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Trang 1 the Theory of Numbers FIFTH EDITION Ivan Niven University of Oregon Herbert S.. Library of Congress Calllloging in Publication Data: Niven, Ivan Morton, Trang 3 This text is in
An Introduction to the Theory of Numbers FIFTH EDITION Ivan Niven University of Oregon Herbert S Zuckerman University of Washington Hugh L Montgomery University of Michigan John Wiley & Sons, Inc New York • Chichester • Brisbane • Toronto • Singapore Acquisitions Editor: Bob Macek Designer: Laura Nicholls Copyediting Supervisor: Gilda Stahl Production Manager: Katherine Rubin Production Supervisor: Micheline Frederick Manufacturing Manager: Denis Clarke Marketing Manager: Susan Elbe Recognizing the importance of preserving what has been written, it is a policy of John Wiley & Sons, Inc to have books of enduring value published in the United States printed on acid-free paper, and we exert our best efforts to that end Copyright© 1960, 1966, 1972, 1980, 1991 by John Wiley & Sons, Inc All rights reserved Published simultaneously in Canada Reproduction or translation of any part of this work beyond that permitted by Sections 107 and 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons Library of Congress Calllloging in Publication Data: Niven, Ivan Morton, 1915- An introduction to the theory of numbers 1 Ivan Niven, Herbert S Zuckerman, Hugh L Montgomery.-5th ed p em Includes bibliographical references (p } Includes index ISBN 0-471-62546-9 1 Number theory I Zuckerman, Herbert S II Montgomery, Hugh L III Title QA241.N56 1991 512'.7-dc20 90-13013 CIP Printed in the United States of America 10 9 8 7 6 5 4 3 Prin&ed and bound by Courier Campaniea, Inc Preface This text is intended for use in a first course in number theory, at the upper undergraduate or beginning graduate level To make the book appropriate for a wide audience, we have included large collections of problems of varying difficulty Some effort has been devoted to make the first chapters less demanding In general, the chapters become gradually more challenging Similarly, sections within a given chapter are progres- sively more difficult, and the material within a given section likewise At each juncture the instructor must decide how deeply to pursue a particular topic before moving ahead to a new subject It is assumed that the reader has a command of material covered in standard courses on linear algebra and on advanced calculus, although in the early chapters these prerequi- sites are only slightly used A modest course requiring only freshman mathematics could be constructed by covering Sections 1.1, 1.2, 1.3 (Theo- rem 1.19 is optional), 1.4 through Theorem 1.21, 2.1, 2.2, 2.3, 2.4 through Example 9, 2.5, 2.6 through Example 12, 2.7 (the material following Corollary 2.30 is optional), 2.8 through Corollary 2.38, 4.1, 4.2, 4.3, 5.1, 5.3, 5.4, 6.1, 6.2 In any case the instructor should obtain from the publisher a copy of the Instructor's Manual, which provides further suggestions con- cerning selection of material, as well as solutions to all starred problems The Instructor's Manual also describes computational experiments, and provides information concerning associated software that is available for use with this book New in this edition are accounts of the binomial theorem (Section 1.4), public-key cryptography (Section 2.4), the singular situation in Hansel's lemma (Section 2.6), simultaneous systems of linear Diophantine equations (Section 5.2), rational points on curves (Section 5.6), elliptic curves (Section 5.7), description of Faltings' theorem (Section 5.9), the geometry of numbers (Section 6.4), Mertens' estimates of prime number sums (in Section 8.1), Dirichlet series (Section 8.2), and asymptotic esti- mates of arithmetic functions (Section 8.3) Many other parts of the books have also been extensively revised, and many new starred problems have v vi Preface been introduced We address a number of calculational issues, most notably in Section 1.2 (Euclidean algorithm), Section 2.3 (the Chinese remainder theorem), Section 2.4 (pseudoprime tests and Pollard rho factorization), Section 2.9 (Shanks' RESSOL algorithm), Section 3.6 (sums of two squares), Section 4.4 (linear recurrences and Lucas pseudoprimes), Section 5.8 (Lenstra's elliptic curve method of factorization), and Section 7.9 (the continued fraction of a quadratic irrational) In the Appendixes we have provided some important material that all too often is lost in the cracks of the undergraduate curriculum Number theory is a broad subject with many strong connections with other branches of mathematics Our desire is to present a balanced view of the area Each subspecialty possesses a personality uniquely its own, which we have sought to portray accurately Although much may be learned by exploring the extent to which advanced theorems may be proved using only elementary techniques, we believe that many such arguments fail to convey the spirit of current research, and thus are of less value to the beginner who wants to develop a feel for the subject In an effort to optimize the instructional value of the text, we sometimes avoid the shortest known proof of a result in favor of a longer proof that offers greater insights While revising the book we sought advice from many friends and colleagues, and we would most especially like to thank G E Andrews, A 0 L Atkin, P T Bateman, E Berkove, P Blass, A Bremner, J D Brillhart, J W S Cassels, T Cochrane, R K Guy, H W Lenstra Jr., D J Lewis, D G Maim, D W Masser, J E McLaughlin, A M Odlyzko, C Pomerance, K A Ross, L Schoenfeld, J L Selfridge, R C Vaughan, S S Wagstaff Jr., H J Rickert, C Williams, K S Williams, and M C Wunderlich for their valuable suggestions We hope that readers will contact us with further comments and suggestions Ivan Niven Hugh L Montgomery Contents Notation xi 1 Divisibility 1 1.1 Introduction 1 1.2 Divisibility 4 1.3 Primes 20 1.4 The Bionomial Theorem 35 Notes on Chapter 1 44 2 Congruences 47 2.1 Congruences 47 2.2 Solutions of Congruences 60 2.3 The Chinese Remainder Theorem 64 2.4 Techniques of Numerical Calculation 74 2.5 Public-Key Cryptography 84 2.6 Prime Power Moduli 86 2.7 Prime Modulus 91 2.8 Primitive Roots and Power Residues 97 2.9 Congruences of Degree Two, Prime Modulus 110 2.10 Number Theory from an Algebraic Viewpoint 115 2.11 Groups, Rings, and Fields 121 Notes on Chapter 2 128 3 Quadratic Reciprocity and Quadratic Forms 131 3.1 Quadratic Residues 131 3.2 Quadratic Reciprocity 137 3.3 The Jacobi Symbol 142 3.4 Binary Quadratic Forms 150 vii viii Contents 3.5 Equivalence and Reduction of Binary Quadratic 180 Forms 155 212 3.6 Sums of Two Squares 163 297 3.7 Positive Definite Binary Quadratic Forms 170 325 Notes on Chapter 3 176 4 Some Functions of Number Theory 4.1 Greatest Integer Function 180 4.2 Arithmetic Functions 188 4.3 The Mobius Inversion Formula 193 4.4 Recurrence Functions 197 4.5 Combinatorial Number Theory 206 Notes on Chapter 4 211 5 Some Diophantine Equations 5.1 The Equation ax + by = c 212 5.2 Simultaneous Linear Equations 219 5.3 Pythagorean Triangles 231 5.4 Assorted Examples 234 5.5 Ternary Quadratic Forms 240 5.6 Rational Points on Curves 249 5.7 Elliptic Curves 261 5.8 Factorization Using Elliptic Curves 281 5.9 Curves of Genus Greater Than 1 288 Notes on Chapter 5 289 6 Farey Fractions and Irrational Numbers 6.1 Farey Sequences 297 6.2 Rational Approximations 301 6.3 Irrational Numbers 307 6.4 The Geometry of Numbers 312 Notes on Chapter 6 322 7 Simple Continued Fractions 7.1 The Euclidean Algorithm 325 7.2 Uniqueness 327 7.3 Infinite Continued Fractions 329 7.4 Irrational Numbers 334 7.5 Approximations to Irrational Numbers 336 7.6 Best Possible Approximations 341 7.7 Periodic Continued Fractions 344 7.8 Pell's Equation 351 7.9 Numerical Computation 358 Notes on Chapter 7 359 8 Primes and Multiplicative Number Theory 360 8.1 Elementary Prime Number Estimates 360 8.2 Dirichlet Series 374 8.3 Estimates of Arithmetic Functions 389 8.4 Primes in Arithmetic Progressions 401 Notes on Chapter 8 406 9 Algebraic Numbers 409 9.1 Polynomials 410 9.2 Algebraic Numbers 414 9.3 Algebraic Number Fields 419 9.4 Algebraic ln!egers 424 9.5 Quadratic Fields 425 9.6 Units in Quadratic Fields 428 9.7 Primes in Quadratic Fields 429 9.8 Unique Factorization 431 9.9 Primes in Quadratic Fields Having the Unique Factorization Property 433 9.10 The Equation x 3 + y 3 = z 3 441 Notes on Chapter 9 445 10 The Partition Function 446 10.1 Partitions 446 10.2 Ferrers Graphs 448 10.3 Formal Power Series, Generating Functions, and Euler's Identity 452 10.4 Euler's Formula; Bounds on p(n) 457 10.5 Jacobi's Formula 463 10.6 A Divisibility Property 467 Notes on Chapter 10 471 11 The Density of Sequences of Integers 472 11.1 Asymptotic Density 473 11.2 Schnirelmann Density and the af3 Theorem 476 Notes on Chapter 11 481 X Contents Appendices 482 A.l The Fundamental Theorem of Algebra 482 500 A.2 Symmetric Functions 484 503 A.3 A Special Value of the Riemann Zeta Function 490 512 A.4 Linear Recurrences 493 522 General References Hints Answers Index Notation Items are listed in order of appearance 7L The set of integers, 4 Q The set of rational numbers, 4 IR The set of real numbers, 4 alb a divides b, 4 a,f'b a does not divide b, 4 akllb aklb but ak+t,rb,4 [x] Integer part, 6, 134, 180 (b,c) The greatest common divisor of b and c, 7 (Alternatively, depending on the context, a gcd(b, c) point in the plane, or an open interval) (bl, bz, •••' bn) The greatest common divisor of b and c, 7 [at, az, · · · • an] The greatest common divisor of the b;, 7 N(a) The least common multiple of the a;, 16 1T(X) Norm of a, 22, 427 f(x)- g(x) Number of primes p,;; x, 26 Asymptotic equivalence, 28 MP (Alternatively, depending on the context, an equivalence relation, 128, 157) N\~ Fermat number, 33 (Alternatively, Fibonacci number, 199) n E A/ Mersenne prime, 33 Subtraction of sets, 34 (%) n is an element of the set A/, 34 NU~ Binomial coefficient, 35 Nn~ Union of sets, 41 tif(x) Intersection of sets, 41 a= b (mod m) k th forward difference, 42 a'¢ b (mod m) a congruent to b modulo m, 48 a not congruent to b modulo m, 48 xi xii Notation c/J(n) Euler's totient function, 50 ii Congruential inverse of a, 52 Determinant of square matrix A, 59 det(A) Number of elements in the set N, 64 card(N) Cartesian product of sets, 68 /I X _/'z X X c./,; Set of pairs (x, y) of real numbers, 68 !Rz Fractional part, {x} = x - [x ], 75 {x} Strong pseudoprime base a, 78 spsp(a) Approximately equal, 85 == Isomorphic, as with groups, 118 - Legendre symbol, 132 (;) Jacobi symbol, 142 (~) Discriminant of a quadratic form, 150 The modular group, 157 d Equivalence of quadratic forms, 157 The class number, 161 r The number of primitive classes, 163 Number of representations of n as a sum of f-g two squares, 163 Number of representations of n as a sum of H(d) two relatively prime squares, 163 h(d) Restricted representations of n as a sum of R(n) two squares, 163 r(n) Number of solutions of x 2 = P(n) -1 (mod n), 163 Number of automorphs of a quadratic N(n) form, 173 The identity matrix, 173 w(f) Number of representations by a quadratic form, 174 I Number of proper representations of n by a R/n) quadratic form, 174 Generalization of N(n), 174 r/n) General linear group, 177 Special linear group, 177 H/n) Number of positive divisors, 188 GL(n, F) Sum of the positive divisors, 188 SL(n, R) Sum of the k th powers of the positive d(n) divisors, 188 u(n) Number of distinct prime factors, 188 uk(n) Total number of primes dividing n, counting multiplicity, 188 w(n) Liouville's lambda function, 192 il(n) A(n)