o c Elementary o I-m Theory -< Number m UNDERWOOD DUDLEY 3 D «-* 03 «^ z c 3 a- H O K«fi (t Elementary Number Theory A Series of Books in Mathematics Editors: R. A. Rosenbaum G. Philip Johnson Elementary Number Theory UNDERWOOD DUDLEY DePauw University W. H. FREEMAN AND COMPANY San Francisco Copyright © 1969 by W. H. Freeman and Company No part of this book may be reproduced by any mechanical, photographic, or electronic process, or in the form of a phonographic recording, nor may it be stored in a retrieval system, transmitted, or otherwise copied for public or private use without written permission of the publisher. Printed in the United States of America Standard Book Number: 7167 0438-2 / Library of Congress Catalog Card Number: 71-80080 9 8 7 6 5 4 Contents Preface vii section 1 Integers 1 2 Unique Factorization 10 3 Linear Diophantine Equations 20 4 Congruences 27 5 Linear Congruences 34 6 Fermat's and Wilson's Theorems 42 7 The Divisors of an Integer 49 8 Perfect Numbers 56 9 Euler's Theorem and Function 63 10 Primitive Roots and Indices 72 11 Quadratic Congruences 82 12 Quadratic Reciprocity 92 13 Numbers in Other Bases 101 14 Duodecimals 109 15 Decimals 115 16 Pythagorean Triangles 122 17 Infinite Descent and Fermat's Conjecture 129 18 Sums of Two Squares 135 19 Sums of Four Squares 143 2 2 20 x - Ny = 1 149 21 Formulas for Primes 156 22 Bounds for x(jc) 164 23 Miscellaneous Problems 177 VI Contents 187 APPENDIX A Proof by Induction 191 APPENDIX B Summation and Other Notations 196 APPENDIX C Quadratic Congruences to Composite Moduli 203 APPENDIX D Tables 209 A Factor Table for Integers Less than 10,000 209 B Table of Squares 216 C Portion of a Factor Table 218 Answers to Exercises 223 Hints for Problems 229 Answers to Problems 243 Index 258 Preface Today, when a course in number theory is offered at all, it is usually taken only by mathematics majors late in their undergraduate careers. There is no reason why number theory should not be more widely taught, since it is, I think, manifestly more valuable to a prospective mathematics teacher than a course in, say, differential equations. This text has been designed for a one- semester or one-quarter course in number theory with minimal prerequisites. The reader is not required to know any mathematics except for the properties of the real numbers and elementary algebra. (A small amount of analysis is helpful, but not necessary, in Sections 21 and 22.) Nevertheless, because the average student does not find number theory easy, I have written detailed proofs and included many numerical examples. The purpose of the examples is to illustrate the theorems and also to try to show the fascination of playing with numbers, which is at the bottom of many of the theorems. I have included here at least an introduction to most of the topics of ele- mentary number theory. In Sections 1 through 5 the fundamental properties of the integers and congruences are developed, and in Section 6 proofs of Fermat's and Wilson's theorems are given. The number-theoretic functions d, a, and <t> are introduced in Sections 7 to 9. Sections 10 to 12 culminate in the quadratic reciprocity theorem. There follow three more or less inde- pendent blocks of material: on the representation of numbers (Sections 13 to 15), diophantine equations (16 to 20), and primes (21 to 22). Because I think that problems and exercises are especially important and interesting in num- ber theory, Section 23 consists of 105 miscellaneous problems. They are ar- ranged, very roughly, in increasing order of difficulty and without regard to topic. — viii Preface For the typical class of inexperienced students, there is more than enough material for one semester. Any of the last three blocks of material mentioned above could be omitted. For that matter, Sections 10 to 12 (which contain the most difficult material in the book) could be left out, since the only fact proved there and used later is that — 1 is a quadratic residue or nonresidue of an odd prime p according as p = 1 or 3 (mod 4). Proofs of theorems are often neither the shortest nor the most elegant known, but are what I thought the most natural. For example, the adaption of Tchebyshev's theorem on the bounds of ir(x) in section 22 is quite long, but it has the advantage of introducing functions and techniques used else- where in number theory. Besides, when the student sees the more elegant proof of the theorem, he will be all the more impressed. In keeping with the idea that the only way to learn mathematics is to do it, I have included more than a thousand exercises and problems. The exercises interrupt the text (as well as some proofs), and the problems appear at the end of each section. In fact, exercises and problems appear in three of the four appendices. The exercises can be used in several ways: the student may do them as he reads the material for the first time, he may return to them later to check on his understanding of material already gone over, or the instructor may include them in his exposition. Some of the exercises and problems are computational and some classical, but some are more or less original, and a few, I think, are startling. Because there are often many methods of attacking a problem in number theory, and because the most fruitful line may not be obvious, hints are given for many problems. These hints appear in a section at the back of the book (some hints are almost complete solutions). A solution that does not use a hint is of course more praiseworthy than one that does, but the amount of inspiration neces- sary to solve some problems is too large to be had to order. Number theory problems can be hard: I. A. Barnett has written (in the November 1966 American Mathematical Monthly), "To discover mathematical talent, there is no better course in elementary mathematics than number theory. Any student who can work the exercises in a modern text in number theory should be encouraged to pursue a mathematical career." Answers are provided for some problems and exercises in separate sections at the back of the book. Although there are more problems than a student could solve in one semester, he is encouraged to treat them, along with their hints, as part of the text to be read with almost equal attention. Often they may be found more inter- esting than the material on which they are based. The book contains four appendices. The first two (Appendix A, Proof by Induction, and Appendix B, Notations) should be read by the student when necessary. Appendix C (Quadratic Congruences with Composite Moduli), is Preface ix included to complete the study of solutions of a quadratic congruence; it could come after section 11. Appendix D consists of three tables that are included both for their inherent fascination and for use in solving numerical problems. The first makes it easy to factor any positive integer less than 10,000, the second gives the first 447 squares, and the third consists of part of a factor table. There are without doubt errors in the problems and in the text. I hope that when the reader finds one, he will feel pleasure at his acuteness instead of annoyance with the author. Corrections will be welcomed. February 1969 UNDERWOOD DUDLEY a SECTION 1 Integers A large part of number theory is devoted to studying the properties of the . , — — . integers integers—that is, the numbers . 2, 1, 0, 1, 2, Usually the 2 are used merely to convey information (3 apples, $32, 17x + 9), with no 2 consideration of their properties. When counting apples, dollars, or x 's, it is immaterial how many divisors 3 has, whether 32 is prime or not, or that 17 can be written as the sum of the squares of two integers. But the integers are so basic a part of mathematics that they have been thought worthy of study for their own sake. The same situation arises elsewhere: the number theorist is comparable to the linguist, who studies words and their properties, independent of their meaning. Another justification for studying the integers is esthetic: many beautiful results can be proved. In this section, and until further notice, lower-case italic letters will invar- iably denote integers. We will take as known and use freely the usual prop- erties of addition, subtraction, multiplication, division, and order for the integers. We also use in this section an important property of the integers— property that you may not be consciously aware of because it is not stated explicitly as often as, say, the associative property of multiplication. It is the least-integer principle: a nonempty set of integers that is bounded below contains a smallest element. It is also true that a nonempty set of integers that is bounded above contains a largest element. will that divides (written a b) if and only if there is an integer We say a b \ -5 8 -24. = 2 12 17 17, | 50, and | that b. | d such ad For example, ] 6, | 60, If a does not divide b, we will write a \ b. For example, 4 \ 2 and 3 \ 4. 2 SECTION 1 Exercise 1. Which integers divide zero? 2. if c, then c. Exercise that \ b Show a and b \ a \ As a sample of the sort of properties that division has, we prove 1.