P1: BINAYA KUMAR DASH/OVY P2: IML/OVY QC: IML/OVY T1: IML bur83147_fm Burton DQ032A-Elementary-v2.cls December 17, 2009 11:59 ELEMENTARY NUMBER THEORY Seventh Edition David M. Burton University of New Hampshire i P1: BINAYA KUMAR DASH/OVY P2: IML/OVY QC: IML/OVY T1: IML bur83147_fm Burton DQ032A-Elementary-v2.cls December 17, 2009 11:59 ELEMENTARY NUMBER THEORY, SEVENTH EDITION Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020. Copyright c 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Previous editions c 2007, 2002, and 1998. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning. Some ancillaries, including electronic and print components, may not be available to customers outside the United States. This book is printed on acid-free paper. 1234567890DOC/DOC 109876543210 ISBN 978-0-07-338314-9 MHID 0-07-338314-7 Vice President & Editor-in-Chief: Martin Lange Vice President EDP & Central Publishing Services: Kimberly Meriwether David Editorial Director: Stewart K. Mattson Sponsoring Editor: Bill Stenquist Developmental Editor: Eve L. Lipton Marketing Coordinator: Sabina Navsariwala-Horrocks Project Manager: Melissa M. Leick Senior Production Supervisor: Kara Kudronowicz Design Coordinator: Brenda A. Rolwes Cover Designer: Studio Montage, St. Louis, Missouri (USE) Cover Image: c Tom Grill/ Getty Images Senior Photo Research Coordinator: John C. Leland Compositor: Glyph International Typeface: 10.5/12 Times Roman Printer: R. R. Donnelley All credits appearing on page or at the end of the book are considered to be an extension of the copyright page. Library of Congress Cataloging-in-Publication Data Burton, David M. Elementary number theory / David M. Burton. —7th ed. p. cm. Includes bibliographical references and index. ISBN 978-0-07-338314-9 (alk. paper) 1. Number theory. I. Title. QA241.B83 2010 512.7’2—dc22 2009044311 www.mhhe.com ii P1: BINAYA KUMAR DASH/OVY P2: IML/OVY QC: IML/OVY T1: IML bur83147_fm Burton DQ032A-Elementary-v2.cls December 17, 2009 11:59 TO MARTHA iii P1: BINAYA KUMAR DASH/OVY P2: IML/OVY QC: IML/OVY T1: IML bur83147_fm Burton DQ032A-Elementary-v2.cls December 17, 2009 11:59 ABOUT THE AUTHOR David M. Burton received his B.A. from Clark University and his M.A. and Ph.D. degrees from the University of Rochester. He joined the faculty of the University of New Hampshire, where he is now Professor Emeritus of Mathematics, in 1959. His teaching experience also includes a year at Yale University, numerous summer insti- tutes for high school teachers, and presentations at meetings of high school teachers’ organizations. Professor Burton is also the author of The History of Mathematics: An Introduction (McGraw-Hill, Seventh edition, 2009) and five textbooks on abstract and linear algebra. In addition to his work in mathematics, he spent 16 years coaching a high school girls’ track and field team. When not writing, he is likely to be found jogging or reading (mainly history and detective fiction). He is married and has 3 grown children and a brown Doberman pinscher. iv P1: BINAYA KUMAR DASH/OVY P2: IML/OVY QC: IML/OVY T1: IML bur83147_fm Burton DQ032A-Elementary-v2.cls December 17, 2009 11:59 CONTENTS Preface viii New to This Edition x 1 Preliminaries 01 1.1 Mathematical Induction 01 1.2 The Binomial Theorem 08 2 Divisibility Theory in the Integers 13 2.1 Early Number Theory 13 2.2 The Division Algorithm 17 2.3 The Greatest Common Divisor 19 2.4 The Euclidean Algorithm 26 2.5 The Diophantine Equation ax + by = c 32 3 Primes and Their Distribution 39 3.1 The Fundamental Theorem of Arithmetic 39 3.2 The Sieve of Eratosthenes 44 3.3 The Goldbach Conjecture 50 4 The Theory of Congruences 61 4.1 Carl Friedrich Gauss 61 4.2 Basic Properties of Congruence 63 4.3 Binary and Decimal Representations of Integers 69 4.4 Linear Congruences and the Chinese Remainder Theorem 76 5 Fermat’s Theorem 85 5.1 Pierre de Fermat 85 5.2 Fermat’s Little Theorem and Pseudoprimes 87 v P1: BINAYA KUMAR DASH/OVY P2: IML/OVY QC: IML/OVY T1: IML bur83147_fm Burton DQ032A-Elementary-v2.cls December 17, 2009 11:59 vi CONTENTS 5.3 Wilson’s Theorem 93 5.4 The Fermat-Kraitchik Factorization Method 97 6 Number-Theoretic Functions 103 6.1 The Sum and Number of Divisors 103 6.2 The M¨obius Inversion Formula 112 6.3 The Greatest Integer Function 117 6.4 An Application to the Calendar 122 7 Euler’s Generalization of Fermat’s Theorem 129 7.1 Leonhard Euler 129 7.2 Euler’s Phi-Function 131 7.3 Euler’s Theorem 136 7.4 Some Properties of the Phi-Function 141 8 Primitive Roots and Indices 147 8.1 The Order of an Integer Modulo n 147 8.2 Primitive Roots for Primes 152 8.3 Composite Numbers Having Primitive Roots 158 8.4 The Theory of Indices 163 9 The Quadratic Reciprocity Law 169 9.1 Euler’s Criterion 169 9.2 The Legendre Symbol and Its Properties 175 9.3 Quadratic Reciprocity 185 9.4 Quadratic Congruences with Composite Moduli 192 10 Introduction to Cryptography 197 10.1 From Caesar Cipher to Public Key Cryptography 197 10.2 The Knapsack Cryptosystem 209 10.3 An Application of Primitive Roots to Cryptography 214 11 Numbers of Special Form 219 11.1 Marin Mersenne 219 11.2 Perfect Numbers 221 11.3 Mersenne Primes and Amicable Numbers 227 11.4 Fermat Numbers 237 12 Certain Nonlinear Diophantine Equations 245 12.1 The Equation x2 + y2 = z2 245 12.2 Fermat’s Last Theorem 252 P1: BINAYA KUMAR DASH/OVY P2: IML/OVY QC: IML/OVY T1: IML bur83147_fm Burton DQ032A-Elementary-v2.cls December 17, 2009 11:59 CONTENTS vii 13 Representation of Integers as Sums of Squares 261 13.1 Joseph Louis Lagrange 261 13.2 Sums of Two Squares 263 13.3 Sums of More Than Two Squares 272 14 Fibonacci Numbers 283 14.1 Fibonacci 283 14.2 The Fibonacci Sequence 285 14.3 Certain Identities Involving Fibonacci Numbers 292 15 Continued Fractions 303 15.1 Srinivasa Ramanujan 303 15.2 Finite Continued Fractions 306 15.3 Infinite Continued Fractions 319 15.4 Farey Fractions 334 15.5 Pell’s Equation 337 16 Some Modern Developments 353 16.1 Hardy, Dickson, and Erd¨os 353 16.2 Primality Testing and Factorization 358 16.3 An Application to Factoring: Remote Coin Flipping 371 16.4 The Prime Number Theorem and Zeta Function 375 Miscellaneous Problems 384 Appendixes 387 General References 387 Suggested Further Reading 390 Tables 393 Answers to Selected Problems 410 Index 421 P1: BINAYA KUMAR DASH/OVY P2: IML/OVY QC: IML/OVY T1: IML bur83147_fm Burton DQ032A-Elementary-v2.cls December 17, 2009 11:59 PREFACE Plato said, “God is a geometer.” Jacobi changed this to, “God is an arithmetician.” Then came Kronecker and fashioned the memorable expression, “God created the natural numbers, and all the rest is the work of man.” FELIX KLEIN The purpose of this volume is to give a simple account of classical number theory and to impart some of the historical background in which the subject evolved. Although primarily intended for use as a textbook in a one-semester course at the undergraduate level, it is designed to be used in teachers’ institutes or as supplementary reading in mathematics survey courses. The work is well suited for prospective secondary school teachers for whom a little familiarity with number theory may be particularly helpful. The theory of numbers has always occupied a unique position in the world of mathematics. This is due to the unquestioned historical importance of the subject: it is one of the few disciplines having demonstrable results that predate the very idea of a university or an academy. Nearly every century since classical antiquity has witnessed new and fascinating discoveries relating to the properties of numbers; and, at some point in their careers, most of the great masters of the mathematical sciences have contributed to this body of knowledge. Why has number theory held such an irresistible appeal for the leading mathematicians and for thousands of amateurs? One answer lies in the basic nature of its problems. Although many questions in the field are extremely hard to decide, they can be formulated in terms simple enough to arouse the interest and curiosity of those with little mathematical training. Some of the simplest sounding questions have withstood intellectual assaults for ages and remain among the most elusive unsolved problems in the whole of mathematics. It therefore comes as something of a surprise to find that many students look upon number theory with good-humored indulgence, regarding it as a frippery on the edge of mathematics. This no doubt stems from the widely held view that it is the purest branch of pure mathematics and from the attendant suspicion that it can have few substantive applications to real-world problems. Some of the worst viii P1: BINAYA KUMAR DASH/OVY P2: IML/OVY QC: IML/OVY T1: IML bur83147_fm Burton DQ032A-Elementary-v2.cls December 17, 2009 11:59 PREFACE ix offenders, when it comes to celebrating the uselessness of their subject, have been number theorists themselves.