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This book is due for return on or before the 
last date shown above, 

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Abacist vs. Atgorismisl 

From Gregor Rds<;h: Margarita Philosophic* 
Strasbourg 1504 


By Oystein Ore 




! 9 4 8 


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. 



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 

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 


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 



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 


3. Factor tables * ' ' 53 

4. For mat' a factorization method - • ■ ■ «4 

5. Euler's factorization method . ™ 

$. The sieve of Eratosthenes 


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 


. yi 


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 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 , ^ 


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 


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 


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 

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 



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- 

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 


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 


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 

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 


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 


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. 


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.) 


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 


; s ■'?'">,. 

a wggggafrjg 

Fig. 1-2. Exchequer Tallies. (From H. Jenkimon, Exchequer Tallies, Archae- 
logica, London, 1011. Courtesy of Society of Antiquaries, London.) 


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 


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: 





10 50 100 






X L C 



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 


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: 

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- 


cative grouping as shown in the illustration. It should be noted 
that the writing is vertical instead of horizontal. 

6 yr 



Example: 3468 

1 — 10 ")" 

2 ^~ 100 ^~ 

3 ^- 1000 -f" 

4 23 


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 


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. 

1-9 I II in — «j in ^ -=- ((« 

10-90 A A A — *A J}) >1 ^ ^ 


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 

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. 



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. 


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. 


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 



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 


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, 



To simplify the writing a subtractive symbol f*~ , lal or minus, is 
applied, as in the example 


= 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 


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: 

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. 


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. 

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 


"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 




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.) 





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 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/ 
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. 





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. 


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. 


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 



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 


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. 


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\. 


Now let o be an arbitrary integer. The multiples of o are all 

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 

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 

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. 


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 


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 


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, 


called the greatest integer contained in a/b. 

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. 


W = 3, [e] = 2, [^J = 4 


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. 


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: 


Theorem 2-2. The square of an odd number is of the form 
8g + 1. 


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 

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 

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 


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 

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 

Qi = a n • 6 n_1 + • • • + a 2 * b + ai 

To determine ai one divides qi by b 

qi = 92 • b + a x 

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. 



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 _! 

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. 


Represent 1,832 to the base 7. One calculates 

7 2 = 49, 7 3 = 343, 7 4 = 2,401 


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 



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 


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 


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 


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. 
















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 



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 







3 ii. 







5 13 







7 15 








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. 


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. 


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. 


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. 

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 




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 


fn—2 — Qn r n—1 i r r, 


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 


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. 


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.) 


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 



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: 


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 

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 


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 — 


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 


[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 

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: 


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 

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) 


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 

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) 


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) 


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 

(«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 

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 

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] 

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 

[[a, b],c] = [a, [b, c]] 


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- 


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. 


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 


























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 









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. 



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- 

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 


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 

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 

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 


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. 


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 


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 


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 


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 

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. 


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 


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 

(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 


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: 


= 45,030 

x 2 — n = 



2x + 1 = 










45,035 499,944 


45,035 499,944 


36 590,015 


37 680,088 


38 770,163 


39 860,240 

45,040 950,319 

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. 


Factor the following numbers by means of Fermat's method: (a) 8,927, 
(6) 57,479, (c) 14,327,581. 


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 


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 

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- 


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 


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 ] 


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. 


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. 


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 


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 


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. 


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. 


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- 

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 


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 

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 


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 


[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 




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 

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. 


LetN be the given number and p t , p 2 , ■ • . , P, the primes less than 
VN. Then 

,w - . ( viv) + 1 = * - nn _ m m 



_pL_-|_... + ... 


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 


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 


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. 


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- 

N = 7 • 19 • 139 • 54,091 

2. The number 

N = 10 9 + 3 9 = 1,000,019,683 


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 

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. 


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 


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) 


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- 


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 


1. Factor the numbers 

10 8 ±3 8 

2. Factor some of the first of the numbers 

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). 


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 



the function ir(x) behaves asymptotically like the function #/log x, 
that is, for large values of x their quotient approaches 1 



oo z/log x 

- = 1 


(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 

tv ^ r dt 

«/2 log t 

The following table indicates the accuracy of the approximation: 

























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. 


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) 


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 


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 

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.) 


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. 



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. 

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. 



Factor Table 


















I 43 






















































































































































































































































































































































































































































































































































































































































































































































Factor Table — (Continued) 








3| 7 
7 3 


3 13 


31 11 

3 .. 

11 3 



11 | 13 




131 | 19 




11 19 
7| 3 

31 17 
3 |23 
7 ' 

3 13 
3 I 7 

7 1 3 131 
17 . . 3 
3 | 17 47 
3 13 

93 | 97 


23 [41 
3 | 13 



3 |43 



' 13 

11 | 3 
3 | 31 

7 37 
3 19 
.. 3 I 11 
31 | 3 
3 11 

23 19 


17 3 

3 13 

3 61 
47 53 


















3 I 7 
7 I 3 

3 59 

. . 3 

. . 29 

3 7 

7 3 


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 


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. 


Cunningham, A. J. C. : Binomial Factorisations Giving Extensive Congruence- 
Tables and Factorisation-Tables, Vols. 1-7, Francis Hodgson, London, 

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). 


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 number 

60 = 2 2 • 3 • 5 


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) 


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 

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 


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 


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 

<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- 


sequence we conclude that when N has the prime factorization 


°(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 


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) 


The sum of the divisors of 

60 = 2 2 • 3 • 5 

<r(60) = (2 3 - 1)(3 + 1)(5 + 1) = 168 

as one can verify by summing the divisors (5-5). Their average is 

^(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), 


we divide by the number v(N) of divisors and find by means of 


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. 


We have seen that the arithmetic mean of the divisors of 60 is 14. Therefore 

#(60) = ff = 4f 

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 


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 

*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 

<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 


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 

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 

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 


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 

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 


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 


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 


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 

Through the Arabs the knowledge of amicable numbers spread 
to Western Europe. They are mentioned in the works of many 


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) 


















As may be seen, each number is obtained by doubling the pre- 
ceding and adding 1. 


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 



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 


J2 3 


2 • 5 • 47 • 359 

[2 4 


[2 5 


2 4 • 23 • 479 

J2 2 


J2 3 


2 4 • 89 • 127 

[2 2 

• 13 • 107 

)2 3 


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) 


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. 


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 


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) 


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. 


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 



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)} 


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 


and cross-cut. The two operations satisfy the same dual set of 
axioms as those mentioned previously : 


a n a = a, a v a = a 


a nb = b n a, a v b = b u a 


a n (6 n c) = (a n b) n c, a u (6 u c) = (a u 6) u c 


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. 


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 

(a, [b, c]) = [(a, 6), (a, c)] (5-33) 


[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 

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. 


There exists another interesting identity, which we shall now 
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 

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- 



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 


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. 


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) 


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. 


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 


(5-37) that are not divisible by p. This is simple since those that 
are divisible by p are the multiples 

V, 2p, -..,-• V (5-39) 


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) 


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, 

pq, 2pq, . . . , —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 



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 


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 


Pt+i, 2p t +i, • ■ • , Pt+i (5-44) 


which have not already been stricken out. A multiple 


ap f+1 , a = 1, 2, . 


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 

numbers in the series 




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) 


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. 


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) 


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 


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. 


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. 


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. 


Escott, E. B.: "Amicable Numbers," Scripta Mathematica, Vol. 12, 61-72 

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. 


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." 



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). 







Ears of wheat 


Hekat measure 




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 


"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 

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 • 





• "fin 
i nnn 




„,™99 S^?0 D 
no 9 i/ avw 




III 9999 III 

70S, 61 


in 999 Sill 


a»« a> n p 

o o> g 

in/-/ 7 

in ' 





u ? 

n-o - t'umt t'^u; 

,9999 ¥¥ i 
' 9999ii • 

106,2 l 

2 6,5 

nn99|jj i 


HI Will* If fe 



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.) 



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. 


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. 


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 


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 

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. 


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 


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 = 


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 


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 > 

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 

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 


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 > 


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 



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 


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 > 

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. 


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 


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* > 


^ 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 


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. 


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 


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 


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 

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 

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 


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 


is an integer and 


8* = 3x + 1 

x = St ! — 


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 > 

| < 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) 


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 — 



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* > 


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 


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 



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 


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 

2 = 90 — x — y — 

so that 

14x + by 

is an integer. Thus we have the reduced equation 
175w = 14z + by 


This shows that 


y = 35w — Sx + - 



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 



and therefore 


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 





V = 1 





«; = 2 





*; = 3 





v = 4 





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, 

19x + 9y + 32 = 100 

This in turn leads us to 

x — 1 
z = 33 - Qx - Sy 


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 

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 


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. 


A later addition requires the solution of second-degree indetermi- 
nate equations. 


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. 


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. 



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 


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 , 


The last remainder is 1 since a and b are relatively prime. We 
shall now obtain a representation (7-1) by a stepwise process 



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 

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 


and substitute 

2 = 20 - 2-9 

from the third division. This gives 

1 = 9-9 - 4-20 


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 

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 



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 


-3 -69 

1 -13 


+ 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. 














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 



alternatingly. This may be executed according to the following 
scheme : 


238 = 




165 = 


•2 + 19 


73 = 


•3 + 16 


19 = 


1+ 3 


16 = 


5+ 1 


3 = 



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 



The algorism is 

1,726 = 

1,229 -1 + 497 


1,229 = 

497 • 2 + 235 


497 = 

235 • 2 + 27 


235 = 



27 = 

19-1 + 8 


19 = 

8-2 + 3 


8 = 

3-2 + 2 


3 = 

2-1 + 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 

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) 


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 

ax + by = c 

to have a solution in integers is that the greatest common divisor 
of a and b divide c. 


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 



17 = 9 

1 +8 



1 + 1 



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 


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 

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. 


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. 


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 


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 



a solution, then there are lattice points on the line for all the 


x , xq dt a, x ± 2a, 

This means that the lattice points that represent solutions lie at 


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 


a c 

l x + l 


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. 


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 


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 


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) 


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 

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). 


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 

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 

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 

Theorem 7-4. The necessary and sufficient condition for the 

a\Xi + • • • + a n x n = c (7-20) 


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). 


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 


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 


3 — 142! — 9x 3 

v — 


is an integer. When this expression is solved for x 3 , 

it follows 


,, o , 3 + 4*! 
x 3 = At 2x 1 + 


3 +Ax x 
u = 


is integral and 

u - 3 

Xi = 2u -\ 


This gives finally that 

u - 3 

v = 


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. 



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 

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. 


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 


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 

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 


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 

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. 


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 

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 


b in S, 

quotient aT 1 = - must also belong to S. Consequently, for any 

ah = -37 

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 


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 

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 


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. 


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. 

Albert, A. A. : College Algebra, McGraw-Hill Book Company, Inc., New York, 

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. 


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 



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 


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, 


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 

°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) 


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 


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) . 




u = 2, 

» = 1 




u = 3, 

v = 2 




u = 4, 

v = 1 




u = 4, 

» = 3 




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 

c-b = 1 

1 Proceedings, Fifth International Mathematical Congress, Cambridge, 1912. 


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. 


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- 


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 



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- 1 


a' 1 


a" 1 




3, 45 


1, 20 




3, 20 


1, 15 










2, 30 


1, 6, 40 




2, 24 




7, 30 


2, 13, 20 

1, 4 

56, 15 


6, 40 



1, 12 





1, 52, 30 

1, 15 





1, 40 

1, 20 







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, 



and so on, which would give infinite expansions, have been 

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 


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 + 


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 



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 



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 


56, 7 

1, 20, 25 [3, 12, 1] 


1, 16, 41 

1, 50, 49 


3, 31, 49 

5, 9, 1 





5, 19 

8, 1 


38, 11 

59, 1 


13, 19 

20, 49 


8, 1 [9, 1] 

12, 49 


1, 22, 41 

2, 16, 1 



1, 15 


27, 59 

48, 49 


2, 41 [7, 12, 1] 

4, 49 


29, 31 

53, 49 



1, 46 [53] 


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 



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: 


















































































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 


a 2 sin 2 a 


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 


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 


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 

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) 


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 


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- 


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 


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 


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 

2nt + t 2 = a + b 

a + b - t 2 

n = 


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, 


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- 

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 


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 


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 


+ k 2 



Therefore the three numbers 

* i. 

form a Pythagorean triangle, and according to the solution we 
have obtained, we can write 

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 



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: 

























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. 


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 = — - > 




By substitution into (8-15) and clearing the fractions, one 

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 


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 

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/ 


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. 


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 


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- 


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 


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 


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. 


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 


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. 


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, 


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 


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 

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\ 


We put this into the second equation in (8-27) and find 

2vi = — - — * — ~ — 
2 2 


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 


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 


v x 2 = k 2 l 2 (8-30) 

The other alternative is that — - — is even, and in this case one 


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) 


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 


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 


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 



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 



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 


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 

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. 


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.) 


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 


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. 


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* 




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 


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 


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) 


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) 

a = c (mod m) 


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) 
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) 


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, • ■ • 


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. 


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 


form a complete system of residues (mod 5) since one has the 

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 

at 7^ aj (mod m) 

since in this case they would all belong to different residue classes. 


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) 


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) 


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) 


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) 


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) 

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) 


The same would hold if one took two polynomials in which the 
corresponding coefficients were congruent. For instance, the 

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 

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. 


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) 


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; 

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 

Almost trivial is 
Theorem 9-4. If 

then one also has 


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^) 

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) 


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 

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 


Let as see how the cancellation rule must be modified. When a 

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) 


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. 
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) 


d~ d\ 

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) 


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 


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) 


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. 


The number 

N = 234,648 s2+3+4 + 6+4 + 8 = 27=0 (mod 9) 
is divisible by 9. 


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. 


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) 


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 

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) 



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. 


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 


ab = —1, c = — 1 (mod 9) 


as one should expect. We shall also check the multiplication (9-19) for the 
modulus 11. Then by (9-15) 


6 = 3-8 + 5-3 = -3 (mod 11) 


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. 




a = 


b = 37, 

By performing the division 

one finds 


= 2,041, 

r = 4,696 


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) 

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 

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, 

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) 


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) 






+ 12 




177 = 78 

+ 5 


= -21 

172 = 





+ 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 : 











- 5 



+ 7 
- 3 
+ 4 
99 = 

here one has 


62 = -2 

(mod 101) 


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 


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. 


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 


Gauss, C. F. : Disquisiliones arithmeticae, Leipzig, 1801. French translation by 
A. C. M. Poullet-Delisle: Recherches arithmetiques, Paris, 1807. 


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) 


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 

x = x (mod m) 

are also solutions and by convention we consider them as a single 

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. 


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 


we wish to make. For the moment we shall be content to have 
introduced the basic concepts. 


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. 


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 

ax — my = b (10-4) 


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 

d = (12, 8) = 4 

does not divide b = 2 so that no solution can exist, as we found 

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) 


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 


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) 

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 

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) 


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) 


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 


19 = 7-2 + 5 


7 = 5-1+2 


5 = 2-2 + 1 


The solution is therefore 

i=59 (mod 140) 

3. Finally in the example 

144a; = 216 (mod 


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 

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) 



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) 

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) 

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) 


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) 

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) 


t = t + u - 



where u is some integer. The resulting general solution of the 
original congruence (10-10) is 



/ , . n \ mn 

x = xq (mod [m, ri\) 

r .. mn 
[m, n] = -— 

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. 

1. When 

x = 7 (mod 42), x = 15 (mod 51) 

there is no solution since 

d = (42, 51) = 3 

7 ^ 15 (mod 3) 


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 ) 


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;. 


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) 


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 

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) 


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 

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 


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 2 ' ' ' m r 

is a number divisible by all other moduls and relatively prime to 
mi. Similarly the number 


— = Wi • • • m^imj+i • • • m r 

is divisible by all moduls except ra*, to which it is relatively prime. 
For each i one can, consequently, solve the linear congruence 

hi — = 1 (mod mi) 610-2G) 


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. 


The example used by Sun-Tse corresponds to the three con- 

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) 

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. 


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 


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 . 


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 

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 



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) 

x a 93 (mod 105) 

for the common solution of the two given congruences, and similarly another 

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 


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 


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) 

189 = 3 3 • 7 

so that we solve the congruence (mod 3 3 ) and (mod 7). One finds, respec- 

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 


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- 


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 

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 

fix) = 100 - 525s -4- 625s 2 = (mod 125) 
fix) = 200 - 725» + 625y 2 = 


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) 


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. 


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. 


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) 


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). 


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) 


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. 


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) 


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 

(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. 


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 

1-2 (p - 1) + 1 

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) 



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 

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 


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) 

#=— 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 

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 


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 

x 2 + 1 = (mod p) 

is solvable and has the roots 

x s ± \^—£- ) ! ( mod V) 

For instance when p = 13 we find 

6! = 5 (mod 13) 

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 


! = ± 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 


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- 

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 ) 

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- 

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 ) 


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 

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) 


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) 


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) 
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 


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 

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, 

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) 


a(xt - x 2 ) = y\ - y 2 (mod p) (11-43) 


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 

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. 


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 


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. 


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, 

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 

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) 


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. 


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 


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 


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) 


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) 

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) 


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 



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. 



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) 


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 


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. 


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 


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) 


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) 


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 


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 


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: 

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. 


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- 

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 

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 


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 


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 


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. 



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 

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- 


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. 


Primitive root 


Primitive root 

























































































- 83 




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. 


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 


<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 ) 


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. 


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) 


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 

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 


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 } 


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. 


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. 


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 


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 


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) 


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 

<p(m) = 192 

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. 


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. 


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) 


^ 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 


3 _ 



This gives us the following table of the numbers with given indices 
(mod 9) for the root r = 2: 














To obtain the index of a given number one rearranges the table to 
make the remainders relatively prime to 9 the primary entry. 














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: 
















































Through rearrangement of the entries in this table, one obtains 
the companion table: 

















































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) 


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. 


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 

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)) 

Ind x = Ind 6 — Ind a (mod <p{m)) 


1. Let us solve the congruence 

7x = 2 (mod 9) 


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- 


havior of the resulting linear congruences (12-46) and (12-48) for 
the indices. 


1. Let us take 

3x 5 = 11 (mod 23) 
Here one finds 

5 Ind x = Ind 11 - Ind 3 (mod 22) 


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) 

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) 



x Ind a = (mod <p(m)) 

and if <p{m) and Ind a have the greatest common factor d, one must 


x = 


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. 

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)? 


2. Construct a table of indices (mod 19) and use it to solve the following 

(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) 


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 

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). 


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 ) 


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 



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 


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- 

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. 


8 = 2 

s =3 











6 -+11 





8 -+11 





11 ->10 


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 


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 


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) 


(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 

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) 

s n = ±1 (mod m) 



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: 























































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. 


Let us only mention, finally, that there are other ways in which 
this problem may be handled. 


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). 


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 



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. 


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 


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- 


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 


it may be written 

a! • 10 71 - 1 + « 2 • 10 n ~ 2 + • ■ • +a n p 

r = 


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 


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 


r = - 


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) 


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) 


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 

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 • • • 


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) 


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 


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 


periodic decimal expansion. When a rational number 

m ( \ i 

— > (w, n) = 1 


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. 


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 


(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). 


Let us take n = 7. The remainders of the successive powers of 10 (mod 7) 

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). 



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. 


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 


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. 


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) 


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 

Very extensive tables have been computed to determine the 
exponents to which the number 10 belongs for various moduls. 


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 


10 6 - 1 = 3 3 • 7 • 11 • 13 • 37 

10 3 - 1 = 3 3 


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 



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? 


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. 


14-1. The converse of Fermat's theorem. Fermat's theorem 
states that for every number a not divisible by the prime p, the 


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) 



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 


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 



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 

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- 



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. 





2,116 s 



2,025 = 




3,481 s 


98 = -45 


900 s 


84 = -59 


256 = 







(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 




1. We take n = 700,001 and use the auxiliary number a - 3. 
of computations takes the form: 

The table 













































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: 











































This shows that the number is composite. 


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.) 


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. 


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. 


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 

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] 


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. 


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) 


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 

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 

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 


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 


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 first few values of t, such that all three numbers (14-17) 
are primes, have been tabulated below. 




































When * is limited so that the least prime pi does not exceed 
10,000, one finds 45 F numbers of the type (14-17). 


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* 



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 

m = (7 -13 -19) (37 -73 -109) 

is an example. 


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 



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 

Sispanov, S.: "Sobre ios numeros pseudo-primos," Boletin matematico, Vol. 
14, 99-106 (1941). 


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 



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 

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 


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. 


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 


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 

b 3 = 2a 3 

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 


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- 


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 

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- 


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 


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. 



Let us recall briefly a few properties of complex numbers. Any 
such number can be written 

a + ib — r (cos <p + i sin <p), 



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) 


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 


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; 


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 — 


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 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 

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 


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 


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 


are conjugate imaginary, and their sum 

e k + <T k = 2 cos — k 
is real. In particular 

v = e + e _1 = 2 cos — > (15-12) 


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) 


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 


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 











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 



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 = 



— j 

-Vii - l 

Pi = 



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 


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 


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. 


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. 



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. 



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 . 


Uhleb, H. S.: "A brief history of the investigations on Mersenne numbers 
and the latest immense primes," Scripta Mathematica, Vol. 18, 122-131 

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 . 


New li?ts of multiply perfect numbers have recently been 
published by B. Franqui and M. Garcia [American Mathematical 



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. 


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 


(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. 


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, 

: 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, 

: 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. 



Abel, N. H., 343 
Abu Kamil, 139, 140 
Abu-1-Hasan Thabit ben Korrah, 

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 


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, 

Brancker, 53 

Briggs, 317 
Brun, V., 84 
Buddha, 4 
Bull, L. S., 141 
Burckhardt, 54 


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 


Dase, 54 

De Moivre, 348-350 
Descartes, Rene, 55, 95, 96, 99 
Dickson, L. E., 199, 358, 359 




Diophantos, 168, 179-185, 187, 193- 
196, 198, 199, 203-205, 208, 211, 
268, 270, 313 

Dirichlet, Lejeune, 79, 80, 206, 207, 


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 


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 


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 


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 


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, 

Lambert, J. H., 53 
Lame, 43, 206 

Lawther, H. P., Jr., 305, 309, 310 
Legendre, 206, 211 



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 


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 


Napier, John, 24 

Napoleon, 199, 209 

Neugebauer, O., 171, 172, 176, 208 

Newton, 55, 209 

Nonius, 195 


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, 

Poullet-Delisle, A. C. M., 233 
Powers, 73 
Proclus, 166 
Ptolemy, King, 340 
Pythagoras, 165, 166 


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 


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 



Thales of Miletus, 174 
Theodoras, Master, 187 
Thomas, J. M., 358 
Thue, A., 268 
Thureau-Dangin, F., 171, 208 

Uhler, 73 
Uspensky, J. V., 39, 40, 43, 359 


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 


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 


Xylander (see Holzmann) 


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, 

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 


Babylonian numerals and system, 2, 
16-18, 36, 37, 172-179, 188, 312, 

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 


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 




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 


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, 

Deficient numbers, 94, 95 
Delian problem (see Doubling of cube) 
De Moivre, theorem of, 348-350 
Demotic numerals, 13 
De revolutionibus orbium coelestium, 

De temporum ratione, 6 
Determination, 212 
De Thiende (see Disme, La) 
Digits, 1, 16 

Diophantine equations or problems, 
165, 184, 193, 204, 207, 208, 
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 


Elements (Euclid), 41, 52, 65, 92, 166 
Eleven, 3 

Equivalence relation, 213 
Eratosthenes, sieve of, 64, 66, 67, 84 



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, 
±1, 302, 303, 309 
universal, 290, 292, 293, 302, 332 


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 
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 


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 


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 

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 


Lattice points, 151, 152 
Lattices, 105, 106 

Least common multiple (l.c.m.), 25, 
45-49, 100-107 



Least common multiple (l.c.m.), asso- 
ciative law for, 48, 103 

Liber Abaci, 20, 117, 122, 187, 228, 

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 


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, 
Multiplicative group, 162, 163, 344 

grouping of numerals, 11, 12, 16 
Multiplicity, 86 

Multiply perfect numbers, 95-96 
Myriad, 4 


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, 

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), 

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 



Numbers, regular, 174, 316 

transcendental, 345 
Numerals, 1 

alphabetic, 13 

Arabic, 19, 24, 117 

Attic, 11, 15 

Babylonian, 2, 16, 18, 172-179, 312, 

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 


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- 
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, 


Quadratic, fields, 163 

forms, 63 
Quadrillion, 5 

Quinary number systems, 2, 3 
Quipu, 8 
Quotient, 30 


Radius vector, 348 

Rational numbers, 158-164, 319, 320 

Ray, 161 



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 


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, 

Talisman, 97 
Tallies, 8, 9 

exchequer, 8, 9 
Telephone cables, splicing of, 302, 

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, 

Tschotu, 15 
Twelve, 3 


Union, 104, 105, 108 


Vigesimal number systems, 2, 18 


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