The Higher Arithmetic – an Introduction to the Theory of Numbers

This page intentionally left blank Now into its eighth edition and with additional material on primality testing, written by J. H. Davenport, The Higher Arithmetic introduces concepts and theorems in a way that does not require the reader to have an in-depth knowledge of the theory of numbers but also touches upon matters of deep mathematical signi? cance. A companion website (www. cambridge. org/davenport) provides more details of the latest advances and sample code for important algorithms. Reviews of earlier editions: ‘. . . the well-known and charming introduction to number theory . . can be recommended both for independent study and as a reference text for a general mathematical audience. ’ European Maths Society Journal ‘Although this book is not written as a textbook but rather as a work for the general reader, it could certainly be used as a textbook for an undergraduate course in number theory and, in the reviewer’s opinion, is far superior for this purpose to any other book in English. ’ Bulletin of the American Mathematical Society THE HIGHER ARITHMETIC AN INTRODUCTION TO THE THEORY OF NUMBERS Eighth edition H. Davenport M. A. , SC. D. F. R. S. late Rouse Ball Professor of Mathematics in the University of Cambridge and Fellow of Trinity College Editing and additional material by James H. Davenport CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www. cambridge. org Information on this title: www. cambridge. org/9780521722360  © The estate of H. Davenport 2008 This publication is in copyright.Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2008 ISBN-13 ISBN-13 978-0-511-45555-1 978-0-521-72236-0 eBook (EBL) paperback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. CONTENTS Introduction I Factorization and the Primes 1. 2. 3. 4. . 6. 7. 8. 9. 10. The laws of arithmetic Proof by induction Prime numbers The fundamental theorem of arithmetic Consequences of the fundamental theorem Euclid’s algorithm Another proof of the fundamental theorem A property of the H. C. F Factorizing a number The series of primes page viii 1 1 6 8 9 12 16 18 19 22 25 31 31 33 35 37 40 41 42 45 46 II Congruences 1. 2. 3. 4. 5. 6. 7. 8. 9. The congruence notation Linear congruences Fermat’s theorem Euler’s function ? (m) Wilson’s theorem Algebraic congruences Congruences to a prime modulus Congr uences in several unknowns Congruences covering all numbers v vi III Quadratic Residues 1. 2. 3. 4. . 6. Primitive roots Indices Quadratic residues Gauss’s lemma The law of reciprocity The distribution of the quadratic residues Contents 49 49 53 55 58 59 63 68 68 70 72 74 77 78 82 83 86 92 94 99 103 103 104 108 111 114 116 116 117 120 122 124 126 128 131 133 IV Continued Fractions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. Introduction The general continued fraction Euler’s rule The convergents to a continued fraction The equation ax ? by = 1 In? nite continued fractions Diophantine approximation Quadratic irrationals Purely periodic continued fractions Lagrange’s theorem Pell’s equation A geometrical interpretation of continued fractionsV Sums of Squares 1. 2. 3. 4. 5. Numbers representable by two squares Primes of the form 4k + 1 Constructions for x and y Representation by four squares Representation by three squares VI Quadratic Forms 1. 2. 3. 4. 5. 6. 7. 8. 9. Introduction Equivalent forms The discriminant The representation of a number by a form Three examples The reduction of positive de? nite forms The reduced forms The number of representations The class-number Contents VII Some Diophantine Equations 1. Introduction 2. The equation x 2 + y 2 = z 2 3. The equation ax 2 + by 2 = z 2 4. Elliptic equations and curves 5.Elliptic equations modulo primes 6. Fermat’s Last Theorem 7. The equation x 3 + y 3 = z 3 + w 3 8. Further developments vii 137 137 138 140 145 151 154 157 159 165 165 168 173 179 185 188 194 199 200 209 222 225 235 237 VIII Computers and Number Theory 1. 2. 3. 4. 5. 6. 7. 8. 9. Introduction Testing for primality ‘Random’ number generators Pollard’s factoring methods Factoring and primality via elliptic curves Factoring large numbers The Dif? e–Hellman cryptographic method The RSA cryptographic method Primality testing revisited Exercises Hints Answers Bibliography IndexINTRODUCTION T he higher arithmetic, or the theory of numbers, is concerned with the properties of the natural numbers 1, 2, 3, . . . . These numbers must have exercised human curiosity from a very early period; and in all the records of ancient civilizations there is evidence of some preoccupation with arithmetic over and above the needs of everyday life. But as a systematic and independent science, the higher arithmetic is entirely a creation of modern times, and can be said to date from the discoveries of Fermat (1601–1665).A peculiarity of the higher arithmetic is the great dif? culty which has often been experienced in proving simple general theorems which had been suggested quite naturally by numerical evidence. ‘It is just this,’ said Gauss, ‘which gives the higher arithmetic that magical charm which has made it the favourite science of the greatest mathematicians, not to mention its inexhaustible wealth, wherein it so greatly surpasses other parts of mathematics. ’ The theory of numbers is generally considered to be the ‘purest’ branch of pure mathematics.It certainly has very few direct applications to other sciences, but it has one feature in common with them, namely the inspiration which it derives from experiment, which takes the form of testing possible general theorems by numerical examples. Such experiment, though necessary in some form to progress in every part of mathematics, has played a greater part in the development of the theory of numbers than elsewhere; for in other branches of mathematics the evidence found in this way is too often fragmentary and misleading.As regards the present book, the author is well aware that it will not be read without effort by those who are not, in some sense at least, mathematicians. But the dif? culty is partly that of the subject itself. It cannot be evaded by using imperfect analogies, or by presenting the proofs in a way viii Introduction ix which may convey the main idea o f the argument, but is inaccurate in detail. The theory of numbers is by its nature the most exact of all the sciences, and demands exactness of thought and exposition from its devotees. The theorems and their proofs are often illustrated by numerical examples.These are generally of a very simple kind, and may be despised by those who enjoy numerical calculation. But the function of these examples is solely to illustrate the general theory, and the question of how arithmetical calculations can most effectively be carried out is beyond the scope of this book. The author is indebted to many friends, and most of all to Professor o Erd? s, Professor Mordell and Professor Rogers, for suggestions and corrections. He is also indebted to Captain Draim for permission to include an account of his algorithm.The material for the ? fth edition was prepared by Professor D. J. Lewis and Dr J. H. Davenport. The problems and answers are based on the suggestions of Professor R. K. Guy. Chapter VIII a nd the associated exercises were written for the sixth edition by Professor J. H. Davenport. For the seventh edition, he updated Chapter VII to mention Wiles’ proof of Fermat’s Last Theorem, and is grateful to Professor J. H. Silverman for his comments. For the eighth edition, many people contributed suggestions, notably Dr J. F. McKee and Dr G. K. Sankaran.Cambridge University Press kindly re-typeset the book for the eighth edition, which has allowed a few corrections and the preparation of an electronic complement: www. cambridge. org/davenport. References to further material in the electronic complement, when known at the time this book went to print, are marked thus:  ¦:0. I FACTORIZATION AND THE PRIMES 1. The laws of arithmetic The object of the higher arithmetic is to discover and to establish general propositions concerning the natural numbers 1, 2, 3, . . . of ordinary arithmetic. Examples of such propositions are the fundamental theorem (I. 4)? hat every nat ural number can be factorized into prime numbers in one and only one way, and Lagrange’s theorem (V. 4) that every natural number can be expressed as a sum of four or fewer perfect squares. We are not concerned with numerical calculations, except as illustrative examples, nor are we much concerned with numerical curiosities except where they are relevant to general propositions. We learn arithmetic experimentally in early childhood by playing with objects such as beads or marbles. We ? rst learn addition by combining two sets of objects into a single set, and later we learn multiplication, in the form of repeated addition.Gradually we learn how to calculate with numbers, and we become familiar with the laws of arithmetic: laws which probably carry more conviction to our minds than any other propositions in the whole range of human knowledge. The higher arithmetic is a deductive science, based on the laws of arithmetic which we all know, though we may never have seen them form ulated in general terms. They can be expressed as follows. ? References in this form are to chapters and sections of chapters of this book. 1 2 The Higher Arithmetic Addition.Any two natural numbers a and b have a sum, denoted by a + b, which is itself a natural number. The operation of addition satis? es the two laws: a+b =b+a (commutative law of addition), (associative law of addition), a + (b + c) = (a + b) + c the brackets in the last formula serving to indicate the way in which the operations are carried out. Multiplication. Any two natural numbers a and b have a product, denoted by a ? b or ab, which is itself a natural number. The operation of multiplication satis? es the two laws ab = ba a(bc) = (ab)c (commutative law of multiplication), (associative law of multiplication).There is also a law which involves operations both of addition and of multiplication: a(b + c) = ab + ac (the distributive law). Order. If a and b are any two natural numbers, then either a is equal to b o r a is less than b or b is less than a, and of these three possibilities exactly one must occur. The statement that a is less than b is expressed symbolically by a < b, and when this is the case we also say that b is greater than a, expressed by b > a. The fundamental law governing this notion of order is that if a b. We propose to investigate the common divisors of a and b.If a is divisible by b, then the common divisors of a and b consist simply of all divisors of b, and there is no more to be said. If a is not divisible by b, we can express a as a multiple of b together with a remainder less than b, that is a = qb + c, where c < b. (2) This is the process of ‘division with a remainder’, and expresses the fact that a, not being a multiple of b, must occur somewhere between two consecutive multiples of b. If a comes between qb and (q + 1)b, then a = qb + c, where 0 < c < b. It follows from the equation (2) that any common divisor of b and c is also a divisor of a.Moreo ver, any common divisor of a and b is also a divisor of c, since c = a ? qb. It follows that the common divisors of a and b, whatever they may be, are the same as the common divisors of b and c. The problem of ? nding the common divisors of a and b is reduced to the same problem for the numbers b and c, which are respectively less than a and b. The essence of the algorithm lies in the repetition of this argument. If b is divisible by c, the common divisors of b and c consist of all divisors of c. If not, we express b as b = r c + d, where d < c. (3)Again, the common divisors of b and c are the same as those of c and d. The process goes on until it terminates, and this can only happen when exact divisibility occurs, that is, when we come to a number in the sequence a, b, c, . . . , which is a divisor of the preceding number. It is plain that the process must terminate, for the decreasing sequence a, b, c, . . . of natural numbers cannot go on for ever. Factorization and the Primes 17 Let us suppose, for the sake of de? niteness, that the process terminates when we reach the number h, which is a divisor of the preceding number g.Then the last two equations of the series (2), (3), . . . are f = vg + h, g = wh. (4) (5) The common divisors of a and b are the same as those of b and c, or of c and d, and so on until we reach g and h. Since h divides g, the common divisors of g and h consist simply of all divisors of h. The number h can be identi? ed as being the last remainder in Euclid’s algorithm before exact divisibility occurs, i. e. the last non-zero remainder. We have therefore proved that the common divisors of two given natural numbers a and b consist of all divisors of a certain number h (the H. C. F. f a and b), and this number is the last non-zero remainder when Euclid’s algorithm is applied to a and b. As a numerical illustration, take the numbers 3132 and 7200 which were used in  §5. The algorithm runs as follows: 7200 = 2 ? 3132 + 936, 3 132 = 3 ? 936 + 324, 936 = 2 ? 324 + 288, 324 = 1 ? 288 + 36, 288 = 8 ? 36; and the H. C. F. is 36, the last remainder. It is often possible to shorten the working a little by using a negative remainder whenever this is numerically less than the corresponding positive remainder. In the above example, the last three steps could be replaced by 936 = 3 ? 324 ? 6, 324 = 9 ? 36. The reason why it is permissible to use negative remainders is that the argument that was applied to the equation (2) would be equally valid if that equation were a = qb ? c instead of a = qb + c. Two numbers are said to be relatively prime? if they have no common divisor except 1, or in other words if their H. C. F. is 1. This will be the case if and only if the last remainder, when Euclid’s algorithm is applied to the two numbers, is 1. ? This is, of course, the same de? nition as in  §5, but is repeated here because the present treatment is independent of that given previously. 8 7. Another proof of t he fundamental theorem The Higher Arithmetic We shall now use Euclid’s algorithm to give another proof of the fundamental theorem of arithmetic, independent of that given in  §4. We begin with a very simple remark, which may be thought to be too obvious to be worth making. Let a, b, n be any natural numbers. The highest common factor of na and nb is n times the highest common factor of a and b. However obvious this may seem, the reader will ? nd that it is not easy to give a proof of it without using either Euclid’s algorithm or the fundamental theorem of arithmetic.In fact the result follows at once from Euclid’s algorithm. We can suppose a > b. If we divide na by nb, the quotient is the same as before (namely q) and the remainder is nc instead of c. The equation (2) is replaced by na = q. nb + nc. The same applies to the later equations; they are all simply multiplied throughout by n. Finally, the last remainder, giving the H. C. F. of na and nb, is nh, wher e h is the H. C. F. of a and b. We apply this simple fact to prove the following theorem, often called Euclid’s theorem, since it occurs as Prop. 30 of Book VII.If a prime divides the product of two numbers, it must divide one of the numbers (or possibly both of them). Suppose the prime p divides the product na of two numbers, and does not divide a. The only factors of p are 1 and p, and therefore the only common factor of p and a is 1. Hence, by the theorem just proved, the H. C. F. of np and na is n. Now p divides np obviously, and divides na by hypothesis. Hence p is a common factor of np and na, and so is a factor of n, since we know that every common factor of two numbers is necessarily a factor of their H. C. F.We have therefore proved that if p divides na, and does not divide a, it must divide n; and this is Euclid’s theorem. The uniqueness of factorization into primes now follows. For suppose a number n has two factorizations, say n = pqr . . . = p q r . . . , where all the numbers p, q, r, . . . , p , q , r , . . . are primes. Since p divides the product p (q r . . . ) it must divide either p or q r . . . . If p divides p then p = p since both numbers are primes. If p divides q r . . . we repeat the argument, and ultimately reach the conclusion that p must equal one of the primes p , q , r , . . . We can cancel the common prime p from the two representations, and start again with one of those left, say q. Eventually it follows that all the primes on the left are the same as those on the right, and the two representations are the same. Factorization and the Primes 19 This is the alternative proof of the uniqueness of factorization into primes, which was referred to in  §4. It has the merit of resting on a general theory (that of Euclid’s algorithm) rather than on a special device such as that used in  §4. On the other hand, it is longer and less direct. 8. A property of the H. C.F From Euclid’s algorithm one can deduce a remarkable property of the H. C. F. , which is not at all apparent from the original construction for the H. C. F. by factorization into primes ( §5). The property is that the highest common factor h of two natural numbers a and b is representable as the difference between a multiple of a and a multiple of b, that is h = ax ? by where x and y are natural numbers. Since a and b are both multiples of h, any number of the form ax ? by is necessarily a multiple of h; and what the result asserts is that there are some values of x and y for which ax ? y is actually equal to h. Before giving the proof, it is convenient to note some properties of numbers representable as ax ? by. In the ? rst place, a number so representable can also be represented as by ? ax , where x and y are natural numbers. For the two expressions will be equal if a(x + x ) = b(y + y ); and this can be ensured by taking any number m and de? ning x and y by x + x = mb, y + y = ma. These numbers x and y will be natura l numbers provided m is suf? ciently large, so that mb > x and ma > y. If x and y are de? ned in this way, then ax ? by = by ? x . We say that a number is linearly dependent on a and b if it is representable as ax ? by. The result just proved shows that linear dependence on a and b is not affected by interchanging a and b. There are two further simple facts about linear dependence. If a number is linearly dependent on a and b, then so is any multiple of that number, for k(ax ? by) = a. kx ? b. ky. Also the sum of two numbers that are each linearly dependent on a and b is itself linearly dependent on a and b, since (ax1 ? by1 ) + (ax2 ? by2 ) = a(x1 + x2 ) ? b(y1 + y2 ). 20 The Higher ArithmeticThe same applies to the difference of two numbers: to see this, write the second number as by2 ? ax2 , in accordance with the earlier remark, before subtracting it. Then we get (ax1 ? by1 ) ? (by2 ? ax2 ) = a(x1 + x2 ) ? b(y1 + y2 ). So the property of linear dependence on a and b is preserved by addition and subtraction, and by multiplication by any number. We now examine the steps in Euclid’s algorithm, in the light of this concept. The numbers a and b themselves are certainly linearly dependent on a and b, since a = a(b + 1) ? b(a), b = a(b) ? b(a ? 1). The ? rst equation of the algorithm was a = qb + c.Since b is linearly dependent on a and b, so is qb, and since a is also linearly dependent on a and b, so is a ? qb, that is c. Now the next equation of the algorithm allows us to deduce in the same way that d is linearly dependent on a and b, and so on until we come to the last remainder, which is h. This proves that h is linearly dependent on a and b, as asserted. As an illustration, take the same example as was used in  §6, namely a = 7200 and b = 3132. We work through the equations one at a time, using them to express each remainder in terms of a and b. The ? rst equation was 7200 = 2 ? 3132 + 936, which tells s that 936 = a ? 2b. The second equation was 3 132 = 3 ? 936 + 324, which gives 324 = b ? 3(a ? 2b) = 7b ? 3a. The third equation was 936 = 2 ? 324 + 288, which gives 288 = (a ? 2b) ? 2(7b ? 3a) = 7a ? 16b. The fourth equation was 324 = 1 ? 288 + 36, Factorization and the Primes which gives 36 = (7b ? 3a) ? (7a ? 16b) = 23b ? 10a. 21 This expresses the highest common factor, 36, as the difference of two multiples of the numbers a and b. If one prefers an expression in which the multiple of a comes ? rst, this can be obtained by arguing that 23b ? 10a = (M ? 10)a ? (N ? 23)b, provided that Ma = N b.Since a and b have the common factor 36, this factor can be removed from both of them, and the condition on M and N becomes 200M = 87N . The simplest choice for M and N is M = 87, N = 200, which on substitution gives 36 = 77a ? 177b. Returning to the general theory, we can express the result in another form. Suppose a, b, n are given natural numbers, and it is desired to ? nd natural numbers x and y such that ax ? by = n. (6) Such an e quation is called an indeterminate equation since it does not determine x and y completely, or a Diophantine equation after Diophantus of Alexandria (third century A . D . , who wrote a famous treatise on arithmetic. The equation (6) cannot be soluble unless n is a multiple of the highest common factor h of a and b; for this highest common factor divides ax ? by, whatever values x and y may have. Now suppose that n is a multiple of h, say, n = mh. Then we can solve the equation; for all we have to do is ? rst solve the equation ax1 ? by1 = h, as we have seen how to do above, and then multiply throughout by m, getting the solution x = mx1 , y = my1 for the equation (6). Hence the linear indeterminate equation (6) is soluble in natural numbers x, y if and only if n is a multiple of h.In particular, if a and b are relatively prime, so that h = 1, the equation is soluble whatever value n may have. As regards the linear indeterminate equation ax + by = n, we have found the condition for it to be soluble, not in natural numbers, but in integers of opposite signs: one positive and one negative. The question of when this equation is soluble in natural numbers is a more dif? cult one, and one that cannot well be completely answered in any simple way. Certainly 22 The Higher Arithmetic n must be a multiple of h, but also n must not be too small in relation to a and b.It can be proved quite easily that the equation is soluble in natural numbers if n is a multiple of h and n > ab. 9. Factorizing a number The obvious way of factorizing a number is to test whether it is divisible by 2 or by 3 or by 5, and so on, using the series of primes. If a number N v is not divisible by any prime up to N , it must be itself a prime; for any composite number has at least two prime factors, and they cannot both be v greater than N . The process is a very laborious one if the number is at all large, and for this reason factor tables have been computed.The most extensive one which is gener ally accessible is that of D. N. Lehmer (Carnegie Institute, Washington, Pub. No. 105. 1909; reprinted by Hafner Press, New York, 1956), which gives the least prime factor of each number up to 10,000,000. When the least prime factor of a particular number is known, this can be divided out, and repetition of the process gives eventually the complete factorization of the number into primes. Several mathematicians, among them Fermat and Gauss, have invented methods for reducing the amount of trial that is necessary to factorize a large number.Most of these involve more knowledge of number-theory than we can postulate at this stage; but there is one method of Fermat which is in principle extremely simple and can be explained in a few words. Let N be the given number, and let m be the least number for which m 2 > N . Form the numbers m 2 ? N , (m + 1)2 ? N , (m + 2)2 ? N , . . . . (7) When one of these is reached which is a perfect square, we get x 2 ? N = y 2 , and consequently N = x 2 ? y 2 = (x ? y)(x + y). The calculation of the numbers (7) is facilitated by noting that their successive differences increase at a constant rate. The identi? ation of one of them as a perfect square is most easily made by using Barlow’s Table of Squares. The method is particularly successful if the number N has a factorization in which the two factors are of about the same magnitude, since then y is small. If N is itself a prime, the process goes on until we reach the solution provided by x + y = N , x ? y = 1. As an illustration, take N = 9271. This comes between 962 and 972 , so that m = 97. The ? rst number in the series (7) is 972 ? 9271 = 138. The Factorization and the Primes 23 subsequent ones are obtained by adding successively 2m + 1, then 2m + 3, and so on, that is, 195, 197, and so on.This gives the series 138, 333, 530, 729, 930, . . . . The fourth of these is a perfect square, namely 272 , and we get 9271 = 1002 ? 272 = 127 ? 73. An interesting algorithm for fact orization has been discovered recently by Captain N. A. Draim, U . S . N. In this, the result of each trial division is used to modify the number in preparation for the next division. There are several forms of the algorithm, but perhaps the simplest is that in which the successive divisors are the odd numbers 3, 5, 7, 9, . . . , whether prime or not. To explain the rules, we work a numerical example, say N = 4511. The ? st step is to divide by 3, the quotient being 1503 and the remainder 2: 4511 = 3 ? 1503 + 2. The next step is to subtract twice the quotient from the given number, and then add the remainder: 4511 ? 2 ? 1503 = 1505, 1505 + 2 = 1507. The last number is the one which is to be divided by the next odd number, 5: 1507 = 5 ? 301 + 2. The next step is to subtract twice the quotient from the ? rst derived number on the previous line (1505 in this case), and then add the remainder from the last line: 1505 ? 2 ? 301 = 903, 903 + 2 = 905. This is the number which is to be divi ded by the next odd number, 7. Now we an continue in exactly the same way, and no further explanation will be needed: 905 = 7 ? 129 + 2, 903 ? 2 ? 129 = 645, 645 ? 2 ? 71 = 503, 503 ? 2 ? 46 = 411, 645 + 2 = 647, 503 + 8 = 511, 411 + 5 = 416, 647 = 9 ? 71 + 8, 511 = 11 ? 46 + 5, 416 = 13 ? 32 + 0. 24 The Higher Arithmetic We have reached a zero remainder, and the algorithm tells us that 13 is a factor of the given number 4511. The complementary factor is found by carrying out the ? rst half of the next step: 411 ? 2 ? 32 = 347. In fact 4511 = 13? 347, and as 347 is a prime the factorization is complete. To justify the algorithm generally is a matter of elementary algebra.Let N1 be the given number; the ? rst step was to express N1 as N1 = 3q1 + r1 . The next step was to form the numbers M2 = N1 ? 2q1 , The number N2 was divided by 5: N2 = 5q2 + r2 , and the next step was to form the numbers M3 = M2 ? 2q2 , N 3 = M3 + r 2 , N 2 = M2 + r 1 . and so the process was continued. It can be deduced from these equations that N2 = 2N1 ? 5q1 , N3 = 3N1 ? 7q1 ? 7q2 , N4 = 4N1 ? 9q1 ? 9q2 ? 9q3 , and so on. Hence N2 is divisible by 5 if and only if 2N1 is divisible by 5, or N1 divisible by 5. Again, N3 is divisible by 7 if and only if 3N1 is divisible by 7, or N1 divisible by 7, and so on.When we reach as divisor the least prime factor of N1 , exact divisibility occurs and there is a zero remainder. The general equation analogous to those given above is Nn = n N1 ? (2n + 1)(q1 + q2 +  ·  ·  · + qn? 1 ). The general equation for Mn is found to be Mn = N1 ? 2(q1 + q2 +  ·  ·  · + qn? 1 ). (9) If 2n + 1 is a factor of the given number N1 , then Nn is exactly divisible by 2n + 1, and Nn = (2n + 1)qn , whence n N1 = (2n + 1)(q1 + q2 +  ·  ·  · + qn ), (8) Factorization and the Primes by (8). Under these circumstances, we have, by (9), Mn+1 = N1 ? 2(q1 + q2 +  ·  ·  · + qn ) = N1 ? 2 n 2n + 1 N1 = N1 . n + 1 25 Thus the complementary factor to the factor 2n + 1 is Mn+1 , as stated in the example. In the numerical example worked out above, the numbers N1 , N2 , . . . decrease steadily. This is always the case at the beginning of the algorithm, but may not be so later. However, it appears that the later numbers are always considerably less than the original number. 10. The series of primes Although the notion of a prime is a very natural and obvious one, questions concerning the primes are often very dif? cult, and many such questions are quite unanswerable in the present state of mathematical knowledge.We conclude this chapter by mentioning brie? y some results and conjectures about the primes. In  §3 we gave Euclid’s proof that there are in? nitely many primes. The same argument will also serve to prove that there are in? nitely many primes of certain speci? ed forms. Since every prime after 2 is odd, each of them falls into one of the two progressions (a) 1, 5, 9, 13, 17, 21, 25, . . . , (b) 3, 7, 11, 15, 19, 23, 27, . . . ; the progression (a) consisting of all numbers of the form 4x + 1, and the progression (b) of all numbers of the form 4x ? 1 (or 4x + 3, which comes to the same thing).We ? rst prove that there are in? nitely many primes in the progression (b). Let the primes in (b) be enumerated as q1 , q2 , . . . , beginning with q1 = 3. Consider the number N de? ned by N = 4(q1 q2 . . . qn ) ? 1. This is itself a number of the form 4x ? 1. Not every prime factor of N can be of the form 4x + 1, because any product of numbers which are all of the form 4x + 1 is itself of that form, e. g. (4x + 1)(4y + 1) = 4(4x y + x + y) + 1. Hence the number N has some prime factor of the form 4x ? 1. This cannot be any of the primes q1 , q2 , . . . , qn , since N leaves the remainder ? when 26 The Higher Arithmetic divided by any of them. Thus there exists a prime in the series (b) which is different from any of q1 , q2 , . . . , qn ; and this proves the proposition. The same argument cannot be used to prove t hat there are in? nitely many primes in the series (a), because if we construct a number of the form 4x +1 it does not follow that this number will necessarily have a prime factor of that form. However, another argument can be used. Let the primes in the series (a) be enumerated as r1 , r2 , . . . , and consider the number M de? ned by M = (r1 r2 . . rn )2 + 1. We shall see later (III. 3) that any number of the form a 2 + 1 has a prime factor of the form 4x + 1, and is indeed entirely composed of such primes, together possibly with the prime 2. Since M is obviously not divisible by any of the primes r1 , r2 , . . . , rn , it follows as before that there are in? nitely many primes in the progression (a). A similar situation arises with the two progressions 6x + 1 and 6x ? 1. These progressions exhaust all numbers that are not divisible by 2 or 3, and therefore every prime after 3 falls in one of these two progressions.One can prove by methods similar to those used above that there ar e in? nitely many primes in each of them. But such methods cannot cope with the general arithmetical progression. Such a progression consists of all numbers ax +b, where a and b are ? xed and x = 0, 1, 2, . . . , that is, the numbers b, b + a, b + 2a, . . . . If a and b have a common factor, every number of the progression has this factor, and so is not a prime (apart from possibly the ? rst number b). We must therefore suppose that a and b are relatively prime. It then seems plausible that the progression will contain in? itely many primes, i. e. that if a and b are relatively prime, there are in? nitely many primes of the form ax + b. Legendre seems to have been the ? rst to realize the importance of this proposition. At one time he thought he had a proof, but this turned out to be fallacious. The ? rst proof was given by Dirichlet in an important memoir which appeared in 1837. This proof used analytical methods (functions of a continuous variable, limits, and in? nite series), an d was the ? rst really important application of such methods to the theory of numbers.It opened up completely new lines of development; the ideas underlying Dirichlet’s argument are of a very general character and have been fundamental for much subsequent work applying analytical methods to the theory of numbers. Factorization and the Primes 27 Not much is known about other forms which represent in? nitely many primes. It is conjectured, for instance, that there are in? nitely many primes of the form x 2 + 1, the ? rst few being 2, 5, 17, 37, 101, 197, 257, . . . . But not the slightest progress has been made towards proving this, and the question seems hopelessly dif? cult.Dirichlet did succeed, however, in proving that any quadratic form in two variables, that is, any form ax 2 + bx y + cy 2 , in which a, b, c are relatively prime, represents in? nitely many primes. A question which has been deeply investigated in modern times is that of the frequency of occurrence of the p rimes, in other words the question of how many primes there are among the numbers 1, 2, . . . , X when X is large. This number, which depends of course on X , is usually denoted by ? (X ). The ? rst conjecture about the magnitude of ? (X ) as a function of X seems to have been made independently by Legendre and Gauss about X 1800.It was that ? (X ) is approximately log X . Here log X denotes the natural (so-called Napierian) logarithm of X , that is, the logarithm of X to the base e. The conjecture seems to have been based on numerical evidence. For example, when X is 1,000,000 it is found that ? (1,000,000) = 78,498, whereas the value of X/ log X (to the nearest integer) is 72,382, the ratio being 1. 084 . . . . Numerical evidence of this kind may, of course, be quite misleading. But here the result suggested is true, in the sense that the ratio of ? (X ) to X/ log X tends to the limit 1 as X tends to in? ity. This is the famous Prime Number Theorem, ? rst proved by Hadamard and de la Vall? e e Poussin independently in 1896, by the use of new and powerful analytical methods. It is impossible to give an account here of the many other results which have been proved concerning the distribution of the primes. Those proved in the nineteenth century were mostly in the nature of imperfect approaches towards the Prime Number Theorem; those of the twentieth century included various re? nements of that theorem. There is one recent event to which, however, reference should be made.We have already said that the proof of Dirichlet’s Theorem on primes in arithmetical progressions and the proof of the Prime Number Theorem were analytical, and made use of methods which cannot be said to belong properly to the theory of numbers. The propositions themselves relate entirely to the natural numbers, and it seems reasonable that they should be provable without the intervention of such foreign ideas. The search for ‘elementary’ proofs of these two theorems was u nsuccessful until fairly recently. In 1948 A. Selberg found the ? rst elementary proof of Dirichlet’s Theorem, and with 28 The Higher Arithmetic he help of P. Erd? s he found the ? rst elementary proof of the Prime Numo ber Theorem. An ‘elementary’ proof, in this connection, means a proof which operates only with natural numbers. Such a proof is not necessarily simple, and indeed both the proofs in question are distinctly dif? cult. Finally, we may mention the famous problem concerning primes which was propounded by Goldbach in a letter to Euler in 1742. Goldbach suggested (in a slightly different wording) that every even number from 6 onwards is representable as the sum of two primes other than 2, e. g. 6 = 3 + 3, 8 = 3 + 5, 10 = 3 + 7 = 5 + 5, 12 = 5 + 7, . . . Any problem like this which relates to additive properties of primes is necessarily dif? cult, since the de? nition of a prime and the natural properties of primes are all expressed in terms of multiplic ation. An important contribution to the subject was made by Hardy and Littlewood in 1923, but it was not until 1930 that anything was rigorously proved that could be considered as even a remote approach towards a solution of Goldbach’s problem. In that year the Russian mathematician Schnirelmann proved that there is some number N such that every number from some point onwards is representable as the sum of at most N primes.A much nearer approach was made by Vinogradov in 1937. He proved, by analytical methods of extreme subtlety, that every odd number from some point onwards is representable as the sum of three primes. This was the starting point of much new work on the additive theory of primes, in the course of which many problems have been solved which would have been quite beyond the scope of any pre-Vinogradov methods. A recent result in connection with Goldbach’s problem is that every suf? ciently large even number is representable as the sum of two numbers, one of which is a prime and the other of which has at most two prime factors.Notes Where material is changing more rapidly than print cycles permit, we have chosen to place some of the material on the book’s website: www. cambridge. org/davenport. Symbols such as  ¦I:0 are used to indicate where there is such additional material.  §1. The main dif? culty in giving any account of the laws of arithmetic, such as that given here, lies in deciding which of the various concepts should come ? rst. There are several possible arrangements, and it seems to be a matter of taste which one prefers. It is no part of our purpose to analyse further the concepts and laws of ? rithmetic. We take the commonsense (or na? ve) view that we all ‘know’ Factorization and the Primes 29 the natural numbers, and are satis? ed of the validity of the laws of arithmetic and of the principle of induction. The reader who is interested in the foundations of mathematics may consult Bertrand Russe ll, Introduction to Mathematical Philosophy (Allen and Unwin, London), or M. Black, The Nature of Mathematics (Harcourt, Brace, New York). Russell de? nes the natural numbers by selecting them from numbers of a more general kind. These more general numbers are the (? ite or in? nite) cardinal numbers, which are de? ned by means of the more general notions of ‘class’ and ‘one-to-one correspondence’. The selection is made by de? ning the natural numbers as those which possess all the inductive properties. (Russell, loc. cit. , p. 27). But whether it is reasonable to base the theory of the natural numbers on such a vague and unsatisfactory concept as that of a class is a matter of opinion. ‘Dolus latet in universalibus’ as Dr Johnson remarked.  §2. The objection to using the principle of induction as a de? ition of the natural numbers is that it involves references to ‘any proposition about a natural number n’. It seems plain the th at ‘propositions’ envisaged here must be statements which are signi? cant when made about natural numbers. It is not clear how this signi? cance can be tested or appreciated except by one who already knows the natural numbers.  §4. I am not aware of having seen this proof of the uniqueness of prime factorization elsewhere, but it is unlikely that it is new. For other direct proofs, see Mathews, p. 2, or Hardy and Wright, p. 21.?  §5. It has been shown by (intelligent! computer searches that there is no odd perfect number less than 10300 . If an odd perfect number exists, it has at least eight distinct prime factors, of which the largest exceeds 108 . For references and other information on perfect or ‘nearly perfect’ numbers, see Guy, sections A. 3, B. 1 and B. 2.  ¦I:1  §6. A critical reader may notice that in two places in this section I have used principles that were not explicitly stated in  §Ã‚ §1 and 2. In each place, a proof by induction co uld have been given, but to have done so would have distracted the reader’s attention from the main issues.The question of the length of Euclid’s algorithm is discussed in Uspensky and Heaslet, ch. 3, and D. E. Knuth’s The Art of Computer Programming vol. II: Seminumerical Algorithms (Addison Wesley, Reading, Mass. , 3rd. ed. , 1998) section 4. 5. 3.  §9. For an account of early methods of factoring, see Dickson’s History Vol. I, ch. 14. For a discussion of the subject as it appeared in ? Particulars of books referred to by their authors’ names will be found in the Bibliography. 30 The Higher Arithmetic the 1970s see the article by Richard K. Guy, ‘How to factor a number’, Congressus Numerantium XVI Proc. th Manitoba Conf. Numer. Math. , Winnipeg, 1975, 49–89, and at the turn of the millennium see Richard P. Brent, ‘Recent progress and prospects for integer factorisation algorithms’, Springer Lecture Notes in Comp uter Science 1858 Proc. Computing and Combinatorics, 2000, 3–22. The subject is discussed further in Chapter VIII. It is doubtful whether D. N. Lehmer’s tables will ever be extended, since with them and a pocket calculator one can easily check whether a 12-digit number is a prime. Primality testing is discussed in VIII. 2 and VIII. 9. For Draim’s algorithm, see Mathematics Magazine, 25 (1952) 191–4. 10. An excellent account of the distribution of primes is given by A. E. Ingham, The Distribution of Prime Numbers (Cambridge Tracts, no. 30, 1932; reprinted by Hafner Press, New York, 1971). For a more recent and extensive account see H. Davenport, Multiplicative Number Theory, 3rd. ed. (Springer, 2000). H. Iwaniec (Inventiones Math. 47 (1978) 171–88) has shown that for in? nitely many n the number n 2 + 1 is either prime or the product of at most two primes, and indeed the same is true for any irreducible an 2 + bn + c with c odd. Dirichlet’s p roof of his theorem (with a modi? ation due to Mertens) is given as an appendix to Dickson’s Modern Elementary Theory of Numbers. An elementary proof of the Prime Number Theorem is given in ch. 22 of Hardy and Wright. An elementary proof of the asymptotic formula for the number of primes in an arithmetic progression is given in Gelfond and Linnik, ch. 3. For a survey of early work on Goldbach’s problem, see James, Bull. American Math. Soc. , 55 (1949) 246–60. It has been veri? ed that every even number from 6 to 4 ? 1014 is the sum of two primes, see Richstein, Math. Comp. , 70 (2001) 1745–9. For a proof of Chen’s theorem that every suf? iently large even integer can be represented as p + P2 , where p is a prime, and P2 is either a prime or the product of two primes, see ch. 11 of Sieve Methods by H. Halberstam and H. E. Richert (Academic Press, London, 1974). For a proof of Vinogradov’s result, see T. Estermann, Introduction to Modern Prime Number Theory (Cambridge Tracts, no. 41, 1952) or H. Davenport, Multiplicative Number Theory, 3rd. ed. (Springer, 2000). ‘Suf? ciently large’ in Vinogradov’s result has now been quanti? ed as ‘greater than 2 ? 101346 ’, see M. -C. Liu and T. Wang, Acta Arith. , 105 (2002) 133–175.Conversely, we know that it is true up to 1. 13256 ? 1022 (Ramar? and Saouter in J. Number Theory 98 (2003) 10–33). e II CONGRUENCES 1. The congruence notation It often happens that for the purposes of a particular calculation, two numbers which differ by a multiple of some ? xed number are equivalent, in the sense that they produce the same result. For example, the value of (? 1)n depends only on whether n is odd or even, so that two values of n which differ by a multiple of 2 give the same result. Or again, if we are concerned only with the last digit of a number, then for that purpose two umbers which differ by a multiple of 10 are effectively the same. The congruence notation, introduced by Gauss, serves to express in a convenient form the fact that two integers a and b differ by a multiple of a ? xed natural number m. We say that a is congruent to b with respect to the modulus m, or, in symbols, a ? b (mod m). The meaning of this, then, is simply that a ? b is divisible by m. The notation facilitates calculations in which numbers differing by a multiple of m are effectively the same, by stressing the analogy between congruence and equality.Congruence, in fact, means ‘equality except for the addition of some multiple of m’. A few examples of valid congruences are: 63 ? 0 (mod 3), 7 ? ?1 (mod 8), 52 ? ?1 (mod 13). A congruence to the modulus 1 is always valid, whatever the two numbers may be, since every number is a multiple of 1. Two numbers are congruent with respect to the modulus 2 if they are of the same parity, that is, both even or both odd. 31 32 The Higher Arithmetic Two congruences can be added, subtracted, or m ultiplied together, in just the same way as two equations, provided all the congruences have the same modulus.If a ? ? (mod m) and b ? ? (mod m) then a + b ? ? + ? (mod m), a ? b ? ? ? ? (mod m), ab ? (mod m). The ? rst two of these statements are immediate; for example (a + b) ? (? + ? ) is a multiple of m because a ? ? and b ? ? are both multiples of m. The third is not quite so immediate and is best proved in two steps. First ab ? ?b because ab ? ?b = (a ? ?)b, and a ? ? is a multiple of m. Next, ? b ? , for a similar reason. Hence ab ? (mod m). A congruence can always be multiplied throughout by any integer: if a ? ? (mod m) then ka ? k? (mod m).Indeed this is a special case of the third result above, where b and ? are both k. But it is not always legitimate to cancel a factor from a congruence. For example 42 ? 12 (mod 10), but it is not permissible to cancel the factor 6 from the numbers 42 and 12, since this would give the false result 7 ? 2 (mod 10). The reason is obvious : the ? rst congruence states that 42 ? 12 is a multiple of 10, but this does not imply that 1 (42 ? 12) is a multiple of 10. The cancellation of 6 a factor from a congruence is legitimate if the factor is relatively prime to the modulus.For let the given congruence be ax ? ay (mod m), where a is the factor to be cancelled, and we suppose that a is relatively prime to m. The congruence states that a(x ? y) is divisible by m, and it follows from the last proposition in I. 5 that x ? y is divisible by m. An illustration of the use of congruences is provided by the well-known rules for the divisibility of a number by 3 or 9 or 11. The usual representation of a number n by digits in the scale of 10 is really a representation of n in the form n = a + 10b + 100c +  ·  ·  · , where a, b, c, . . . re the digits of the number, read from right to left, so that a is the number of units, b the number of tens, and so on. Since 10 ? 1 (mod 9), we have also 102 ? 1 (mod 9), 103 ? 1 (mod 9), and so on. Hence it follows from the above representation of n that n ? a + b + c +  ·  ·  · (mod 9). Congruences 33 In other words, any number n differs from the sum of its digits by a multiple of 9, and in particular n is divisible by 9 if and only if the sum of its digits is divisible by 9. The same applies with 3 in place of 9 throughout. The rule for 11 is based on the fact that 10 ? ?1 (mod 11), so that 102 ? +1 (mod 11), 103 ? 1 (mod 11), and so on. Hence n ? a ? b + c ?  ·  ·  · (mod 11). It follows that n is divisible by 11 if and only if a ? b+c?  ·  ·  · is divisible by 11. For example, to test the divisibility of 9581 by 11 we form 1? 8+5? 9, or ? 11. Since this is divisible by 11, so is 9581. 2. Linear congruences It is obvious that every integer is congruent (mod m) to exactly one of the numbers 0, 1, 2, . . . , m ? 1. (1) r < m, For we can express the integer in the form qm + r , where 0 and then it is congruent to r (mod m). Obviously there are othe r sets of numbers, besides the set (1), which have the same property, e. . any integer is congruent (mod 5) to exactly one of the numbers 0, 1, ? 1, 2, ? 2. Any such set of numbers is said to constitute a complete set of residues to the modulus m. Another way of expressing the de? nition is to say that a complete set of residues (mod m) is any set of m numbers, no two of which are congruent to one another. A linear congruence, by analogy with a linear equation in elementary algebra, means a congruence of the form ax ? b (mod m). (2) It is an important fact that any such congruence is soluble for x, provided that a is relatively prime to m.The simplest way of proving this is to observe that if x runs through the numbers of a complete set of residues, then the corresponding values of ax also constitute a complete set of residues. For there are m of these numbers, and no two of them are congruent, since ax 1 ? ax2 (mod m) would involve x1 ? x2 (mod m), by the cancellation of the factor a (permissible since a is relatively prime to m). Since the numbers ax form a complete set of residues, there will be exactly one of them congruent to the given number b. As an example, consider the congruence 3x ? 5 (mod 11). 34 The Higher ArithmeticIf we give x the values 0, 1, 2, . . . , 10 (a complete set of residues to the modulus 11), 3x takes the values 0, 3, 6, . . . , 30. These form another complete set of residues (mod 11), and in fact they are congruent respectively to 0, 3, 6, 9, 1, 4, 7, 10, 2, 5, 8. The value 5 occurs when x = 9, and so x = 9 is a solution of the congruence. Naturally any number congruent to 9 (mod 11) will also satisfy the congruence; but nevertheless we say that the congruence has one solution, meaning that there is one solution in any complete set of residues. In other words, all solutions are mutually congruent.The same applies to the general congruence (2); such a congruence (provided a is relatively prime to m) is precisely equivalent to the con gruence x ? x0 (mod m), where x0 is one particular solution. There is another way of looking at the linear congruence (2). It is equivalent to the equation ax = b + my, or ax ? my = b. We proved in I. 8 that such a linear Diophantine equation is soluble for x and y if a and m are relatively prime, and that fact provides another proof of the solubility of the linear congruence. But the proof given above is simpler, and illustrates the advantages gained by using the congruence notation.The fact that the congruence (2) has a unique solution, in the sense explained above, suggests that one may use this solution as an interpretation b for the fraction a to the modulus m. When we do this, we obtain an arithmetic (mod m) in which addition, subtraction and multiplication are always possible, and division is also possible provided that the divisor is relatively prime to m. In this arithmetic there are only a ? nite number of essentially distinct numbers, namely m of them, since two numbers w hich are mutually congruent (mod m) are treated as the same.If we take the modulus m to be 11, as an illustration, a few examples of ‘arithmetic mod 11’ are: 5 ? 9 ? ?2. 3 Any relation connecting integers or fractions in the ordinary sense remains true when interpreted in this arithmetic. For example, the relation 5 + 7 ? 1, 5 ? 6 ? 8, 1 2 7 + = 2 3 6 becomes (mod 11) 6 + 8 ? 3, because the solution of 2x ? 1 is x ? 6, that of 3x ? 2 is x ? 8, and that of 6x ? 7 is x ? 3. Naturally the interpretation given to a fraction depends on the modulus, for instance 2 ? 8 (mod 11), but 2 ? 3 (mod 7). The 3 3 Congruences 35 nly limitation on such calculations is that just mentioned, namely that the denominator of any fraction must be relatively prime to the modulus. If the modulus is a prime (as in the above examples with 11), the limitation takes the very simple form that the denominator must not be congruent to 0 (mod m), and this is exactly analogous to the limitation in ordina ry arithmetic that the denominator must not be equal to 0. We shall return to this point later ( §7). 3. Fermat’s theorem The fact that there are only a ? nite number of essentially different numbers in arithmetic to a modulus m means that there are algebraic relations which are satis? d by every number in that arithmetic. There is nothing analogous to these relations in ordinary arithmetic. Suppose we take any number x and consider its powers x, x 2 , x 3 , . . . . Since there are only a ? nite number of possibilities for these to the modulus m, we must eventually come to one which we have met before, say x h ? x k (mod m), where k < h. If x is relatively prime to m, the factor x k can be cancelled, and it follows that x l ? 1 (mod m), where l ? h ? k. Hence every number x which is relatively prime to m satis? es some congruence of this form. The least exponent l for which x l ? (mod m) will be called the order of x to the modulus m. If x is 1, its order is obviously 1. To illustrate the de? nition, let us calculate the orders of a few numbers to the modulus 11. The powers of 2, taken to the modulus 11, are 2, 4, 8, 5, 10, 9, 7, 3, 6, 1, 2, 4, . . . . Each one is twice the preceding one, with 11 or a multiple of 11 subtracted where necessary to make the result less than 11. The ? rst power of 2 which is ? 1 is 210 , and so the order of 2 (mod 11) is 10. As another example, take the powers of 3: 3, 9, 5, 4, 1, 3, 9, . . . . The ? rst power of 3 which is ? 1 is 35 , so the order of 3 (mod 11) is 5.It will be found that the order of 4 is again 5, and so also is that of 5. It will be seen that the successive powers of x are periodic; when we have reached the ? rst number l for which x l ? 1, then x l+1 ? x and the previous cycle is repeated. It is plain that x n ? 1 (mod m) if and only if n is a multiple of the order of x. In the last example, 3n ? 1 (mod 11) if and only if n is a multiple of 5. This remains valid if n is 0 (since 30 = 1), and it remains valid also for negative exponents, provided 3? n , or 1/3n , is interpreted as a fraction (mod 11) in the way explained in  §2. 36 The Higher ArithmeticIn fact, the negative powers of 3 (mod 11) are obtained by prolonging the series backwards, and the table of powers of 3 to the modulus 11 is n =†¦ ?3 ? 2 ? 1 0 1 2 3 4 5 6 . . . 9 5 4 1 3 9 5 4 1 3 †¦ . 3n ? . . . Fermat discovered that if the modulus is a prime, say p, then every integer x not congruent to 0 satis? es x p? 1 ? 1 (mod p). (3) In view of what we have seen above, this is equivalent to saying that the order of any number is a divisor of p ? 1. The result (3) was mentioned by Fermat in a letter to Fr? nicle de Bessy of 18 October 1640, in which he e also stated that he had a proof.But as with most of Fermat’s discoveries, the proof was not published or preserved. The ? rst known proof seems to have been given by Leibniz (1646–1716). He proved that x p ? x (mod p), which is equivalent to (3), b y writing x as a sum 1 + 1 +  ·  ·  · + 1 of x units (assuming x positive), and then expanding (1 + 1 +  ·  ·  · + 1) p by the multinomial theorem. The terms 1 p + 1 p +  ·  ·  · + 1 p give x, and the coef? cients of all the other terms are easily proved to be divisible by p. Quite a different proof was given by Ivory in 1806. If x ? 0 (mod p), the integers x, 2x, 3x, . . . , ( p ? )x are congruent (in some order) to the numbers 1, 2, 3, . . . , p ? 1. In fact, each of these sets constitutes a complete set of residues except that 0 has been omitted from each. Since the two sets are congruent, their products are congruent, and so (x)(2x)(3x) . . . (( p ? 1)x) ? (1)(2)(3) . . . ( p ? 1)(mod p). Cancelling the factors 2, 3, . . . , p ? 1, as is permissible, we obtain (3). One merit of this proof is that it can be extended so as to apply to the more general case when the modulus is no longer a prime. The generalization of the result (3) to any modulus was ? rst given b y Euler in 1760.To formulate it, we must begin by considering how many numbers in the set 0, 1, 2, . . . , m ? 1 are relatively prime to m. Denote this number by ? (m). When m is a prime, all the numbers in the set except 0 are relatively prime to m, so that ? ( p) = p ? 1 for any prime p. Euler’s generalization of Fermat’s theorem is that for any modulus m, x ? (m) ? 1 (mod m), provided only that x is relatively prime to m. (4) Congruences 37 To prove this, it is only necessary to modify Ivory’s method by omitting from the numbers 0, 1, . . . , m ? 1 not only the number 0, but all numbers which are not relatively prime to m.There remain ? (m) numbers, say a 1 , a2 , . . . , a? , Then the numbers a1 x, a2 x, . . . , a? x are congruent, in some order, to the previous numbers, and on multiplying and cancelling a1 , a2 , . . . , a? (as is permissible) we obtain x ? ? 1 (mod m), which is (4). To illustrate this proof, take m = 20. The numbers less than 20 and relati vely prime to 20 are 1, 3, 7, 9, 11, 13, 17, 19, so that ? (20) = 8. If we multiply these by any number x which is relatively prime to 20, the new numbers are congruent to the original numbers in some other order.For example, if x is 3, the new numbers are congruent respectively to 3, 9, 1, 7, 13, 19, 11, 17 (mod 20); and the argument proves that 38 ? 1 (mod 20). In fact, 38 = 6561. where ? = ? (m). 4. Euler’s function ? (m) As we have just seen, this is the number of numbers up to m that are relatively prime to m. It is natural to ask what relation ? (m) bears to m. We saw that ? ( p) = p ? 1 for any prime p. It is also easy to evaluate ? ( p a ) for any prime power pa . The only numbers in the set 0, 1, 2, . . . , pa ? 1 which are not relatively prime to p are those that are divisible by p. These are the numbers pt, where t = 0, 1, . . , pa? 1 ? 1. The number of them is pa? 1 , and when we subtract this from the total number pa , we obtain ? ( pa ) = pa ? pa? 1 = pa? 1 ( p ? 1). (5) The determination of ? (m) for general values of m is effected by proving that this function is multiplicative. By this is meant that if a and b are any two relatively prime numbers, then ? (ab) = ? (a)? (b). (6) 38 The Higher Arithmetic To prove this, we begin by observing a general principle: if a and b are relatively prime, then two simultaneous congruences of the form x ? ? (mod a), x ? ? (mod b) (7) are precisely equivalent to one congruence to the modulus ab.For the ? rst congruence means that x = ? + at where t is an integer. This satis? es the second congruence if and only if ? + at ? ? (mod b), or at ? ? ? ? (mod b). This, being a linear congruence for t, is soluble. Hence the two congruences (7) are simultaneously soluble. If x and x are two solutions, we have x ? x (mod a) and x ? x (mod b), and therefore x ? x (mod ab). Thus there is exactly one solution to the modulus ab. This principle, which extends at once to several congruences, provided that the moduli ar e relatively prime in pairs, is sometimes called ‘the Chinese remainder theorem’.It assures us of the existence of numbers which leave prescribed remainders on division by the moduli in question. Let us represent the solution of the two congruences (7) by x ? [? , ? ] (mod ab), so that [? , ? ] is a certain number depending on ? and ? (and also on a and b of course) which is uniquely determined to the modulus ab. Different pairs of values of ? and ? give rise to different values for [? , ? ]. If we give ? the values 0, 1, . . . , a ? 1 (forming a complete set of residues to the modulus a) and similarly give ? the values 0, 1, . . . , b ? 1, the resulting values of [? , ? constitute a complete set of residues to the modulus ab. It is obvious that if ? has a factor in common with a, then x in (7) will also have that factor in common with a, in other words, [? , ? ] will have that factor in common with a. Thus [? , ? ] will only be relatively prime to ab if ? is relatively prime to a and ? is relatively prime to b, and conversely these conditions will ensure that [? , ? ] is relatively prime to ab. It follows that if we give ? the ? (a) possible values that are less than a and prime to a, and give ? the ? (b) values that are less than b and prime to b, there result ? (a)? (b) values of [? ? ], and these comprise all the numbers that are less than ab and relatively prime to ab. Hence ? (ab) = ? (a)? (b), as asserted in (6). To illustrate the situation arising in the above proof, we tabulate below the values of [? , ? ] when a = 5 and b = 8. The possible values for ? are 0, 1, 2, 3, 4, and the possible values for ? are 0, 1, 2, 3, 4, 5, 6, 7. Of these there are four values of ? which are relatively prime to a, corresponding to the fact that ? (5) = 4, and four values of ? that are relatively prime to b, Congruences 39 corresponding to the fact that ? (8) = 4, in accordance with the formula (5).These values are italicized, as also are the corresponding values of [? , ? ]. The latter constitute the sixteen numbers that are relatively prime to 40 and less than 40, thus verifying that ? (40) = ? (5)? (8) = 4 ? 4 = 16. ? ? 0 1 2 3 4 0 0 16 32 8 24 1 25 1 17 33 9 2 10 26 2 18 34 3 35 11 27 3 19 4 20 36 12 28 4 5 5 21 37 13 29 6 30 6 22 38 14 7 15 31 7 23 39 We now return to the original question, that of evaluating ? (m) for any number m. Suppose the factorization of m into prime powers is m = pa q b . . . . Then it follows from (5) and (6) that ? (m) = ( pa ? pa? 1 )(q b ? q b? 1 ) . . . or, more elegantly, ? (m) = m 1 ? For example, ? (40) = 40 1 ? and ? (60) = 60 1 ? 1 2 1 2 1 p 1? 1 q †¦. (8) 1? 1 3 1 5 = 16, 1 5 1? 1? = 16. The function ? (m) has a remarkable property, ? rst given by Gauss in his Disquisitiones. It is that the sum of the numbers ? (d), extended over all the divisors d of a number m, is equal to m itself. For example, if m = 12, the divisors are 1, 2, 3, 4, 6, 12, and we have ? (1) + ? (2) + ? (3) + ? (4) + ? (6) + ? (12) = 1 + 1 + 2 + 2 + 2 + 4 = 12. A general proof can be based either on (8), or directly on the de? nition of the function. 40 The Higher ArithmeticWe have already referred (I. 5) to a table of the values of ? (m) for m 10, 000. The same volume contains a table giving those numbers m for which ? (m) assumes a given value up to 2,500. This table shows that, up to that point at least, every value assumed by ? (m) is assumed at least twice. It seems reasonable to conjecture that this is true generally, in other words that for any number m there is another number m such that ? (m ) = ? (m). This has never been proved, and any attempt at a general proof seems to meet with formidable dif? culties. For some special types of numbers the result is easy, e. g. f m is odd, then ? (m) = ? (2m); or again if m is not divisible by 2 or 3 we have ? (3m) = ? (4m) = ? (6m). 5. Wilson’s theorem This theorem was ? rst publis