
problem has since never quite ceased, and Fermat's problem has
without question the distinction of being the mathematical
problem for which the greatest number of incorrect proofs have
been published. (See Supplement.)
Bibliography
Bonconpagni, B.: Scritti di Leonardo Pisano, matematico del secolo decimo
terza, Rome, 1857-1862.
Carmichael, R. D.: Diophantine Analysis, John Wiley & Sons, Inc., New
York, 1915
208 NUMBER THEORY AND ITS HISTORY
Colebrook, H. T. : Algebra with Arithmetic and Mensuration from the Sanscrit
of Brahmegupta and Bhascara, London, 1817.
Fermat, P.: Oeuvres, publiees par les soins de M. M. Paul Tannery et Charles
Henry, 4 vols , Paris, 1891-1912.
Heath, T. L. : Diophantos of Alexandria; A Study in the History of Greek
Algebra, second edition, with a supplement containing an account of
Fermat's theorems and problems connected with Diophantine problems by
Euler, Cambridge University Press, London, 1910.
Mordell, L. J.: Three Lectures on Fermat's Last Theorem, Cambridge Uni-
versity Press, London, 1921.
Neugebauer, O. : Mathematische Keilschrifttexte, 3 vols., Verlag Julius Springer,
Berlin, 1935-1937.
and A. Sachs: Mathematical Cuneiform Texts, American Oriental
Series, Vol. 29, New Haven, 1945.
Thureau-Dangin, F. : Textes mathematiques babyloniens, Leiden, 1938.
Vandiver, H. S. : "Fermat's Last Theorem, Its History, and the Nature of the
Known Results Concerning It," American Mathematical Monthly, Vol. 53,
555-578 (1946).
Woepke, F. : Extrait du Fakhri, Paris, 1853.
CHAPTER 9
CONGRUENCES
9-1. The Bisquisitiones arithmeticae. Who were the greatest
mathematicians of all times? If one should put this question to a
gathering of mathematicians, there would of course be disagree-
ment, but a considerable number would undoubtedly state as their
choices: Archimedes, Newton, and Gauss. Among these, Gauss
is the only one whose work made an essential contribution to the
theory of numbers. One could go further and state that while
Fermat was the father of number theory as a systematic science,
Gauss inspired the modern phase of the subject. His most
important work on the properties of numbers is the Disquisitiones
arithmeticae, which appeared in 1801 when he was twenty-four
years of age.
Carl Friedrich Gauss (1777-1855) was the son of a bricklayer
who on the whole was quite opposed to the idea of an advanced
education for the boy. The young Gauss was, however, a preco-
cious child whose ability so overwhelmed his teachers that as a
fourteen-year-old boy he was presented to Carl Wilhelm Ferdinand,
the Duke of Brunswick. The duke financed his education and
granted him a small pension on which he lived until the tragic
death of the duke in 1806, on the flight from Napoleon's armies.
The next year Gauss was appointed director of the university
observatory in Gottingen. Here he lived until his death, secluded
and reserved, caring little for students and pupils, indifferent to
honors, but bringing forth from time to time some masterpiece <f|
mathematical creation. His contemporaries looked up to him
with awe and universally acclaimed him the princeps maihe*
209
210
NUMBER THEORY AND ITS HI STORY
maticorum. His contributions covered practically all fields of
mathematics, pure and applied, including mechanics, astromony,
physics, geodesy, and statistics.
The Disquisiliones arithemeticae has often been pronounced the
greatest among his many great works, both in results and in the
depth of its new ideas. Many problems, some of them previously
Fif;
Carl Fried rich Gauss (1777-1855).
attacked in vain by prominent mathematicians, here received their
solution for the first time. In the opening sections Gauss intro-
duces a new calculus, the theory of congruences, that almost immedi-
ately gained general acceptance and ever since has put its stamp
on all terminology in number theory. The subsequent chapters
will be applied to the discussion of various aspects of this theory.
In a devoted statement Gauss dedicates the Disquisitiones to
his patron, the Duke of Brunswick and Lunebourg, praising him
particularly because he had been willing to lend his support also
to those parts of science "which appear most abstract and with
less application to ordinary usefulness, because in the depth of
your wisdom, able to profit by all which tends to the happiness
CONGRUENCES 211
and prosperity of society, you have felt the intimate and necessary
liaison which unites all sciences." In the introduction Gauss
mentions earlier investigations in the theory of numbers, particu-
larly those by Euclid and Diophantos as well as those by Fermat,
Euler, Lagrange, and Legendre. He relates that he began his
research in the theory of numbers when he was eighteen years old
and that he had been so attracted to these questions that a con-
siderable part of the Disquisitiones had been completed before he
became familiar with the results of other mathematicians. But
"reading the works of these men of genius I was not late in recog-
nizing that I had employed the greater part of my meditations on
things known for a long time; but animated by a new ardor in
following their steps, I exerted myself to advance further the
cultivation of number theory." In the final presentation he
included many of his earlier and previously known results to give
a systematic view of the whole field.
9-2. The properties of congruences. Gauss introduces his
congruences through the following definition: Two integers a and
b shall be said to be congruent for the modulus m when their difference
a — b is divisible by the integer m. This he expresses in the
symbolic statement
a = b (mod m) (9-1)
When a and b are not congruent, they are called incongruent for
the modulus m and this is written
a ^ b (mod m)
These terms, as one sees, are derived from Latin, congruent mean-
ing agreeing or corresponding while modulus signifies little measure.
The latter term is often shortened to modul.
Let us illustrate the definitions by a few examples. One has
for instance
26 = 16 (mod 5)
since the difference 26 — 16 = 10 is divisible by 5; also
12 = 39 (mod 9)
212 NUMBER THEORY AND ITS HISTORY
since 12 - 39 = -27 is divisible by 9, while
3 ^ 11 (mod 7)
because 3 — 11= —8 is not divisible by 7. Gauss uses the
examples ^ ^ _ g (mod fi)
-7 = 15 (mod 11)
-7 ^ 15 (mod 3)
One can state the congruence (9-1) slightly differently by saying
that b is congruent to a when it differs from a by a multiple of m
b = a + km (9-2)
There are certain basic properties of congruences, which we shall
enumerate. The first is
1. Determination. For any pair of integers a and b one has one
or the other of the alternatives
a = b (mod m), a ^ b (mod m)
In other words, either the difference a — b is divisible by m or
it is not. The second property is equally trivial :
2. Reflexivity. One has
a = a (mod m)
This states only that a — a = is a multiple • m of any
number m.
3. Symmetry. When
a = b (mod m)
then one also has
b = a (mod m)
This is clear since when the difference a — b is divisible by m
so is 6 — a. The last property of this kind is
4. Transitivity. When
a = b (mod m), b = c (mod m)
then
a = c (mod m)
CONGRUENCES 213
To prove it we need only observe that
a — c = (a — 6) + (6 — c)
is divisible by m according to the first two congruences.
These four properties 1-4 show that the congruences for some
given modul define a relation between any two numbers of a type
that in mathematics is called an equivalence relation. The best-
known example of such a relation is the ordinary equality
a = b
It may be of interest to observe that the equality may itself be
considered to be a congruence, namely for the modulus 0, since
according to (9-2) the congruence
a = b (mod 0)
signifies that a = b. This artificial terminology is not in use.
There is, however, another relation that may be expressed
conveniently by means of congruences. As one sees immediately
from the definition of a congruence, the fact that a number a is
divisible by a number m may be stated
a = (mod m)
For example, one has
6 = (mod 2), 35 = (mod 5), 13 ^ (mod 7)
The even numbers n are characterized by
n = (mod 2)
Problems.
Verify the congruences
1. 40 ^ 13 (mod 9) 4. 11 = 23 (mod 12)
2. 7 ^ 99 (mod 13) 5. 132 s (mod 11)
3. 3 4 = 1 (mod 5) 6. 7 2 = 1 (mod 8)
9-3. Residue systems. When an integer a is divided by
another m, one has
a = km + r (9-3)
214 NUMBER THEORY AND ITS HISTORY
where the remainder r is some positive integer less than m. Thus
for any number a there exists a congruence
a = r (mod m)
where r is a unique one among the numbers
0, 1, 2, . . . , to - 1 (9-4)
For this reason the set (9-4) is called a complete residue system
(mod m). One has for instance
35 s 2 (mod 11), -11 = 5 (mod 8)
On the other hand, all numbers a that are congruent to a given
remainder r in (9-4) will be of the form (9-3), where k is an arbi-
trary integer. Since these are the numbers that correspond to
the same remainder r when divided by m, we say that they form a
residue class (mod m). There are m residue classes (mod m). For
a given remainder r the residue class to which it belongs consists
of the numbers
r, r ± m, r ± 2m, • • •
According to our definition the congruence
a = b (mod m)
signifies that the numbers a and b differ by a multiple of m;
consequently the congruence can also be expressed in the terms
that a and b belong to the same residue class (mod m).
Note : We have previously (Sec. 7-3) introduced the term modul
as a set of numbers closed with respect to addition and subtraction.
This is a somewhat different concept from the congruence modul
just defined; in the following we shall use the term only for con-
gruences so that no confusion can arise. The two concepts are,
however, closely related. We showed in theorem 7-6 that a
modul as a set of integers consisted of all multiples
0, ± to, ± 2m, • ■ •
CONGRUENCES 215
of an integer m, so that this set is the zero residue class (mod m),
i.e., the set of all numbers a for which the congruence
a = (mod m)
is fulfilled.
Examples.
1. For the modulus m = 2 there are two remainders, and 1, and the
corresponding residue classes are
•••, -4, -2, 0, 2, 4, ...
••-, -3, -1, 1, 3, 5, •••
consisting respectively of the even and odd numbers.
2. When m = 3 there are three residue classes
• • • , -6, -3, 0, 3, 6, •••
••-, -5, -2, 1, 4, 7, •••
■••,. -4, -1, 2, 5, 8, •••
3. Prove that all numbers in a residue class have the same g.c.d. with the
modulus m.
,_ There are many other residue systems such that every number
is congruent (mod m) to a single one among them. We may
( recall, for instance, that in the division process we sometimes
found it convenient to use least absolute remainders. In general
we shall say that m numbers
«i, «2, • • . , a m (9-5)
form a complete system of residues if every number is congruent to
some Ojf. One sees that to obtain such a system one must pick one
Of from each of the m residue classes. To examine whether the
numbers (9-5) form a complete residue system, one can verify
that they aie congruent to the numbers (9-4) in some order. For
jiST&ance, the numbers
32, -1, 8, 20, 11
216 NUMBER THEORY AND ITS HISTORY
form a complete system of residues (mod 5) since one has the
congruences
32 = 2, -1=4, 8 = 3,
20 = 0, 11 = 1 (mod 5)
Another way of determining that the m numbers (9-5) form a com-
plete system of residues would be to show that no two of them are
congruent
at 7^ aj (mod m)
since in this case they would all belong to different residue classes.
Problems.
1. Show that the numbers
-3, 14, 3, 12, 37, 50, -1
form a complete residue system (mod 7).
2. Do the numbers
5, 12, -3, -4, 9, 22
form a complete residue system (mod 6)?
9-4. Operations with congruences. We began by emphasizing
that some of the basic properties of congruences are the same as
those of ordinary equality. We shall now pursue this analogy
further and establish that one can operate with congruences
according to rules that in many ways resemble those used in
combining equations. A little later on we shall show that several
important applications of congruences depend on this fact.
This first property we mention is:
Theorem 9-1. Congruences for the same modul may be added
and subtracted. If
a = b; c = d (mod m) (9-6)
then
a + c = b + d; a — c = b — d (mod w)
To prove, for instance, the first one of these congruences, it is
sufficient to observe that the difference
a + c - (6 + d) = (a-b) + (c- d)
CONGRUENCES 217
is divisible by m according to the two given congruences (9-6).
As an example, we may take
5 = 32; 11 = -7 (mod 9)
By addition and subtraction one finds the new congruences
16 = 25; 6 = -39 (mod 9)
which are also seen to be correct.
By repeated application of the addition rule, it follows that one
can add an arbitrary set of congruences for the same modulus.
For instance, from the three congruences
47 = -5 (mod 13)
11 = 37 (mod 13)
1 = -25 (mod 13)
one obtains by addition of the numbers on both sides
59 = 7 (mod 13)
Another application of the addition theorem results in:
Theorem 9-2. A congruence may be multiplied by an arbi-
trary integer. From
a = b (mod m)
it follows that
ka = kb (mod m)
Clearly the new congruence has been obtained by adding the given
congruence to itself k times. For instance, from
3 = 7 (mod 4)
one concludes by multiplication with 5 that
15 = 35 (mod 4)
The next result is:
Theorem 9-3. Two congruences may be multiplied together.
From the congruences in (9-6) one obtains
ac = bd (mod m) (9-7)
218 NUMBER THEORY AND ITS HISTORY
This may be derived in various ways; one can, for instance,
multiply the first congruence (9-6) by c and the second by b so that
ac = be = bd (mod m)
One can also express the difference between the two sides in the
congruence (9-7) in the form
ac — bd = (a — b)c + b(c — d)
showing that it is divisible by m. To illustrate, the multiplication
of the two congruences
3 s 14, 9 = -2 (mod 11)
gives
27 = -28 (mod 11)
Again the multiplication rule may be applied to several con-
gruences. In particular, a congruence may be multiplied by itself
any number of times so that
a = b (mod m)
implies
a n = b n (mod m)
for any exponent n.
Since any of the operations of addition, subtraction, and multi-
plication when applied to congruent numbers will give congruent
results, we conclude that any algebraic expression constructed by
repeated use of these operations will give congruent results when
congruent values are substituted. For instance, since
-2 = 3 (mod 5)
the polynomial
Six) = x 3 - Sx + 6
must give congruent results when —2 and 3 are substituted.
One finds actually
/(-2) = 14 = 9 = /(3) (mod 5)
CONGRUENCES 219
The same would hold if one took two polynomials in which the
corresponding coefficients were congruent. For instance, the
polynomials
f(x) = x 3 - 8x + 6, g{x) = ±x 3 - 3x 2 - 2x - 3
have congruent coefficients (mod 3), namely,
1=4, = -3, -8 = -2, 6 = -3 (mod 3)
Thus the two values x = — 2 and x = 1, which are congruent
(mod 3), must give congruent values when substituted in/(x) and
g(x), respectively. One sees that
/(-2) = 14 = -4 = g(l) (mod 3)
Analogous results must hold if one takes expressions with several
variables.
These rules for the computation with congruences are, as we
have seen, quite simple and analogous to those for equations.
Nevertheless, the reader who makes his beginning steps with this
somewhat unfamiliar and strange calculus will need a little time
and several examples to gain the necessary confidence in the
method. After some experience it will become clear how much the
notion of congruences facilitates certain kinds of considerations
in number theory.
Examples.
1. Let us determine the smallest positive remainder (mod 17) of the number
37 when raised to the thirteenth power.
Problems of this kind are quite common in the theories we shall discuss in
the next chapter. Clearly one could compute the large number 37 13 and find
its remainder when divided by 17. However, by congruences we proceed in
much simpler fashion as follows. We observe first that
37 = 3 (mod 17)
By squaring this congruence, one finds
37 2 = 9 (mod 17)
and by repetition
37 4 a 81 = -4 (mod 17)
220 NUMBER THEORY AND ITS HISTORY
Squaring again, one finds
37 8 = 16 = -1 (mod 17)
By multiplying the congruences for the first, fourth, and eighth powers of 37
one obtains
37 13 = 37 • 37 4 • 37 8 = 3(-4)(-l) = 12 (mod 17)
so the remainder is 12.
2. Compute the remainder of the expression
A = 531xV \ x = 31, y = 2
for the modulus 7.
One finds for the modulus 7
x = 31 = 3, y z = 8 = 1
x 2 = 9 = 2, y 9 = 1
531 = -1, y 11 a 4 ■ 1 = 4
Ah -1-2-4= -8 a6 (mod 7)
/(x) = 3x 7 - 41a; 2 - 91a;
Consequently
3. Let
and find /(ll) (mod 13). !
One sees that
41 = 2, 91 = (mod 13)
so that for any x
f(x) = 3a; 7 - 2x 2 (mod 13)
Furthermore in this case
x = 11 = -2
a; 2 =4
x 4 = 16 = 3
x 1 = -2 • 4 • 3 =. 2
4 , „ o ( mod 13 )
so that
/(ll) ^3-2-2-4= -2 = 11 (mod 13)
So far we have indicated only those rules for congruences
corresponding to those that are familiar for equations. We shall
now supplement this by deriving a number of properties for con-
gruences that do not have an analogue among the properties of
equations.
Almost trivial is
Theorem 9-4. If
then one also has
CONGRUENCES 221
a = b (mod m)
a = 6 (mod d)
where d is any divisor of the modulus m.
Clearly, if a — b is divisible by ra, it is divisible by any divisor
d of m. For instance, one has
23 = - 1 (mod 12)
and therefore
23 = -1 (mod 4), 23 = -1 (mod 3)
Another fact that is often used in computations with congruences
is the following:
Theorem 9-5. When a congruence holds for two different
moduls, it holds for their least common multiple. If
a = b (mod mi), a = b (mod m^)
then
a = b (mod M), M — [mi, m 2 ]
Conversely, the last congruence implies each of the first two.
The proof is an immediate consequence of the fact that when the
difference a — b is divisible both by mi and m 2 , it is divisible by
their l.c.m. M. Clearly the rule extends to an arbitrary number
of moduls. In the example
37 = 109 (mod 8), 37 = 109 (mod 12)
it follows that
37 = 109 (mod 24)
The converse is a consequence of theorem 9-4.
Let us state separately a special application of theorem 9-5 that
appears commonly:
Theorem 9-6. When a set of congruences
a = b (mod m*) i = 1, 2, • • • , k (9-8)
222 NUMBER THEORY AND ITS HISTORY
holds where the moduls ra* are relatively prime in pairs, then one
also has the same congruence for the product of the moduls
a = b (mod miw 2 • • • m&) (9-9)
and conversely from (9-9) each of the congruences (9-8) follows.
We need only to recall that the l.c.m. of relatively prime numbers
is equal to their product.
Theorem 9-6 is often useful in reducing the study of congruences
to the case of moduls that are powers of primes. If the modul has
the prime factorization
m = pi ai p 2 ai • ' ' Vk ak
then the congruence
a = b (mod m) (9-10)
implies each one of the congruences
a = b (mod Pi tti ) (9-11)
and these, in turn, together imply (9-10). One has for instance
730 = 10 (mod 180)
and therefore also
730 = 10 (mod 2 2 ), (mod 3 2 ), (mod 5)
and conversely this system of congruences is equivalent to the
original.
The final rules we wish to establish refer to the division of a
congruence by a number. We have seen in theorem 9-2 that in a
congruence both sides may be multiplied by the same integer.
Now let us consider conversely when one can cancel a common
factor on both sides. This is not always possible as the following
example shows. In the congruence
36 ss 92 (mod 8)
the numbers on both sides are divisible by 4, but if this factor is
canceled, there remains
9 = 23 (mod 8)
which is incorrect
CONGRUENCES 223
Let as see how the cancellation rule must be modified. When a
congruence
ak = bk (mod m)
holds, it means that the difference ak — bk must be divisible by m
so that
(a - b)k = Im (9-12)
where I is some integer. We assume that k and m have the g.c.d.
d = (k, m) and divide (9-12) by it to obtain
(a - b) - = I -
a a
Here the two numbers k/d and m/d are relatively prime, and since
the product on the left is divisible by m/d, one concludes that a — b
must be divisible by m/d, in other words
We can therefore state:
Theorem 9-7. In a congruence
ak = bk (mod m)
the common factor k can be canceled
«- 6 ( mod f)
provided the modulus is divided by the greatest common divisor
d of k and w.
In the previous example
36 = 92 (mod 8)
cancellation by 4 gives, according to this rule,
9 = 23 (mod 2)
Similarly, in the congruence
220 = 1,180 (mod 96)
224 NUMBER THEORY AND ITS HISTORY
both sides have the common factor k = 20, and d = (20, 96) = 4.
Consequently, after cancellation with 20, there remains
11 s 59 (mod 24)
Again, the theorem 9-7 has two special cases that are so im-
portant we mention them separately :
Theorem 9-8. In a congruence
ak = bk (mod m)
the factor k may be canceled
a = b (mod m)
provided k is relatively prime to the modul m. For instance, in
the congruence
27 = 102 (mod 25)
one can cancel by 3
9 = 34 (mod 25)
since 3 is relatively prime to the modul.
Finally:
Theorem 9-9. If in a congruence
a = b (mod m)
the three numbers a, b, and m are divisible by a number d, then
i( mod f)
a
d~ d\
Problems.
1. Add, subtract, multiply, and square the two congruences
31 s-7, 3 = 22 (mod 19)
and check the results.
2. Compute the least positive residue of each of the numbers
(a) 2 U (mod 17) (b) ll 35 (mod 13)
(c) 2 21 (mod 11) (d) 3 100 (mod 5)
CONGPMENCES 225
3. Compute the residues of /(2) and/(13) (mod 12) when
fix) = 73x 9 - lllx 7 + 32x - 14
4. Find the residue (mod 19) of the expression
B = Slx 2 y + 17y 4 x 5 , x = 11, y = 24
5. Compute the remainders of the numbers
2! = 1-2, 3! = 3-2-1, 4! = 4 • 3 • 2 • 1, ...
and in general the remainder of n! for the modulus n + 1 up to n = 10 and
try to establish a general rule.
6. In the following congruences cancel the common factors on both sides :
(a) 284 = 1,224 (mod 48)
(6) 45 m 150 (mod 7)
(c) 168 = -48 (mod 72)
9-5. Casting out nines. Until now we have mainly compiled
rules for handling congruences, and the time has come to touch
upon some simple applications to illustrate their usefulness.
Towards the end of the first section of the Disquisitiones Gauss
points out how one can, by means of congruences, derive general
methods for checking numerical computations. Such checks are
of ancient origin and may have been obtained from India by the
Arabs together with the Hindu numerals. They occur in many
of the Arab reckoning manuals, for instance, in the influential
works of al-Khowarizmi and al-Karkhi, and so they came into gen-
eral use in Europe in the Middle Ages.
These checks were particularly useful at a time when familiarity
with arithmetic manipulations was not as widespread nor as
thorough as at present. Furthermore, in computations on the
abacus or casting on the lines, once the calculation was completed
there remained no permanent record whose details could be re-
checked. Nowadays these control methods, even the simplest
and the most common one, casting out nines, have largely gone
out of use and are no longer explained in the elementary texts in
arithmetic. Occasionally we check our computations by the
inverse operations, for instance, subtraction by adding the sub-
tracted number to the difference, or division by multiplying back
226 NUMBER THEORY AND ITS HISTORY
again, but in most cases the check is performed simply by going
over each individual step in the calculation again. However as
anyone who has spent some time at numerical computations will
realize, one is apt to succumb to the same pitfalls in repeating the
procedure the second or third time, or in mechanical computation
the machine may fail in the same manner as previously. Con-
sequently for any large-scale computation it is, if not absolutely
necessary, at least very desirable to have some independent check-
ing method for the results.
Since we perform our computations in the decadic system of
numbers, we shall limit our considerations to such systems, but,
as we will see, there is no difficulty in extending the results to
arbitrary base numbers other than 10. Let
N = (a n , a n _i, • • • , ai, a ) (9-13)
= a«10 n + a n _ilO n ~ 1 H 1- a 2 100 + a x \Q + a
denote a number written in the decadic system so that the digits
ai may have values from to 9. It is simple to find the remainder
of N when divided by divisors of the base number. For instance,
since 2 divides 10 and all powers of 10, it follows from (9-13) that
N = a (mod 2)
Consequently, N is divisible by 2 only when the last digit is
divisible by 2, hence when a has one of the values «o = 0, 2, 4, 6, 8.
Similarly, since 4 divides 100 and all higher powers of 10, one has
N = ttilO + a (mod 4)
so that N is divisible by 4 only when the number represented by
the two last digits is divisible by 4. For example, the number
N = 7,342 = 42 = 2 (mod 4)
is not divisible by 4. Equally simple and familiar are the rules for
divisibility by 5 or 25. One sees that
N = a (mod 5)
CONGRUENCES 227
so that N is divisible by 5 only when a = or 5; one also has
AT s£ ailO + a (mod 25)
so a number is divisible by 25 only when it ends in 00, 25, 50, or 75.
More interesting are the rules one can derive for the remainders
and divisibility by other numbers that are relatively prime to 10.
We begin by considering the number N in (9-13) for the modul
m = 9. Since one has
10 = 1 (mod 9)
it follows that
10 2 = 1, 10 3 = 1, • • • (mod 9)
so that we find from (9-13)
N = a + a x -\ (- a n (mod 9) (9-14)
This congruence expresses the basis for the process of casting out
nines. It shows that by division with nine a number has the same
remainder as the sum of the digits. One may notice that on an
abacus or by computations on the lines this sum of the digits is a
number that appears naturally since it is the number of counters
or jetons that one uses to represent the number.
When the rule (9-14) is applied to find the remainder of a
number with respect to the divisor 9, the sum of the digits may
itself be a fairly large number, which one can reduce further by
repeated application of the same rule. For instance,
N = 39,827,437 = 3 + 9 + 8 + 2 + 7 + 4 + 3 + 7
= 43 = 4 + 3 = 7 (mod 9)
One concludes immediately from the congruence (9-14): A
number is divisible by 9 only if the sum of its digits is divisible by 9.
Example.
The number
N = 234,648 s2+3+4 + 6+4 + 8 = 27=0 (mod 9)
is divisible by 9.
228 NUMBER THEORY AND ITS HISTORY
Since the congruence (9-14) also will hold for the divisor 3 of 9,
exactly the same rules as for 9 apply for the remainders and
divisibility by 3. For instance, a number is divisible by 3 only
when the sum of the digits is divisible by 3.
Example.
N = 874,326 ^8 + 7 + 4 + 3 + 2 + 6 = 30 =3 (mod 9)
is a number divisible by 3 but not by 9.
Another number for which simple divisibility rules can be
established is m = 11. In this case one verifies that
10= -1, 10 2 = 1, 10 3 = -1,
10 4 = 1, • • • (mod 11)
and one concludes from (9-13)
N = a — «i + a 2 — a s + • • • (mod 11) (9-15)
Example.
N = 39,827,437 =7-3+4-7 + 2-8 + 9-3 = 1 (mod 11)
so that this number is not divisible by 11.
Rules for remainders and divisibility by other numbers have
been derived but they are less simple than those we have already
obtained. Leonardo Fibonnaci in his Liber abaci, in addition to
the rules for 9 and 11, also gives a rule for the number 7. As we
have seen, these rules depend essentially on the behavior of the
powers of 10 for the chosen modulus. When m = 7, one obtains
successively
10 = 3, 10 2 = 2, 10 3 =-1, 10 4 =-3,
10 5 = -2, 10 6 = 1 (mod 7)
Consequently one has
N = a + 3ai + 2a 2 — a 3 — 3a 4
- 2a 5 + a 6 • • • (mod 7) (9-16)
CONGRUENCES 229
Example.
N = 39,827,437 = +7 + 3-3 + 2-4-7
-3-2-2-8 + 9+3-3 = 13 =6 (mod 7)
shows that this number is not divisible by 7.
We shall now turn to the application of these residue rules to
give checks for the correctness of arithmetic operations. These
methods are based on the idea that when an operation of addition,
subtraction, or multiplication has been performed on certain
integers, the result must be correct also when considered as a
congruence for an arbitrary modulus. For instance, let
c = ab (9-17)
be a product obtained by the multiplication of two numbers a and
b. Then the congruence
c = ab (mod m) (9-18)
must hold for any modulus m. By selecting m as a number for
which the residues may easily be computed by means of the pre-
ceding rules, the congruence (9-18) may be verified without much
effort. If it should fail to hold, the multiplication (9-17) is not
correct. On the other hand, if the congruence is fulfilled, the
result (9-17) is not necessarily correct, but the chance of an error
is considerably reduced.
When casting out nines, one uses the modulus m = 9.
Examples.
1. Let us take the multiplication (9-17) when
a = 8,297, b = 3,583, c = 29,728,151 (9-19)
Here one finds
a = 8 + 2 + 9 + 7 = 26 e-1
&=3+5+8 + 3 = 19sl (mod 9)
cs2 + 9+7 + 2 + 8 + l+5 + l=35= -1
Consequently
ab = —1, c = — 1 (mod 9)
230 NUMBER THEORY AND ITS HISTORY
as one should expect. We shall also check the multiplication (9-19) for the
modulus 11. Then by (9-15)
o=7-9+2-8s3
6 = 3-8 + 5-3 = -3 (mod 11)
csl-5+l-8+2-7+9-2=2
and therefore
a& = 3(-3) = 2, c = 2 (mod 11)
2. Let us assume that in (9-17) one has
a = 7,342, & = 2,591, c = 19,032,122 (9-20)
By casting out nines, one finds
a = 7, 6=8, c = 2 (mod 9)
which checks, since
ab = 2 (mod 9)
But when one uses the modul 11, one finds from (9-20)
a = 5, 6=6, c = 10 (mod 11)
and this indicates that there must be an error in the multiplication since
ab = 30 = 8 ^ c (mod 11)
By performing the multiplication a second time one finds that the correct value
should have been
c* = 19,023,122
This illustrates the fact that casting out nines will not catch the rather com-
mon error of two digits having been interchanged.
These checking methods may be used analogously for addition
and subtraction but they are of lesser importance since these
operations may be so easily repeated. On the other hand, for
division the checks are quite convenient. When an integer a is
divided by 6 with the incomplete quotient q and the remainder r,
the relation
a = qb + r
must hold for every modulus.
CONGRUENCES
Example.
Let
a =
76,638,123,
b = 37,
By performing the division
one finds
1
= 2,041,
r = 4,696
231
Casting out nines gives
a = 0, 6 = -1, q = -2, r = -2 (mod 9)
and this is correct since
? fe + r = (_2)(-l) -2=0 (mod 9)
similarly
a = 1, 6=4, ? = -5, r = -1 (mod 11)
which again checks, since
qb + r = 4(-5) -1 = 1 (mod 11)
With large figures it is more efficient to take larger moduls for
check purposes. One may, for instance, use m = 99 since in this
case
10 2 =1, 10 4 = 1, • • • (mod 99)
and from (9-13) one obtains
N = a + 10a! + a 2 + 10a 3 -\ (mod 99)
This means that one finds the remainder (mod 99) by splitting JV
up into two digit numbers and taking their sum. For instance,
when
N = 7,342,948
one has
N = 48 + 29 + 34 + 7 = 19 (mod 99)
Similarly one finds for the modul m = 101
N = a + 10ai - (a 2 + 10a 3 ) + (a 4 + 10a 5 ) - • • • (mod 101)
hence in the example we just used
TV = 48 - 29 + 34 - 7 = 46 (mod 101)
232 NUMBER THEORY AND ITS HISTORY
Let us check the multiplication
728,223 X 5,535,064 = 4,030,760,911,272
by these two moduls. For the three numbers one finds in the
given order (mod 99)
23
64
72
+82
+50
+ 12
+72
+53
+91
177 = 78
+ 5
+60
= -21
172 =
73
+07
-26
+03
+ 4
249 = 51
The fact that
(-21) (-26) = 51 (mod 99)
points to a correct result. Had one used the modulus 101, the
check would have looked as follows :
23
64
72
-82
-50
-12
+72
+53
+91
13
- 5
-60
62
+ 7
- 3
+ 4
99 =
here one has
13
62 = -2
(mod 101)
-2
In number theory one often runs into computations with very
large numbers, exceeding even the capacity of the machines, so
that multiplications and divisions may have to be performed in
installments. Furthermore, there may be chains of computations
where the result of one step enters into the next. This, for
instance, is the case in some of the methods for deciding whether
CONGRUENCES 233
a number is a prime or not. Under such circumstances it is
particularly important to have efficient checks, and one introduces
moduls for even higher numbers than those just mentioned. It is
convenient to take m = 999, m = 9,999, . . . since this leads to a
simple addition of the digits of the numbers in groups of three,
four, and so on, or one may also take m = 1,001, . . . and add and
subtract such groups of digits alternatingly. By means of adding
machines these checks may be performed with relatively small
effort in comparison with the work involved in a complete repetition
of the operation.
Problems.
1. In Arab and medieval European arithmetics one finds checks for to = 7,
9, 11 and also for m = 13 and m = 19. Determine the form of the residue
rules for the two last moduls.
2. Why does to = 17 not give a simple rule?
3. Try to give a criterion for the divisibility of a number by 37.
4. Check the following multiplications and divisions by several of the rules
established above:
(a) 14,745 X 19,742 = 291,095,790
(6) 52,447 X 81,484 = 4,279,531,348
(c) 24,726,928,309 = 3,569,644 X 6,927 + 4,321
(d) 41,587 2 = 1,729,478,569
5. If a is a number in the decadic system and b the number with the same
digits in the reverse order, prove that a — b is divisible by 9.
6. What rule corresponds to the simultaneous use of the moduls 7, 11, and
13?
Bibliography
Gauss, C. F. : Disquisiliones arithmeticae, Leipzig, 1801. French translation by
A. C. M. Poullet-Delisle: Recherches arithmetiques, Paris, 1807.
CHAPTER 10
ANALYSIS OF CONGRUENCES
10-1. Algebraic congruences. The congruences, as we have
already indicated several times, have many properties in common
with equations and this analogy we shall now pursue further. In
equation theory one tackles the problem of finding the roots of an
algebraic equation
fix) =
i.e., the numbers x that satisfy this condition, where fix) is some
given polynomial. Similarly, in the theory of congruences one
can propose the problem of finding those integers x that fulfill a
certain congruence
fix) = (mod m) (10-1)
for some modul. Since we deal only with integers, we must in this
case suppose that f(x) is a polynomial with integral coefficients.
As an example, let us take the congruence
f(x) = x 3 4- 5x - 4 s (mod 7) (10-2)
It is satisfied when x = 2 since
/(2) = 14 = (mod 7)
As for equations, we say that x = 2 is a root or solution of the
congruence. But since congruent values of x will give congruent
values of the polynomial, as we mentioned earlier, any value of x
congruent to 2 (mod 7) must also be a solution. In the theory of
congruences it is therefore agreed to consider all values
x = 2 (mod 7)
234
ANALYSIS OF CONGRUENCES 235
as one solution. In the general case (10-1) the situation is the
same. When a solution x — xq has been found, all values x for
which
x = x (mod m)
are also solutions and by convention we consider them as a single
solution.
As a consequence, to find all solutions of a congruence (10-1)
we need only try all values 0, 1, . . . , m — 1 [or the numbers in any
other complete residue system (mod m)] and determine which of
them satisfy the congruence; this gives us the total number of
different solutions. In the example (10-2) we should try the
numbers from to 6 or, more conveniently, the numbers from —3
to +3. One finds that there is only the single solution x = 2.
The number of solutions of a congruence may vary considerably;
there may be none or the number may even greatly exceed the
degree of the congruence.
Examples.
1. The congruence
x 2 + 5 = (mod 11)
has no solutions as one establishes by trying the eleven values 0, ±1, ±2, ±3,
±4, ±5.
2. The congruence
has the two solutions
x 3 - 2x + 6 = (mod 5)
x = 1, x = 2 (mod 5)
x 3 s (mod 27)
3. The congruence
has nine solutions
x = 0, x s ±3, x = ±6, x = ±9, x = ±12 (mod 27)
The theory of algebraic congruences is an interesting but quite
complicated and difficult field, in which many investigations have
been made during the last century. A few essential facts about
special congruences will be derived subsequently since they enter
into some of the applications of the theory of congruences that
236 NUMBER THEORY AND ITS HISTORY
we wish to make. For the moment we shall be content to have
introduced the basic concepts.
Problems.
Find all solutions of the following algebraic congruences:
1. x 2 = 5 (mod 11) 5. x 3 - 3x 2 - 3 =* (mod 13)
2. x 2 = 4 (mod 15) 6. x 4 - 2x + 5 = (mod 7)
3. x 2 = 1 (mod 32) 7. x 3 - 3x 2 + 7x + 2 = (mod 12)
4. x 3 = (mod 25) 8. x 10 = 1 (mod 11)
10-2. Linear congruences. The simplest congruences are those
of first degree, or linear congruences.
ax = b (mod m) (10-3)
Before we proceed to the general method for solving such con-
gruences we shall give a few examples to illustrate that there are
various possibilities which may occur.
Examples.
1. The congruence
7x s 3 (mod 12)
has the single solution
x = 9 (mod 12)
as one concludes by trying out the integers from to 11.
2. The congruence
12x = 2 (mod 8)
is found to have no solution.
3. Finally the congruence
6x = 9 (mod 15)
has three solutions
x = 4, x = 9, x s 14 (mod 15)
We now return to the general linear congruence (10-3). Accord-
ing to the definition of congruences, this equation means that there
shall exist some integer y such that
ax — b = my
or
ax — my = b (10-4)
ANALYSIS OF CONGRUENCES 237
This shows that the solution of a congruence (10-3) is equivalent
to the solution of a linear indeterminate equation (10-4), and
since we have already analyzed such equations quite exhaustively,
the results may be transferred directly to congruences.
We recall first that an indeterminate equation (10-4) has a
solution only if the greatest common divisor of the coefficients of x
and y also divides the constant term b. Therefore we can state:
Theorem 10-1. A linear congruence (10-3) is solvable only
when the greatest common divisor
d = (a, m)
divides b.
In the first example given above
d = (7, 12) = 1
divides b = 3 so that the congruence is solvable. Similarly in the
third example
d = (6, 15) = 3
divides b = 9 so that there are solutions. But in the second
example
d = (12, 8) = 4
does not divide b = 2 so that no solution can exist, as we found
directly.
Let us consider the general indeterminate equation (10-4) and
suppose that d divides b so that it is solvable. We can cancel d
in each term and obtain
d m b
- x - - y = - (10-5)
add
This equation, as one sees, corresponds to the linear congruence
In (10-5) the coefficients of x and y are now relatively prime, and
we can solve the equation by means of our previous methods. We
238 NUMBER THEORY AND ITS HISTORY
recall that if x and yo is an arbitrary solution of (10-5), the
general solution has the form
m , . m .
x = x ° ~d ' V = y ° 1 '
where t is an arbitrary integer. This gives us as the general
solution of the congruences (10-6) and (10-3)
(7Yl\
mod — I (10-7)
There is one further remark that must be made. In connection
with congruences we agreed that the different solutions were the
numbers satisfying the congruence and not congruent to each other
(mod m). The numbers (10-7) are not all congruent (mod m).
If we select x , as we may, to be a positive integer less than m/d,
all the numbers
xo, *o + ^, aJo + 2-,---, xo + (d-l)- (10-8)
satisfy the congruence and are incongruent (mod m), since they
are less than m. Then the d numbers (10-8) define different
solutions of the original congruence (10-3). To summarize:
Theorem 10-2. A congruence
ax = b (mod m)
is solvable only if the greatest common divisor d = (a, m) divides
b, and when this is the case there are d solutions given by (10-8).
When a and m are relatively prime, the congruence has a single
solution.
In the first example above we had d = 1 so that there was one
solution. In the third example we had d = 3, and when this
factor was canceled the congruence became
2x = 3 (mod 5)
with the general solution
x = 4 (mod 5)
ANALYSIS OF CONGRUENCES 239
Corresponding to (10-8) it follows that since the modul in the
original congruence was 15, one has altogether three solutions
x = 4, x = 9, x = 14 (mod 15)
Examples.
1. The congruence
has no solution since
does not divide b = 8.
2. The congruence
36x s 8 (mod 102)
d = (36, 102) = 6
19a; = 1 (mod 140)
has a single solution since d = 1. To obtain it we solve the equation
19x - 140y = 1
by means of our previous procedure based upon Euclid's algorism
140 = 19 • 7 + 7
59
19 = 7-2 + 5
8
7 = 5-1+2
3
5 = 2-2 + 1
2
1
The solution is therefore
i=59 (mod 140)
3. Finally in the example
144a; = 216 (mod
360)
one has d = 72, and this number divides b = 216 so that there are 72 different
solutions (mod 360). When the factor d is canceled in the congruence, there
remains
2x =Z (mod 5)
which has the solution
x = 4 (mod 5)
The 72 solutions of the original congruence are as in (10-8)
x = 4, x = 9, x = 14, • • • , x = 359 (mod 360)
240 NUMBER THEORY AND ITS HISTORY
Problems.
Solve the congruences
1. 7x s 3 (mod 10) 4. 20x = 7 (mod 15)
2. 15x a 9 (mod 12) 5. 315x = 11 (mod 501)
3. 221x = 111 (mod 360) 6. 360x = 3,072 (mod 96)
10-3. Simultaneous congruences and the Chinese remainder
theorem. It is often required to find a number that has pre-
scribed residues for two or several moduls. As an example, let us
suppose that we wish to determine an integer x such that
x = 5 (mod 11), x = 3 (mod 23) (10-9)
The first condition (10-9) states that
x = 5+ lit
where t is some integer. In order that x shall satisfy the second
congruence in (10-9), one must have
5 + lit = 3 (mod 23)
or
llt= -2 (mod 23)
The solution of this congruence is obtained most simply by multi-
plying both sides by 2 so that
22* s -t= -4 (mod 23)
or
t s 4 (mod 23)
The general form for t is therefore
t = 4 + 23w
where u is some integer. When this is substituted into the ex-
pression for x, one obtains
x = 5 + 11(4 + 23w) = 49 + 11 • 23 • u
po that the general solution of the two congruences (10-9) is
x = 49 (mod 11 • 23)
ANALYSIS OF CONGRUENCES 241
The method used in this example is applicable in the general
case of two congruences
x = a (mod m), x = b (mod n) (10-10)
From the first of these, follows
x = a + mt
and the second shows that t must satisfy the condition
a + mt = b (mod n)
or
mt = b — a (mod n) (10-11)
According to the general rules we just derived, this linear con-
gruence in t can only have a solution when the greatest common
divisor d — {m, n) divides b — a; in other words, the condition
a = b (mod d)
must be fulfilled. When this is the case the congruence (10-11)
may be divided by d
*'- — ( mod s) (10 " 12)
Let t Q be some particular solution of this congruence and
xq = a + mto
the resulting special solution of (10-10). The general solution of
(10-12) is then
t = tt
so that we can write
*( mod s)
n
t = t + u -
d
242 NUMBER THEORY AND ITS HISTORY
where u is some integer. The resulting general solution of the
original congruence (10-10) is
or
since
/ , . n \ mn
x = xq (mod [m, ri\)
r .. mn
[m, n] = -—
a
is the least common multiple of m and n.
When one considers a set of algebraic congruences for several
moduls and x is a number satisfying all of them, it is clear that if
one adds any multiple of the l.c.m. of all moduls to x , the resulting
number will also be a solution. Therefore, with several moduls
it is agreed that the number of different solutions is given by the
incongruent solutions for the l.c.m. of the moduls.
We summarize as follows:
Theorem 10-3. Two simultaneous congruences
x = a (mod m), x = b (mod n)
are solvable only when
a = b (mod (w, n))
and then there is a single solution
x = xq (mod [m, n])
which may be found by the method given above.
Examples.
1. When
x = 7 (mod 42), x = 15 (mod 51)
there is no solution since
d = (42, 51) = 3
and
7 ^ 15 (mod 3)
ANALYSIS OF CONGRUENCES 243
2. In the example
x = 3 (mod 14), x = 7 (mod 16)
one has d = 2, and the condition for solvability is fulfilled; the answer is
x s 87 (mod 112)
When several simultaneous congruences are given
x = a\ (mod mi), x = a 2 (mod ra 2 ), a; = a 3 (mod ra 3 )
(10-13)
the solution may be found by repeated applications of the method
given above. One combines the first congruences and finds a
single congruence
x = xq (mod [mi, m 2 ])
which can replace them in (10-13). This in turn is solved in con-
junction with the third, and so on. One sees that if there exists a
solution of the congruences (10-13), there is only a single one,
with respect to a modul that is the l.c.m. of all moduls m;.
Example.
The following example is taken from the Disquisitiones:
x = Y7 (mod 504), x = -4 (mod 35), x = 33 (mod 16)
When the two first congruences are solved, one finds
x = 521 (mod 2,520)
and when this is combined with the third, the result is
x = 3,041 (mod 5,040)
Many puzzle questions belong mathematically to the type of
problems solved by simultaneous congruences. In Chap. 6 we
mentioned the ancient problem of the woman with a basket of
eggs. When the eggs were taken out two at a time, there was one
left; similarly, when they were taken out 3, 4, 5, and 6 at a time,
there was always one egg left, while at seven at a time, the count
came out even. In mathematical terms this means that
x = 1 (mod 2, 3, 4, 5, 6)
x = (mod 7)
244 NUMBER THEORY AND ITS HISTORY
where x is the number of eggs. Since, according to the first con-
gruences, the number x — 1 is divisible by all numbers 2, 3, 4, 5,
and 6, it is divisible by their l.c.m., which is 60, and the conditions
become
x = l (mod 60), x = (mod 7)
When these simultaneous congruences are solved, one finds
x = 301 (mod 420)
The smallest number of eggs the basket could have contained is
therefore x — 301.
The simple condition for the solvability of two simultaneous
congruences given in theorem 10-3 can be extended to an arbitrary
number of congruences as follows:
Theorem 10-4. The necessary and sufficient condition for a
set of simultaneous congruences
x = di (mod mi) i _= 1, 2, • • ♦ ,"r (10-14)
to have a solution is that for any pair
di = aj (mod (m iy ray)) (10-15)
and in this case, there is a single solution for the modulus
M r = [mi, • - • , m r ]
which is the l.c.m. of the given ones.
We observe first that if the congruences (10-14) are to have a
solution, any pair of them must be solvable so that according to
theorem 10-3 it is necessary that the conditions (10-15) be ful-
filled. To prove that these conditions are sufficient for the
existence of a solution, we shall use the induction procedure.
Theorem 10-3 states that the result is true for two congruences.
We suppose, therefore, that the result is true when there are
r — 1 congruences and from this we deduce it for r congruences.
According to this assumption, there exists a solution
xq = at (mod mi) i = 1, 2, • • • , r — 1 (10-16)
ANALYSIS OF CONGRUENCES 245
of the r — 1 first congruences in (10-14), and any other such solu-
tion x must be of the form
x = x (mod Mr-i), M r _ x = [mi, • • • , m r _ x \ (10-17)
To have a solution of all congruences (10-14), one must at the
same time satisfy
x = a r (mod m r ) (10-18)
From theorem 10-3 we conclude again that the congruences
(10-17) and (10-18) can have a common solution only when
x = a r (mod (M r -i, m r )) (10-19)
and in this case there is a single solution for the modulus
[M r _ u m r ] = M r
It remains, therefore, to show that the condition (10-19) is
fulfilled for the x we have found. We observe that according to
theorem 5-9
(ilf r _i, m r ) = ([mi, • • • , m r _i], m r ) = [(mi, m r ), • • • , (m r _i, m r )\
so that the congruence (10-19) is equivalent to the set of con-
gruences (theorem 9-5)
x = a r (mod (m;, m r )) i = 1, 2, . . . , r — 1
But these congruences are true, since one finds from (10-16) and
(10-15)
x = a,i (mod (mi, m r )), a» = a r (mod (m;, m r ))
The special case where all moduls in the simultaneous con-
gruences (10-14) are relatively prime in pairs occurs in many
applications. According to theorem 10-4, there is a unique solu-
tion to these congruences for a modul that is equal to the product
of all the given ones. Gauss introduces a special procedure already
used previously by Euler for the determination of the solution.
The method, however, is ancient and occurs in the works of several
early mathematicians. The first known source is the Arithmetic
of the Chinese writer Sun-Tse, probably around the beginning of
246 NUMBER THEORY AND ITS HISTORY
our era, and the resulting formula is often termed the Chinese
remainder theorem.
We begin by forming the product
M = mim 2 - - • m r
of the relatively prime moduls in the set of congruences (10-14).
When M is divided by mi, the quotient
M
— = m 2 ' ' ' m r
mi
is a number divisible by all other moduls and relatively prime to
mi. Similarly the number
M
— = Wi • • • m^imj+i • • • m r
mi
is divisible by all moduls except ra*, to which it is relatively prime.
For each i one can, consequently, solve the linear congruence
M
hi — = 1 (mod mi) 610-2G)
m,%
The Chinese remainder theorem may then be stated:
Theorem 10-5. Let a set of simultaneous congruences (10-14)
be given for which the moduls m% are relatively prime. For each
r . one determines 6; through the linear congruence (10-20). The
solution of the set of congruences is then
M M M
x = ci-bi f- a 2 & 2 1 r- a r h — (mod M) (10-21)
The verification of the solution (10-21) is immediate. For
instance, to see that it satisfies the first congruence (10-14) we
ecail that m\ divides all M/mi except M/m\ so that
X s aibi — ss oi (mod mi)
The other congruences (10-14) fellow by similar arguments.
ANALYSIS OF CONGRUENCES 247
The example used by Sun-Tse corresponds to the three con-
gruences
x s 2 (mod 3), x = 3 (mod 5), x = 2 (mod 7)
Here ilf = 105 and
^ = 35 , ^ = 21, *-16
mi m2 W3
The set of linear congruences
35&i s 1 (mod 3), 216 2 = 1 (mod 5), 156 3 = 1 (mod 7)
has the solutions
61 =2, 62 = 1, h = 1
so that one finds, according to (10-21),
x = 2 • 2 • 35 + 3 • 1 • 21 + 2 • 1 • 15 = 233 (mod 105)
or
x = 23 (mod 105)
Congruences represent a very convenient tool in many calendar
questions, such as determination of Easter dates, the day of the
week of a particular date, and so on. Gauss illustrates the Chinese
remainder theorem on a problem to find the years that have a
certain period number with respect to the solar and lunar cycle
and the Roman indiction. Similar problems with respect to the
planetary cycles occur earlier by Brahmagupta.
In the formula (10-21) for the solution of congruences for rela-
tively prime moduls, the multipliers
h M
ra t -
depend only on the numbers m{. If, therefore, one has to solve
several sets of congruences, all with the same moduls, the ex-
pression (10-21) is particularly convenient since one need not
recalculate the multipliers for each set.
As an example, let us take a play problem Leonardo discusses
in the Liber abaci. A person is requested to think of some number.
248 NUMBER THEORY AND ITS HISTORY
Then he is asked what the remainders of the number are when it
is divided by 5, 7, and 9, and on the basis of this information the
number is divined.
Let us denote the unknown number by x and the three re-
mainders by a\, a 2 , and a 3 so that
x = oi (mod 5), x = a 2 (mod 7), x = a 3 (mod 9)
The moduls are relatively prime, and one has
M = 5-7-9 = 315
M „ M Ar M oc
— = 63, — = 45, — = 35
mi ra 2 m 3
The linear congruences
63&! = 1 (mod 5), 456 2 = 1 (mod 7), 356 3 = 1 (mod 9)
have the solutions
b x =2, 6 2 = 5, b 3 = 8
so that the Chinese remainder formula (10-21) yields
x = 126a! + 225a 2 + 280a 3 (mod 315)
From this expression one obtains x, according to the remainders
a>\, a<2, «3 indicated. Only when the number is required to be less
than 315 is there a unique solution.
We conclude these investigations with a remark that will be
applied later. Let us suppose that in solving a problem for a
modul mi the number x to be determined has Si admissible values
x = ai, a 2 , • • • , a Sl (mod Wi)
Similarly for the modul m 2 there are s 2 values
x = 61, 6 2 , • • • , b S2 (mod m 2 )
When wi and m 2 are relatively prime, each value a* (mod mi) may
be combined with an arbitrary value bj (mod m 2 ) so that there
exists a total of S1S2 admissible values (mod mim 2 ). In general,
one sees that if there are r moduls mi, m2, . . . , m r , all relatively
ANALYSIS OF CONGRUENCES 249
prime, and si,S2, . . . , s r possible values for x for each modul, there
will be S\S2 ' • ' s r possible values for x for the product modul
wiiw 2 • • • m T .
Problems.
1. The basket of eggs problem is often given in the form: When the eggs
are taken out 2, 3, 4, 5, 6 at a time, there remain respectively 1, 2, 3, 4, 5 eggs,
while the number comes out even when they are taken out seven at a time.
Find the smallest number of eggs there could have been in the basket. Brah-
magupta discusses such a problem and makes the comment that it is a popular
one.
2. Ancient Chinese problem. Three farmers divide equally the rice they
have raised in common. They went to different markets where various basic
weights were used, at one place 83 pounds, at another 110 pounds, and at the
third 135 pounds. Each sold as much as he could in full measures, and when
they came home one had 32 pounds left, another 70 pounds, and the third 30
pounds. How much rice had they raised together?
3. Ancient Chinese problem. Four labor gangs take over the construction
of a dam, each contracting to take the same total number of workdays. The
first gang consists of two men, the second has three, the third six, and the
fourth twelve men. They complete their work as far as possible in full work
by each gang, and then there remains one workday for one man for the first
gang, two for the second, and five for the third and fourth. How many work-
days did the whole project involve?
4. Regiomontanus. Find a number such that
x = 3 (mod 10), x = 11 (mod 13), x = 15 (mod 17)
5. Euler. Find a number such that
x = 3 (mod 11), x s 5 (mod 19), x = 10 (mod 29)
10-4. Further study of algebraic congruences. The methods
we have just developed for simultaneous congruences may be
applied to find the solutions of several algebraic congruences. If,
for instance, one has two congruences
f(x) = (mod m), g(x) = (mod n)
and one wishes to find those x's which satisfy both at the same
time, each congruence may be solved separately, and the two sets
of roots for the moduls m and n may be combined as simultaneous
congruences.
250 NUMBER THEORY AND ITS HISTORY
Example.
1. Let us take
x 3 - 2x + 3 = (mod 7)
2x 2 = 3 (mod 15)
The first congruence is found to have a single solution
x = 2 (mod 7)
whila ths second has two solutions
x = ±3 (mod 15)
The simultaneous congruences
x = 2 (mod 7), x = 3 (mod 15)
give
x a 93 (mod 105)
for the common solution of the two given congruences, and similarly another
solution
x = 72 (mod 105)
is obtained from
x = 2 (mod 7), x = -3 (mod 15)
2. We wish to find some x such that
12x = 3 (mod 15), lOx = 14 (mod 8)
The first congruence, according to the theory of linear congruences, has the
three solutions
x = 4, x = 9, x = 14 (mod 15)
while the second has two solutions
3 = 3, x S3 7 (mod 8)
When these are combined, one obtains six solutions
x a 19, x = 79
x = 39, x = 99 (mod 120)
x = 59, x = 119
satisfying both congruences for the l.c.m. of the moduls.
By means of the results for simultaneous congruences, one can
reduce the solution of an algebraic congruence to the case where
the modul is a power of a prime. Let m be some number and
m = pi ai p r ar
ANALYSIS OF CONGRUENCES 251
its factorization into prime factors. An algebraic congruence
fix) = (mod m) (10-22)
will hold only for those x's that at the same time satisfy each of
the congruences
f(x) = (mod Pi ai ) i = 1, 2, • • • , r (10-23)
To solve the congruence (10-22) we can therefore determine the
solutions of each congruence (10-23) separately and use the
Chinese remainder theorem to obtain the values that satisfy them
simultaneously.
Examples.
1. Let us take the congruence
a; 3 - 7x 2 + 4 = (mod 88)
Since 8 and 11 are the prime powers in the modul 88, we solve the congruence
(mod 8) and (mod 11). In the first case one obtains three solutions
x = 2, x=3, x = 6 (mod 8)
and in the second case there is a single solution
x = 4 (mod 11)
When these solutions are combined, one finds three solutions
x ss 23, x s= 50, x = 70 (mod 88)
of the original congruence.
2. Let us take
5x 2 + 1 = (mod 189)
Here
189 = 3 3 • 7
so that we solve the congruence (mod 3 3 ) and (mod 7). One finds, respec-
tively,
x = ±4 (mod 27), x = ±2 (mod 7)
and the combination of these gives four solutions of the given congruence
x = ±23, x s ±58 (mod 189)
We have just seen how the solution of general algebraic con-
gruences may be derived from congruences with a prime-power
252 NUMBER THEORY AND ITS HISTORY
modulus. One can go one step further and give a method to solve
congruences for prime-power moduls by means of congruences for
a prime modul. We shall make the procedure clear on two ex-
amples.
Examples.
1. Let us first consider the congruence
fix) = x 2 - 7x + 2 e= (mod 5) (10-24)
By trial one finds the solutions
x = 3 and x = — 1 (mod 5) (10-25)
In the second step we take the same congruence (10-24) for the modulus 25
fix) = x 2 - 7x + 2 = (mod 5 2 ) (10-26)
It is clear that a solution of this congruence must be found among the numbers
(10-25) so that we can put
x = 3 + 5t or x = —1 + 5u (10-27)
To determine t and u we obtain from (10-26) by substitution of these values
respectively
f{x) = -10 - 5t + 25t 2 = (mod 25)
f(x) = 10 - 45m + 25u 2 =
and this reduces to
t s= —2, u = -2 (mod 5)
Therefore, according to (10-27), we can write
t = -2 + 5s, x = -7 + 25s (10-28)
u = — 2 + 5v, x = — 11 + 25v
so that the only solutions of the congruence (10-28) are
x = — 7, x = - 11 (mod 25)
In the third step we take the congruence (10-24) for a modul that is the
third power of 5
fix) = x 2 - 7x + 2 = (mod 125) (10-23)
The solutions x must be of the form (10-28), and when they are substituted, one
obtains
fix) = 100 - 525s -4- 625s 2 = (mod 125)
fix) = 200 - 725» + 625y 2 =
ANALYSIS OF CONGRUENCES 253
as the conditions s and v must satisfy. This reduces to
s = 4, v = 2 (mod 5)
and it follows that
x = 93, x s 39 (mod 125)
are the only solutions of the congruence (10-29). The same process may be
repeated indefinitely to obtain the solutions for moduls that are arbitrarily
high powers of 5.
2. We begin by observing that the congruence
fix) = x z - Sx 2 + 21s - 11 s (mod 7) (10-30)
has the two solutions
x 35 2, x = 3 (mod 7)
To find the solution of the congruence
f(x) = x 3 - Sx 2 + 21s - 11 = (mod 7 2 ) (10-31)
we have to substitute
x = 2 + It, x = 3 + 7s (10-32)
respectively. By the substitution of the second expression into (10-31), the
condition reduces to the impossible congruence
7=0 (mod 49)
so that we find no s that will give an x satisfying (10-31). When the first
expression (10-32) is substituted, the congruence reduces to
t = -1 (mod 7)
and correspondingly,
x = —5 (mod 49)
is the only solution of (10-31). When this method is applied to the same
congruence (10-30) (mod 7 3 ), one finds a single solution
x = 93 (mod 343)
We shall conclude this study of algebraic congruences by es-
tablishing a few results that extend the analogies between equations
and congruences. The first is:
Theorem 10-6. When an algebraic congruence of degree n
fix) = (mod m)
has a solution
x = a,\ (mod m)
254 NUMBER THEORY AND ITS HISTORY
then one can write
S(x) = (x - ai)A(x) (modm) (10-33)
where Si (x) is a polynomial of degree n — 1.
To prove the theorem we divide /(a;) by x — a x and find
Six) = (x- ai )Si(x) + R
where R is some integer and the degree of Si (x) is one less than the
degree of /Or). By putting x = a\ in this identity, we obtain
/(ai) = R = (mod m)
so that the congruence (10-33) follows.
Examples.
1. Let us take the third-degree congruence
/(x) = x 3 - 7x 2 + 4 e= (mod 88)
We have already found that this congruence has three solutions
x = 26, x = 59, x = 70 (mod 88)
The division of /(x) by x — 26 yields
x 3 - 7x 2 + 4 = (x - 26) (x 2 + 19x + 494) + 12,848
and since 12,848 is divisible by 88, one has
x 3 - 7x 2 + 4 = (x - 26) (x 2 + 19x + 494) (mod 88)
The reader may determine the corresponding decompositions for the other
roots of the congruence.
2. In the example of second degree
Six) = 3x 2 + 7x - 2 = (mod 23)
one finds the root
x = 3 (mod 23)
and the decomposition
3x 2 + 7x - 2 = (x - 3)(3x + 16) (mod 23)
The remaining two theorems, it should be noted, hold only for
congruences for a prime modul.
ANALYSIS OF CONGRUENCES 255
Theorem 10-7. When the congruence
f(x) = (mod p) (10-34)
of nth degree for a prime modul p has r different roots
x = a-i, x = a 2 , - • • , x = a r (mod p)
one can write
f(x) = (x - a\)(x - a 2 ) - • • (x — a r )f r {x) (mod p) (10-35)
where f r (x) is a polynomial of degree n — r.
The proof is based upon theorem 10-6, which shows first that
one can write
f(x) = (x- aJMx) (mod p) (10-36)
where /i (x) is of degree n — 1 . But since a 2 is also a root of
(10-34), we must have
f(a 2 ) = (a 2 - Oi)/i(a 2 ) = (mod p)
Here we use the fact that when a prime divides a product it must
divide one of the factors. The difference a 2 — a\ is not divisible
by p since a\ and a 2 were different roots, so that we conclude
/i (02) = (mod p)
According to theorem 10 -6 we can write again
fi(x) = (x - a 2 )f 2 (x) (mod p)
where f 2 (x) is of degree n — 2; hence from (10-36)
f(x) = (x - a x ) (x - a 2 )f 2 (x) (mod p)
For the third root a 3 of (10-34), one finds
/(a 3 ) = (o 3 - «i) a 3 - a 2 )/ 2 (a 3 ) = (mod p)
and one concludes similarly that
/ 2 (a 3 ) = (modp)
This gives
f 2 (x) = (x - a 3 )f 3 (x) (mod p)
256 NUMBER THEORY AND ITS HISTORY
where f 3 (x) is of degree n — 3, and
f(x) = (x - ai)(x - a 2 )(x - a 3 )f 3 (x) (modp)
The process may be continued until one arrives at the general
decomposition (10-35).
Examples.
1. In the congruence of fourth degree
x 4 - 5x 3 - 5x - 1 = (mod 7)
we have the roots
x = 2, x = 3 (mod 7)
and corespondingly the decomposition
x 4 - 5x 3 - 5x - 1 = (x - 2)(x - 3)(x 2 + 1) (mod 7)
2. The congruence
x 4 - 1 = (mod 5)
has the roots
x = 1, x = 2, x = 3, x = 4, (mod 5)
and therefore
x 4 - 1 = (x - l)(x - 2)(x - 3)(x - 4) (mod 5)
3. The congruence
3x 2 + 1 = (mod 19)
has the solutions
x == ±5 (mod 19)
and correspondingly one finds
3x 2 + 1 s 3(x - 5)(x + 5) (mod 19)
Our last result is due to the French mathematician Lagrange
(1768), as Gauss observes in this connection in the Disquisitiones.
Lagrange, of course, does not use the congruence terminology but
the content of his theorem is as follows:
Theorem 10-8. A congruence
f(x) = (mod p)
ANALYSIS OF CONGRUENCES 257
for a prime modul p cannot have more different solutions than its
degree, except in the trivial case where all coefficients in f{x) are
divisible by p.
Let us suppose that the degree of /(#) is n and that
x = ai, x = a 2 , . . . , x = a n (mod p)
are n different solutions of the congruence. From theorem 10-7
we conclude that
fix) = (x — ax){x — a 2 ) • • • (x — a n )F (mod p)
where F is of zero degree, hence some integer. If there were some
further solution
x = a n+ i (mod p)
we would have
f(a n+ i) =s (a n+1 - ax) • • • (a w+1 - a n )^ = (mod p)
Here none of the differences a n+ i — a; are divisible by p since we
deal with different solutions (mod p). The conclusion is that
F = (mod p)
and therefore identically
f(x) = (mod p)
which means that all coefficients mf(x) are divisible by p.
Problems.
1. Find the common solutions to the congruences
(a) 3x 2 - 7 s (mod 17)
5x 2 - 2z - 3 = (mod 12)
(6) 3x == 11 (mod 23)
50x s 2 (mod 32)
(c) x 2 + 5 = (mod 27)
3x + 1 = (mod 10)
258 NUMBER THEORY AND ITS HISTORY
2. Solve the congruences
(o) x 3 - 3x - 8 = (mod 60)
(6) x 2 + 11 = (mod 180)
(c) x 2 + 2x + 7 = (mod 75)
3. Solve the following congruences and find the corresponding congruence
factorizations:
(a) x % - x 2 - 2x s (mod 5)
(6) x 3 + x 2 - 2 s (mod 5)
(c) x 2 + 1 s (mod 13)
4. Solve the congruences in problem 3 for the second and third powers
of the moduls.
CHAPTER 11
WILSON'S THEOREM AND ITS CONSEQUENCES
11-1. Wilson's theorem. In the Meditationes Algebraicae by
Edward Waring, published in Cambridge in 1770, one finds, as we
have already mentioned, several announcements on the theory of
numbers. One of them is the following: For any prime p the
quotient
1-2 (p - 1) + 1
V
is an integer.
This result Waring ascribes to one of his pupils John Wilson
(1741-1793). Wilson was a senior wrangler at Cambridge and
left the field of mathematics quite early to study law. Later he
became a judge and was knighted. Waring gives no proof of
Wilson's theorem until the third edition of his Meditationes, which
appeared in 1782. Wilson probably arrived at the result through
numerical computations. Among the posthumous papers of
Leibniz there were later found similar calculations on the re-
mainders of n\, and he seems to have made the same conjecture.
The first proof of the theorem of Wilson was given by J. L. La-
grange in a treatise that appeared in 1770.
We shall prefer to give Wilson's theorem in the now usual con-
gruence form:
Theorem 11-1. For any prime p one has
(p _ i)! = -1 (mod p) (11-1)
The theorem is easily verified for small values of p.
11 = -1 (mod 2), 2! s -1 (mod 3), 4! s -1 (mod 5)
259
260 NUMBER THEORY AND ITS HISTORY
Before we proceed to the general proof we shall indicate its main
idea in the special case p — 19. For this modulus one has the
congruences
2 • 10 = 1, 7 • 11 = 1
3 • 13 = 1, 8 • 12 = 1
4-5=1, 9 • 17 = 1
6 • 16 = 1, 14 • 15 = 1
and also
1 = 1, 18 • 18 s 1
When the first group of congruences is multiplied together and the
numbers rearranged, it follows that
2-3 16 • 17 = 1 (mod 19)
This congruence is multiplied by
18 s= - 1 (mod 19)
and we obtain
18! s -1 (mod 19)
as required by Wilson's theorem.
The proof in the general case proceeds along the same lines. Let
p be some prime and a one of the numbers
1, 2, . . . , p - 1 (11-2)
The linear congruence
ax = 1 (mod p)
as we have seen, has a single solution
x = b (mod p)
In this manner there corresponds to every a in (11-2) a unique b,
such that
ab = 1 (mod p)
and clearly b corresponds to a in the same way. This shows that
the numbers in (11-2) can be divided into pairs a, b whose product
WILSON'S THEOREM AND ITS CONSEQUENCES 261
is congruent to 1 (mod p). The two numbers in a pair x, x can
only be equal when
x 2 = 1 (mod p)
This can be written
(x - l)(x+ 1) = (modp)
so that it can occur only when
x = 1 (mod p)
or
#=— 1 = p — 1 (mod p)
that is, when x=lorx = p— 1. In multiplying the numbers
(11-2) together to form (p — 1)!, the pairs with different a and
b will give products congruent to 1 (mod p), while the two re-
maining factors 1 and p — 1 have a product that is congruent to
— 1. This completes the proof of Wilson's congruence (11-1).
One may ask what happens for other moduls. One computes
easily
1! = -1 (mod 2), 6! = -1 (mod 7)
2!=-l (mod 3), 7!- (mod 8)
3!= 2 (mod 4), 8!= (mod 9)
4! = -1 (mod 5), 9! = (mod 10)
5! = (mod 6), 10! = -1 (mod 11)
The general result is:
Theorem 11-2. For a composite number n one has
(n - 1)! = (modn)
except when n — 4.
The proof is quite simple. Let us write
n — pq
where p is a prime. If p is not equal to q, both p and q occur as
factors in the product (n — 1) ! so that it is divisible by n. When
262 NUMBER THEORY AND ITS HISTORY
p — q and n — p 2 , the factors p and 2p occur in (n — 1 ) !, provided
p > 2. In the remaining case when p = 2, n = 4, there is an
exception to the rule, as we found above.
Theorem 11-2 shows that Wilson's congruence (11-1) will hold
for the primes and for no other numbers.
We shall deduce certain consequences from Wilson's congruence.
In the product
(p- i)! = i.2 — ^• E y J: <p-2)<p-i)
one has the following congruences:
p- 1= -1, p-2 = -2, ...,
p + 1 _ p - 1
(mod p)
2 2
for the series of factors. Consequently
2i±( P - lV
(p- 1)!=. (-1) 2 (^1-2 ^-j (modp)
and when this is substituted in Wilson's congruence, one finds
(i • 2 nr 1 ) 2 s ( ~ 1} ^" (mod p) (11_3)
Let us discuss this result a little further. When p is a prime of
the form 4n + 1,
(-1) 2 = (_l)2n+l = _l
so that (11-3) takes the form
(l • 2 E -^ L ) + X s ° ( mod P)
and we can state:
Theorem 11-3. When p is a prime of the form 4n + 1, the
congruence
x 2 + 1 = (mod p)
WILSON'S THEOREM AND ITS CONSEQUENCES 263
is solvable and has the roots
x s ± \^—£- ) ! ( mod V)
For instance when p = 13 we find
6! = 5 (mod 13)
and
5 2 + 1 = (mod 13)
In the second case when p is a prime of the form An + 3, the
congruence (11-3) reduces to
(i • 2 ^T/ ~ l ~ ° (mod p)
This may be written
and one concludes:
Theorem 11-4. For any prime p of the form An + 3, one has
one of the congruences
(<Hr)
! = ± 1 (mod p) (11-4)
One finds for the lowest primes
1! = 1 (mod 3), 3! = -1 (mod 7), 5! = -1 (mod 11)
9! == _i ( m od 19), 11! s 1 (mod 23)
There exists a complicated rule determining whether one shall
use +1 or -1 in the congruence (11-4).
11-2. Gauss's generalization of Wilson's theorem. In the third
section of the Disquisitiones Gauss indicates, without giving the
details of the proof, how the theorem of Wilson can be extended
264 NUMBER THEORY AND ITS HISTORY
to arbitrary moduls. Before we proceed to this theorem, it is
necessary to carry through a certain auxiliary investigation.
We wish to determine the number of solutions of the congruence
x 2 s 1 (mod m) (11-5)
for a given modul m. As we have already observed previously, we
can solve the congruence first for prime-power moduls and then
obtain the solution for a general modul by the Chinese remainder
theorem. Therefore, let p be a prime and let us study the con-
gruence
x 2 = 1 (mod p a ) (11-6)
This may be written
(x - l)(x + 1) = (mod p a ) (11-7)
If p > 2, only one of these factors can be divisible by p so that one
has either
x = 1 (mod p a )
or
x = — 1 (mod p a )
and we conclude:
For a prime p > 2, the congruence (11-6) has two solutions
x = ±1 (mod p a )
The case when p = 2 is slightly more complicated. For a = 1
in (11-6), the congruence becomes
x 2 = 1 (mod 2)
which has a single solution x = 1 (mod 2). For a = 2, the con-
gruence
x 2 5= 1 (mod 4)
has two solutions # = ±1 (mod 4). Finally let a > 2. If one
of the factors in (11-7) is divisible by 2, so is the other, but only
one of them can be divisible by 4 or a higher power of 2. If x + 1
is divisible only by 2 to the first power, one must have
x = 1 (mod 2" -1 )
WILSON'S THEOREM AND ITS CONSEQUENCES 265
and this represents two different solutions
x = 1, isl+ 2"- 1 (mod 2 a )
of the original congruence. Similarly when x — 1 contains only
the first power of 2, one finds the solutions
x = -1, a; = -1 - 2 a_1 (mod 2 a )
The four solutions are different (mod 2 a ), as one easily checks. To
sum up, for p = 2, the congruence (11-6) has four solutions
x= ±1, x=±(l + 2"" 1 ) (mod 2 a )
except when a = 2, when there are two solutions
x = ± 1 (mod 4)
and when a = 1, when there is a single solution
x = 1 (mod 2)
It remains to determine the number of solutions of the general
congruence (11-5). We decompose the modul into its prime
factors
m = 2 a p/ 1 • • • p/ r (11-8)
and solve the congruence (11-6) for the moduls 2 a and pf\ The
Chinese remainder theorem shows that the solutions of (11-5) are
obtained by selecting a particular solution for each of the prime
powers and combining them by simultaneous congruences. When
m is not divisible by 2, hence when a = 0, each of the congruences
(mod pt^) has 2 roots so that we obtain a total of 2 T solutions.
When a = 1 the congruence (mod 2) has a single solution so that
there will still be 2 r solutions. When a = 2 the congruence
(mod 4) has two solutions, and the total number in this case is
2 r+1 . Finally, when a > 2, the congruence (mod 2") has four
solutions so that the total number of solutions of (11-5) is 2 r+2 .
Thus we can state:
Theorem 11-5. The congruence
x 2 = 1 (mod m)
266 NUMBER THEORY AND ITS HISTORY
where the modul has the prime decomposition
m =* 2°7> 1 /31 • • • pf 7
will have
2 r solutions when a = or a = 1
2 r+1 solutions when a = 2
2 r+2 solutions when a > 2
This completes our preparations for Gauss's generalization of
Wilson's theorem:
Theorem 11-6. If one forms the product P of the remainders
relatively prime to the number to, then
P s= ±1 (mod to) (11-9)
In this congruence one has the value +1 in all cases, with the
following exceptions where —1 appears:
1. to — 4.
2. to = p@ is a power of an odd prime.
3. m = 2pP is twice the power of an odd prime.
When to = 2 it is immaterial whether one uses +1 or — 1.
The result, as one sees, implies Wilson's theorem. For the
smallest composite moduls one finds
1 • 3 = - 1 (mod 4), l-2-4-5-7-8=-l (mod 9)
1 • 5 = -1 (mod 6), 1 • 3 • 7 • 9 = -1 (mod 10)
1 • 3 • 5 • 7 s= l (mod 8), 1 • 5 • 7 • 11 = 1 (mod 12)
The proof is based on the same principle as our proof of Wilson's
theorem. The set of all positive integers less than and relatively
prime to to are paired as before. To each such number a one
can find a unique b for which
ab = 1 (mod to)
WILSON'S THEOREM AND ITS CONSEQUENCES 267
Let us suppose first that a and b are different. Then in computing
the remainder of the product P in (11-9), they may be disregarded
since their product is congruent to 1 (mod m). In P, therefore,
we need only to consider the product of those numbers a that
belong to a pair a, a with equal components, i.e., the solutions of
the congruence
x 2 = 1 (mod m) (11-10)
If a is a solution of this congruence, so is — a and these two numbers
represent different remainders (mod m) since
a = —a (mod m)
can only occur in the trivial case m = 2. In forming the remainder
of the product P, we multiply a and —a and note that
a(—a) = —a 2 = —1 (mod m)
Each pair of roots a and —a therefore contributes a factor — 1 to
the remainder of P (mod m) and the congruence (11-9) follows.
The remainder + 1 must be used when there is an even set of roots
a and —a, hence when the number of roots of (11-10) is divisible
by 4; otherwise one must use — 1. But in theorem 11-5 this
number of roots has been determined, and one verifies that the
only cases in which it is not divisible by 4 are exactly those that
have been enumerated above in Gauss's theorem.
We notice further, according to theorem 11-5, that the integers
m for which the negative sign must be used in (11-9) are those for
which the congruence
x 2 ss 1 (mod m)
has only two solutions and clearly these must be
x = ±1 (mod m)
Problems.
Check the theorem of Gauss for the composite numbers below 25.
11-3. Representations of numbers as the sum of two squares.
We have already mentioned certain results regarding the repre-
sentation of numbers as the sum of two squares, both in discussing
268 NUMBER THEORY AND ITS HISTORY
Fermat's notes on Diophantos and in connection with the factori-
zation of numbers. We are now ready to give the proofs for some
of the basic facts. There are numerous ways in which this theory
may be treated. We shall prefer a method based on a simple
theorem on congruences given by the Norwegian mathematician
Axel Thue (1863-1922), who is known for his important contri-
butions to the newer theory of Diophantine equations.
The theorem of Thue is of interest also because it gives a simple
application of a mathematical method known as Dirichlet's box
principle: If one has n boxes and more than n objects to distribute
in them, at least one of the boxes must contain more than one
object.
This statement sounds extremely trivial, but, neverthless, it has
important applications to various questions in number theory.
The theorem of Thue that we wish to prove is the following:
Theorem 11-7. Let p be a prime and k the least integer greater
than Vp. Then for any integer a not divisible by p, one can find
numbers x and y belonging to the set 1, 2, . . . , k — 1 such that
xa = ±y (mod p) (11-11)
Before we proceed to the proof, let us take as an example the
case where p = 23 and k = 5. For a — 9 and a = 10 one finds,
respectively,
3a = 4, 2a = -3 (mod 23)
The reader may give a complete set of such congruences for all
remainders a (mod 23).
To prove the theorem let us take all numbers
ax — y, x, y — 0, 1, • • •, k — 1 (11-12)
and classify them according to their remainders (mod p). There
are altogether k 2 > p such numbers (11-12), so that according to
Dirichlet's box principle at least two of them must have the same
remainder, hence be congruent (mod p). Let us suppose that
ax\ — Vi = a%2 — 2/2 (mod p)
or
a(xt - x 2 ) = y\ - y 2 (mod p) (11-43)
WILSON'S THEOREM AND ITS CONSEQUENCES 269
The absolute values of the differences
xi - x 2 , 2/1-2/2
again belong to the set 0, 1, 2, . . . , k — 1. But the value is
excluded, because, for instance, y x = y 2 would involve, according
to (11-13),
a(xi — x 2 ) = (mod p)
from which one concludes that x x = x 2 , consequently x x = x 2 ,
contrary to the assumption that we are dealing with two different
numbers in (11-12). In (11-13) we have, therefore, a congruence
of the desired form (11-11) when one, if necessary, adjusts the sign
so that the coefficient of a is positive.
In the next step we show:
Theorem 11-8. A prime p is representable as the sum of two
squares if the congruence
a 2 + 1 = (mod p) (11-14)
is solvable.
To prove this result we take a solution a of the congruence
(11-14) and determine the two numbers x and y according to
theorem 11-7. We multiply the congruence (11-14) by x 2 and
obtain
x 2 a 2 + x 2 s= y 2 + x 2 = (mod p)
so that
x 2 + y 2 = tp (11-15)
for some suitable positive integer t. But x and y were chosen
such that
x 2 S (k - l) 2 < p, y 2 ^ (k- l) 2 < p
and we conclude from (11-15)
pt < p + p = 2p
or t < 2. The only possibility is, therefore, t = 1 and
p = x 2 + y 2
is the desired representation of p as the sum of two squares.
270 NUMBER THEORY AND ITS HISTORY
When theorem 11-8 is combined with theorem 11-3 we arrive
at the key result:
Theorem 11-9. Any prime of the form 4w + 1 can be repre-
sented as the sum of two squares.
The prime 2 is evidently the sum of two squares, but no prime
of the form An -\- 3 can be the sum of two squares, because such a
sum is never of the form 4n + 3, a fact we have already mentioned
in Chap. 4 in connection with the factorization method based on
the representation of a number as a sum of two squares. At that
time we also showed that a prime can have but a single such
representation, and we gave the representation of all primes below
100 as the sum of two squares. Extensive tables of this kind have
been computed.
Theorem 11-10. Let JV be a positive integer and n 2 its greatest
square factor so that
N = N n 2
The necessary and sufficient condition for N to be representable
as the sum of two squares is that N contain no prime factors of
the form An + 3.
In Chap. 8-5 in discussing Fermat's notes on Diophantos's Arith-
metics, we proved that on the basis of theorem 11-9 it follows that N
can be represented as the sum of two squares provided the prime
factors of iVo were 2 or of the form An + 1.
There remains, after what we have just said, only to prove that
if No has a prime factor p = An + 3, no representation of N as the
sum of two squares can exist. This we achieve by showing that
a decomposition
N = x 2 + y 2 (11-16)
must lead to a contradiction. Let us suppose that in (11-16)
the greatest common divisor of x and y is d so that
x — dx\, y — dy\
We divide (11-16) by d 2 and find
?L = f 2 N = x 1 2 + y 1 2 (H-17)
d d
WILSON'S THEOREM AND ITS CONSEQUENCES 271
where d 2 must divide n 2 , and x\ and y x are relatively prime. Since
No is divisible by po, we obtain from (11-17) the congruence
xi 2 + Vx = (mod po) (11-18)
One of the numbers x\ and ?/i, for instance X\, is not divisible by
po, so that we can find some z such that
X\Z = 1 (mod po)
When (11-18) is multiplied by z 2 , it follows that
( Vl z) 2 +1^0 (mod po)
This, however, according to theorem 11-8, would show that p
must be the sum of two squares, notwithstanding the fact that it
is a prime of the form 4w + 3. We have therefore established the
impossibility of a representation (11-16).
Problem. Represent all primes of the form 4n + 1 between 500 and 600
as the sum of two squares.
CHAPTER 12
EULER'S THEOREM AND ITS CONSEQUENCES
12-1. Euler's theorem. In a letter to Frenicle de Bessy dated
October 18, 1640, Fermat writes as follows:
It seems to me after this that I should tell you the foundation on which
I support the demonstrations of all which concerns geometric progressions,
namely:
Every prime number measures [divides] infallibly one of the powers
minus unity in any progression, and the exponent of this power is a divisor
of the given prime number minus one; and after one has found the first
power which satisfies the condition, all those whose exponents are multiples
of the first satisfy the condition.
Fermat uses the example of the powers of three
1, 2, 3, 4, 5, 6
3, 9, 27, 81, 243, 729
where the first line gives the exponents. He points out that 3 3 — 1
is the first such expression that is divisible by 13 and that the
exponent 3 divides 13 — 1 = 12 so that 3 12 — 1 is divisible by 13.
Fermat continues: "And this proposition is generally true for
all series and all prime numbers. I would send you the demon-
stration, if I did not fear it being too long."
Unfortunately Fermat 's correspondents never seemed particu-
larly interested in demanding information about proofs for his
results.
In congruence terminology Fermat states that for any number
a and any prime p, there exists some smallest exponent d such that
a d - 1 s (mod p)
272
EULER'S THEOREM AND ITS CONSEQUENCES 273
and d divides p — 1, hence
a P-i _ i = o (mod p)
It should be observed that in this theorem one must place the
obvious restriction that a shall not be divisible by p.
For this result, as well as for several other of Fermat's theorems,
Euler was the first to publish a proof; it appeared in the Proceed-
ings of the St. Petersburg Academy in 1736. Considerably later
(1760), Euler gave a more general theorem of the same kind, which
we shall prefer to deduce first and then let Fermat's theorem follow
as a special case.
To formulate Euler's theorem we first recall the definition of
Euler's ^-function, which we studied in Chap. 5. For any positive
integer m we denoted by <p(m) the number of remainders from 1
to w that were relatively prime to m. For this function of m,
we found the expression
, (B) _ B (l-I)...(l-I) (12-1)
where p\, p 2 , • • • , Pr are the different primes dividing m. Euler's
theorem is then:
Theorem 12-1. For any number a that is relatively prime to
m one has the congruence
a <f(rn) ^ 1 ( mod m ) ( 12 -2)
Before we proceed to the proof let us consider some examples.
Examples.
1. Let to = 60 and a = 23. One finds
*(60) = 60(1 -i)(l - i)d - i) = 16
and to verify the congruence (12-2) we compute for the modulus 60
23 2 = -11
23 4 s 121 a 1
23 16 = 1
274 NUMBER THEORY AND ITS HISTORY
2. Let us take m = 11 and o = 2, hence
*(11) = 11(1 -tV) = 10
We have
2 6 = 32 s -1
2 10 & 1 (mod 11)
To prove Euler's congruence (12-2) we denote the <p(m) re-
mainders less than and relatively prime to m by
n, r 2 , . . . , Mm) (12-3)
We multiply each of them by the given number a relatively prime
to m and divide each product by w:
r { a — qm + r\ (12-4)
where r\ is the least positive remainder. Here r\ must be rela-
tively prime to m because in (12-4) a common factor of m and r[
would divide r^a and this is impossible, since r* and a are both
relatively prime to m. The number r< is, therefore, also one of
the remainders (12-3).
We shall prefer to write the relations (12^4) as congruences
na = r'i (mod m) (12-5)
Two different remainders r t - and ry in (12-3) cannot give rise to
the same r\ in (12-5) because the congruence
na = Tjd (mod m)
implies, since a may be canceled, that
Ti = rj (mod m)
or n = rj. We conclude that in (12-5) the remainders r* and r£
both run through the whole set (12-3).
Let us illustrate the situation on the example where m = 20
and a = 7. Here
<p(20) =20(l-J)(l--fr) =8
EULER'S THEOREM AND ITS CONSEQUENCES 275
and the relatively prime residues are the eight numbers
1, 3, 7, 9, 11, 13, 17, 19
One finds
la = 7, 11a s 17
" ' X ' 13a " U (mod 20)
7a s 9, 17a = 19
9a = 3, 19a = 13
When all these congruences are multiplied together, one obtains
a 8 • 1 • 3 • 7 • 9 • 11 • 13 • 17 • 19
= 1-3 -7 -9 -11 -13 -17 -19 (mod 20)
The product of the remainders is relatively prime to the modul
and so may be canceled to give
a 8 = 1 (mod 20)
The general proof is quite analogous. We multiply all <p(m)
congruences (12-5) and find
a v(m) rir 2 • • • r , (TO) s r[r' 2 • • • r' v(m) (mod to) (12-6)
Since the numbers n and r< form the set (12-3) of remainders,
their products are equal. Furthermore, the r/s are relatively
prime to to so that in (12-6) the product can be canceled and
there remains Euler's congruence (12-2).
The subsequent sections are all based on Euler's congruence.
At this point we shall mention only one minor application to our
familiar problem of solving linear congruences, or equivalently,
linear indeterminate equations.
Theorem 12-2. When a and to are relatively prime, the solu-
tion of the linear congruence
ax = b (mod w) (12-7)
is given by the formula
x = ba* (m)-1 (mod to) (12-8)
276 NUMBER THEORY AND ITS HISTORY
For the proof we multiply both sides of the congruence (12-7)
by a <p{ - m) ~ 1 and find from Euler's congruence
€^ m) x = x = bo?^' 1 (mod m)
Examples.
1. In the congruence
7x = 5 (mod 24)
we have <p(24) = 8 and therefore according to (12-8)
x = 5 ■ 7 7 (mod 24)
To compute the smallest remainder of x we notice that
7 2 = lf 7 6 = lf 7 7 = 7 (mod 24)
and
x = 5 ■ 7 = 11 (mod 24)
2. Let us take the congruence
Ux = 9 (mod 20)
Here <p(29) = 28 so that
x a 9 • ll 27 (mod 29)
One computes
11 2 = 5
11* = 25 s -4
lis s 16 (mod 29)
ll 16 = 256 s -5
ll 27 = ll 16 • ll 8 • ll 2 • 11 s (-5) • 16 • 5 • 11 = 8
and
x
s 9 • 8 = 14 (mod 29)
The formula (12-8) is interesting because it gives the solution
of the linear congruence in explicit form. However, for con-
gruences that involve fairly large numbers, one finds that in
regard to simplicity of computations it is definitely inferior to our
previous method based on the linear indeterminate equations.
EULER'S THEOREM AND ITS CONSEQUENCES 277
Problems.
1. Verify Euler's congruence in the following examples:
to = 81, a — 5
to = 120, a = 7, o = 19
to = 59, a = 2
2. Solve the following congruences by Euler's theorem:
7x = 2 (mod 24)
4x = 3 (mod 49)
17a; = 41 (mod 620)
3. Compare the work involved by solving the congruence
311a; m 19 (mod 203)
by Euler's theorem and by linear indeterminate equations.
12-2. Fermat's theorem. The theorem of Euler will now be
applied to certain important special cases. We assume first that
the modul m — p a is a power of a prime. Then
<p(m) = p a (l --) = p a - <p a - y
and the numbers relatively prime to m are those that are not
divisible by p. Euler's theorem states therefore:
Theorem 12-3. If the number a is not divisible by the prime
p, one has
a p a -p a - 1 = i ( m od p a ) (12-9)
By specializing further to the case where the modul m = p is a
prime, we arrive at Fermat's original theorem:
Theorem 12-4. When p is a prime and a some number not
divisible by p, then
aP- 1 = 1 (mod p) (12-10)
278 NUMBER THEORY AND ITS HISTORY
As an example, let us take p = 101 and a = 2. One computes
2 4 3 16
2 8 s 256 = 54
2 16 = (54) 2 = - 13 (mod 101)
2 32 = 169 = _33
2 6* = (-33) 2 s -22
and
2 ioo ^ 2 64 . 2 32 , 2 4 s (-22) (-33)16 = 1 (mod 101)
According to Fermat's theorem the algebraic congruence
x p-i -1=0 (mod p)
has the p - 1 different solutions 1, 2, . . . , p - 1. When theorem
10-7 on algebraic congruences is applied to this case, it leads
immediately to
Theorem 12-5. For any prime modulus p, one has
x p-i _ i = (x - i)( x - 2) • • • (x - (p - 1)) (modp) (12-11)
For instance when p = 5
(x- l)(x- 2)(x- 3)(x-4)
= (z- l)(x-2)(x + 2)(x+l)
= (a; 2 - l)(x 2 - 4) = x 4 - 1 (mod 5)
When one compares the coefficients of the powers of x on both
sides of the congruence (12-11), one obtains various congruences
relating to the numbers 1, 2, . . . , p - 1. If we take the constant
form in which x does not occur, one has on the left the number — 1
and on the right the product
(-l)(-2) • • • [-(p - 1)] = (-l^fo ~ 1)1
so that
(-lF^Cp - 1)! = -1 (modp)
This, as one sees, is the same as Wilson's congruence.
EULER'S THEOREM AND ITS CONSEQUENCES 279
Format's congruence (12-10) holds for all numbers a except
those divisible by p. However, it is possible to formulate- this
result in such a manner that it is valid for every number without
exception. When the congruence (12-10) is multiplied by a, one
finds
a? = a (mod p)
for every a not divisible by p. Clearly this congruence holds also
for those a that are divisible by p, since then both sides are divisible
by p. Therefore, we can restate Fermat's theorem as follows:
Theorem 12-6. For a prime p one has
aP = a (mod p) (12-12)
for any a.
As a minor application of this theorem let us show:
Theorem 12-7. In the decadic system a number and its fifth
power have the same final digit.
In terms of congruences we shall have to establish that
a 5 = a (mod 10)
for any a. But by Fermat's congruence one has
a 5 = a (mod 5)
and trivially one finds
a b = a (mod 2)
Problems.
1. Verify Fermat's congruence (12-10) for
p = 71, a = 3, V = 59, a = 2
2. Verify the congruence (12-11) for p = 11 and p = 13.
3. Show that for any number n not divisible by 2 and 3 one has
n 2 = 1 (mod 24)
12-3. Exponents of numbers. Euler's theorem shows that to
any number a that is relatively prime to the modul m there must
exist some exponent n such that
a n = 1 (mod m) (12-13)
280 NUMBER THEORY AND ITS HISTORY
The least positive exponent n for which this congruence (12-13)
holds we shall call the exponent to which a belongs (mod m). For
small moduls it may be determined directly by finding the residue
(mod m) of the various powers of a.
For instance, let m = 30 and a = 7. One finds successively
a 2 = -11, a 3 = 13, a 4 = 1 (mod 30)
so that 7 belongs to the exponent 4 (mod 30).
In connection with the initial statement of his theorem, which
we quoted, Fsrmat points out in the example that the number 3
belongs to the exponent 3 (mod 13) and the exponent 3 divides
p - 1 = 12.
In the study of the properties of the exponents to which a number
belongs, wo begin by mentioning :
Theorem 12-8. If the number a belongs to the exponent n
(mod m) and if N is some other number such that
,iV
then n di vides N.
We divide N by n
ar = I (mod m) (12-14)
N = qn + r, ^ r < n
and deduce from (12-13) and (12-14)
Q N _ aqn +r = (a ny a r ^ ^ ^ j ^ %)
Since n was the smallest positive exponent such that the congru-
ence (12-13) would hold, we conclude that r = and n divides N.
When theorem 12-8 is applied to Euler's congruence, there
follows immediately:
Theorem 12-9. The exponent n to which a number a belongs
(mod m) divides <p(m).
This theorem brings up the important question: When a divisor
n of <p{m) is given, does there exist some number a belonging to
the exponent n (mod m) ?
In general this is not true, as one can show by examples. For
instance, let us take m = 15 and <p(m) = 8. The numbers that
EULER'S THEOREM AND ITS CONSEQUENCES 281
are relatively prime to m are congruent to one of the eight numbers
±1, ±2, ±4, ±7
One finds that all of them satisfy the congruence
x 4 = 1 (mod 15)
so that there is no number belonging to the exponent n = 8.
On the other hand, let us compute the exponents to which the
various numbers belong (mod 13). Here «?(13) = 12 and this
number has the divisors
1, 2, 3, 4, 6, 12
One verifies that there are numbers belonging to each of these
exponents (mod 13), namely:
Numbers
Exponent Belonging
1 1
2 12
3 3,9
4 5,8
6 4, 10
12 2, 6, 7, 11
This table illustrates the general result :
Theorem 12-10. For a prime modul p there exist numbers
belonging to every divisor n of p — 1.
The proof will be based on the following theorem, which throws
light on the interrelation between the numbers that belong to the
same exponent.
Theorem 12-1 1 . Let a be a number belonging to the exponent n
for the prime modul p. Then the powers
a n , a r \ ... , o r " n > (12-15)
represent all numbers belonging to the same exponent where
ri = 1, r 2 , • • • , r, w (12-16)
are the positive remainders less than and relatively prime to n.
282 NUMBER THEORY AND ITS HISTORY
For instance, in the example above, the number 2 belongs to
the exponent n — 12 (mod 13). The remainders less than and
relatively prime to 12 are
1, 5, 7, 11
and one finds as in the table that
2, 2 5 = 6, 2 7 = 11, 2 11 = 7, (mod 13)
are the numbers belonging to the exponent 12 (mod 13).
To prove theorem 12-11 we notice first that a satisfies the con-
gruence
x n s 1 (mod p) (12-17)
But for any i one has
(«T = (a n y = 1 (mod p)
so that the n numbers
1, a, a 2 , ... , a n_1 (12-18)
satisfy the same congruence (12-17). Furthermore the n powers
(12-18) are incongruent (mod p) because from a congruence
a 1 = a } (mod p)
would follow
a %—j = j (mod p)
with a smaller exponent i — j than n. We conclude therefore from
the theorem of Lagrange (theorem 10-8) that the numbers (12-18)
are all the roots of the congruence (12-17) of nth degree.
Consequently, the numbers belonging to the exponent n (mod p)
are to be found in (12-18). But all numbers a r — a r , where r
has a common factor d with n, must belong to a smaller exponent
since
n r
a r d = (a n )" d = 1 (mod p)
As possible numbers belonging to the exponent n there are left
only those a r = a r where r is relatively prime to n, that is, where
r is one of the remainders (12-16). Let n r be the exponent to
EULER'S THEOREM AND ITS CONSEQUENCES 283
which a r belongs (mod p). Since a r is a root of the congruence
(12-17), n r must divide n. But conversely from
{a r ) nr = 1 (mod p)
it follows that n divides r • n r , therefore, since (r,n) = 1, n also
divides n r , so that n r = n. Ihis completes the proof of theorem
12-11.
The deduction of theorem 12-10 from theorem 12-11 is based
on the following reasoning. We denote by
m = 1, n 2 , n 3 , • • • , n„ = p - 1 (12-19)
the set of all divisors of p - 1, and for any such divisor m let N f
be the number of integers belonging to the exponent m (mod p).
Since any integer not divisible by p belongs to some exponent
(mod p), we must have
Nt + AT 2 + N 3 + • • • + N v = p - 1 (12-20)
Theorem 12-11 states that for any divisor m we have only two
alternatives: either there is no integer belonging to this exponent
(mod p) so that N t = 0, or we have Ni = <p(m) where <p is Euler's
function. But in discussing Euler's ^-function we derived the
theorem (theorem 5-12) that the sum of the ^-functions of the
divisors of a number was equal to the number itself. When
applied to the divisors (12-19) of p - 1, this gives
*>(ni) + *>(n 2 ) H r- v(n,) = p - 1 (12-21)
By comparison of (12-20) and (12-21), one sees that both sums
cannot have the same value p - 1 except when each N { takes the
Value <p(n { ). Thus for every divisor m of p - 1, there exist
numbers belonging to it.
The proof shows that one can supplement theorem 12-10 as
follows:
Theorem 12-12. For every divisor n of p - 1, there exist
<p(n) numbers belonging to the exponent n for the prime modul p.
One may verify this result on the table of exponents (mod 13)
given above.
284 NUMBER THEORY AND ITS HISTORY
Problems.
1. Find the exponents to which the relatively prime remainders belong for
the following moduls:
to = 40, to = 30, to = 20, to = 7
2. Check theorem 12-12 on the residues for the following primes:
p = 11, p = 17, p = 19
12-4. Primitive roots for primes. The highest possible exponent
to which a number can belong (mod m) is <p(m). If a number
belongs to this maximal exponent, we shall call it a primitive root
for the modul m. Not every modul has primitive roots, as we
have already mentioned; for instance, for the modul m = 15, one
finds that every remainder relatively prime to 15 will satisfy the
congruence
z 4 = 1 (mod 15)
and yet ^(15) = 8. The determination of those moduls for which
primitive roots can exist is one of the problems to be taken up in
the following discussion.
From theorem 12-12 we conclude immediately:
Theorem 12-13. For a prime modul p there exist <p(p — 1)
primitive roots.
From the previous table of the exponents to which the various
remainders belong (mod 13), we see that
2, 6, 7, 11
are the primitive roots (mod 13).
To find the primitive roots of a modul if they exist, one must
usually proceed by trial and error, although there are certain rules
that may facilitate the search. Often one of the small numbers
2, 3, 5, or 6 may turn out to be a primitive root. The following
table gives the smallest positive primitive root for all primes
below 200.
Extensive tables of primitive roots for primes have been com-
EULER'S THEOREM AND ITS CONSEQUENCES 285
puted. The first of these, the Canon ariihmeticus (1839) by K. G.
J. Jacobi, included primitive roots for all primes below 1,000.
More recent tables by Kraitchik, Cunningham, and others give
primitive roots for all primes up to 25,000 and even beyond.
Prime
Primitive root
Prime
Primitive root
2
1
89
3
3
2
97
5
5
2
101
2
7
3
103
5
11
2
107
2
13
2
109
6
17
3
113
3
19
2
127
3
23
5
131
2
20
2
137
3
31
3
139
2
37
2
149
2
41
6
151
6
43
3
157
5
47
5
163
2
53
2
167
5
59
2
173
2
61
2
179
2
67
2
181
2
71
7
191.
19
73
5
193
5
79
3
197
2
- 83
2
199
3
12-5. Primitive roots for powers of primes. We shall now
tackle the question of finding all moduls for which there exist
primitive roots. One of the main results in this direction is
Theorem 12-14. There exist primitive roots for all powers of
a prime p > 2.
286 NUMBER THEORY AND ITS HISTORY
The proof of this theorem is quite long, and we shall take it in
several steps. First we show:
There exist primitive roots when the modul is a square of a
prime.
Since
<p(P 2 ) = p(p ~ 1)
every number a not divisible by p satisfies the congruence
a p(p-i) s x (mod p2)
We shall take some primitive root r (mod p) and examine when
it may be a primitive root (mod p 2 ) . Let r belong to the exponent
d (mod p 2 ) so that
r d = 1 (mod p 2 )
is the congruence with the smallest possible exponent that r satisfies
(modp 2 ). Then d divides p(p — 1). On the other hand,
r d = 1 (mod p)
and r is a primitive root (mod p) so that d is a multiple of p — 1.
The two possibilities for d are, therefore, either d = p — 1
or d = p(p — 1). In the latter case r is a primitive root (mod p 2 ).
Thus we are concerned only with the alternative where
r^ 1 = 1 (mod p 2 ) (12-22)
Clearly, when r satisfies this congruence (12-22) it is not a primitive
root (mod p 2 ), but the situation may be remedied by using
ri = r + p
instead of r. Since
r\ = r (mod p)
the new number is also a primitive root (mod p). Furthermore,
by the binomial expansion we have
r/ -. . ^ + t^l^ p+ (p - »& ~ 2 V 3f)2 + ...
J- i. ' ^
so that
ri* -1 s r*- 1 + p(p - I)!*" 2 (mod p 2 )
EULER'S THEOREM AND ITS CONSEQUENCES 287
From the congruence (12-22) follows further
ri P-i _ i == ( p _ l) pr P- 2 (mod p 2 )
This shows that n does not satisfy the condition (12-22), con-
sequently it is a primitive root (mod p 2 ).
We have established that a primitive root r (mod p 2 ) is a
primitive root (mod p) such that the congruence (12-22) does
not hold, that is,
r p-i = 1 + tp (12-23)
where t is not divisible by p. We shall use this to show:
A number r with these properties is a primitive root for any
power p a .
For the proof it is necessary to establish the auxiliary fact that
one always has
r pa ~ l (p - 1} = 1 + t a • p a (12-24)
where the integer t a is not divisible by p. We have assumed this
to be true for a = 1, and may therefore use induction to prove it
in general. We raise both sides of (12-24) to the pth power and
expand the right-hand side by the binomial theorem to obtain
r p«(p-l) = ! + V . ta . p a + rtLzJl . ta * . p 2« + . . .
1 1.^5
where the subsequent terms contain p to powers with exponents at
least equal to 3a. When we write this expression in the form
r P a (p— l) _]__!_ r> a+1
+ E_i.C-^ +
■]
we shall have to show that the number t a+1 represented by the
bracket is not divisible by p. According to the induction as-
sumption t a is not divisible by p; the second term is divisible by
p (it is integral since p is an odd prime; this is the only place in
the proof of theorem 12-14 that this fact is used), and finally all
terms in the bracket not written out explicitly are also divisible
by p since 3a > a + 1 ; thus t a is not divisible by p as desired.
288 NUMBER THEORY AND ITS HISTORY
To prove that r is a primitive root for all powers of p we again
proceed by induction and assume that r is a primitive root (mod p a )
and show that it is a primitive root (mod p a+1 ).
Let us suppose that r belongs to the exponent d (mod p a+1 ).
r d s 1 (mod p a+1 )
We know that d is a divisor of
<p(p a+1 ) = p a (p - 1)
But since
r d = 1 (mod p a )
and r is a primitive root (mod p a ), d is a multiple of p a-1 (p — 1),
so that only the two values
d = p^Hp - 1) = <p(p a ), d = p a (p - 1) = <p(p a+1 )
are possible for d. But the first of these is excluded by the con-
dition (12-24), and r is a primitive root (mod p a+1 ).
As an example let us determine a primitive root for the powers
of 7. One finds that 3 is a primitive root (mod 7) and since
3 6 - 1 ^ (mod 7 2 )
it is a primitive root (mod 7 2 ) and therefore for all higher powers
of 7.
The powers of the prime 2 have been excluded from our con-
siderations and here the situation is different. For the modul
p = 2 there is the primitive root r = 1, and r = 3 (mod 2 2 ) is a
primitive root. But for the modul 8 and for higher powers of 2,
there are no primitive roots. The numbers relatively prime to
the modulus are in this case the odd numbers
The square is
so that for all of them
a = An d= 1
a 2 = 1 ± 8n + 16n 2
a 2 = 1 (mod 8)
EULER'S THEOREM AND ITS CONSEQUENCES 289
while <p(S) = 4. Again from
a 2 = 1 + St
one finds by successive squarings
a 4 = 1 (mod 16)
a 8 = 1 (mod 32)
a 2 " -2 = 1 (mod 2 a ) (12-25)
^(2 a ) = 2 a_1
and in general
Since
the congruence (12-25) shows that there can be no primitive roots
for the higher powers of 2.
The congruence (12-25) implies that the highest exponent to
which a number can possibly belong (mod 2 a ) is 2 a ~ 2 , when a ^ 3.
It is not difficult to find numbers belonging to this maximal ex-
ponent, for instance, a = 3 is one of them. This may be seen
from the following sequence of congruences, where each is obtained
from the preceding by squaring :
3 = -1+4 (mod 8)
3 2 == 1 + 8 (mod 16)
3 4 = 1 + 16 (mod 32)
= 1 + 2 a_1 (mod 2 a )
Since <p(2 a ) is a power of 2, every number belongs to an exponent
that is also a power of 2. But our last congruence shows that
when 3 is raised to the power 2 a_3 one still does not have the
remainder 1 (mod 2 a ). The power with the exponent 2 a_2 is
therefore the first with this property. It may be noted that 3 is a
primitive root (mod 2) and (mod 4) so that it belongs to the
290 NUMBER THEORY AND ITS HISTORY
greatest possible exponent for any power of 2. Let us summarize
the results:
Theorem 12-15. Among the powers of 2 only 2 and 4 have
primitive roots. For all higher powers of 2 a , a ^ 3, every odd
number satisfies the congruence
2«- = j<p ( 2«) s ! (mod 2 a }
a
The number 3 is a primitive root (mod 2) and (mod 4) and belongs
to the highest possible exponent 2 a ~ 2 when a ^ 3.
Problems.
1. Find primitive roots for the powers of all odd primes up to p = 19.
2. Find numbers other than 3 that belong to the exponent 2 a ~ 2 (mod 2 a ).
12-6. Universal exponents. After we have investigated the
existence of primitive roots for powers of primes, it is relatively
simple to find all numbers divisible by two or more different primes
for which there can be primitive roots. Let
m = pY . . . (12-26)
be some such number. For an arbitrary a relatively prime to m,
we know by Euler's theorem that
o^ (pa) = 1 (mod p a ) (12-27)
for any prime power p a occurring in (12-26). The least common
multiple of the various exponents in these congruences (12-27)
we shall denote by
M = [p*- 1 (p - 1), q*- 1 (q - 1), • • • ] (12-28)
Then clearly also
a M = 1 (mod p a )
for every prime power p a in (12-26), honce
a M = 1 (mod m) (12-29)
for every a relatively prime to m.
EULER'S THEOREM AND ITS CONSEQUENCES 291
If a primitive root is to exist (mod m), M cannot be less than
<p(m). But we know that
<p(m) = <p(p a ) v tf) • • •
and this product can only be equal to the l.c.m. (12-28) of its
factors when these are relatively prime. Ordinarily this is not
the case since all numbers <p(p a ) are even, with the single exception
0,(2) = 1. We conclude that only when m has the special form
m = 2p a
is there a possibility for a primitive root. But in this case such a
root is easily obtained. We notice
<p(m) = <p(2) • <p(p a ) =<p(p a )
so that when r is a primitive root (mod p a ), we have
,#(«) = i ( mo d p «) (12-30)
and <p(m) is the smallest exponent with this property. When r
is odd, the congruence (12-30) holds also (mod 2), and therefore
(mod m) ; consequently r is a primitive root (mod m). If r should
happen to be even, it may be replaced by
r x = r + p a
which is an odd primitive root (mod p a ).
To recapitulate our various results on primitive roots we state:
Theorem 12-16. There exist primitive roots only for the three
following classes of moduls:
1. m = p a is the power of an odd prime.
2. m = 2p a is the doubh of the power of an odd prime.
3. m = 2 and m = 4.
The same class of numbers as those that have primitive roots
we have encountered earlier in connection with Gauss's extension
292 NUMBER THEORY AND ITS HISTORY
of Wilson's theorem. On the basis of theorem 12-16 we can re-
formulate Gauss's theorem as follows:
Theorem 12-17. The product P of all positive remainders less
than and relatively prime to a number m satisfies the congruence
P = ± 1 (mod m)
where the sign —1 occurs if and only if there exists a primitive
root (mod m).
According to a remark which we made in connection with
theorem 11-6, we conclude also that a primitive root (mod m)
can exist only if the congruence
x 2 = 1 (mod m)
has no other roots than
x = ±1 (mod m)
We shall mention another problem closely related to those we
have discussed. We know that for a given modul m there exist
exponents M such that
a M = 1 (mod w)
for every a relatively prime to m; for instance <p(m) is such an
exponent according to Euler's congruence. We may call such a
number M a universal exponent (mod m), and among these there
is a minimal universal exponent X(m). This number X(m) is of
importance in several questions in number theory, and we shall
now show how it may be determined.
For a power p a of an odd prime, one must have
Hp a ) = <p(p a )
because there exist primitive roots belonging to the exponent
<p(p a )-
For the powers of 2 the expression for the minimal universal
exponent is slightly more complicated. Since there are primitive
roots (mod 2) and (mod 4), one has
X(2) = 1, X(4) = 2
EULER'S THEOREM AND ITS CONSEQUENCES 293
For the higher powers theorem 12-15 shows that
X (2«) = 2 a ~ 2 = %<p(2 a ), oc ^ 3
If one prefers to include the three cases in a single statement,
one can write
[{$ = a, when a ^ 3
X(2 a ) = 2 /3_2 -j/3 = 3, when a = 2 (12-31)
[p = 2, when a = 1
We now turn to the general case and prove:
Theorem 12-18. Let m be an integer with the factorization
m = 2 ao • p x ai • p 2 a2 • • • (12-32)
into prime powers. The minimal universal exponent of m is the
least common multiple
N = [X(2-), <p(pi ai ), <p(P2 a2 ), ' • •] (12-33)
of the corresponding minimal exponents of the prime powers in
(12-32). Furthermore, there exist numbers belonging to this
exponent (mod m).
Since N, according to its definition, is divisible by the smallest
universal exponent for each prime power p a occurring in (12-32),
we conclude that one has
a N = 1 (mod p a )
for any a not divisible by p, consequently also
a N = 1 (mod m) (12-34)
for any a relatively prime to m.
To complete the proof of theorem 12-18, it is sufficient to show
that there exists some number a for which N as defined is the
smallest exponent such that the congruence (12-34) holds. This
may be achieved by taking a to satisfy the congruences
a = 2, (mod2 ao ), a = n (modpi" 1 ),
a^r 2 (mod p 2 " 2 ), • • • (12-35)
294 NUMBER THEORY AND ITS HISTORY
where r u r 2 , . . . are primitive roots of the prime powers pi ai ,
P2 12 , •••• According to the Chinese remainder theorem, one can
always determine some a fulfilling all conditions (12-35). For
the various prime powers occurring in the decomposition (12-32)
of ra, the number a must belong respectively to the exponents
X(2«°), <p{p^), <p(p 2 °*), ...
consequently the smallest exponent to which a belongs (mod m)
is their l.c.m. N = X(m).
The number X(ra), sometimes called the indicator of m, is a
divisor of <p(m), but X(m) may be considerably smaller than <p(m)
for composite m. For instance, let us take
m = 720 = 2 4 • 3 2 • 5
Here
<p(m) = 192
while
X(m) = [2 2 , *>(9), *>(5)] = [4, 6, 4] = 12
As a consequence every number not divisible by 2, 3, or 5 satisfies
the congruence
a 12 = 1 (mod 720)
We may notice that X(m) according to its definition is even when
m > 2.
Problems.
1. Find the indicator X(m) for the following numbers:
(a) m = 385 (6) m = 144 (c) m = 5! (d) m = 10!
In each case try to find a number belonging to the exponent X(m) (mod to).
2. Find a primitive root for the moduls
(a) to = 54 (6) to = 50 (c) to = 68
12-7. Indices. We have solved the problem of finding all
moduls for which there exist primitive roots. One of the reasons
for placing emphasis on this question is that when such a root
exists, one can introduce a curious theory reminiscent of logarithms,
which in important cases facilitates the study of congruences.
EULER'S THEOREM AND ITS CONSEQUENCES 295
Let r be a primitive root for some modul m. We consider the
SeneS 1, r, r 2 , ..., r^" 1 (12-36)
of the <p(m) first powers of r. None of these can be congruent
(mod m). Let us assume, for instance, that i > j and
r l = r } (mod m)
Since r is relatively prime to m, we can cancel a power of r and
obtain i _ j
r J = 1 (mod m)
and this contradicts the fact that r as a primitive root belongs to
the exponent <p(m) (mod m).
The <p(w) numbers (12-36) are all incongruent and relatively
prime to m so that in some order they must be congruent to the
<p(m) relatively prime remainders (mod m). Therefore, for any
number a relatively prime to m, one can find a unique exponent i
such that
a = r l (mod m) (12-37)
where
^ i ^ <p(m) - 1 (12-38)
This exponent i we shall call the index of the number a for the root r
(mod m) and denote it by
i = Ind r (a) (12-39)
Very often the root r remains the same throughout some dis-
cussion, and it may then be dropped in the notation (12-39).
Among the special cases we notice particularly
Ind r 1=0, Ind r r = 1
As an example let us take the modul m = 9. Here <p(m) = 6
and r = 2 is a primitive root. We compute the various powers of
r and find 1 2 4 = 7
2, 2 5 = 5 (mod 9)
2 2 = 4, 2 6 = 1
2
3 _
296
NUMBER THEORY AND ITS HISTORY
This gives us the following table of the numbers with given indices
(mod 9) for the root r = 2:
Index
1
2
3
4
5
Number
1
2
4
8
7
5
To obtain the index of a given number one rearranges the table to
make the remainders relatively prime to 9 the primary entry.
Number
1
2
4
5
7
8
Index
1
2
5
4
3
In a similar example let us take m = 23. A primitive root is
r = 5, and through a reduction of the powers of 5 (mod 23), one
finds the table:
Index
1
2
3
4
5
6
7
8
9
10
Number
1
5
2
10
4
20
8
17
16
11
9
Index
11
12
13
14
15
16
17
18
19
20
21
Number
22
18
21
13
19
3
15
6
7
12
14
Through rearrangement of the entries in this table, one obtains
the companion table:
Number
1
2
3
4
5
6
7
8
9
10
11
Index
2
16
4
1
18
19
6
10
3
9
Number
12
13
14
15
16
17
18
19
20
21
22
Index
20
14
21
17
8
7
12
15
5
13
11
EULER'S THEOREM AND ITS CONSEQUENCES 297
In the definition (12-37) of the index, we assumed that it was
limited to the range given in (12-38). It is practical to drop
this convention and permit several indices for the same number.
If for two numbers i\ and i 2 , we have simultaneously
a == r ix = r h (mod m) (12-40)
it follows that
r i\-h = i ( mo d. m)
Since r belongs to the exponent <p(m) (mod m) this congruence is
possible only when i\ — i 2 is a multiple of <p(m), hence
tist2 (mod <p (m)) (12-41)
Conversely, when the congruence (12-41) is fulfilled it is easily
seen that (12-40) must hold. Therefore, in dealing with indices,
two indices shall be considered to be the same when they are
congruent (mod v(m)) and relations between indices may be
treated as congruences (mod <p{m)).
The defining congruence (12-37) for the index of a number can
be written in the form
a = r Indr(a) (mod m) (12-42)
This is entirely analogous to the definition of logarithms to some
base g by the common rule
a = g l0 ^ a
and the indices could well have been called congruence logarithms.
In regard to congruences, they have applications similar to those
of the logarithms for leal numbers. The idea of indices goes
back to Euler, but Gauss gives the first systematic discussion in
the third section of the Disquisitiones.
The basic law for logarithms is expressed in the formula for the
logarithm of a product
log (rib) = log a + log b
For indices one has analogously
Ind (rib) = Ind a + Ind b (mod <p(m)) (12-43)
298 NUMBER THEORY AND ITS HISTORY
The proof is immediate : From
a = r Inda^ h _ ^nd 6 ( mod m )
one obtains by multiplication
a-b = r Ind (a ' 6) = r Lnda + lndb (mod m)
from which the rule (12-43) follows.
Examples.
Let us take
w = 23, <p{m) = 22, a = 15, 6 = II
so that
ah = 165 s 4 (mod 23)
Here one finds from our previous table
Ind a = 17, Ind b = 9, Ind (ab) = 4
and this agrees with the rule (12-49) since
17 + 9 s 4 (mod 22)
The formula (12-43) can be extended to an arbitrary number
of factors. When applied to n equal factors a, one has the power
rule
Ind a n = n Ind a (mod <p(m)) (12-44)
Among the applications of the index theory, let us consider first
the solution of a linear congruence
ax = b (mod ra)
where a and b are relatively prime to the modul m. By taking-
indices, one finds according to (12-43)
Ind a -\- Ind x = Ind b (mod <p(m))
or
Ind x = Ind 6 — Ind a (mod <p{m))
Examples.
1. Let us solve the congruence
7x = 2 (mod 9)
EULER'S THEOREM AND ITS CONSEQUENCES 299
By means of the table of indices (mod 9), one concludes
Ind x = Ind 2 - Ind 7 s 1 - 4 s 3 (mod 6)
and the number corresponding to this index is
x = 8 (mod 9)
2. Next we solve the congruence
17s ss 9 (mod 23)
by means of the table of indices (mod 23). One finds
Ind x s Ind 9 - Ind 17 = 10 - 7 = 3 (mod 22)
and the number with this index is
x = 10 (mod 23)
It is seen from the examples that in the index calculus it is
convenient to have a double set of tables, one with the number
as entry giving the indices, and another with the indices as entries
giving the corresponding numbers. In this respect the indices
are not as easy to handle as the logarithms, since the values of the
logarithms occur in order and one can use the same table to find
both the logarithms and the antilogarithms.
Congruences of the type
ax n = b (mod m) (12-45)
may be solved readily by means of indices. By the previous rules
(12-43) and (12-44), one finds
Ind a + n Ind x = Ind b (mod <p(m))
so that Ind x is the solution of the linear congruence
n Ind x = Ind b — Ind a (mod <p(rn)) (12-46)
Similarly, for an exponential congruence
ab x = c (mod m) (12-47)
one finds the solution from
x Ind 6 = Ind c - Ind a (mod p(m)) (12-48)
It should be noted that the congruences (12-45) and (12-47) may
have one or several or even no solutions, depending on the be-
300 NUMBER THEORY AND ITS HISTORY
havior of the resulting linear congruences (12-46) and (12-48) for
the indices.
Examples.
1. Let us take
3x 5 = 11 (mod 23)
Here one finds
5 Ind x = Ind 11 - Ind 3 (mod 22)
or
5 Ind x = -7 (mod 22)
This congruence has a single solution
Ind x = 3 (mod 22)
and from the table one finds
x = 10 (mod 23)
2. We wish to find integers x such that
3x u = 2 (mod 23)
This leads to
14 Ind x = Ind 2 - Ind 3 = -14 (mod 22)
and one finds two solutions
Ind x s 10, Ind x = 21 (mod 22)
which correspond to the values
x = 9, x = 14 (mod 23)
3. There is no solution to the congruence
13* = 5 (mod 23)
because it leads to
x Ind 13 = Ind 5 (mod 22)
or
14x = 1 (mod 22)
which is impossible.
Let us mention finally that the indices may be used to determine
the exponent to which a number a belongs (mod m). This
exponent x is the smallest positive solution of
a x = 1 (mod m)
EULER'S THEOREM AND ITS CONSEQUENCES 301
Consequently
x Ind a = (mod <p(m))
and if <p{m) and Ind a have the greatest common factor d, one must
have
<p(m)
x =
d
For instance, the number 3 has the index 16 and since (16, 22) = 2
it belongs to the exponent 11 (mod 23).
The index theory is valid only for moduls with primitive roots.
This difficulty may be evaded by various methods, by generali-
zation of the theory or by relying on the fact that a congruence
can be reduced to prime-power moduls p a and that for such moduls
there exist primitive roots when p > 2. A more serious defect,
in comparison with logarithms, is that the tables of indices must
be computed separately for each modul. Gauss, in a supplement
to the Disquisitiones, gives a table of indices for the possible moduls
up to 100. A monumental achievement is the Canon ariihmeticus
of K. G. J. Jacobi, which contains dual sets of index tables for all
prime powers up to 1,000. By means of these tables, a large
number of congruence problems for moderate-sized moduls may
be solved with great ease.
Problems.
1. Solve the following problems by means of indices:
(a) 7x = 13 (mod 23)
(b) 4x = 19 (mod 23)
(c) Sx 7 = 11 (mod 23)
(d) llx 3 s 2 (mod 23)
(e) 5 • 7 X = 2 (mod 23)
(/) 7 • ll 1 = 15 (mod 23)
(g) To which exponents do 11, 13, and 15 belong (mod 23)?
302 NUMBER THEORY AND ITS HISTORY
2. Construct a table of indices (mod 19) and use it to solve the following
problems:
(a) 3x s 17 (mod 19)
(6) 3x 4 = 4 (mod 19)
(c) 13x 7 = 2 (mod 19)
(d) 3 • 5 X = 1 (mod 19)
(e) To which exponents do the numbers 2, 3, 4, and 5 belong (mod 19)?
12-8. Number theory and the splicing of telephone cables. We
have established previously that for each modul m there exist
numbers r that belong to the greatest possible exponent X(m),
the universal exponent or indicator (mod m). This may be stated
in the form that there exist numbers r relatively prime to m such
that the remainders (mod w) of the powers
r, r 2 , r 3 , ... (12-49)
avoid the value + 1 as long as possible.
We shall now discuss a problem of a very similar character. This
time we shall try to determine a number r such that the remainders
(mod m) of the series (12-49) avoid both values ±1 as long as
possible. For some number r, relatively prime to m, let us say
that t is its ±l-expownt when it is the smallest possible exponent
for which either one of the congruences
r' s ±1 (modm) (12-50)
is fulfilled. Our problem is then to determine the largest possible
±l-exponent X (m) that may occur (mod w). We notice that
the ± 1-exponent of a number can at most be equal to the exponent
to which the number belongs (mod m). The general result is:
Theorem 12-19. The maximal ± 1-exponent X (m) for a
number m > 2 has the value
Xo(m) = ^X(m) (12-51)
when there is a primitive root (mod m), and otherwise
Xo(m) = X(m) (12-52)
EULER'S THEOREM AND ITS CONSEQUENCES 303
Let us suppose first that there exists a primitive root (mod m).
We noticed in Sec. 12-6 that a primitive root can only exist when
the congruence
x 2 £= 1 (mod m)
has no other roots than
x = ±1 (mod m)
For a modul with a primitive root one has X(ra) = <p{m) and
we recall that <p(m) is even when m > 2. From Euler's theorem
follows
r <p(m) = ( r -^(m))2 s x (mcd m )
for any r relatively prime to m, consequently
r Wjn) = ±1 ( mod TO )
The maximal ±1 exponent X (m) can therefore at most be equal
to \(p(m). On the other hand it cannot be less than this number,
because if t ia the ±1 exponent of a primitive root r, it follows
from (12-50) that
r 2t = 1 (mod m) (12-53)
hence 2t is divisible by <p(m) and t is divisible by %<p(m). Thus
when there exists a primitive root, the maximal ±1 exponent
Xo(m) is determined by (12-51).
To complete the proof of theorem 12-19 we must deduce that
^o( m ) = M m ) when there is no primitive root.
Let us take some number r belonging to the exponent X(m). If
the ±l-exponent of r should be t < X(m), the congruence (12-53)
would hold and 2t would be divisible by X(w). This is only
possible when
t = ^X(m)
and then one must have
mm) = _ 1 ( mod m ) (12-54)
Therefore, if we can find a number r belonging to the exponent
X(ra) such that the congruence (12-54) does not hold, we will have
proved the equality (12-52).
304 NUMBER THEORY AND ITS HISTORY
The reasoning must again, as in so many number-theory inves-
tigations of this kind, be separated into several cases.
The prime factorization of to may be
to = 2>i> 2 a2 • • • (12-55)
and we suppose first that there are at least two odd primes pi and
p 2 . Among the ^-functions of the various odd prime powers in to,
let <p(pi ai ) be one that is divisible by the smallest power of 2 and
let us select a number Pl that belongs to the exponent |<?(Pi ai )
(mod pi ai )- The square p x = n 2 of a primitive root r x can be
used. A number r can be defined by the Chinese remainder
theorem by the following set of congruences
r = 3 (mod 2 ao ), r = pi (mod pf 1 ), r = r 2 (mod p 2 a2 ), • • •
where the subsequent numbers r 2 , r 3 , . . . are primitive roots for the
corresponding prime powers. The exponent to which r belongs
(mod m) is equal to the least common multiple
rx(2-), h(Vi ai ), *(P2 a2 ), <p(P3 a3 ), ...]
The manner in which pi was chosen insures that this number is
equal to A (to). But by the definition of r one has also
r i*(Pi«i) = pi i<p(pi ai ) = i (mod pf 1 )
and since i<p(pi ai ) divides J\(m) this implies
/\(m) = 1 (mod pi ai )
which is incompatible with the congruence (12-54).
In the next cases, the number w has the form
to = 2 ao • pi ai (12-56)
with «i ^ 1 and here a ^ 2 since there shall be no primitive root.
For the special cases
to = 4 • p! 011 , to = 8 • pi" 1
one obtains
X(m) = [2, *(?!«)! = <P(Vi ai )
EULER'S THEOREM AND ITS CONSEQUENCES 305
This shows that the number r defined by the congruences
r = 1 (mod 4), r = ri (mod pi ai )
where r x is a primitive root (mod pi ai ), must belong to the ex-
ponent X(m) (mod m). Furthermore, the congruence (12-54)
cannot hold since
r i\(m) = X (mod 4)
Fig. 12-1. Cross section of one layer of cable.
When we suppose that a Q ^ 4 in (12-56), we define our r
by the congruences
r = 3 (mod 2 a °), r = r x (mod p^ 1 )
and as in Sec. 12-5, it follows that r belongs to the exponent
X(m) (mod m). In this case the number A(2 a °), and therefore
X(m), is divisible at least by the second power of 2; from
3 2 = 1 (mod 8)
one concludes that
r lUm) s 3 ^X(m) _ x ( mod g)
which again shows that the congruence (12-54) is not fulfilled.
The same argument with r = 3 is applicable in the final case
where m is a power of 2.
H. P. Lawther, Jr., in an article in the American Mathematical
Monthly, has pointed out how such a theory may be applied to
introduce a systematic method for splicing telephone cables. The
306
NUMBER THEORY AND ITS HISTORY
cables for long-distance telephone service are manufactured in
concentric layers of insulated wires or conductors. The cable is
produced in sections of approximately uniform length, each perhaps
1,000 feet long, and the line is made up of a succession of such
sections spliced end to end. At the splices the order of the wires
should be mixed up considerably to minimize interference and
cross talk, and it is particularly desirable to avoid having two
s
Fig. 12-2. Splicing arrangement.
conductors that are adjacent in one section adjacent in sections
following closely afterward.
For practical purposes the splicing scheme cannot be compli-
cated, and it would seem that the following three rules would
embody the utmost in simplicity:
1. The same splicing directions should be used at each inter-
section.
2. The wires in one concentric layer are spliced to those in the
corresponding layer in the next section.
3. When some wire in one section &\ is spliced to a wire in the
next section S 2 , the adjacent wire in Si should be spliced
to the wire in S 2 that is obtained from the one last spliced
by counting forward a fixed number s. (See Fig. 12-2 for
the case s = 2.)
The number s may be called the spread of the splicing rule.
For instance, if one takes a layer in a cable in which there are 11
wires, the following table gives the splicings for the spreads s = 2
and s = 3.
EULER'S THEOREM AND ITS CONSEQUENCES 307
8 = 2
s =3
1-*1
1-»1
2^3
2-+4
3->5
3->7
4->7
4->10
5-»9
5-+2
6 -+11
6->5
7->2
7->8
8-+4
8 -+11
9-+6
9-*3
10^8
10->6
11 ->10
ll->9
The numbers in each of the right-hand columns are obtained
from the first 1 by adding 2 or 3 each time. Since the arrangement
of the wires is circular, the numbers are reduced (mod 11). One
sees that the first wire in Si always corresponds to the first wire
in S 2 ; this is only an expression for an agreement to let the count
in the second section start from the first wire spliced.
In general the splicing table for a spread s will run
1-*1
2->l + s
3 -+ 1 + 2s (mod m) (12-57)
1 + (i - 1)8
where the numbers on the right shall be reduced to their smallest
positive remainders (mod m), when m is the number of wires in a
cylindrical layer.
It should be observed that the correspondences (12-57) impose
a condition limiting the number of acceptable spreads s. As a
consequence of the rather trivial fact that two wires in Si always
are spliced to two distinct wires in >S 2 , one concludes that all
308 NUMBER THEORY AND ITS HISTORY
remainders of the numbers on the right in (12-57) must be
different. In other words, a congruence
1 -f- is == 1 + js (mod m)
or
(i - j) s = (mod m) (12-58)
will not be possible except when i = j. This is the case only
when the spread s is relatively prime to m; because if s had a common
factor d with m, the congruence (12-58) would be fulfilled when-
ever i andj differed by m/d or a multiple of this number.
Now let us repeat the splicing process. By the first operation
the ith. wire in Si was connected with the wire number
*2 = 1 + (*' _ 1 ) s
in S 2 . By the second splice this wire leads to wire number
h = 1+ {h ~ l)s = 1 + (i - 1> 2
and when this is continued the ith wire in the first section, after
n splices, will be connected with the wire numbered
i n+1 = 1 + (i - l)s n (mod m) (12-59)
in the (n + 1) section.
The object of our splicing arrangement was to produce a scatter
in the distribution of the connections such that two wires that
were adjacent in some section would stay separated as long as
possible in the subsequent sections. Let us examine how our
scheme behaves in this respect. Two adjacent wires in some
section may be numbered i and i + 1. According to (12-59),
after n splices, they become connected with the wires numbered
respectively
1 + (i - l)s n , 1 + is n (mod m)
These two wires are adjacent only if their difference is congruent
to ±1 (mod m), that is, when
1 + is n - (1 + (i - IK) = ±1 (mod m)
or
s n = ±1 (mod m)
EULER'S THEOREM AND ITS CONSEQUENCES
309
This basic condition leads us directly back to our previous
number-theory investigations. It shows that in order to keep
onetime adjacent wires separated as far as possible, one must
select the spread s as a number whose ±l-exponent (mod m)
has the maximal value X (m). In the preceding we have made all
necessary preparations for this problem. We know how such a
spread s may be determined and also how to compute the value of
the corresponding maximal ±l-exponent X (m).
In the paper by Lawther, one finds a table giving the value of
X (ra) for all ra's up to m = 139, as well as a suitable spread s.
For the first small values the table is as follows:
TO
Xo
s
m
Xo
s
5
2
2
13
6
2
6
1
5
14
3
3
7
3
3
15
4
2
8
2
3
16
4
3
9
3
2
17
8
3
10
2
3
18
3
5
11
5
2
19
9
2
12
2
5
20
4
3
The reader may check some of these results and compute the
values of X for some of the higher moduls m.
Let us make the final observation that when the modul is a
prime p, theorem 12-19 shows that
, v- 1
is the number of sections for which no two wires are adjacent
more than once. In comparison with the number m of wires, this
is the best possible result obtainable by any method. To see this,
one need only notice that when one starts with some wire it can
only remain separated from adjacent ones as long as it is possible to
find a new pair of wires between which it can be placed at each
splice, and altogether there are only \ (m - 1) such different pairs.
310 NUMBER THEORY AND ITS HISTORY
Let us only mention, finally, that there are other ways in which
this problem may be handled.
Bibliography
Cunningham, A. J. C, H. J. Wood all, and T. G. Creak: Haupt-exponenls,
Residue-indices, Primitive Roots and Standard Congruences, London, 1922.
Jacobi, K. G. J. : Canon arithmeticus, sine tabulae quibus exhibentur pro singulis
numeris primis vel primorum potestatibus infra 1000 numeri ad datos indices
et indices ad daios numeros pertinentes, Berlin, 1839.
Kraitchik, M. : Recherches sur la theorie des nombres, Paris, 1924.
Lawther, Jr., H. P.: "An Application of Number Theory to the Splicing of
Telephone Cables," American Mathematical Monthly, Vol. 42, 81-91 (1935).
CHAPTER 13
THEORY OF DECIMAL EXPANSIONS
13-1. Decimal fractions. In the opening chapter we discussed
the number systems and number representations. When the
basic group was b, we could write every integer uniquely by means
of the powers of & in the form
N = (c n , Cn_]_, • • • , ci, c )b = c n -b n
+ c n _i • b n ~ l + • • • + a • b + c
where the c/s can take values from to b — 1. From a mathe-
matical point of view, it may appear simple to extend this process
and use negative powers of b to represent the fractions, as we do
with our decimal fractions. Historically, however, this does not
seem to have been an easy step. In dealing with the fractions
those that are encountered first and the ones that are used most
commonly are the very simplest
112 13
2) 3> 3> 4=> 4> • • •
and it may not appear natural even now to put them in the strait
jacket of decimal notation. Furthermore, such representation may
even involve conceptual difficulties; for instance, ^ is a much more
easily understood and explicable concept than the infinite series
10 " 10* ' 10"
0.333 •••=— + -2 + ^3 +
These remarks may explain, at least in part, why the decimal
fractions made their appearance relatively late in the history of
the number concept and why a similar construction was not
achieved in other number systems. As an exception, one should
311
312 NUMBER THEORY AND ITS HISTORY
mention the Babylonian sexagesimal system. As we have already
stated, it was based on a positional principle but without an
absolute determination of values. Thus the symbol ff may
mean 2 or 2 X 60 or 2 X 60 2 and so on, but it can also signify
1 1
2 X — » 2 X — , » • • •
60 60 2
Although there was no indication of the integral part of a number or
even the power a certain symbol may denote, the system included
almost all the practical advantages of the decimal notation. For
instance, the common tables of inverses could be considered to
give the expansions according to the negative powers of 60. The
Babylonian preeminence in extensive and accurate computations
must have been induced by this technical perfection of their
number system. Probably this is one of the contributory reasons
for the persistent use of sexagesimal notations in many later
Greek and Arab mathematical texts, particularly in the field of
astronomy. We may recall that even Leonardo Pisano, the
ardent protagonist of the Hindu positional numbers, when con-
fronted with the necessity of computing a root of a cubic equation
with great accuracy in one of his tournament problems, resorts
to sexagesimal computations and gives the answer in this nota-
tion as
x= l°22 I 7 II 43 m 33 IV 4 v 40 VI
There are initial steps towards decimal notation and computa-
tions to be found among Arabic and European mathematicians
by the fifteenth century. These attempts usually appeared
through a desire to eliminate fractional computations by the
simplest possible operation, namely, the multiplication of the
numbers by powers of 10. For instance, to compute V2 one
would prefer to take the square root of the number
1,000^2 = ^2,000,000
and proceed to calculate with integers.
THEORY OF DECIMAL EXPANSIONS 313
The first work in which one finds computations with decimal
fractions is a collection of reckoning examples by Christoff Rudolff ,
which appeared in Augsburg in 1530. He uses a decimal notation
similar to the modern one, with a bar to separate the integral and
fractional parts. In spite of the fact that Rudolff's books, The
Rules of Coss and Reckoning Manual, enjoyed great contemporary
popularity, little is known about his life except that he spent most
of his time in Vienna as an "amateur of the liberal arts," as he
characterizes himself in the foreword to one of his books.
The first systematic presentation of the rules of operations on
decimal fractions was given in 1585 in a brief treatise in Dutch,
De Thiende, written by Simon Stevin. A French translation
under the title La Disme followed shortly afterward. Simon
Stevin (1548-1620) was a curious, many-sided genius who com-
bined his strong scholarly leanings with great ability for practical
constructive and organizational tasks. He made inventions of
various kinds, became quartermaster general of the Dutch army
during the critical period of the Spanish wars, and also was put
in charge of the system of dikes and waterworks. He reorganized
the system of governmental bookkeeping, computed interest
tables, and proposed reforms in weights and measures suggestive
of the metric system. His writings on the art of fortifications
are well known, but at the same time he translated Diophantos into
French and published books on arithmetic, geometry, mechanics,
and hydrostatics. He seems to have been the first to formulate
explicitly the law of the triangle or parallelogram of forces
in mechanics.
La Disme has the subtitle "Teaching how all computations that
are met in business may be performed by integers alone without
the aid of fractions." The pamphlet opens as follows:
To astrologers, surveyors, measurers of tapestry, gaugers, stereometers
in general, mintmasters and to all merchants Simon Stevin sends greeting:
A person who contrasts the small size of this book with your greatness,
my most honorable sirs to whom it is dedicated, will think my idea absurd,
especially if he imagines that the size of this volume bears the same ratio
to human ignorance that its usefulness has to men of your outstanding
314 NUMBER THEORY AND ITS HISTORY
ability; but in so doing he will have compared the extreme terms of the
proportion which may not be done. Let him rather compare the third
term with the fourth.
What is it that is here propounded? Some wonderful invention?
Hardly that, but a thing so simple that it scarce deserves the name in-
vention; for it is as if some stupid country lout chanced upon great
treasure without using any skill in the finding. If any one thinks that,
in expounding the usefulness of decimal numbers, I am boasting of my
cleverness in devising them, he shows without doubt that he has neither
the judgment nor the intelligence to distinguish simple things from
difficult, or else that he is jealous of a thing that is for the common good.
However this may be, I shall not fail to mention the usefulness of these
numbers, even in the face of this man's empty calumny. But, just as
the mariner who has found by chance an unknown isle, may declare all
its riches to the king, as, for instance, its having beautiful fruits, pleasant
plains, precious minerals, etc., without its being imputed to him as conceit;
so may I speak freely of the great usefulness of this invention, a usefulness
greater than I think any of you anticipates, without constantly priding
myself on my achievements. 1
The introduction of the decimals was an innovation for which
the time was ripe, and after the publication of La Disme one sees
a rapid increase in their use. In numerical tables the advantages
of decimals over ordinary fractions are particularly evident.
Shortly before the appearance of La Disme, Stevin had computed
and published a set of interest tables, and one may well conjecture
that this had led him to an appreciation of the simplicity of decimal
procedures. The invention of logarithms and the introduction of
logarithmic tables shortly afterwards undoubtedly contributed
greatly to the progress of decimals.
Stevin presaged another practical innovation in a statement
toward the end of the pamphlet
In view of the great usefulness of the decimal division, it would be a
praiseworthy thing if the people would urge having this put into effect so
that in addition to the common divisions of the measures, weights and
money, that now exist, the state would declare the decimal division of the
1 Translation by V. Sanford, in D. E. Smith, A Source Book in Mathe-
matics.
THEORY OF DECIMAL EXPANSIONS 315
large units legitimate to the end that he who wished might use them. It
would further this cause also, if all new money should be based on this
system of primes, seconds, thirds, etc. [tenths, hundreds, thousands]. If
this is not put into operation as soon as we might wish, we have the con-
solation that it will be of use to posterity, for it is certain that if men of
the future are like men of the past, they will not always be neglectful of
a thing of such great value.
Not all of Stevin's reform proposals had the same ultimate
success. In one of his works on geography he makes strong
propaganda for Dutch as a world language, and he demonstrates
its supremacy over other languages in the briefness of expression
and in its superabundance of words of one syllable.
13-2. The properties cf decimal fractions. In computing with
decimals it soon became apparent that the numbers fall into
several distinct categories with respect to their expansions. For
some the expansion might break off after a finite number of terms;
for others it might be infinite. In the latter case the digits could
progress without any discernible law, or from a certain point on
they might repeat themselves periodically, forming circulating
decimals. Certain rules for the expansion of rational numbers
into decimals had already been propounded by earlier mathe-
maticians, but not until Gauss were those indispensable tools of
number theory created that were required for a systematic explor-
ation. In the sixth and next to last section of the Disquisitiones,
Gauss considers the use of his results in certain applications, among
them the properties of decimal expansions of fractions.
The characterization of the numbers with a finite decimal
expansion is quite simple; the main idea has in reality been pre-
sented already in our discussion of the Babylonian mathematical
tablets. As a general remark for the subsequent considerations,
let us observe that when dealing with the properties of decimal
expansions, we shall usually disregard the integral part of a number
since it has no influence on the decimals.
When some number has a finite expansion
ai a 2 a n
r = 0, a u a 2 , • • • , a n = — -f- j^ i r 1Qn
316 NUMBER THEORY AND ITS HISTORY
it may be written
a! • 10 71 - 1 + « 2 • 10 n ~ 2 + • ■ • +a n p
r =
<2
Therefore r is a rational nurriber whose denominator q in the
reduced form contains only prime factors of 10, that is, 2 and 5.
For instance,
0.1375 = i™ = 11 - "
10,000 80 2 4 • 5
Conversely let us take a fraction
P
r =
2 a -5 /3
If, for example, a ^ /?, we may multiply both numerator and
denominator by 5 a_ ^ so that
5 a -Pp
r =
10 a
and this shows that r can be written with a finite decimal expansion.
Theorem 13-1. The numbers with a finite decimal expansion
are the fractions
r = ^r^ ( 13 -!)
It is not difficult to see that when r is in reduced form (13-1),
the number of decimals in the expansion is the larger of the two
exponents a and (3.
In general, let us say that a number is regular with respect to
some base number b when it can be expanded in the corresponding
number system with a finite number of negative powers of b.
Through exactly the same argument as before, one concludes
that the regular numbers are the fractions
V
r = -
THEORY OF DECIMAL EXPANSIONS 317
where q contains no other prime factors than those that divide b.
Then there exists some exponent n such that q divides b n , and
when r is reduced and n is the smallest exponent that can be used
for this purpose, the number of "decimal" terms is exactly n.
Since we are mainly interested in the number-theory properties
that regulate the expansions into decimal fractions, we shall sup-
pose that the reader is familiar with the essential principles of
decimals. In particular, we know that every real number can be
expressed by a decimal fraction, and this expansion can be per-
formed in only one way. The single exception to this last state-
ment occurs for numbers with a finite expansion. Here one can
diminish the last digit by one unit and continue with an infinite
series of nines, as for instance, in the examples
1.00 ••• = 0.999 • • • , 0.375 = 0.374999 • • •
Since we have already completed the discussion of numbers with
finite expansions this anomaly need not concern us here.
We shall examine the decimal expansions one obtains for tie
various reduced fractions
r = -, (m,n) = 1 (13-2)
n
where the denominator n is some fixed integer. Gauss calls the
decimal sequence of a number, disregarding the integral part, the
mantissa of the number. This is a term that was first introduced
by Briggs in connection with his logarithms to the base 10, and
he uses it in the sense of a "minor part" or "appendix." As we
have already stated, the mantissa of a number is the main object
of our subsequent studies.
Two fractions m/n and l/n in (13-2) have the same mantissa
when their difference
m I _ ,
n n
is an integer, in other words when
m = I (mod n)
318 NUMBER THEORY AND ITS HISTORY
Therefore, to obtain the various mantissas for the fractions
(13-2), we can limit ourselves to those fractions in which the
numerator m is one of the <p(n) remainders less than and relatively-
prime to n. For instance, for the denominator n = 18, the
fractions
1 5 _7_ li II 1.1.
T8> T8' 18' 18' 18' 18
will exhaust all possibilities for the mantissas.
Another simple but essential remark is the following: When
the mantissa of the fraction m/n is known, one finds the mantissa
of the fraction 10 • m/n by dropping the first digit on the left.
For instance
i = 0.142857 • • •
has the mantissa 142857 . . . , while the mantissa of hf- is 428571 ....
On the basis of these observations, it is not difficult to prove
that the decimal expansion of a rational number m/n is always
periodic, i.e., that the mantissa after a certain number of terms
consists of groups of digits that keep repeating themselves indefi-
nitely. For instance,
5 7 = 0.3454545 • • •
loo
We consider the series of fractions
m 10m 10 2 m , .
— } ) > . . . \16-o)
n n n
Here the mantissa of the first fraction will produce those of all
the subsequent ones by leaving out successively one, two, and so
on, digits from the left. But in the unlimited sequence of numer-
ators of the fractions (13-3)
m, 10m, 10 2 w, . . .
the numbers cannot all be incongruent (mod n). Consequently,
there exists some first exponent s such that
m ■ 10 s = m • 10 s+t (mod n) (13-4)
THEORY OF DECIMAL EXPANSIONS 319
and the two fractions
m- 10 s m • 10 8_H
have the same mantissa. This means that leaving out s digits
and s + t digits from the mantissa of m/n will produce the same
sequence. We conclude, therefore, that in the decimal expansion
of m/n the digits will repeat themselves periodically after s terms,
in groups of length t.
Since in (13-2) we made the assumption that m and n were
relatively prime, the factor m in (13-4) can be canceled to give
the equivalent congruence
10 s = 10 s+< (mod n)
Conversely, let us take some periodic decimal fraction
r = 0.aia 2 • • • a s b x b 2 • • • b t b x b 2 • • • b t • • • (13-5)
where the period has the length t and begins after the s first terms.
When (13-5) is multiplied by 10 s and 10 s+< one obtains
10 S+ V = a x a 2 ■ - • a s bib 2 • ■ • b t .hb 2 ■ • • b t • • •
10V = aia 2 • • • a s .b\b 2 • • • b t • • •
The two numbers on the right have the same mantissa so that
their difference A is an integer. By subtracting one from the
other, we find
(10 S "M _ 10 «) r = A
and we conclude that
A m
r =
l s+t _ 10 s
n
is rational. Furthermore, it is clear that 10 s and 10 s+ * are the
smallest powers of 10 such that r • 10 s and r • 10 s+ ' have the same
mantissa. We conclude therefore:
Theorem 13-2. Any periodic decimal fraction represents a
rational number, and conversely any rational number has a
320 NUMBER THEORY AND ITS HISTORY
periodic decimal expansion. When a rational number
m ( \ i
— > (w, n) = 1
n
is expanded, the period begins after s terms and has a length t,
where s and t are the smallest numbers such that
10 s = W +t (mod n) (13-6)
This condition shows the further interesting fact that the length
of the period as well as the point at which it starts depends only
on the denominator n and not on the numerator m.
Examples.
1. Let us consider the expansions with denominator 18. For the powers
of 10 (mod 18), one finds simply
1, 10, 10 2 = 10
so that s = 1 and t = 1. This is confirmed by the decimal fractions
J^ = .055 • ■ ■ , H = - 611 • • •
A-.277--., H--722-.-
^ = .388.-., i| = .944...
2. In a second illustration we take the fractions with denominator 84. The
remainders of the powers of 10 (mod 84) are found to be
1, 10, 16, -8, 4, 40, -20, -32, 16
The period therefore begins after two terms and has the length 6; for instance,
|f = .44 047619 047619
When the denominator n of the fraction to be expanded has no
factors 2 or 5, the conditions become simpler. Since n is relatively
prime to 10, we can cancel in the congruence (13-6) to obtain
10* = 1 (mod n) (13-7)
This shows that we have s = and the period starts with the first
decimal, or as one sometimes says, the expansion is purely periodic.
The length of the period is the exponent to which 10 belongs
THEORY OF DECIMAL EXPANSIONS 321
(mod n). Conversely, it is clear that a congruence (13-7) can
hold only when n and 10 are relatively prime so that we may say:
Theorem 13-3. The decimal expansion of an irreducible
fraction m/n is purely periodic if and only if n has no prime factors
2 and 5, and in this case the length of the period is equal to the
divisor t of <p(n) to which 10 belongs (mod n).
Example.
Let us take n = 7. The remainders of the successive powers of 10 (mod 7)
are
1, 3, 2, 6, 4, 5, 1
This shows that 10 belongs to the exponent 6 (mod 7), or, in other words, 10
is a primitive root (mod 7). All fractions with denominator 7 have periods of
length 6; for instance,
f = .285711 - • •
We return for a moment to the general case where the denom-
inator n in the fraction may have factors 2 and 5, and write
n = n {) -2 a -5 , (n , 10) - 1
The preceding result may be used to reformulate the criteria in
theorem 13-2. When one multiplies the fraction m/n by 10 M ,
where m is the larger of the exponents a and /3, the resulting fraction
2 n.-a . 5 m-0 . m
10" - =
n no
has a denominator relatively prime to 10 and n is the lowest
exponent by means of which this can be achieved. Theorem 13-3
gives us then:
Theorem 13-4. When the denominator of a fraction m/n has
the form
n = wo • 2 a • 5", (n , 10) = 1
the period in the decimal expansion of m/n begins after n terms,
where ju is the larger of a and 0, and the length of the period is the
exponent to which 10 belongs (mod no).
322 NUMBER THEORY AND ITS HISTORY
Examples.
Let us reconsider our previous examples from this new point of view. When
n = 18 = 2 • 9,_the period must begin after the first term and have the length
1 since
10 = 1 (mod 9)
When n = 84 = 2 2 • 21, the period starts after the second decimal and has the
length 6 since one finds that 10 belongs to the exponent 6 (mod 21).
We have mentioned that in the expansion of an irreducible
fraction m/n in decimals all periods begin at the same point and
have the same length for a given n. Let us discuss briefly the
interrelation between the various periods defined by n. We may
multiply m/n by a suitable power of 10 so that the expansion
becomes purely periodic, or, equivalently, n shall be assumed
relatively prime to 10.
The general situation can best be explained on the basis of
some examples.
Examples.
1. We first take the decimal expansions of fractions with denominator 7.
Since 10 belongs to the exponent 6 (mod 7), the period is 6 and
\ = .142857 • • •
This fraction is multiplied by the successive powers of 10 and the integral parts
discarded. There results a set of decimal fractions whose mantissas are
derived from that of \ by leaving out one, two, and so on, digits on the left.
Since the powers of 10 (mod 7) have the remainders
1, 3, 2, 6, 4, 5, 1
that
j- = .142857 • • •
f = .857142
f = .428571 • • •
-f- = .571428
f- = .285714 • • • ,
f = .714285
All possible mantissas for the denominator 7 can therefore be obtained from
one of them by permuting the digits in the period cyclically. It is evident
that the same situation will prevail whenever 10 is a primitive root (mod n).
An index table in which the numbers are arranged according to their indices
will give the information as to which period appears for a prescribed fraction.
THEORY OF DECIMAL EXPANSIONS 323
2. Next let us take an example where 10 is not a primitive root (mod n),
for instance, n = 13. In this case the remainders of the powers of 10 (mod 13)
are
1, 10, 9, 12, 3, 4, 1
and from the expansion of T V one consequently finds
T V = .076923 • • • ,
f f = .923076
if = .769230 • • • ,
T 3 3 = .230769
T 9 3 = .692307 • • •
T 4 3 = .307692
Here we have only half of the twelve reduced proper fractions with denominator
13. The numerator 2 is not among them so that we multiply 2 by the
powers of 10 (mod 13). The remainders are
2, 7, 5, 11, 6, 8, 2
and, correspondingly, one has the cyclic family of expansions
T 2 3 = .153846 • • • , H = .846153 • • •
^ = .538461 • • • , T 6 3 = .461538 • • •
5 _
T3" -
.384615 ■ • • , T 8 3 = -615384
In the general case the situation is analogous. When the
number 10 belongs to the exponent t, the <p(n) mantissas of the
fractions with the denominator n will fall into cp (n)/t families, the
periods are of length t and within each family the mantissas are
obtained by cyclical permutations as above. For instance, 10 be-
longs to the exponent 5 (mod 41) ; hence, the period of the fractions
with denominator 41 is equal to 5 and the mantissas fall into 8
cyclical classes.
There exist tables that give the classes of mantissas for all
numbers not divisible by 2 or 5 up to certain limits. Gauss in an
appendix to the Disquisitiones gave such a table, which he later
enlarged. The most complete tables of this kind are due to
H. Goodwin and include the mantissas for all denominators up
to 1,024, but these tables are now so rare that they are practically
unavailable.
Very extensive tables have been computed to determine the
exponents to which the number 10 belongs for various moduls.
324 NUMBER THEORY AND ITS HISTORY
In arranging these tables one makes use of the fact, which is
easily proved, that if 10 belongs to the exponent a (mod m) and
the exponent b (modn), then for the least common multiple of
m and n as modul, it belongs to an exponent that is the l.c.m. of
a and b. It is sufficient, therefore, to tabulate the exponents to
which 10 belongs for the various prime powers p a . For
p a < 10,000, such tables have been constructed by A. J. C.
Cunningham and coworkers. For prime moduls, still more
extensive tables with p < 120,000 have been computed; one
should mention particularly those by W. Shanks.
One may be interested in determining the denominators that
yield short periods. Such a study is facilitated through the
factorization of the various numbers 10 fc — 1 into prime factors.
For the first few exponents one obtains
10 - 1 = 3 2
10 5 - 1 = 3 2 • 41 • 271
10 2 - 1 = 3 2
11
10 6 - 1 = 3 3 • 7 • 11 • 13 • 37
10 3 - 1 = 3 3
37
10 7 - 1 = 3 2 • 239 • 4,649
10 4 - 1 = 3 2
11 • 101
10 8 - 1 = 3 2 • 11 • 73 • 101 • 137
10 9 - 1
= 3 4 • 37 • 333,667
From this table one concludes, for instance, that 7 and 13 are the
only primes whose periods have the length 6, while 239 and 4,649
have the period 7, and so on.
In the preceding we have limited ourselves exclusively to the
case where the base of the number system is 10, but it is evident
that the results we have obtained are valid, with very small
modifications, for arbitrary base numbers. For example, when
examining the Babylonian tables of inverses, one may wish to
know which denominators correspond to short periods in the
sexagesimal system. The following prime factorizations yield
this information:
60 - 1 = 59 60 3 - 1 = 7 • 59 • 523
60 2 - 1 = 59 • 61 60 4 - 1 = 13 • 59 • 61 • 277
60 5 - 1 = 11 • 59 • 1,198,151
THEORY OF DECIMAL EXPANSIONS 325
Problems.
1. Find the length of the decimal period for fractions with the denominators
n = 17, n = 31, n = 39, n = 43
and find the corresponding families of mantissas.
2. At which point does the period begin when n = 10!?
3. Find all numbers whose periods are 6 and 12 in a number system with
the base 2.
4. In which number systems does a prime power p a give a period of length 2?
Bibliography
Cunningham, A. J. C, H. J. Wood all, and T. G. Creak: Haupt-exponents,
Residue-indices, Primitive Roots and Standard Congruences, London, 1922.
Goodwin, H. : A Table of Circles arising from the Division of a Unit or Any
Other Whole Number by All the Integers from 1 to 1024, being all the pure
decimal quotients that can arise from this source, London, 1823.
: A Tabular Series of Decimal Quotients of All Proper Vulgar Fractions
of which, when in their lowest terms, neither the numerator nor the denominator
is greater than 1000, London, 1823.
Stevin, Simon: La Disme. Translation in D. C. Smith, A Source Book in
Mathematics, McGraw-Hill Book Company, Inc., New York, 1929.
CHAPTER 14
THE CONVERSE OF FERMAT'S THEOREM
14-1. The converse of Fermat's theorem. Fermat's theorem
states that for every number a not divisible by the prime p, the
congruence
a p_1 = 1 (mod p)
is satisfied. It is natural to investigate conversely whether the
fact that some congruence of this kind holds implies that the
modul is a prime.
In general such a conclusion is not valid. There exist numbers
a and n such that
a"- 1 = 1 (modn), a j£ 1 (modn) (14-1)
without n being a prime. Several writers have made this observa-
tion; for instance, F. Sarrus (1819) noted the congruence
2 340 = 1 (mod 341)
where the modul 341 = 11-31 is composite. Another example is
3 90 = 1 (mod 91), 91 = 7-13
and numerous other instances may be given.
However, by imposing additional restrictions on the number
a in the congruence (14-1), it is possible to express a converse
form of the theorem of Fermat. This observation was first made
and applied by the French specialist in number theory E. Lucas
(1876). His original theorem was
Theorem 14-1. When for some number a the congruence
a B_1 = 1 (mod n) (14-2)
326
THE CONVERSE OF FERMAT'S THEOREM 327
holds, while no similar congruence with a lower exponent
a 1 = 1 (mod n), n - 1 > t > (14-3)
is fulfilled, the modul n is a prime.
On the basis of our previous results the proof is immediate. The
condition of the theorem states that the number a belongs to the
exponent n — 1 (modn). But the highest exponent to which a
can belong is <p(w). We recall further that
„(„)_„(!_!). ..(,_£)
where the pi's are the different primes dividing n so that
/ 1 \ n
(pin) ^ n ( 1 ) = n ^ n — 1
\ Pi/ Pi
This shows that one can have
(p(n) = n — 1
only when n = pi is a prime.
When it comes to the actual verification that a number is a
prime, Lucas's theorem is not very practical in the form in which
it stands. However through a few further remarks it may be
effectively improved upon.
First, if a congruence (14-3) should hold for some exponent t,
the number a will belong to an exponent d less than n — 1.
According to the congruence (14-2), d would divide n — 1.
Instead of investigating all congruences (14-3), therefore, it is
sufficient to examine whether such a congruence can hold when
the exponent is a proper divisor of n — 1.
Second, one need not consider all divisors t of n — 1, because
when the congruence (14-3) holds for some t, it must be fulfilled
for all multiples of t. This leads us to reformulate the theorem of
Lucas as follows:
Theorem 14-2. Let n be some integer and
q x , q 2 , • • ■ , q s
328 NUMBER THEORY AND ITS HISTORY
the different prime factors dividing n — 1. If for some number a
the congruence
a n-i = i (mod n)
holds, while none of the congruences
n-\
a
1 (mod n) i = 1, 2, • • • , s (14-4)
are fulfilled, the number n is a prime.
To prove this statement it suffices to observe that any divisor
of n — 1 must divide one of the maximal divisors
n — 1 n — 1
so that when a congruence (14-3) holds for some exponent t
dividing n — 1, at least one of the congruences (14-4) must be
satisfied.
To apply the theorem one selects some number a, usually small,
for instance, a = 2 or a — 3, and computes the remainder of the
power a n ~ x for the modul n. If the congruence (14-2) should
not be fulfilled, one concludes that n is not a prime. The method
itself is quite practical. However, it has the disadvantage that
when it has been used to decide that some particular number is
composite, one is left in the rather curious position of having no
clue to what the factors may be.
In the other alternative, if it should turn out that the congruence
(14-2) is true for some number a, the prime factors of n — 1
must be found and the congruences (14-4) examined. If none
of them hold, n is a prime; when one of them is fulfilled, the
method gives no final decision and one can try the same procedure
for some other number a. When there are few different prime
factors of n — 1, the number of congruences (14-4) to be investi-
gated is small. This is the case for many of the larger special
numbers to which the method has been applied.
Lucas's converse form of Fermat's theorem involves the com-
THE CONVERSE OF FERMAT'S THEOREM
329
putation of the remainders (mod n) of the high powers of a and
it is essential that the work be organized in the most effective
manner. One suitable procedure will be illustrated first on the
very simple example n = 143 and a = 2.
A
B
C
142
2,116 s
-29
71
2,025 =
23
46
35
3,481 s
49
98 = -45
17
900 s
42
84 = -59
8
256 =
-30
4
16
2
4
1
(mod 143)
The left-hand column is constructed first. It begins with the
top entry n — 1 = 142, and each successive number is the quo-
tient of the preceding when divided by 2. These entries are the
exponents of the various powers to which a = 2 shall be raised in
the second column. Here one proceeds from the bottom upward.
The lowest entries are a raised to the powers 1, 2, 4, and 8, respec-
tively. To obtain the 17th power, the preceding entry, —30, is
squared and reduced (mod n) to give 42 as the remainder of the
16th power; this is multiplied by 2 and entered in the third column
as the remainder —59 of the 17th power. Similarly by squaring
— 59, one has the remainder 49 of the 34th power, which is doubled
and entered in the third column for the 35th power. Since finally
the computations show that the 142nd power is not congruent to 1,
the conclusion is that the number 143 is composite.
This example, although trivial, gives the key to the general
setup, which is well adapted to machine computation. The
third column is largely superfluous and it has been included above
only for greater clarity of explanation.
We shall give some examples that illustrate the power of the
method.
330
NUMBER THEORY AND ITS HISTORY
Examples.
1. We take n = 700,001 and use the auxiliary number a - 3.
of computations takes the form:
The table
A
B
A
B
700,000
1
683
564,086
350,000
700,000
341
111,831
175,000
197,937
170
2,222
87,500
182,523
85
686,051
43,750
413,705
42
546,504
21,875
70,917
21
238,260
10,937
481,554
10
59,049
5,468
37,250
5
243
2,734
426,023
2
9
1,367
282,506
1
3
This shows that our number is likely to be a prime. The prime factors
of 700,000 are 2, 5, and 7 and the corresponding quotients
350,000, 140,000, 100,000
By similar computations one obtains
3 350, ooo s 700,000
3100,000 = 59^336 (mod 700,001)
3140,000 = 425,344
Since none of these remainders is 1 we conclude that 700,001 is a prime.
2. We want to determine the character of the number 373,831. We use
a = 2 and obtain the following table of residues:
A
B
A
B
373,830
104,740
365
333,424
186,915
110,115
182
261,278
93,457
327,152
91
323,778
46,728
182,017
45
236,742
23,364
313,895
22
82,163
11,682
140,353
11
2,048
5,841
345,890
5
32
2,920
111,522
2
4
1,460
124,779
1
2
730
205,672
This shows that the number is composite.
THE CONVERSE OF FERMAT'S THEOREM 331
Both of the last two examples are trivial in the sense that the
numbers examined are within the limits of the prime tables. The
importance of the method lies, of course, in the fact that it is
applicable to numbers of arbitrary size.
D. H. Lehmer and P. Poulet have contributed a valuable
adjunct to the method of testing primality by means of the con-
verse of Fermat's theorem. Through various ingenious devices
they have constructed tables containing the composite numbers n
up to 100,000,000 for which the congruence
2 n_1 = 1 (mod n) (14-5)
is fulfilled, and for each such n a prime factor is given. The
tables of Poulet are somewhat simpler to use than those of Lehmer,
which leave out numbers n with prime factors not exceeding 313.
To determine whether a number n within the limit of the tables
is a prime, one checks first by Poulet's tables whether it is one of
the exceptional composite numbers for which (14-5) holds. When
this is not the case, n is a prime if and only if the congruence
(14-5) is satisfied, and through our previous method the test can
be performed fairly quickly. (See Supplement.)
Problems.
Check by the converse theorem of Fermat whether the following numbers
are composite or prime:
1. w = 2 16 + 1 3. n = 300,301
2. n = 1,111,111 4. n = 1,234,567
14-2. Numbers with the Fermat property. We have mentioned
in the last section that for certain composite numbers n there may
exist numbers a for which
a" 1 ' 1 = 1 (mod n) (14-6)
Much more remarkable is the fact that one can find numbers n
that are not prime such that Fermat's congruence (14-6) is satis-
fied for every number a relatively prime to n. Numbers of this
kind shall be said to have the Fermat property or, for short, we
may call them F numbers.
332 NUMBER THEORY AND ITS HISTORY
The existence of F numbers was first pointed out by R. D.
Carmichael (1909). The smallest among them is
561 = 3 • 11 • 17 (14-7)
and they are on the whole quite rare. Below 2,000 there are
only two others, namely,
1,105 = 5-13-17, 1,729 = 7-13-19
Relatively few investigations on F numbers have been made, and
little is known about them beyond the properties we shall deduce
in the following.
From the definition of the universal exponent or indicator \{n)
of the number n, we conclude first that a congruence (14-6) can
hold for all a's relatively prime to n only when the exponent
n — 1 is divisible by X(n). This leads to the basic criterion for
an F number:
Theorem 14-3. The necessary and sufficient condition for
a number n to have the Fermat property is that
n = l(modX(n)) (14-8)
where X(n) is the universal exponent (mod n).
To illustrate the application of this theorem let us verify that
the number 561 in (14-7) actually is an F number. We recall the
formula for X(n) and find
X(561) = [„(3), *>(11), *(17)] = [2, 10, 16] = 80
and corresponding to (14-8) one has
561 = 1 (mod 80)
Certain simple properties of the F numbers flow directly from
the criterion in theorem 14-3:
Theorem 14-4. A number with the Fermat property is odd
and equal to a product of different prime factors
n = p x p 2 • • -p s (14-9)
where the number of primes is at least three.
THE CONVERSE OF FERMAT'S THEOREM 333
These results were given first by Carmichael and for the examples
of F numbers given above they are evidently fulfilled.
For the proof of theorem 14-4 we observe that according to the
congruence (14-8), the two numbers n and X(n) cannot have any
common factor. The number X(n), as we have mentioned several
times before, is always even when n > 2, consequently n is odd.
Next let p a be the highest power of some prime p that divides n.
The indicator \{n) is divisible by
<p(p") =?r- 1 (p-i)
so that if a > 1, both n and X(n) would have the factor p. This
establishes that an F number has the form (14-9). It remains to
prove that n cannot be the product of two different primes. In
that case one would have
n — P1P2
and the indicator would be
X(n) = [pi - 1, p 2 - 1]
To fulfill the congruence (14-8) the number
n - 1 = P1P2 - 1 = (Pi - 1)P2 + P2 — 1
must be divisible by pi — 1. This is possible only when pi — 1
divides p 2 — 1, and in the same manner one concludes that p 2 — 1
must divide pi — 1. Consequently
Pi - 1 = p 2 - 1
and the two prime factors would be equal, contrary to our previous
conclusion.
On the basis of theorem 14-4 the condition (14-8) for an F
number may be reformulated. When n is a product (14-9) of
different prime factors, the corresponding indicator is the least
common multiple
X(n) = [pi - 1, • • • , p s - 1]
334 NUMBER THEORY AND ITS HISTORY
The single congruence (14-8) may therefore be replaced by the
family of congruences
n = l (modp,--l) * = 1,2,...,« (14-10)
The F numbers, as we stated, are quite scarce. In the tables by
Poulet, which were mentioned in the preceding section, the com-
posite numbers n were listed for which the congruence
2 n-i = x ( mo d n )
holds. The F numbers must be found among these entries; they
have been especially marked with an asterisk and below 100,000,-
000 one counts a total of 250.
A method for constructing F numbers has been given by J.
Chernick (1939) and a similar method by S. Sispanov (1941). The
three examples we quoted all contain three prime factors, and we
shall discuss this case in some detail.
Let
n = V1P2P3 (14-11)
be such an F number. Our first observation is contained in the
lemma :
Any two of the three numbers
Pl - 1, p 2 ~ 1, Ps - 1 (14-12)
have the same greatest common divisor.
For instance, in the example
1,729 = 7-13-19
one finds
(6, 12) = (6, 18) = (12, 18) = 6
To prove the lemma it is sufficient to show that the g.c.d. of
any pair of the numbers in (14-12), for instance,
d = (pi - 1, p 2 - 1)
divides the two others, namely,
(pi - 1, p 3 - 1), (P2 - 1, Ps - 1)
THE CONVERSE OF FERMAT'S THEOREM 335
or, equivalently, that d divides p 3 — 1. But from the definition
of d, it follows that
Pisl, p 2 = 1 (mod d)
so that according to (14-10) and (14-11)
n = P1P2P3 = V3 = 1 (mod d)
as required.
We use the lemma to write
pi — 1 = dP 1} p 2 — 1 = dP 2 , 7>3 — 1 = ^P 3
or
pi = 1 + dP 1} p 2 = 1 + dP 2 , p 3 = 1 + dP 3 (14-13)
where the numbers Pi, P 2 , P 3 are relatively prime in pairs. This
yields
X(n) = [pi - 1, p 2 - 1, Pa - 1] = [^1, dP 2 , dP 3 ] = dP x P 2 P 3
and the basic condition (14-8) for an F number takes the form
P1P2P3 = 1 (mod dP x P 2 P 3 )
Here we substitute the values (14-13) and expand the left-hand
product to obtain
d 3 P 1 P 2 P 3 + d 2 (PiP 2 + P1P3 + P2P3)
+ d{P x + P 2 + P 3 ) + 1 = 1 (mod dP x P 2 P 3 )
This reduces first to
d\P x P 2 + PiPs + P2P3) + d(Pi + P 2 + Pa)
= (mod dP x P 2 P 3 )
and then again to
dCPxPa + P X P 3 + P 2 P 3 ) s - (P x + P 2 + P 3 )
(mod PiP 2 P 3 ) (14-14)
To construct P numbers from this condition we proceed as
follows. Three positive numbers Pi, P 2 , and P 3 , relatively prime
in pairs, are selected. The suitable values for d corresponding to
338 NUMBER THEORY AND ITS HISTORY
them are solutions of the linear congruence (14-14). Clearly the
coefficient of d is relatively prime to the modul so that there
exists a unique smallest positive solution d , and the corresponding
general solution becomes
d = d + tP x P 2 P 3 (14-15)
Since we assume that d is positive, t runs through the series
0, 1, 2, ... . When the value (14-15) for d is substituted in
(14-13), we find
Pl = 1 + Pi^o + tP 1 2 P 2 P 3
p 2 = 1 + P 2 d + tP x P 2 2 P z t = 0, 1, 2, • • •
p 3 = 1 + P 3 do + tP^Ps 2 (14-16)
To make the product of these numbers an F number, only one
condition remains to be fulfilled : they must all be primes. When
the successive values of t are introduced and the character of the
resulting three numbers checked against a table of primes, it is
usually possible to derive a large set of numbers with the Fermat
property.
Example.
The simplest example is
Pi = 1, P 2 = 2, P 3 = 3
and the congruence (14-14) becomes
lid = -6 (mod 6)
with the smallest positive solution do = 6. These numerical values, when
substituted in the formulas (14-16) give
pi = 7 + 64, p 2 = 13 + 12t, p 3 = 19 + 18* (14-17)
as the possible expressions for the three primes defining the F numbers. For
t = Owe find the first F number of this type
n = 7 • 13 • 19 = 1,729
an example to which we have already referred.
THE CONVERSE OF FERMAT'S THEOREM
337
The first few values of t, such that all three numbers (14-17)
are primes, have been tabulated below.
t
Pi
P2
Vz
7
13
19
5
37
73
109
34
211
421
631
44
271
541
811
50
307
613
919
54
331
661
991
55
337
673
1,009
1,514
9,091
18,181
27,271
When * is limited so that the least prime pi does not exceed
10,000, one finds 45 F numbers of the type (14-17).
Example.
In a second example we take
Pi = 1, P 2 = 2, P 3 = 5
The congruence (14-14) becomes
I7d = -8 (mod 10)
and its least positive solution is d = 6. From (14-16) one deduces the
corresponding primes
pi = 7 + HW, p 2 = 13 + 20i, p 3 = 31 + 50*
For t = 0, one finds the F number n = 7 ■ 13 • 31.
A similar procedure may be devised to determine the F numbers
with four or more prime factors. Let us mention only the example
n = P1P2P3P4
where the primes belong to the series
Vl = 7 + Qt, p 2 = 13 + 12*, p 3 = 19 + 18*,
p 4 = 37 + 36*
(14-18)
338 NUMBER THEORY AND ITS HISTORY
The smallest F number of this type
n = 7 • 13 • 19 • 37
is obtained for t = 0, and with the limitation p\ < 10,000 one
finds 13 more of them.
As we remarked earlier, the literature on numbers with the
Fermat property is scant and there are several natural questions
still awaiting solution. It is not even known whether there are
infinitely many F numbers, although this seems probable. To
illustrate the difficulty, let us take the F numbers that are the
product of three primes of the form (14-17). It is known by
the theorem of Dirichlet that one can find an unlimited number of
values of t for which one of the numbers in (14-17), for instance,
Pi, becomes a prime. But whether one can find infinitely many
values of t such that all three numbers simultaneously become
primes is a problem beyond the present power of number theory.
The F numbers by their definition have a basic property in
common with the primes, but in other respects they behave quite
differently. For instance, one F number may divide another, as
in the example
m = 7 • 13 • 19, n 2 = 7 • 13 • 19 • 37
or, more generally, any F number with four prime factors (14-18)
is divisible by the F number with the three factors (14-17). An
F number can even be the product of two other F numbers; for
instance,
m = (7 -13 -19) (37 -73 -109)
is an example.
Problems.
Find the general form of three prime factors of an F number corresponding
to the following values and determine one or more examples in each case
1. Pi = 1, P 2 = 3, P 3 = 4
2. Pi = 1, P 2 = 2, P 3 = 15
3. Pi = 2, P 2 = 3, P 3 = 5
THE CONVERSE OF FERMAT'S THEOREM 339
Bibliography
Carmichael, R. D.: "Note on a New Number Theory Function," Bulletin
American Mathematical Society, Vol. 16, 232-238 (1910).
: "On Composite Numbers P Which Satisfy the Fermat Congruence
a P ~ l = 1 mod P," American Mathematical Monthly, Vol. 19, 22-27 (1912).
Chernick, J.: "On Fermat's Simple Theorem," Bulletin American Mathe-
matical Society, Vol. 45, 269-274 (1939).
Lehmer, D. H.: "On the Converse of Fermat's Theorem," American Mathe-
matical Monthly, Vol. 43, 347-354 (1936).
Poulet, P. : "Table des nombres composes verificant le theoreme de Fermat
pour le modul 2 jusqu'a 100,000,000," Sphinx (Brussels), Vol. 8, 42-52
(1938).
Sispanov, S.: "Sobre ios numeros pseudo-primos," Boletin matematico, Vol.
14, 99-106 (1941).
CHAPTER 15
THE CLASSICAL CONSTRUCTION PROBLEMS
15-1. The classical construction problems. Quite early, proba-
bly in the fifth century b.c, Greek mathematical investigations
led to the study of some geometric construction problems that
have remained landmarks in the history of mathematics. Three
of these have acquired a particular fame .
1. The squaring of the circle. When a circle is given, this prob-
lem requires that a square shall be constructed whose area is
equal to that of the circle. The same difficulty is involved in the
construction of a straight distance equal to the circumference of
the circle.
2. The trisection of the angle. This problem demands a method
for dividing an arbitrary angle into three equal parts.
3. The doubling of the cube. This problem is sometimes known
as the Delian problem. According to tradition, it arose when the
Athenians sought the assistance of the oracle at Delos to gain
relief from a devastating epidemic. They were advised to double
the size of the altar of Apollo, cubical in shape.
These problems enjoyed popular fame among the Greeks; we
know, for instance, according to a letter from the mathematician
Eratosthenes to King Ptolemy of Egypt, that Euripides mentions
the Delian problem in one of his tragedies, now lost. Greek
geometers also focused their interest on several other construction
problems; we mention especially the construction of regular
polygons, which we shall discuss in some detail subsequently,
and the Platonic bodies or regular polyhedrons.
These problems had an inspiring influence and added new
340
THE CLASSICAL CONSTRUCTION PROBLEMS 341
aspects to Greek geometry. Special higher curves were introduced
and their properties studied, and by such means the trisection of
the angle and the duplication of the cube could be accomplished.
However, in the strict Greek sense of construction by compass
and ruler, the problems remained unsolved in spite of strenuous
and ingenious efforts by the Greek and later geometers.
In the seventh and last section of the Disquisitiones arithmeticae,
Gauss turns to the problem of constructing regular polygons.
It may seem out of order to introduce a geometric topic in a work
on number theory; hence Gauss feels obliged to explain: "The
reader may be surprised at encountering such an investigation
which at first view appears wholly dissimilar to it; but the exposi-
tion will show very clearly the actual relation between this topic
and the transcendental arithmetic."
We shall present a brief account of Gauss's principal results on
the construction of regular polygons. For this purpose it is
necessary to touch upon some of the principles of geometric con-
struction in general. When it is required that a construction shall
be performed by compass and ruler, it is assumed that each of
these two instruments shall be used only for a single, specific
operation:
1. With the compass, circles with given center and radius can
be traced.
2. With the ruler, a straight line can be drawn through two
given points.
In these statements it is tacitly included that one can draw
circles with arbitrarily large radii and that straight lines can be
prolonged indefinitely. Any points or lengths one can deduce
from given geometric quantities by a finite number of these two
operations are said to have been constructed by compass and ruler.
It is not permissible to apply the two instruments in any other
way; for instance, markings on a ruler cannot be utilized. There
exists, for instance, a very simple solution of the trisection problem
by means of a ruler with two fixed marks.
While we are on the subject of geometric constructions, let us
342 NUMBER THEORY AND ITS HISTORY
take a small step out of the direct path and mention the rather
interesting fact that any construction that can be performed by
compass and ruler can be made by compass alone, and also, if
a fixed circle has been drawn, the constructions may be achieved
by ruler only.
After the basic rules for the construction by means of compass
and ruler have been clarified, the next move in the analysis of the
construction problems consists in bringing them in relation to
the theory of algebraic equations. It must be emphasized that
the subsequent presentation is essentially expository and that
some of the most important steps can only be stated here without
any attempts to give proofs.
To determine which quantities can be constructed, let us assume
that one performs the geometric operations within a coordinate
system in the plane and examine the algebraic operations involved
in each step. When two points are given by means of their
coordinates, the coefficients of the equation of the straight line
passing through them can be computed rationally from the coordi-
nates; dually, when the coefficients of the two straight lines are
known, the coordinates of their intersection point can be deter-
mined rationally from them. The calculation of the intersection
points of a circle and a straight line, or of a circle with another
circle, leads to a second-degree equation. The coordinates, there-
fore, are obtained as the sum of a rational expression in the known
coefficients of the equations and the square root of such an expres-
sion. The distance between two points is also expressible as a
square root.
Since all other constructions can be composed of a series of
these simple operations, we conclude from our observations that
those magnitudes that can be constructed from given ones may
be computed algebraically by repeated applications of the four
arithmetic operations and by extracting square roots. But the
converse is also true. When a and b are two given lengths, one
obtains the distances a ± b as well as ab and a/b by elementary
constructions while the square root Va is the result of taking the
me?*n proportional of 1 and a.
THE CLASSICAL CONSTRUCTION PROBLEMS 343
To summarize: The geometric quantities that are constructible
from known data by means of compass and ruler correspond
algebraically to those expressions that may be deduced from given
numbers by repeated use of the four rational operations and
square root extraction.
Through this analysis we have succeeded in transferring the
construction problems to questions in the theory of equations,
since it is relatively easy to show that each constructible expression
is the root of an algebraic equation whose coefficients are rational
in the given quantities. One way of finding such an equation is
the following: Our constructible expression contains a number
of square roots. Related expressions can be deduced from it by
changing the ± signs in front of each of these radicals in all possible
ways. The equation whose roots are all these quantities is an
equation of the desired kind. For instance, the quantity x = aV3
satisfies the equation
x 2 ~ 3a 2 =
The expression
x = l + Vs - V5
is a root of the equation of fourth degree
[(x - l) 2 - 3] 2 - 5 = x 4 - 4x 3 + 8x - 1 =
This transformation makes it clear that to decide on the possi-
bility of solving a construction problem, one must examine first
whether the quantity to be found satisfies an algebraic equation,
and second whether this equation has a constructible solution, or,
as one prefers to say in equation theory, whether it is solvable by
square roots.
The further problem of solving an equation by square roots or,
more generally, by radical expressions could not be tackled
until the discovery by the two young geniuses, N. H. Abel (1802-
1829), Norwegian, and E. Galois (1811-1832), French, of the
principles underlying the solution of algebraic equations. It is
impossible to discuss these theories here; we will only say that
they have been fundamental in the history of the newer phases of
344 NUMBER THEORY AND ITS HISTORY
algebra and gave rise to the all-pervading mathematical concept of
groups. For our purpose a few very simple facts will suffice.
We have seen that any constructible expression satisfies an
algebraic equation with coefficients that are rational in the given
quantities. There may be several such equations, but among them
there is one of minimal degree, which cannot be factored further
with rational coefficients, and it divides all other equations of the
same kind. From the theory of Galois it follows that for this
minimal equation to be solvable by square roots it must have
very special properties. One of these is that its degree must be
a power of two.
We return to the three classical construction problems, and
consider first the duplication of the cube. The given cube may
have the side a and the doubled cube the side b. Since the volume
of one cube is to be the double of the other, they must fulfill the
condition
b 3 = 2a 3
or
b = v^-a
Therefore, the problem is essentially to construct the number
x = ^2, which is a root of the equation
x 3 - 2 =
This equation cannot be factored into rational factors, and since
its degree is not a power of 2, we conclude that a cube cannot be
doubled by means of a construction with compass and ruler.
The impossibility of a general construction for the trisection of
the angle can be deduced by a similar argument. An angle a can
be constructed when one knows cos a (or sin a) because a occurs
in the right triangle with the hypotenuse 1 and one leg equal to
cos a. Conversely, when an angle is given, its cosine or sine
can be constructed. The problem of trisection of an angle may
therefore be expressed: It is required to construct the number
x = cos - when the number a = cos a is known. By means of
o
THE CLASSICAL CONSTRUCTION PROBLEMS 345
the elementary trigonometric formula
cos 39 = 4 cos 3 — 3 cos
one finds
-\ 3 " «
cos a = 4 ( cos - ) — 3 cos -
a = 4 ( cos i)
and this may be rewritten as a cubic equation
4a: 3 — 3x — a =
In general, one cannot decompose this equation further into
factors whose coefficients depend rationally on a; there are numer-
ous values. of a, in fact infinitely many rational values of a, within
any interval such that the equation cannot be factored rationally.
Since the equation is cubic, we conclude that a general construc-
tion for trisecting an angle by compass and ruler cannot be found.
The quadrature of the circle is a problem on a different level of
difficulty. It is equivalent to finding a construction for the
number t, the proportion between the circumference and the
diameter of a circle. There is no algebraic equation that is
naturally associated with this problem, and the final result is
actually to the effect that not only is ir not constructive but it is
a transcendental number, i.e., not the root of any algebraic equation
with rational coefficients. The proof was found in 1882 by the
German mathematician F. Lindemann, and it was based on
methods devised previously by the French mathematician C.
Hermite, who, in 1873, showed that the number e, the base of
the natural logarithms, is a transcendental number.
The detailed proofs of the impossibility of constructing solutions
to the three classical problems leave nothing to be desired in
regard to mathematical stringency. Nevertheless, every mathe-
matician has received and undoubtedly will continue to receive
new and ingenious constructions purporting to be exact solutions.
Usually they have been tested by the inventor on large-scale
drawings and the proof of the pudding lies in the eating: no percep-
tible error has been found. All of these constructions are, need-
346 NUMBER THEORY AND ITS HISTORY
less to say, approximations with errors that are small but definite
and computable by elementary trigonometry. Some of the
published constructions are of interest due to their simplicity and
great accuracy. Any one of them can be improved upon by
further complications. Mathematically, an accuracy that leaves
no errors observable by the naked eye is not impressive. For
instance, t would be constructive if one were permitted to cut
off its expansion after the first hundred or first thousand decimals;
even by trial and error, the stage of error not perceptible to the
eye is reached in a few steps.
15-2. The construction of regular polygons. We now come to
the principal topic of this chapter, Gauss's investigations on the
construction of regular polygons. A regular polygon with n sides
has its vertices equidistant on a circle. The size of the circle is
unessential so that we shall assume that its radius is r = 1. Since
each of the sides of the polygon corresponds to a central angle
360° _ 2t
n n
the problem is to divide a full angle of 360° into n equal parts.
Any angle can be bisected, so that when a regular polygon with
n sides has been obtained, one can successively construct one
with 2n, 4n, . . . , in general, with 2 M n sides. On the other hand,
from a polygon with 2n sides, one can draw one with n sides by
joining every second vertex by a side. Consequently, if one so
desires, one can limit the considerations only to regular polygons
with an odd number of sides. From the fact that regular polygons
with 3, 4, and 5 sides can be constructed, it follows that all polygons
with
2", 3-2", 5-2"
sides are obtainable.
It is evident that if one has a polygon with n sides and a is
a divisor of n, say, n = ab, a polygon with a sides can be derived
by taking every bth vertex. More interesting is the fact that the
basic result on linear indeterminate equations under certain circum-
THE CLASSICAL CONSTRUCTION PROBLEMS 347
stances permits us to proceed the other way and obtain polygons
with a larger number of sides.
From polygons with a and with b sides, where a and b are rela-
tively prime, a polygon with ab sides is obtainable.
To prove the statement we recall that one can find such integers
x and y that
ax — by = 1
Division by ab gives
JL _ - _ U.
ab b a
or
360° 360° 360°
— — = x—7 y
ab b a
This shows that the central angle for a polygon with ab sides is the
difference between two multiples of the central angles of the poly-
gons with a and b sides. For instance, from the polygons with
3 and 5 sides, a polygon with 15 sides is constructive. One
concludes also that it would suffice to study the construction of
polygons for which the number of sides is an odd prime power.
The construction of a polygon with n sides, or equivalently, an
angle 2ir/n, may be achieved by using one of the trigonometric
functions of the angle, for instance,
cos — or sin — (15-1)
n n
By means of the law of cosines one finds the expression
/ 2^
Sn = ^2-2cos-
for the side of the polygon. Instead of dealing with these quanti-
ties directly, Gauss takes a step that at the time was an innova-
tion: Imaginary or complex numbers are introduced to solve a
problem that essentially concerns real quantities.
348
NUMBER THEORY AND ITS HISTORY
Let us recall briefly a few properties of complex numbers. Any
such number can be written
a + ib — r (cos <p + i sin <p),
V=i
(15-2)
where a and b are the coordinates in the complex plane, r the
radius vector or absolute value, while <p is the angle or amplitude
Fig. 15-1.
Fig. 15-2.
that the radius vector makes with the real axis. A complex
number (15-2) is multiplied by another
a\ -f- ib\ = ri(cos<pi + isin^)
according to the rule
(a + ib) (ai + ibx) = rr^cos (<p + vi) + i sin (<p + ^?i)]
When this result is applied to a product of n equal factors (15-2),
one derives the formula
(a + ib) n = r n (cos n<p + i sin rup)
(15-3)
known as the theorem of de Moivre.
Gauss assumes that a circle with radius 1 and center at the zero
point has been drawn in the complex plane. In this circle he
inscribes a regular polygon with n sides such that one vertex
THE CLASSICAL CONSTRUCTION PROBLEMS 349
lies on the positive real axis at the point x = 1 (see Fig. 15-1).
Then the next vertex will correspond to the complex number
e = cos - — V i sin — (15-4)
n n
and the subsequent ones to
4tt , . . 4x
e 2 = cos h i sin — , • • • ,
n n
e = cos — (n - 1) + i sin — (n - 1) (15-5)
n n
The theorem of de Moivre shows that these numbers are powers of e
eo = e o = l, €l = e, e 2 = eV • • , *„_! = e"" 1 (15-6)
From the same formula (15-3) we conclude further
e n = ( cos — + i sin — ) = cos 2tt + i sin 2tt = 1
\ n n /
so that
e n = 1
This result establishes that e as well as all its powers (15-6) must
be roots of the algebraic equation
x n - 1 = (15-7)
For this reason one calls the roots (15-6) the nth roots of unity,
while the equation (15-7) is known as the equation of the division
of the circle or the cyclotomic equation.
The two trigonometric functions (15-1), on which the construc-
tion of the regular polygon depends, occur as the components of
the nth roots of unity (15-4). When they can be expressed by
square root operations, the same is true for e, recalling that
i = \/~^l. Therefore, if the regular polygon with n sides can be
constructed with compass and ruler, the corresponding cyclotomic
equation (15-7) can be solved by square roots.
Equation (15-7) is not the equation of minimal degree that the
nth root of unity e satisfies, since it can be factored rationally;
350 NUMBER THEORY AND ITS HISTORY
for instance, x — 1 is an obvious factor. In general, the roots
(15-6) of the cyclotomic equation fall into two groups. Some
of them cannot be roots of unity for a smaller exponent than n;
these are usually called the primitive roots. Others do satisfy
equations of the type (15-7) with lower exponents and these are
the nonprimitive roots. We may remark that these primitive roots
constitute a different, although related, concept from the primitive
roots we introduced previously when studying the residue classes of
the integers for some modul.
It is simple to decide when a root of unity
. 2irk , . . 2irk ... „ _
Ck = e = cos 1- t sin (15-8)
n n
is primitive. If it should satisfy an equation
x* = 1, t < n
the theorem of de Moivre shows that
. 2irkt . 2irkt
ek = cos 1- % sin = 1
n n
This is only possible if the amplitude
n ht
2t —
n
is an integral multiple of 2x; in other words the number kt must
be divisible by n. When k is relatively prime to n, the smallest t
that will satisfy this condition is t = n, while a smaller t can be
found when k and n have a common factor. Thus we have :
An nth root of unity e^, defined in (15-8), is primitive only
when k is relatively prime to n.
Since the roots in (15-6) that are primitive correspond to those
numbers k that are less than and relatively prime to n, we can
state further:
The number of primitive nth roots of unity is equal to (p(n),
where <p denotes Euler's function.
THE CLASSICAL CONSTRUCTION PROBLEMS 351
The subsequent step in the algebra of the roots of unity is to
demonstrate that the <p(n) primitive roots satisfy an equation
with rational coefficients of degree <p(n), and that this equation
cannot be factored further rationally. Again we must abstain
from giving a proof. To this minimal equation for the primitive
roots, we apply our previous criterion limiting the degree of an
equation solvable by square roots. This produces the interesting
result:
For the equation of the nth roots of unity to be solvable by
means of square roots, it is necessary that <p(ri) be a power of 2.
This places a strong restriction on the number n. To analyze
its implications, let
n = 2>i ai • • • Vr ar (15-9)
be the prime factorization of n. The number
<p(n) = 2 Q ^V ai ~~ 1 (Pi -I)'" Pr^HPr - 1)
can be a power of 2 only when each of its factors is such a power.
One concludes first that none of the odd prime factors pi can
occur, so that all exponents a x , . . . , a r in (15-9) must be equal
to 1. Second, the numbers pi — 1 are powers of 2; hence the odd
primes dividing n are of the form
Vi = 2* + 1 (15-10)
But these are actually the Fermat primes, which were examined
in Chap. 4. We found that such a number as (15-10) cannot be
a prime except when the exponent is itself a power of 2 and that
the Fermat primes, therefore, are defined by an expression
F t = 2 2t + 1
We mentioned also that so far the study of these numbers has
revealed only five Fermat primes, namely,
Fq = 3, F x = 5, F 2 = 17, F s = 257, F 4 = 65,537
Through these observations we have arrived at Gauss's funda-
mental result:
A regular polygon with n sides can be constructed by compass
352 NUMBER THEORY AND ITS HISTORY
and ruler only when the number n is of the form
n = 2 a pip 2 ' • -Vr
where the prime factors are Fermat primes.
Our previous discussion has been directed towards showing
that this condition is necessary. Gauss proves conversely that
it is also a sufficient one, by demonstrating that a polygon with
p sides can be constructed when p is a Fermat prime. In this
case he finds that the equation for the primitive pth roots of
unity can be solved by a series of second-degree equations. We
shall not go through the details of the general proof, but only
consider a couple of examples sufficient to clarify the underlying
principles.
Problem.
Find all polygons with less than 100 sides that can be constructed with
compass and ruler.
15-3. Examples of constructible polygons. When p is a prime,
the number of primitive pth roots of unity must be <p(p) = p — 1,
and clearly the only nonprimitive root is x = 1. Since they all
satisfy the equation
x p - 1 =
the primitive ones are the roots of
xP ~ ] = x p - x + x p - 2 H \-x 2 + x+l=0 (15-11)
x — i
These roots, as we mentioned, are all some power of
2tt . . 2tt
e = cos V i sin —
V V
We notice further that the two roots
ft 27T 27T
e = cos — fc + isin — k
V V
e 7 ' - * = e = cos — k — i sin — k
p p
THE CLASSICAL CONSTRUCTION PROBLEMS 353
are conjugate imaginary, and their sum
e k + <T k = 2 cos — k
V
is real. In particular
2t
v = e + e _1 = 2 cos — > (15-12)
P
and this number may serve to construct the polygon.
For the smallest Fermat prime p = 3, (15-11) becomes
x 2 + x + 1 =
We substitute x = e and obtain after division by 6
v + 1 = 6 + t~ l + 1 =
According to (15-12), this gives
2tt 1
cos ¥ ---
and for the side of the polygon, one finds the value
s 3 = V3
The next Fermat prime is p = 5. Here e satisfies the equation
e 4 + e 3 + e 2 + e + 1 =
and division by e 2 yields
e 2 + e~ 2 + e + e- 1 + 1 = (15-13)
From
we obtain, by squaring,
V = € + € _1
v 2 _ 2 = e 2 + e~ 2
When these values are substituted into (15-13), it follows that
rj is the root of the second-degree equation
^ 2 + ^-l=0
354 NUMBER THEORY AND ITS HISTORY
The solution of this equation is
VI - 1
where we have taken the plus sign for the square root since 77 is
positive according to (15-12). We obtain further
2tt VI - 1
cos — =
5 4
and the side of the pentagon is computed to be
S5 = i V 10 - 2v"5
In our last example we take the Fermat prime p = 17, and in
this case the principles of the general theory emerge more clearly.
When the number is excluded, the sixteen other remainders
(mod 17) may be written
±1, ±2, ±3, ±4, ±5, ±6, ±7, ±8 (15-14)
We shall first divide these numbers into two classes
±1, ±2, ±4, ±8 (15-15a)
±3, ±5, ±6, ±7 (15-156)
The numbers in (15-1 5a) are known as the quadratic residues
(mod 17) ; they are obtained by squaring the numbers in (15-14)
and taking the remainders (mod 17). The remaining numbers
in (15-14), which have been put in the set (15-156) are the
quadratic nonresidues. One should notice that the numbers in
(15-156) can be derived from those in (15-15a) by multiplication
with some nonresidue, for instance, 3.
Second, the remainders (15-14) shall be distributed into four
classes, each of four numbers
(15-16o)
(15-166)
(15-16c)
(15-16d)
±1,
±4
±2,
±8
±3,
±5
±6,
±7
THE CLASSICAL CONSTRUCTION PROBLEMS 355
Here the first set consists of the remainders of the fourth powers
of the numbers in (15-14) or, equivalently, of the squares of the
numbers in (15-1 5a). They are called the biquadratic residues.
To obtain (15-166), the numbers in (15-16a) are multiplied by
some number in (15-15a) not already in (15-16o), for instance, 2.
The numbers in (15-16c) follow from (15-16a) through multipli-
cation by some number not in the two preceding groups, and
(15-16d) is derived similarly.
Through a third division the remainders (15-14) fall into eight
sets of two numbers. Here we use the basic group ±1, namely,
the residues of the eighth powers of the numbers in (15-14). By
this process the first set (15-16a) splits into ±1 and ±4, and
the other sets in (15-16) are divided similarly.
After these preliminaries we turn to the solution of (15-11)
f or p = 17. When the root x = e is substituted and the equation
divided by e 8 , it follows that
e + 6 -l + € 2 + e -2 + . . . + e 8 + 6 "8 = _1 (15-17)
At this stage Gauss introduces two quantities which he calls
the first periods
p = e + e - 1 + e2 + € - 2 + 64 + e - 4 + e s + e - 8 |
pi = e 3 + e" 3 + e 5 + e~ 5 + e 6 + e" 6 + e 7 + e' 7 ]
These periods, as one sees, are the sums of the roots whose expo-
nents are the numbers in the two classes (15-1 5a) and (15-156).
Both periods are real, for instance
2t 4t , n 8x 16tt
p = 2 cos — + 2 cos — + 2 cos — + 2 cos — —
1/ 17 1< 1<
and it is readily checked that p is positive and pi negative.
The periods (15-18) are the roots of an equation of second degree
r(x) = (x - p)(x - pi) = x 2 - (p + pi)x + ppi (15-19)
which we shall show has rational coefficients. From (15-17)
and (15-18) one concludes immediately
P + pi = — 1
356
NUMBER THEORY AND ITS HISTORY
The computation of the product of the two periods (15-18) is
somewhat more cumbersome. By direct multiplication one
obtains 8 • 8 = 64 products, each a power of e. One verifies
that every term e k with k ^ occurs equally often and, since
there are 16 different powers, each of them appears four times.
According to (15-17) we conclude that
PPi = -4
These investigations show that the equation (15-19) has the form
x 2 + x - 4 =
and its roots are
p =
17
1
— j
-Vii - l
Pi =
(15-20)
(15-21)
In the next step we study the second periods
0l = e 2 + e- 2 + e 8 + e~ 8
a2 = e 3 + € "3 + e 5 + e -5
^ = € 6 + -6 + e 7 + -7
Here the various sets (15-16) serve as the exponents for the terms
in periods. The periods are all real, for instance,
2tt 8tt n
a = cos — + cos — >
Similarly one finds that o- 2 is positive while a x and o- 3 are negative.
The four periods (15-21) in pairs, a, a x and o- 2 , <r 3 , are roots of
second-degree equations whose coefficients can be expressed by
the first periods. From (15-21) and (15-18) one sees that
<t + o\ = P, 02 + 03 = Pi
and by multiplication from (15-17)
ere - !
0'2 O '3 = — 1
THE CLASSICAL CONSTRUCTION PROBLEMS 35?
so that the two quadratic equations are
x 2 - px - I = 0, x 2 - Pl x - 1 =
For the solution of these equations, one finds by our previous
remark about the signs of the periods
P + vV + 4 p - Vp^Tl
a = } di =
Pi + Vpi 2 + 4 _ pi - vV + 4
*2 = g ' *3 - ~ 2
Here one can substitute the values (15-20) for p and p x to obtain
the explicit expressions for the second periods; for instance,
a = i(\/l7 - 1 + ^34 - 2N/17),
ff2 = |(-Vl7 - 1 + ^34 + 2\/l7) (15-22)
Finally the third-order periods should be computed. We shall
need only two of them, namely,
, = e + e" 1 = 2cos^> ,1 = e 4 + e~ 4 = 2 cos ^ (15-23)
where the exponents are taken from (15-16a). Again these two
quantities satisfy a second-degree equation whose coefficients
can be expressed by means of the second periods. From (15-23)
one obtains .
V + Vi = °"> VVi = °2
and the equation is
X 2 — (XX + 02 =
The expressions (15-23) show that v > Vi, and therefore the solu-
tion of the quadratic equation gives us
77 = 2 cos — =
2ir o- + V(T 2 - 4ff
17 2
Here the values (15-22) may be substituted, and the final formula
in terms of square roots will emerge. The reader may compute
358 NUMBER THEORY AND ITS HISTORY
the length of the side s 17 of the regular polygon with 17 sides.
Various fairly simple methods of construction have been devised.
After having completed these investigations, towards the end
of the Disquisitiones, the young Gauss states with justifiable pride:
There is certainly good reason to be astonished that while the division
of the circle in 3 and 5 parts having been known already at the time of
Euclid, one has added nothing to these discoveries in a period of two
thousand years and that all geometers have considered it certain that,
except for these divisions and those which may be derived from them
(divisions into 2 M , 15, 3 • 2 M , 5 • 2 M , 15 • 2 M parts), one could not achieve
any others by geometric constructions.
It has been told that Gauss proposed, perhaps not too seri-
ously, that a polygon with 17 sides be inscribed on his grave,
emulating the tombstone of Archimedes, which was decorated by
a figure of a sphere and the circumscribed cylinder, suggesting
his formula for the area of a sphere. OnjGauss's simple grave in
Gottingen there is no such polygon, but it does appear on the
monument in his native town of Brunswick.
Gauss's results on the construction of regular polygons by
compass and ruler represent a great achievement, but the final
solution of the problem is not yet in sight. Gauss transfers the
whole question to number theory, to the determination of the
Fermat primes. Whether there exist any others than the five we
have mentioned, no one knows. It is possible that the new
electronic computing devices may be of assistance in discovering
others. But the general problems, for instance, the question
whether there might be an infinite number of Fermat primes,
lie beyond the reach of the present methods of number theory.
Bibliography
Dickson, L. E.: Modern Algebraic Theories, Benj. H. Sanborn & Co., New
York, 1930.
MacDuffee, C. C: An Introduction to Abstract Algebra, John Wiley & Sons,
Inc., New York, 1940.
Rouse-Ball, W. W. : Mathematical Recreatians and Essays. Revised by H.
S. M. Coxeter, eleventh edition, The Macmillan Company, New York, 1939.
Thomas, J. M. : Theory of Equations, McGraw-Hill Book Company, Inc., New
York, 1938.
SUPPLEMENT
RECENT NUMERICAL RESULTS
The predictions about the usefulness of the electronic com-
puters for calculations in number theory have been amply fulfilled
since the manuscript for this book was first prepared. Several
interesting computations, vastly beyond the range of ordinary
calculating machines have been carried out by means of various
types of electronic computers and more will undoubtedly follow.
Since such studies are not income-producing tasks many have
been achieved in the hours of the night where the machines
would otherwise have been idle. All have been repeated and
checked, sometimes on different computers with new programming
and other operators, since men and machines tend to fall into
errors of habit and constitution.
Mersenne primes. Among the most formidable of these calcu-
lations is the search for new Mersenne primes
M p = 2? - 1
Professor D. H. Lehmer has made a wide sweep for possible
primes M p and he announced in 1952 and 1953 that the values
corresponding to
p = 521, 607, 1,279, 2,203, 2,281
are primes. For the last two of these Mersenne primes, after
coding and preparations, the actual running time of the SWAC
calculator amounted to 59 and 66 minutes, respectively. In
comparison with these giants the Mersenne prime M m given on
page 73 appears quite puny. The Mersenne prime M 2 ,28i is a
number with 687 digits. Professor H. S. Uhler has taken the
trouble of calculating explicitly the new perfect numbers corre-
sponding to these primes.
359
359a NUMBER THEORY AND ITS HISTORY
Various conjectures have been made about Mersenne primes,
but as far as they have been checked none of them seems to have
any validity. It has been noticed, for instance, that when the
Mersenne primes
V = 3, 7, 31, 127
are used as exponents in M p they give new Mersenne primes.
This is no general rule, however, since Professor R. M. Robinson
recently announced that calculations carried out on an ILLIAC
machine by D. J. Wheeler show that for the Mersenne prime
p — 8,191 the corresponding M p is not a prime.
In this connection let us also mention that the Fermat numbers
F n = 2 2n + 1
have been further examined, but no new primes have been found.
One of the latest results is an actual factor of Fi .
Bibliography
Uhleb, H. S.: "A brief history of the investigations on Mersenne numbers
and the latest immense primes," Scripta Mathematica, Vol. 18, 122-131
(1952).
Uhler, H. S.: "On the 16th and 17th perfect numbers," Scripta Mathe-
matica, Vol. 19, 128-131 (1953).
Bang, T.: "Store primtal (Large primes)," N ordisk M atematisk Tidskrift,
Vol. 2, 157-168 (1954).
Robinson, R. M.: "Mersenne and Fermat numbers," Proceedings of the
American Mathematical Society, Vol. 5, 842-846 (1954).
Odd perfect numbers. A great variety of results have been
obtained on the possible forms of odd perfect numbers, particu-
larly by A. Brauer and H. J. Kanold. It has also been shown
by Kanold [Journal fur die reine und angewandte Mathematik,
Vol. 186, 25-29 (1944)] that there are no odd perfect numbers
below 1.4 X 10 14 . One of my students, J. B. Muskat, has
informed me that he has been able to raise this bound to 10 18 .
GENERALIZATIONS OF THE PERFECT NUMBERS
New li?ts of multiply perfect numbers have recently been
published by B. Franqui and M. Garcia [American Mathematical
SUPPLEMENT
359b
Monthly, Vol. 60, 169-171 (1953)] and also by A. L. Brown
[Scripta Mathematica, Vol. 20, 103-106 (1954)] adding more
than 200 such numbers to those previously known. In this con-
nection let us also mention a list by P. Poulet [Scripta Mathe-
matica, Vol. 14, 77 (1948)] over new couples of amicable numbers.
Another generalization of the perfect numbers has been sug-
gested by the author. It is not difficult to prove that for a perfect
number n the harmonic mean H(n) of the divisors of n as defined
in Sec. 5.1 is always an integer, while for other numbers this is
only rarely the case. Thus the integers with integral harmonic
mean for the divisors may be considered a generalization of the
perfect numbers. They seem to share the property that they
are all even. This has been checked by M. Garcia for all such
numbers up to 10,000,000.
Bibliography
Ore, O.: "On the averages of the divisors of a number," American
Mathematical Monthly, Vol. 55, 615-619 (1948).
Garcia, M.: "On numbers with integral harmonic mean," American
Mathematical Monthly, Vol. 61, 89-96 (1954).
Prime tables. Based on calculations by Kulik, Poletti, and
Porter a new prime table covering the 11th million has been
published by N. G. Beeger (Amsterdam, 1951). There are
61,938 primes in the 11th million. It may also be mentioned
that lists of prime twins up to 200,000 have been prepared by
E. S. Selmer and G. Nesheim [Det Kongelige Norske Videnskabers
Selskabs, Forhandlinger Trondheim, Vol. 15, 95-98 (1942)] and
up to 300,000 by H. Tietze [Sitzungsberichte der mathematisch-
naturwisse'nschaftlichen Klasse der bayerischen Akadamie der
Wissenschaften zu Munchen, 57-72 (1947)].
We indicated in Sec. 14.1 how tables of the composite numbers
n, satisfying the congruence
2 «-i = l(mod n) (1)
were an essential aid to the factorization of large numbers.
D. H. Lehmer [American Mathematical Monthly, Vol. 56, 300-309
359c NUMBER THEORY AND ITS HISTORY
(1949)] has extended these tables to n = 200,000,000 by means
of the Army Ordnance ENIAC computer. Sierpinsky pointed
out that there is an infinity of composite numbers satisfying (1),
and later Lehmer and P. Erdos proved that there is an infinity
of such numbers with any given number of prime factors [Ameri-
can Mathematical Monthly, Vol. 56, 623-624, (1949)].
Fermat's problem. The studies on Fermat's problem have
been continued quite intensively, both in theory and by means of
high-speed computers. Among the most notable contributions
are those by Professor H. S. Vandiver and the numerical calcula-
tions carried out by D. H. Lehmer, Emma Lehmer, and J. Self-
ridge at his suggestion. It is shown that the equation
x n + y n = z n n > 2 (2)
can have no nonzero integral solutions for any value of n < 2,521.
In this case the extensive calculations have had the ideal effect
of bringing in new points of view also in regard to the theoretical
problems. "Thanks to SWAC for special exponents we have
really come to grips with the Fermat problem," as Vandiver
expresses it [Proceedings of the National Academy of Science, Vol.
40, 474-480 (1954)]. (Word has just been received from
Professor Vandiver that a series of calculations on Fermat's
theorem by C. A. Nicol and J. Selfridge has been completed. It
has been verified that Fermat's theorem is true for all exponents
up to n = 4,000.)
A number of improved estimates of the lowest possible values
of the numbers x, y, and z in a possible solution of (2) have also
been made, particularly by Oblath and Inkeri. On the basis of
Vandiver's calculations and new estimating methods by Duparc
and Wijngaarden these values could again be raised essentially.
However, there does not seem to be much point in continuing
along these lines; Oblath observes in connection with his own
estimates that x and y would have to exceed a number which at
good speed would take more than two centuries to write and a
strip of 4,000 miles to print.
GENERAL BIBLIOGRAPHY
For a complete, encyclopedic account of the history of the discoveries in
number theory up to 1918 the reader is referred to:
Dickson, L. E.: "History of the Theory of Numbers," 3 vols., Carnegie
Institution of Washington Publication 256, 1919-1923.
A review of the existing table material on number-theory questions can be
found in:
Lehmer, D. H.: "Guide to the Tables in the Theory of Numbers," National
Research Council Bulletin 105, Washington, 1941.
A considerable selection of translations and reproductions of essential con-
tributions to number theory is contained in :
Smith, D. E. : A Source Book in Mathematics, McGraw-Hill Book Company,
Inc., New York, 1929.
The following is a list of English books on number theory for the reader who
wishes to pursue the subject further :
Carmichael, R. D.: Theory of Numbers, John Wiley & Sons, Inc., New York,
1914.
: Diophantine Analysis, John Wiley & Sons, Inc., New York, 1915.
Dickson, L. E. : Introduction to the Theory of Numbers, University of Chicago
Press, Chicago, 1929.
-.Studies in the Theory of Numbers, University of Chicago Press, Chicago,
1930.
: Modern Elementary Theory of Numbers, University of Chicago Press,
Chicago, 1939.
Hardy, G. H. and E. M. Wright: An Introduction to the Theory of Numbers,
Oxford University Press, New York, 1938.
Ingham, A. E. : The Distribution of Prime Numbers, Cambridge University
Press, London, 1932.
Uspensky, J. V., and M. A. Heaslet: Elementary Number Theory, McGraw-
Hill Book Company, Inc., New York, 1939.
Wright. H. N. : First Course in Theory of Numbers, John Wiley & Sons, Inc.,
New York, 1939.
359eZ
NAME INDEX
Abel, N. H., 343
Abu Kamil, 139, 140
Abu-1-Hasan Thabit ben Korrah,
98,99
Ahmes, 116-118
Albert, A. A., 39, 164
Alcuin, 94, 121
Al-Karkhi, 185-187, 225
Al-Khowarizmi, Mohammed ibn
Musa, 19, 20, 187, 225
Archibald, R. C, 141
Archimedes, 5, 140, 209, 358
Aristotle, 26
Aryabhata, 122
Athelard of Bath, 20
B
Bachet, Claude, Sisur de M5ziriac,
125, 126, 132, 133, 141, 196, 198
Barlow, 93, 131
Bede, Venerable, 6
Bell, E. T., 39
Bertelsen, 69, 77
Bhaskara, 26, 122, 123, 129, 136, 138,
193, 208
Birkhoff, G., 39, 164
Bonconpagni, B., 207
Brahmagupta, 26, 122, 193, 208, 247,
249
Brancker, 53
Briggs, 317
Brun, V., 84
Buddha, 4
Bull, L. S., 141
Burckhardt, 54
C
Cajori, F., 24
Cardanus, 98, 195
Carmichael, R. D., 96, 207, 332, 333,
339, 359
Cataldi, 73
Chace, A. B., 141
Charlemagne, 94, 121
Chaucer, 20
Chernac, 54
Chemick, J., 334, 339
Chuquet, Nicolas, 98
Colebrook, H. T., 141, 208
Copernicus, Nikolaus, 21
Coxeter, H. S. M., 358
Crelle, 54
Creak, T. G., 310, 325
Cunningham, A. J. C, 85, 96, 285,
310, 324, 325
D
Dase, 54
De Moivre, 348-350
Descartes, Rene, 55, 95, 96, 99
Dickson, L. E., 199, 358, 359
361
362
NUMBER THEORY AND ITS HISTORY
Diophantos, 168, 179-185, 187, 193-
196, 198, 199, 203-205, 208, 211,
268, 270, 313
Dirichlet, Lejeune, 79, 80, 206, 207,
268
E
Eratosthenes, 64, 66, 67, 75, 84, 340
Escott, E. B., 100, 115
Etienne de la Roche, 98
Euclid, 41Jf., 52, 65, 79, 92, 94, 174,
211, 358
Euler, Leonhard, 59-64, 73, 74, 78,
81, 84, 93, 100, 110, 126-128, 131,
132, 138, 141, 198, 199, 206, 208,
211, 245, 249, 272, 273, 277, 297
Euripides, 340
F
Felkel, 54
Ferdinand, Carl Wilhelm, Duke of
Brunswick, 209, 210
Fermat, Pierre de, 54-59, 62, 63, 69,
73-75, 80, 95, 96, 98, 99, 166, 194,
196, 198, 199, 203-209, 211, 268,
270-273, 277, 280
Fermat, Samuel, 196
Fibonacci (see Leonardo)
Frederic II, Emperor, 187, 193
Frederic II, King of Prussia, 60
Frenicle de Bessy, 59, 60, 74, 96, 272
G
Galois, E., 343, 344
Gauss, C. F., 74, 75, 209-212, 225,
233, 245, 247, 256, 263, 266, 267,
291, 297, 301, 315, 317, 323, 341,
346-348, 351, 352, 355, 358
Gelon, King, 5
Gerbert, Pope Sylvester II, 20
Glaisher, J. W. L., 54, 95, 113, 115
Goldbach, Christian, 60, 81, 84, 85
Goodwin, H., 323, 325
H
Hadamard, J., 78
Hardy, G. H., 39, 40, 199, 359
Heaslet, M. A., 39, 40, 43, 359
Heath, T. L., 208
Henry, Charles, 208
Heraclitus, 195
Hermite, C, 345
Hilbert, D., 199
Hill, G E., 24
Holzman, 194-196
Hopper, V. F., 40
Iamblichus of Chalcis, 97
Ibn Khaldun, 97
Ingham, A. E., 359
Jacobi, K. G. J., 285, 301, 310
Jenkinson, 9re.
John of Halifax, 21
John of Palermo, 188, 193
Jumeau, Andre, 96
K
Karpinski, L. C, 24
Kraitchik, M., 76, 85, 285, 310
Kronecker, 43
Kulik, J. P., 54
Kummer, E., 206, 207
Lagrange, J. L., 199, 211, 256, 259,
282
Lambert, J. H., 53
Lame, 43, 206
Lawther, H. P., Jr., 305, 309, 310
Legendre, 206, 211
NAME INDEX
363
Lehmer, D. H., 54, 75n., 331, 339, 359
Lehmer, D. N., 54, 66, 85, 96
Leibniz, 55, 259
Leonardo Fibonacci (Pisano),20, 117,
118, 122, 185, 187, 188, 191-193,
196, 202, 207, 228, 247, 312
Lindemann, F., 345
Littlewood, J. E., 199
Lucas, E., 73, 96, 326-328
M
MacDuffee, C. C, 40, 164, 358
MacLane, S., 39, 164
Mahaviracarya, 122, 131, 141
Manning, H. P., 141
Martin, A., 169
Mason, D. E., 96
Meissel, 69, 77
Mendelsohn, I., 175
Mersenne, 53, 55, 59, 69, 71-73, 75,
92-96, 99
Mills, W. H., SOn.
Mohammed ibn Musa al-Khowarizmi
(see Al-Khowarizmi)
Mordell, L. J., 208
N
Napier, John, 24
Napoleon, 199, 209
Neugebauer, O., 171, 172, 176, 208
Newton, 55, 209
Nonius, 195
P
Pacioli, Luca di Burgo, In.
Paganini, Nicolo, 100
Pascal, 55, 74
Pell, John, 53
Pervouchine, 73
Peter II, 60
Philo Judaeus, 27
Pipping, W., 84
Pisano (see Leonardo)
Plato, 26, 166
Pliny, 6
Poulet, P., 71, 96, 100, 115, 331, 334,
339
Poullet-Delisle, A. C. M., 233
Powers, 73
Proclus, 166
Ptolemy, King, 340
Pythagoras, 165, 166
R
Rahn, 53
Rangacarya, M., 141
Recorde, Robert, 21
Regiomontanus, 141, 194, 249
Reisch, Gregor, 22
Rhind, Henry, 116
Riemann, G. F. B., 78
Robert of Chester, 20
Roberval, 198
Rohnius (see Rahn)
Rouse-Ball, W. W., 24, 358
Rudolff, Christoff, 122, 195, 313
S
Sachs, A., 176, 208
Sacrobosco (see John of Halifax)
Sanford, V., 314rc.
Sarrus, F., 326
Seelhoff, 73
Shanks, W., 324
Shodja, C. Aslam (see Abu Kamil)
Sispanov, S., 334, 339
Smith, D. E., 7, 24, SUn., 325, 359
Stevin, Simon, 24, 313-315, 325
Stiefel, Michael, 98, 195
Sun-tse, 245, 247
Sylvester II, Pope (see Gerbert)
Tannery, Paul, 208
Tartaglia, 98
364
NUMBER THEORY AND ITS HISTORY
Thales of Miletus, 174
Theodoras, Master, 187
Thomas, J. M., 358
Thue, A., 268
Thureau-Dangin, F., 171, 208
U
Uhler, 73
Uspensky, J. V., 39, 40, 43, 359
V
Valee-Poussin, C. J., de la, 78
Vandiver, H. S., 208
Venerable Bede (see Bede)
Villefranche (see Etienne de la Roche)
Vinogradoff, I., 85, 199
W
Waring, Edward, 199, 259
Welser, Marcus, 14
Wilson, John, 259
Woepke, F., 208
Wolfskehl, 207
Woodall, H. J., 310, 325
Wright, E. M., 39, 40, 359
Wright, H. N., 359
X
Xylander (see Holzmann)
SUBJECT INDEX
Abacists, 21, 185
Abacus, 14, 15, 225, 227
Absolute value, 28, 348
Absorption law, 103
Abundant numbers, 94, 95
Al-Fakhri, 185, 187, 193, 208
Algebra (Euler), 60, 126, 128, 132,
138, 141, 206
Algebra, 20, 123
syncopated, 181
Algebraic congruences, 234, 235,
249-258
Algebraic numbers, 206, 207
Algorism, 20, 21
(See also Euclid's algorism)
Algorismus, 21
Aliquot parts, 86, 91, 98
Al-Jabr wal-Muqabalah, 19, 20, 187
Al-Kafi fil hisab, 185
Amicable numbers, 27, 96-100
Amplitude, 348, 350
Apices, 20
Arabic numerals, 19, 24, 117
Arithmetic (Sun-Tse), 245
Arithmetic, 180
Arithmetic mean, 90, 91
Arithmetic series, 79
Arithmetics (Diophantos), 180-185,
194, 199, 270
Associative law, 48, 103
Astrology, 28
Attic numerals, 11, 15
Average, 90
B
Babylonian numerals and system, 2,
16-18, 36, 37, 172-179, 188, 312,
324
Base (of number systems), 3
Bija-Ganita, 123, 136, 193
Binary number systems, 2, 37
Billion, 5
Book of Precious Things in the Art of
Reckoning, The, 139
Brahma-Sphuta-Siddhanta, 122, 193
Brahmi numerals, 19
Bureau, 15
C
Calculations, 14, 15
Calculus, 15, 55
Canon arithmeticus, 285, 301
Casting, on the lines, 15, 225, 227
out nines, 15, 225, 227, 229-231
Cattle problem, 140
Checks, 15
numerical, 225-233
Chinese remainder theorem, 240,
246-248, 251, 264, 265, 294, 304
Chinese- Japanese numerals, 11, 12,
16, 17
Cipher, 11, 20
Ciphered numerals, 12-13, 19
365
366
NUMBER THEORY AND ITS HISTORY
Closed systems of numbers, 158-163
Cogita Physico-malhematica, 71
Columna Rostrata, 4
Commutative law, 103
Complete Introduction to Algebra (see
Algebra, Euler)
Complex numbers, 158-163, 347, 348
Composite numbers, 50, 51, 331
Comptoir, 15
Congruences, 210^.
algebraic, 234, 235, 249-258
linear, 236-240, 275, 276, 298, 299
root of, 234#.
simultaneous, 240-250
Congruent, 21 1#.
Congruum, 191, 202
Construction by compass and ruler,
341-345, 349, 351, 352
Cossica, or Cossick Art, 195
Counters, 1, 15, 21, 227
Counting process, 1, 8
Cross-cut, 105
Cuttaca, 122
Cycle, solar, lunar, planetary, 247
Cyclotomic equation, 349
D
Decadic number systems, 1, 3, 12, 33,
34, 226, 233, 279
Decimal, circulating, 315
Decimal expansion, 311-325
finite, 315-317
periodic, 318-325
purely periodic, 320, 321
Decimal fractions, 21, 36, 311-326
Decimal number systems, 1, 3, 10, 34,
36
Deficient numbers, 94, 95
Delian problem (see Doubling of cube)
De Moivre, theorem of, 348-350
Demotic numerals, 13
De revolutionibus orbium coelestium,
21
De temporum ratione, 6
Determination, 212
De Thiende (see Disme, La)
Digits, 1, 16
Diophantine equations or problems,
165, 184, 193, 204, 207, 208,
268
Dirichlet, box principle of, 268
theorem of, 79, 80, 338
Disme, La, 24, 313, 314, 325
Disquisitiones arilhmeticae, 74, 209-
211, 225, 233, 243, 256, 263, 297,
301, 315, 323, 341, 358
Distributive laws, 106, 108
Divisible, 29, 213
Divisibility, 25, 226-228
Division, 25, 30
Division lemma, 44
Divisor, common, 41, 47
greatest common (gc.d.), 25, 41,
47, 48, 100-107, 154
Divisors of a number, 28, 29, 86
arithmetic mean of, 90, 91
average of, 90
geometric mean of, 87, 88, 91
harmonic mean of, 90, 91
maximal, 328
number of, 86
product of, 87
proper, 30
sum of, 88, 89, 95
trivial, 29
Doubling of cube (Delian problem),
340, 341
Dual, 103, 105, 106
Duplication, 38
Dyadic number systems, 2, 37-39
E
Elements (Euclid), 41, 52, 65, 92, 166
Eleven, 3
Equivalence relation, 213
Eratosthenes, sieve of, 64, 66, 67, 84
SUBJECT INDEX
367
Euclid's algorism, 41^45, 100, 122,
142, 146, 193
Euler's congruence or theorem, 272-
280, 290, 292, 303
^function, 109-115, 273, 283, 350
Exchequer, 8, 9, 15
Exponent, to which a number belongs,
280-284
±1, 302, 303, 309
universal, 290, 292, 293, 302, 332
F
Factor, 29
Factor tables, 53, 54, 82, 83, 85
Factorization method, Euler's, 59-64
Fermat's, 54-58, 62
Fermat numbers, 74, 75
primes, 69, 75, 205, 351-354, 358
(See also Numbers, with Fermat
property)
Fermat's theorem, 272, 273, 277-280,
326, 339
converse of, 326, 328, 331, 339
Fermat's last theorem, 203-208
Field, 163, 164
quadratic, 163
Finger numbers, 5-7
Finite decimal expansion, 315-317
F numbers {see Numbers, with Fer-
mat property)
Foil, 8
Forms, quadratic, 63
G
Greatest common divisor (g.c.d.), 25,
41, 47, 48, 100-107, 154
associative law for, 48, 103
Greek numerals, 11, 13, 15, 28
Ground of Artes, The, 21
H
Harmonic mean, 90, 91
Hebrew numerals, 28
Herodianic numerals, 11, 15
Hieratic numerals, 13
Hieroglyphic numerals, 10, 13
Hindu- Arabic numerals, 15, 19, 21,
24, 312
Hindu-Brahmi numerals, 13
Historical Prolegomenon, 97
Horoscope, 97
Hundred, 3
I
Ideals, 207
Idempotent law, 103
Incongruent, 211
Indeterminate problems, 120, 182
linear, 120^., 142ff., 160, 184, 193.
237, 276
Indicator, 110, 294, 302, 333
Indices, 294-301, 310, 322, 325
Infinite descent, method of, 198, 200
Integers, 28
greatest, contained in a number, 31
group of, 1, 2, 10-13
Integral logarithm, 77
Intersection, 105, 108
Ganita-Sara-Sangraha, 123, 141
Gauss's generalization of Wilson's con-
gruence, 263, 266, 267, 291, 292
Gematry (Gematria), 28
Geometric mean, 87, 88, 91
Geometric progression, 89, 272
Gobar numerals, 20
Goldbach's conjecture, 81, 84, 85
Jetons, 15, 20, 227
Join, 104
L
Lattice points, 151, 152
Lattices, 105, 106
Least common multiple (l.c.m.), 25,
45-49, 100-107
368
NUMBER THEORY AND ITS HISTORY
Least common multiple (l.c.m.), asso-
ciative law for, 48, 103
Liber Abaci, 20, 117, 122, 187, 228,
247
Liber quadratorum, 188, 203
Lilavati, 26, 122, 123
Linear congruences, 236-240, 275,
276, 298, 299
Local value, principle of, 16
Logarithms, 77, 294, 297, 299, 301,
314, 345
Logistics, 181
Lucas, theorem of, 326-328
M
Mantissa, 317-319, 322, 323, 325
Mayan numerals, 2, 4, 18
Mean, arithmetic, 90, 91
geometric, 87, 88, 91
harmonic, 90, 91
Mediation, 38
Meditationes algebraicae, 199, 259
Meet, 104
Mersenne primes, 69, 71-73, 92-94
Milliard, 5
Million, 4
Modul, of congruences, 21 1$".
of numbers, 159-163, 214
Modulus (see Modul)
Multiples, 29
common, 45, 48
least common (l.c.m.), 25, 45-49,
100-107
Multiplicative group, 162, 163, 344
grouping of numerals, 11, 12, 16
Multiplicity, 86
Multiply perfect numbers, 95-96
Myriad, 4
N
Neo-Platonic, 97
Neo-Pythagorean, 26
Nim, 39
Nonresidue, quadratic 354
Number, 1
of the Beast, 28
Number systems, 2, 3, 21, 34, 311
Babylonian sexagesimal, 2, 16-18,
36, 37, 172-179, 188, 312, 324
binary, 2, 37
closed, 158-163
decadic, 1, 3, 12, 33, 34, 226, 233,
279
decimal, 1, 3, 10, 34, 36
duodecimal, 2
dyadic, 2, 37-39
positional, 16
quinary, 2, 3
ternary, 37
triadic, 37
vigesimal, 2, 18
(See also Numerals)
Number theory (Barlow), 93, 131
Number theory, analytic, 76, 78, 85
Numbers, abundant, 94, 95
algebraic, 206, 207
amicable, 27. 96-100
associated, 28, 29
base, 34
binomial, 70
classification of, 158
complex, 158-163, 347, 348
composite, 50, 51, 331
deficient, 94, 95
even, 29, 31
Fermat, 74, 75
with Fermat property (F numbers),
331-339
finger, 5-7
imaginary, 347
Mersenne, 75
multiply perfect, 95, 96
natural, 25, 28
Ixld, 31
perfect, 27, 71, 91, 95
prime, 50-85, 93, 359
rational, 158-164, 319, 320
real, 158-163
SUBJECT INDEX
369
Numbers, regular, 174, 316
transcendental, 345
Numerals, 1
alphabetic, 13
Arabic, 19, 24, 117
Attic, 11, 15
Babylonian, 2, 16, 18, 172-179, 312,
324
Brahmi, 19
Chinese- Japanese, 11, 12, 16, 17
Chinese mercantile, 12
ciphered, 12-13, 19
Demotic, 13
Gobar, 20
Greek, 11, 13, 15, 28
Hebrew, 28
Herodianic, 11, 15
Hieratic, 13
hieroglyphic, 10, 13
Hindu-Arabic, 15, 19, 21, 24, 312
Hindu-Brahmi, 13
Mayan, 2, 4, 18
with multiplicative grouping, 11,
12, 16
positional, 16, 18, 19, 312
Roman, 8, 10, 14, 21
with simple grouping, 10, 11, 16, 18
Sumerian, 16
Numerical checks, 225-233
Numerology, 25-27, 39, 91, 94, 98
O
On the Creation of the World, 27
Palatine Anthology, 132, 180
Papyrus Rhind, 116, 117, 119, 141
Parts, 91
Periodic decimal expansion, 318-325
Perfect numbers, 27, 71, 91, 95
Periods, first, second, third (Gaus-
sian), 355-357
Platonic bodies, 340
Plimpton Library tablet, 170, 175,176
Polygonal numbers, 180
Polyhedrons, regular, 340
Polygons, regular, 340
construction of, 74, 340, 346-358
Porisms, 180
Prime, even, 50
factorization, 50, 51
factors, determination of, 52
number theorem, 77, 78
numbers, 50-85, 93, 359
relatively, 41, 50
tables, 66, 67, 69, 76, 85
twins, 76
Primes, distribution of, 75, 76, 78
Fermat, 69, 75, 205, 351-354, 358
infinitude of, 65
Mersenne, 69, 71-73, 92-94
Primitive roots (see Roots, primitive)
Primitive solutions, 167-169, 178,190,
191, 200
Problems (see Indeterminate prob-
lems)
Problemes plaisans, 125, 141
Pulverizer, 122, 193
Puzzles, 116|f.
Pythagorean school, 26, 27, 97, 165,
166, 174, 195
triangle and theorem, 165-170,
177-184, 189, 190, 200, 202,
203
Q
Quadratic, fields, 163
forms, 63
Quadrillion, 5
Quinary number systems, 2, 3
Quipu, 8
Quotient, 30
R
Radius vector, 348
Rational numbers, 158-164, 319, 320
Ray, 161
370
NUMBER THEORY AND ITS HISTORY
Real numbers, 158-163
Reckoning coins, 15
Reckoning Manual, 313
Reflexivity, 212
Reformation, 28
Regular numbers, 174, 316
Remainder, 30
least absolute, 32, 43, 215
least positive, 30
Republic, 26
Residue, biquadratic, 355 .
class, 214-216
quadratic, 354
systems, 213-216
Riemann's hypothesis, 78
Ring, 161-163
Roman indication, 247
Roots, primitive, for a modul, 284-
302, 310, 325
of unity, 349-352
nonprimitive, 350
primitive, 350, 352
Roman numerals, 8, 10, 14, 21
Rules of Coss, 313
Russian multiplication, 38, 39
S
Sand Reckoning, The, 5
Score, 2, 8
Self-dual, 107
Set operations, 105, 108
Seven, 27, 117
Simultaneous congruences, 240-250
Soroban, 15
Spread, 306-309
Squaring of the circle, 340, 345
Stock, 8
Structures, 105
Subtraction principle, 10
Sum, 105, 108
of three or four squares, 198
of two squares, 60-63, 196-198,
203. 267-271
Sumerian numerals, 16
Surd, 195
Swanpan, 15
Symmetry, 212
Syncopated algebra, 181
Tablets, cuneiform, Babylonian, 17,
170-179
Talisman, 97
Tallies, 8, 9
exchequer, 8, 9
Telephone cables, splicing of, 302,
305-310
Ten, 3
Thousand, 3
Totient, 110
Transitivity, 212
Trattato de numeri perfelti, 73
Trigonometric functions, 179
Trillion, 5
Trisection of the angle, 340, 341, 344,
345
Tschotu, 15
Twelve, 3
U
Union, 104, 105, 108
V
Vigesimal number systems, 2, 18
W
Wilson's congruence or theorem,
259-263, 266, 278
Gauss's generalization of, 263, 266,
267, 291, 292
Zero, 6, 18-20
residue class, 215
Zeta function, Riemann's, 78