Table Of Content

FUNDAMENTALS ©F N IM IE R f f lü l® ’ WILLIAM J. LeVEQUE Claremont Graduate School ▲ VT ADDISON-WESLEY PUBLISHING COMPANY V Reading, Massachusetts • Menlo Park, California London • Amsterdam • Don Mills, Ontario • Sydney The portrait of Adolph Hurwitz on page 225 is reproduced by permission of the Library of Eidgenössische Technische Hochschule, Zurich, Switzerland. All the other portraits in this book are from the David Eugene Smith Collection, Columbia University Libraries and are reprinted by permission of Columbia University. Copyright © 1977 by Addison-Wesley Publishing Company, Inc. Philippines copyright 1977 by Addison-Wesley Publishing Company, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. Published simultaneously in Canada. Library of Congress Catalog Card No. 76-55645. ISBN 0-201-04287-8 BCDEFGHIJKL-MA-798 I started to write the present book as a second edition of my Topics in Number Theory, Volume 1, but I soon became aware that the changes were too extensive to allow it to be considered the same book, especially because of the back-references from Volume 2 which would become inappropriate. The two books are still quite similar, and I have carried over some passages verbatim; nevertheless, the differences are both substantial and significant. Perhaps the most striking change is the introduction of the language of abstract algebra. Many students will already have met the concepts of group, ring, field, and domain, but this is not assumed. All terms used are defined and examples are given, and the book is intended to be self-contained in this respect. I have taught the material to mixed classes, containing students who had had algebra and students who had not, and did not find that students of the first kind had any great advantage. Second, there is a new emphasis on the history of number theory. In part, this reflects my own growing interest in the history of mathematics, but it also recognizes what I take to be more concern among students with the human side of mathematics. A closely related change is the inclusion of many notes and bibliographical references, selected with an eye to their readability as well as their relevance. The reader acquainted with my earlier book will find a number of new topics: factorization and primality of large integers, p-adic numbers, algebraic number fields, Brun’s theorem on twin primes, and the transcendence of e, to mention some. The number of problems has also been substantially increased. The importance of problem-solving as an integral part of a first course in number theory can hardly be overemphasized, and I hope that the additional problems will prove to be useful. Hints are supplied with the more difficult ones. As was true of its predecessor, this book is directed primarily toward under graduate majors and beginning graduate students in mathematics. No post calculus prerequisite is supposed, although here and there comments are in cluded for the benefit of students with additional background. iv Preface All the portraits except that of Hurwitz were selected from the superb David Eugene Smith collection in the Columbia University Library; Hurwitz’s came from the Eidgenössische Technische Hochschule in Zürich. The Factor Table on pp. 266-267 is a reproduction of a small portion of Burkhardt’s Table des Diviseurs of 1814-17, as it was reprinted in G. S. Carr’s Formulas and Theorems in Pure Mathematics, Second Edition, Chelsea Publishing Com pany, New York 1886/1970. Permission for all these reproductions is gratefully acknowledged. My first draft benefited from the criticism of a number of friends. In particu lar, H. W. Gould supplied a detailed critique, and J. Brillhart provided au thoritative advice on computational matters, as well as general comments. I am indebted to R. J. LeVeque for preparing the computer plots on pp. 268-269. April 1977 W.J. L. Claremont, California Contents Introduction What is number theory?............................ 1 Algebraic properties of the set of integers . 8 Types of proofs, and some examples . 13 Representation systems for the integers 18 The early history of number theory 25 Unique Factorization and the GCD The greatest common divisor 31 Unique factorization in other domains 35 The linear Diophantine equation 40 The least common multiple . 44 Congruences and the Ring Zm Congruence and residue classes............................................. 47 Complete and reduced residue systems; Euler’s ^-function . 51 Linear congruences............................................................... 58 Higher-degree polynomial congruences............................ 63 The p-adic fields.................................................................... 70 Primitive Roots and the Group Um Primitive roots.................................. 79 The structure of Um ....................... 86 n\h power residues....................... 90 An application to Fermat’s equation 93 v vl Contents Chapter 5 Quadratic Residues 5.1 Introduction...........................................................................................97 5.2 Quadratic residues of primes, and the Legendre symbol.............................99 5.3 The law of quadratic reciprocity.............................................................103 5.4 The Jacobi sym bol....................................................................................109 5.5 Factorization of large integers...................................................................113 Chapter 6 Number-Theoretic Functions and the Distribution of Primes 6.1 Introduction...............................................................................................121 6.2 The Mobius function....................................................................................127 6.3 The function [x]..........................................................................................131 6.4 The symbols “<9”, “o”, and .............................................133 6.5 The sieve of Eratosthenes.........................................................................139 6.6 Sums involving prim es..............................................................................143 6.7 The true order of n (x )..............................................................................148 6.8 Primes in arithmetic progressions..............................................................154 6.9 Bertrand’s hypothesis..........................................................................159 6.10 The order of magnitude of cp, 0, and r ..................................................163 6.11 Average order of m agnitude...................................................................167 6.12 Brun’s theorem on twin primes...................................................................173 Chapter 7 Sums of Squares 7.1 Preliminaries...............................................................................................179 7.2 Primitive representations as a sum of two squares..................................181 7.3 The total number of representations........................................................184 7.4 Sums of three squares...............................................................................187 7.5 Sums of four squares . . 187 7.6 Waring’s problem ....................................................................................189 Chapter 8 Quadratic Equations and Quadratic Fields 8.1 Legendre’s theorem....................................................................................191 8.2 Pell’s equation...............................................................................................198 8.3 Algebraic number fields and algebraic integers........................................207 8.4 Arithmetic in quadratic fields...................................................................212 Chapter 9 Diophantine Approximation 9.1 Farey sequences and Hurwitz’s theorem..................................................219 9.2 Best approximations to a real number........................................................226 9.3 Infinite continued fractions.........................................................................234 9.4 Quadratic irrationalities..............................................................................239 9.5 Applications to Pell’s equation and to factorization..................................245 Contents vii 9.6 Equivalence of numbers..............................................................................248 9.7 The transcendence of e ..............................................................................255 Bibliography...............................................................................................260 Appendix.....................................................................................................265 Factor Table..........................................................................................266 Computer-Plotted Graphs.........................................................................268 Table of Indices....................................................................................270 Greek Alphabet....................................................................................274 List of Symbols....................................................................................275 Index...........................................................................................................277 1 Introduction 1.1 WHAT IS NUMBER THEORY? This could serve as a first attempt at a definition: it is the study of the set of integers 0, ±1, ±2,..., or some of its subsets or extensions, proceeding on the assumption that integers are interesting objects in and of themselves, and dis regarding their utilitarian role in measuring. This definition might seem to include elementary arithmetic, and in fact it does, except that the concern now is to be with more advanced and more subtle aspects of the subject. A quick review of ele mentary properties of the integers is incorporated with some other material, which may or may not be new to the reader, in Sections 1.2 and 1.3. To get some idea of what the subject comprises, let us go back to the seven teenth century, when the modern epoch opened with the work of Pierre de Fermat \_fair-mali]. One of Fermat’s most beautiful theorems is that every positive integer can be represented as the sum of the squares of four integers, for example, 1 = l2 + o2 + 02 + 02, 2 = l2 + l2 + 02 + 02, 4 = l2 + l2 + l2 + l2 = 22 + 02 + 02 + 02, 7 = 22 + l2 + l2 + l2, 188951 = 3712 + 2262 + 152 + 32. He announced this theorem in 1636, but the first published proof of it was given by Joseph-Louis Lagrange in 1770. It could serve as the ideal example of a theorem in number theory: it is elegant and immediately comprehensible; it reveals a subtle and unexpected relationship among the integers; it is the best theorem of its kind (7 cannot be represented with fewer than four squares); and it says something about an infinite class of integers. The last is an important qualification, as it distinguishes between theorems and numerical facts. It is a fact, and perhaps even an interesting one, that 1729 is the smallest positive integer having two distinct representations as the sum of two cubes (103 + 93 and 123 + 13), but this would hardly be called a theorem since it can be verified by examining the finite set 1, 2, 3,.. ., 1729. On the other hand, the assertion that there are only finitely 1