Editor-in-Chief· Deirdre Lynch Senior Acquisitions Editor: William Hoffman Associate Editor: Caroline Celano Marketing Manager: Jeff Weidenaar Marketing Assistant: Kendra Bassi Senior Managing Editor: Karen Wernholm Production Project Manager: Beth Houston Project Manager: Paul C Anagnostopoulos Composition and Illustration: Windfall Software, using ZzTEX Manufacturing Manager: Evelyn Beaton Photo Research: Maureen Raymond Senior Cover Designer: Beth Paquin Cover Design: Nancy Goulet, Studio;wink Cover Image: Gray Numbers, 1958 (collage)© Jasper Johns (b 1930) I Private Collection I Licensed by VAGA, New York, N.Y Photo Credits: Grateful acknowledgment is made to the copyright holders of the biographical photos, listed on page 752, which is hereby made part of this copyright page Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks W here those designations appear in this book, and Addison Wesley was aware of a trademark claim, the designations have been printed in initial caps or all caps Library of Congress Cataloging-in-Publication Data Rosen, Kenneth H Elementary number theory and its applications I Kenneth H Rosen - 6th ed p cm Includes bibliographical references and index ISBN-13: 978-0-321-50031-1 (alk paper) ISBN-10: 0-321-50031-8 (alk paper) Number theory-Textbooks I Title QA241.R67 2011 512.7'2 -dc22 2010002572 Copyright © 2011, 2005, 2000 by Kenneth H Rosen 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 For information on obtaining permission for use of material in this work, please submit a written request to Pearson Education, Inc., Rights and Contracts Department, 500 Boylston Street, Suite 900, Boston, MA 02116, fax your request to (617) 848-7047, or e-mail at http://www.pearsoned.com/legal/permissions.htm 10-C\V-14 13 12 11 10 Addison-Wesley is an imprint of PEARSON - ISBN 10: 0-321-50031-8 www.pearsonhighered.com ISBN 13: 978-0-321-50031-1 Preface My goal in writing this text has been to write an accessible and inviting introduction to number theory Foremost, I wanted to create an effective tool for teaching and learning I hoped to capture the richness and beauty of the subject and its unexpected usefulness Number theory is both classical and modem, and, at the same time, both pure and applied In this text, I have strived to capture these contrasting aspects of number theory I have worked hard to integrate these aspects into one cohesive text This book is ideal for an undergraduate number theory course at any level No formal prerequisites beyond college algebra are needed for most of the material, other than some level of mathematical maturity This book is also designed to be a source book for elementary number theory; it can serve as a useful supplement for computer science courses and as a primer for those interested in new developments in number theory and cryptography Because it is comprehensive, it is designed to serve both as a textbook and as a lifetime reference for elementary number theory and its wide-ranging applications This edition celebrates the silver anniversary of this book Over the past 25 years, close to 100,000 students worldwide have studied number theory from previous editions Each successive edition of this book has benefited from feedback and suggestions from many instructors, students, and reviewers This new edition follows the same basic approach as all previous editions, but with many improvements and enhancements I invite instructors unfamiliar with this book, or who have not looked at a recent edition, to carefully examine the sixth edition I have confidence that you will appreciate the rich exercise sets, the fascinating biographical and historical notes, the up-to-date coverage, careful and rigorous proofs, the many helpful examples, the rich applications, the support for computational engines such as Maple and Mathematica, and the many resources available on the Web Changes in the Sixth Edition The changes in the sixth edition have been designed to make the book easier to teach and learn from, more interesting and inviting, and as up-to-date as possible Many of these changes were suggested by users and reviewers of the fifth edition The following list highlights some of the more important changes in this edition ix x Preface • New discoveries This edition tracks recent discoveries of both a numerical and a theoretical nature Among the new computational discoveries reflected in the sixth edition are four Mersenne primes and the latest evidence supporting many open conjectures The Tao-Green theorem proving the existence of arbitrarily long arithmetic progressions of primes is one of the recent theoretical discoveries described in this edition • Biographies and historical notes Biographies of Terence Tao, Etienne Bezout, Norman MacLeod Ferrers, Clifford Cocks, and Waclaw Sierpinski supplement the already extensive collection of biographies in the book Surprising information about secret British cryptographic discoveries predating the work of Rivest, Shamir, and Adleman has been added • Conjectures The treatment of conjectures throughout elementary number theory has been expanded, particularly those about prime numbers and diophantine equations Both resolved and open conjectures are addressed • Combinatorial number theory A new section of the book covers partitions, a fascinating and accessible topic in combinatorial number theory This new section introduces such important topics as Ferrers diagrams, partition identies, and Ramanujan's work on congruences In this section, partition identities, including Euler's important results, are proved using both generating functions and bijections • Congruent numbers and elliptic curves A new section is devoted to the famous congruent number problem, which asks which positive integers are the area of a right triangle with rational side lengths This section contains a brief introduction to elliptic curves and relates the congruent number problem to finding rational points on certain elliptic curves Also, this section relates the congruent number problem to arithmetic progressions of three squares • Geometric reasoning This edition introduces the use of geometric reasoning in the study of diophantine problems In particular, new material shows that finding rational points on the unit circle is equivalent to finding Pythgaorean triples, and that finding rational triangles with a given integer as area is equivalent to finding rational points on an associated elliptic curve • Cryptography This edition eliminates the unnecessary restriction that when the RSA cryptosystem is used to encrypt a plaintext message this message needs to be relatively prime to the modulus in the key Preface • xi Greatest common divisors Greatest common divisors are now defined in the first chapter, as is what it means for two integers to be relatively prime The term Bezout coefficients is now introduced and used in the book • Jacobi symbols More motivation is provided for the usefulness of Jacobi symbols In particular, an expanded discussion on the usefulness of the Jacobi symbol in evaluating Legendre symbols is now provided • Enhanced exercise sets Extensive work has been done to improve exercise sets even farther Several hundred new exercises, ranging from routine to challenging, have been added Moreover, new computational and exploratory exercises can be found in this new edition • Accurancy More attention than ever before has been paid to ensuring the accuracy of this edition Two independent accuracy checkers have examined the entire text and the answers to exercises • Web Site, www.pearsonhighered.com/rosen The Web site for this edition has been considerably expanded Students and instructors will find many new resources they can use in conjunction with the book Among the new features are an expanded collection of applets, a manual for using comptutional engines to explore number theory, and a Web page devoted to number theory news Exercise Sets Because exercises are so important, a large percentage of my writing and revision work has been devoted to the exercise sets Students should keep in mind that the best way to learn mathematics is to work as many exercises as possible I will briefly describe the types of exercises found in this book and where to find answers and solutions • Standard Exercises Many routine exercises are included to develop basic skills, with care taken so that both odd-numbered and even-numbered exercises of this type are included A large number of intermediate-level exercises help students put several concepts together to form new results Many other exercises and blocks of exercises are designed to develop new concepts • Exercise Legend Challenging exercises are in ample supply and are marked with one star ( *) indicating a difficult exercise and two stars (* *) indicating an extremely difficult exercise There are xii Preface some exercises that contain results used later in the text; these are marked with a arrow symbol (> ) These exercises should be assigned by instructors whenever possible • Exercise Answers The answers to all odd-numbered exercises are provided at the end of the text More complete solutions to these exercises can be found in the Student's Solutions Manual that can be found on the Web site for this book All solutions have been carefully checked and rechecked to ensure accuracy • Computational Exercises Each section includes computations and explorations designed to be done with a com putational program, such as Maple, Mathematica, PARIIGP, or Sage, or using programs written by instructors and/or students There are routine computational exercises students can to learn how to apply basic commands (as described in Appendix D for Maple and Mathematica and on the Web site for PARI/GP and Sage), as well as more open-ended questions designed for experimentation and creativity Each section also includes a set of programming projects designed to be done by students using a programming language or the computational program of their choice The Student's Manual to Computations and Explorations on the Web site provides answers, hints, and guidance that will help students use computational tools to attack these exercises Web Site Students and instructors will find a comprehensive collection of resources on this book's Web site Students (as well as instructors) can find a wide range of resources at www.pearsonhighered.com/rosen Resources intended for only instructor use can be ac cessed at www.pearsonhighered.com/irc; instructors can obtain their password for these resources from Pearson • External Unks The Web site for this book contains a guide providing annotated links to many Web sites relevant to number theory These sites are keyed to the page in the book where relevant material is discussed These locations are marked in the book with the icon (J For convenience, a list of the most important Web sites related to number theory is provided in Appendix D • Number Theory News The Web site also contains a section highlighting the latest discoveries in number theory • Student's Solutions Manual Worked-out solutions to all the odd-numbered exercises in the text and sample exams can be found in the online Student's Solution Manual Preface • xiii Student's Manual for Computations and Explorations A manual providing resources supporting the computations and explorations can be found on the Web site for this book This manual provides worked-out solutions or partial solutions to many of these computational and exploratory exercises, as well as hints and guidance for attacking others This manual will support, to varying degrees, different comptutional environments, including Maple, Mathematica, and PARl/GP • Applets An extensive collection of applets are provided on the Web site These applets can be used by students for some common computations in number theory and to help understand concepts and explore conjectures Besides algorithms for comptutions in number theory, a collection of cryptographic applets is also provided These include applets for encyrp tion, decryption, cryptanalysis, and cryptographic protocols, adderssing both classical ciphers and the RSA cryptosystem These cryptographic applets can be used for individ ual, group, and classroom activities • Suggested Projects A useful collection of suggested projects can also be found on the Web site for this book These projects can serve as final projects for students and for groups of students • Instructor's Manual Worked solutions to all exercises in the text, including the even-numbered execises, and a variety of other resources can be found on the Web site for instructors (which is not available to students) Among these other resources are sample syllabi, advice on planning which sections to cover, and a test bank How to Design a Course Using this Book This book can serve as the text for elementary number theory courses with many different slants and at many different levels Consequently, instructors will have a great deal of flexibility designing their syllabi with this text Most instructors will want to cover the core material in Chapter (as needed), Section 2.1 (as needed), Chapter 3, Sections 4.1-4.3, Chapter 6, Sections 7.1-7.3, and Sections 9.1-9.2 To fill out their syllabi, instructors can add material on topics of interest Generally, topics can be broadly classified as pure versus applied P ure topics include Mobius inversion (Section 7.4), integer partitions (Section 7.5), primitive roots (Chapter 9), continued fractions (Chapter 12), diophantine equations (Chapter 13), and Guassian integers (Chapter 14) Some instructors will want to cover accessible applications such as divisibility tests, the perpetual calendar, and check digits (Chapter 5) Those instructors who want to stress computer applications and cryptography should cover Chapter and Chapter They may also want to include Sections 9.3 and 9.4, Chapter 10, and Section 11.5 xiv Preface After deciding which topics to cover, instructors may wish to consult the following figure displaying the dependency of chapters: /I� I 12 /i� I 13 14 � � /9 10 11 Although Chapter may be omitted if desired, it does explain the big-0 notation used throughout the text to describe the complexity of algorithms Chapter 12 depends only on Chapter 1, as shown, except for T heorem 12.4, which depends on material from Chapter Section 13.4 is the only part of Chapter 13 that depends on Chapter 12 Chapter 11 can be studied without covering Chapter if the optional comments involving primitive roots in Section 9.1 are omitted Section 14.3 should also be covered in conjunction with Section 13.3 For further assistance, instructors can consult the suggested syllabi for courses with different emphases provided in the Instructor's Resource Guide on the Web site Acknowledgments I appreciate the continued strong support and enthusiam of Bill Hoffman, my editor at Pearson and Addison-Wesley far longer than any of the many editors who have preceded him, and Greg Tobin, president of the mathematics division of Pearson My special grati tude goes to Caroline Celano, associate editor, for all her assistance with development and production of the new edition My appreciation also goes to the production, marketing, and media team behind this book, including Beth Houston (Production Project Manager), Maureen Raymond (Photo Researcher), Carl Cottrell (Media Producer), Jeff Weidenaar (Executive Marketing Manager), Kendra Bassi (Marketing Assistant), and Beth Paquin (Designer) at Pearson, and Paul Anagnostopoulos (project manager), Jacqui Scarlott (composition), Rick Camp (copyeditor and proofreader), and Laurel Muller (artist) at Windfall Software I also want to reiterate my thanks to all those who supported my work on the first five editions of this book, including the multitude of my previous edi tors at Addison Wesley and my management at AT&T Bell Laboratories (and its various incarnations) Preface xv Special thanks go to Bart Goddard who has prepared the solutions of all exercises in this book, including those found at the end of the book and on the Web site, and who has reviewed the entire book I am also grateful to Jean-Claude Evard and Roger Lipsett for their help checking and rechecking the entire manuscript, including the answers to exercises I would also like to thank David W right for his many contributions to the Web site for this book, including material on PARI/GP, number theory and cryptography applets, the computation and exploration manual, and the suggested projects Thanks also goes to Larry Washington and Keith Conrad for their helpful suggestions concerning congruent numbers and elliptic curves Reviewers I have benefited from the thoughtful reviews and suggestions from users of previous edi tions, to all of whom I offer heartfelt thanks Many of their ideas have been incorporated in this edition My profound thanks go to the reviewers who helped me prepare the sixth edition: Jennifer Beineke, Western New England College David Bradley, University of Maine-Orono Flavia Colonna, George Mason University Keith Conrad, University of Connecticut Pavel Guerzhoy, University of Hawaii Paul E Gunnells, University of Massachusetts-Amherst Charles Parry, Virginia Polytechnic Institute and State University Holly Swisher, Oregon State University Lawrence Sze, California State Polytechnic University, Pomona I also wish to thank again the approximately 50 reviewers of previous editions of this book They have helped improve this book throughout its life Finally, I thank in advance all those who send me suggestions and corrections in the future You may send such material to me in care of Pearson at math@pearson.com Kenneth H Rosen Middletown, New Jersey This page intentionally left blank 742 Index Gross, 60 Indices, 368, 636 639 Gynecologist,62 Induction,mathematical,23-27 lnduction,strong,25 Hadamard,Jacques,79,80 Inductive step,23 Hajratwala,Nayan,263 Inequality,Bonse's,91 Hanoi,tower of,28 Infinite continued fraction,491 Hardy,G H., 2,78, 92, 254,278 Infinite descent,531,535 Harmonic series,27 Infinite simple continued fraction,491 Haros, C.,101 Infinitude of primes, 70-71, 76, 101, 102, Hashing, 204-206 124,125,133-134 double,205-206 Initial term of a geometric progression,10 function, 204 Integer,6 quadratic, 429 abundant,267 Hashing function,202 composite,70 Hastad broadcast attack,328,330 deficient,267 Hellman,M E.,318,324,333 Eisenstein,597 Hensel,Kurt, 173 Gaussian,579 Hensel's lemma,173 k-abundant,267 Heptadecagon,146 k-perfect,267 Heptagonal number,21 order of,347-348 Heron triangle,574 palindromic,195 Hex,48,49 powerful,120 Hexadecimal notation,48,49 rational,579 Hexagonal number,21 sequences,11 Highly composite,253 square-free,120 Hilbert,David,122,478 superperfect,268 Hilbert prime,121 Integers,6 Hill,Lester S.,305,306 Gaussian,579 Hill cipher, 305-309 most wanted,ten,133 Home team,203 Intel,86,89,266 House of Wisdom,57 International fixed calendar,201 Horses,same color,28 International Mathematical Olympics, 87, Hundred fowls problem,143 325 Hurwitz,Alexander,262 International Standard Book Number,210 Hyperinflation,534 International Standard Serial Number, Hypothesis,Riemann,83 215 Internet,239,261,624 IBM 360 computer,262 Interpolation, Lagrange,359 IBM 7090 computer,262 Inverse,additive,605 Identity, Inverse of an arithmetic function,247 Bezout, 95-96 Inverse of a matrix modulo Rogers-Ramanujan,287 Inverse modulo m, m, 182 Identity elements,605 Inversion,Mobius,272-274 ILLIAC, 262 Involutory matrix, 185 Inclusion-exclusion, principle of, 77, Irrational number,6,118-119 613-614 quadratic,503-506,579 ,J2, 6-7,119 Incongruent,145 Irrationality of Index arithmetic,368-371 ISBN, 210 Index of coincidence,303 ISBN-10,210,211 Index of an integer, 368, 636-639 ISBN-13,210,212 Index of summation,16 Iterated knapsack cipher,336 Index system, 377 lwaniec,Henryk,89 178 Index Jackpot,265 Law, Jacobi,Carl G J.,443,597 associative,605 Jacobi symbol,443 cancellation,605 reciprocity law for,446 447 Jeans,J.H.,225 Jigsaw puzzle,28 743 commutative,605 distributive,605 trichotomy,606 Julian calendar,197 Law of quadratic reciprocity,418,430-438 Julius Caesar,197,292 Leap year,197 Jurca,Dan,262 Least common multiple, finding using prime factorizations,116 k-abundant number,267 k-perfect number,267 of more than two integers,123 of two integers,116 Kaprekar,D R., 53 Least nonnegative residue,147 Kaprekar constant,52 Least nonnegative residues,148 Kasiski,F., 302 Least positive residue,147 Kasiski test,302 Least primitive root for a prime,358 Kayal,N., 75 Least-remainder algorithm,111 Key,292 Leblanc, M (pseudonym of Sophie agreement protocol,338 Germain),531 common,338-339 Legendre,Adrien-Marie,79,417,418,531 decryption,292 Legendre conjecture,89-90 encryption,292 Legendre symbol,417 exchange,338-339 Lehmer,Derrick,249,259,518 for hashing,204 Lehmer,Emma,262 master, 342 Lemma, public,322 Gauss's,420 Keyspace,292 Hensel's, 173 Keystream,310 Thue's,551 Knapsack ciphers,331-336 weakness in,335 Knapsack problem,334 multiplicative,336-337 Lemmermeyer,Franz,431 Lenstra,Arjen,130 Lenstra,H.,75 Letters,frequencies of,295-296 Knuth,Donald,62,63 Lifting solutions,173 Kocher,Paul,329 Linear combination,94 Kronecker, Leopold, 174, 434, 451, 452 greatest common divisor as a, 94-97, 107-109,110 Kronecker symbol,451 Linear congruence,157 kth power residue,372 Linear congruences,systems of,162,178 Kummer,Ernst,452,478,531-532 Linear congruential method,395-396 Linear diophantine equation,137 Lagarias,Jeffrey,84 in more than two variables,140 Lagrange,Joseph,217,218,350,355,359, nonnegative solutions,142 506-507,531,542,546,549,555 Linear homogeneous recurrence relation, 33 Lagrange interpolation,359 Liouville, Joseph,247,248,476 Lagrange's theorem Liouville's function,247 on continued functions,506-507 Little theorem,Fermat's,219 on polynomial congruences,355 Littlewood,J.E.,78,84,92,254 Lame,Gabriel,105,106,531 Lobsters,142,169 Lame's theorem,105-106 Logarithm,discrete,368 Landau,Edmund,62,89-90 Logarithmic integral,79 Largest known primes,73-74 Logarithms modulo p, 368 Largest number naturally appearing,84 Lowest terms,94 744 Index Lucas,Edouard,30,34,259,261,379 Lucas converse of Fermat's little theorem, 379 Lucas numbers,34 l\1iller's test, 228-229,373 l\1ills,W H.,74 l\1ills formula,74 l\1inimal universal exponent,386 Lucas-Lehmer test, 259-260 l\1inims,order of the,258 Lucifer,310 l\1inimum-disclosure proof,461-462 Lucky numbers,77 l\1IPS-years,129 l\1oats,Gaussian,588 l\1acl\1ahon,Percy,286 l\1obius,A F., 271 l\1acTutor History of l\1athematics Archives, l\1obius function, 270-271 625 l\1obius inversion,272-274 MAD Magazine, 63 l\1obius strip,271 l\1agic square,186 l\1odified division division, 41 l\1ahavira, 141 l\1odular arithmetic,148 l\1angoldt function, 276 l\1odular exponentiation algorithm, 151-152 l\1anhattan project,15 l\1aple,615-619 Gaussian integer package,618 complexity of,152-153 l\1odular inverses,159 l\1odular square roots,423-424 l\1arkov's equation,542 l\1odulus,145 l\1aster key,342,359 l\1onkeys,156,168 l\1aster Sun,162 l\1onks,28 Mathematica, 619-623 l\1onographic cipher,292 l\1athematical induction,23-26 l\1onte Carlo method,15,187 origins of,24 second principle,25 l\1athematics,Prince of,146 l\1orrison,l\1.A.,518 l\1ost wanted integers,133 l\1r Fix-It, 87 l\1atrices,congruent,180-181 l\1ultinomial coefficient,614 l\1atrix,involutory,185 l\1ultiple,36 l\1atrix multiplication,67 least common,116 l\1aurolico,Francesco, 24 l\1ultiple precision,55 l\1aximal ± 1-exponent,408 l\1ultiplication, l\1ayans,45 algorithm for,57 l\1ean, complexity of,64-65 arithmetic,29 geometric, 29 matrix,67 l\1ultiplicative function,239,240 l\1erkle, R C.,333 l\1ultiplicative knapsack problem,336-337 l\1ersenne,l\1arin,128, 258 l\1utually relatively prime,98 l\1ersenne numbers, 258,428 l\1ysteries of the universe,301 double,268 l\1ersenne primes,73-74,258-266,382,396, 428,624 search for,261-265, 624 l\1ertens,Franz,274 Namaigiri,254 National Institute of Standards and Technology,310 Nicely,Thomas,86,89 l\1ertens conjecture,276 Nickel, Laura, 262 l\1ertens function,274,276 Nicomachus,162 l\1essage expansion factor,403 Nim,52 l\1ethod, Noll,Landon,262 Kasiski's,302 Nonresidue,quadratic,416 l\1onte Carlo,187 Norm,121 l\1ethod of infinite descent,531,535 l\1iddle-square method,394 l\1ihailescu,Preda,537 of complex number,578 Notation, Arabic,30, 56 Index big-0,61 p-adic,173 binary,48 pseudorandom,393-398 binary coded decimal,51 random,393 decimal,48 duodecimal,60 hexadecimal,48 ten most wanted,133 Number of divisors function,250,634 multiplicativity of,251 octal,48 Number system,positional,45 one's complement,51 Number theory,definition of,1 product,19-20 combinatorial,277 summation,16-19 elementary,definition of,3 two's complement,51 NOVA,534,625 745 Number T heory Web,625 Numerals,Hindu-Arabic,56 NOVA Online-The Proof, 625 Number, Octal notation,48 abundant, 267 Odd number,39 algebraic,7 Odd perlect number,266,268 Carmichael,227,228,388-389 Odlyzko,Andrew,84 composite,70 Oliveira e Silva,Tomas,84 congruent,560 One-time pad,311 Cullen,234 One-to-one correspondence,11 deficient,267 One's complement representation, 51 double Mersenne,268 Ono,Kenneth,287 even,39 Operation,bit, 61 everything is,522 Orange, Prince of,555 Fermat,131-133,353,414,428 Order of an integer,348 Fibonacci,30 Ordered set,6,606 generalized Fibonacci,35 Origin of, heptagonal,21 mathematical induction,24 hexagonal,21 the word "algebra," 57 irrational,6 k-abundant,267 the word "algorithm," 56 Origins of mathematical induction,24 k-perlect,267 Lucas,34 Pad,one-time,311 lucky,77 p-adic numbers,173 Mersenne,258 Pair,amicable,267 most wanted,133 Pairwise relatively prime,98-99 odd,39 Palindromic integer,195 odd perlect,266 Parameterization,527 pentagonal,21 Parity check bit,209 perlect,256 Parity theorem,Euler,283 pseudorandom,393-398 Partial key disclosure attack on RSA,328 random,15, 393 Partial quotient,482 rational,6 Partial remainder,59 Sierpinski,384 Partition,277 superperlect,268 conjugate,279 t-congruent,574 function,278 tetrahedral, 21 restricted,278 transcendental,7,452, 476-478 self-conjugate,279 triangular, 19,20 Ulam,15 Numbers, lucky,77 unrestricted,278 Parts,aliquot,268 Pascal,Blaise,609-610 Pascal's identity,609 746 Index Pascal's triangle,610 Primality test,71,379-381 Pell,John,554 Pocklington's,381 Pell's equation,553-558 probabilistic,231,459 Pentagonal number,21,284 Pentagonal number theorem,Euler's,284 Pentagonal numbers,generalized,286 Proth's,382 Prime, in arithmetic progressions,73 Pentium,54,86,89,129,262,263,266 definition of,70 Pepin's test,438-439 Eisenstein,597 Perfect number,256,266 Fermat,131-132 even,256-257 Gaussian,582 odd,266,268 Hilbert,121 Perfect square, last two decimal digits,135 modulo p, 416 Period, of a base b expansion,4 74 largest known,73-74 Mersenne, 73-74, 258-266, 382, 396, 428,624 power,91 relatively, 39 of a continued fraction,516 size of the nth,84 length of a pseudorandom number Sophie Germain,75 generator,396 Wilson,224 Periodic base b expansion,473 Prime number theorem,79-83 Periodic cicada,122 Prime Pages, The, 624 Periodic continued fraction,503 Prime power,91 Perpetual calendar, 197-200 PrimeNet,262,266 Phyllotaxis, 31 Prime-power factorization,113 Jl'' 6,499 Pigeonhole principle 8,9 Pintz,Janos,86 Pirates,169 using to find greatest common divisors, 115 using to find least common multiples,116 Primes, Plaintext,292 in arithmetic progressions,73 Pocklington,Henry,381 infinitude of, 70 71, 76, 101, 102,124, Pocklington's primality test,381 125,133-134 Poker,electronic,340 341,429 distribution of,79-90 Pollard,J.M.,128,129,187,221 finding,71-72 Pollard, p - factorization,221 rho factorization, 187-189 Polygon,regular,134 Polygraphic cipher,300,308 formula for,74 gaps,84-85 largest known,73-74 primitive roots of,357 twin,86 Polynomial,cyclotomic,276-277 PRIMES is in P,75 Polynomial congruences, solving, Primitive Pythagorean triple,522,536,561 171-177,355-356 Polynomial time algorithm,75 Primitive root,350,635 Primitive root, Polynomials,congruence of,156-157 method for constructing,359 Pomerance,Carl,75,129 modulo primes, 354-358, 635 Positional number system,45 modulo prime squares,360 362 Potrzebie system,63 modulo powers of primes,362-365 Power,prime,91 Power generator,401 of unity,276,441 Prince of Orange,555 Power residue,372 Principle, pigeonhole,8-9 Powerful integer,120 Principle of inclusion-exclusion,77,613-614 Powers,R E.,518 Principle of mathematical induction,23-26 Pre-period,473 second,25 Index Private-key cryptosystem,321 middle-square,394 Prize, 1/ P, 480 for factorizations,130 power,401 for finding large primes,265 pure multiplicative,396 for proving the Riemann hypothesis,83 quadratic congruential,402 for settling Beal's conjecture,537 square,397-398 Wolfskehl,534 Probabilistic primality test,231,459 Solovay-Strassen,459 747 Pseudorandom numbers,393-399,480 Ptolemy II, 72 Public-key cipher,321-323 Probing sequence,206 Public-key cryptography,321-329, 402-403 Problem, Public-key cryptosystem,321-322 coconut,156 Pulvizer,the,102 congruent number,560 Pure multiplicative congruential method, discrete logarithm,368-369,372 396 397 hundred fowls,143 Purely periodic continued fraction,511-512 knapsack,331 Puzzle,141,143,162 multiplicative knapsack,336 337 jigsaw, 28 Waring's,549 tower of Hanoi,28 Problems,Landau,89-90 Pythagoras, 522 Product,Dirichlet,247 Pythagorean theorem,522 Product cipher,299 Pythagorean triple,522,561 Product notation,19-20 Progression, primitive,522,524, 561,603 Pythagoreans,522 arithmetic,10 geometric,10,17-18 Project, Quadratic character of -1,419-420 Quadratic character of 2,421-422 Cunningham,133 Quadratic congruential generator, 402 Manhattan,15 Quadratic hashing,429 Proof, minimum-disclosure,461-462 Quadratic irrationality,504,579 reduced,512 primality,74-75 Quadratic nonresidue,416 zero-knowledge,461-462 Quadratic reciprocity law,418, 430-438 Property, different proofs of,431 reflexive,146 Euler's version of,431-432 symmetric,147 Gauss's proofs of,431 transitive,147 history of, 430-431 well-ordering,6,606 proof oL434-437,441,442 Proth,E.,382 Quadratic residue,416 Proth's primality test,382 Quadratic residues Protocol, cryptographic,338 failure,328 key agreement protocol,338 chain of,429,430 consecutive,428 Quadratic residues and primitive roots,417 Quadratic sieve,129 Prover,in a zero-knowledge proof,462 Queen of mathematics,146 Pseudoconvergent,502 Quotient,37 Pseudoprime,225-227 Euler,453-455 Fermat, 224 partial, 482 strong,229,456 Pseudorandom number generator,393-399 Rabbits,30 discrete exponential,401 Rabin, Michael, 329 Fibonacci,400-401 Rabin cryptosystem,329,429 linear congruential,395 Rabin's probabilistic primality test,231 748 Index rad function,125,538-539 Restricted partitions,278 Radix,48 Riemann,George F riedrich,80,83,232 Ramanujan,Smivasa,253,254, 286-287 Riemann hypothesis,83 Ramanujan congruences,287 Riemann hypothesis,generalized,231 Random numbers,15,393 Riesel,Hans, 262 Ratio,common,10 Right triangle, Rational integer, 579 integer, 560 Rational number,6 rational, 560 Rational numbers, Rijndael algorithm,310 countability of,11-12 Rational point, Rivest,Ronald,324 Robinson,Raphael,262 on curve, 526 Rogers,Leonard James,287 on elliptic curve,568 Rogers-Ramanujan identities,287 on unit circle,526-528 Root,primitive,350 Real number,base b expansion of,469 471 Real numbers, of unity,276 Root of a polynomial modulo equivalent,502 m, 350 Root of unity,441 uncountability of,478 479 Reciprocity law, primitive,276,441 Roman numerals,45 for Jacobi symbols,446 447 quadratic,418,430 438 Recurrence relation, Romans, 45 Round-robin tournament, 202-203 RSA cryptosystem,323-328,354,390,500, linear homogeneous, 621,625 attacks on implementations of,328-329 35 for the partition function,286 cycling attack on,354 Recursive definition,26-27 digital signatures in,339-340 Reduced quadratic irrational,512 Hastad broadcast attack on,328,330 Reduced residue system,235 partial key disclosure attack on,328-329 Reducing modulo security of,326-327 m, 147 Reflexive property,146 Regular polygon,constructability,134,146 Relatively prime,39,93 mutually,98 pairwise, 98-99 Wiener's low encryption exponent attack, 328,500-501 RSA factoring challenge,130 RSA Labs,130,625 cryptography FAQ,625 Remainder,37 RSA-129,129,130 Remainder,partial,59 RSA-130,130 Representation, RSA-140,130 one's complement,51 RSA-155,130 two's complement,51 RSA-200,129,130 Zeckendorf,34 Rule for squaring an integer with final digit 5, Repunit,195 base b, 195 60 Rumely, Robert,75 Residue, cubic,378 Sarrus,P.F., 225 kth power,372 Saxena,N., 75 least nonnegative,147 Scottish Cafe,15 quadratic,416 Second principle of mathematical induction, system,reduced,235 Residues, absolute least,148 25 Secret sharing,342-343 Security of RSA,326-327 complete system of,148 Seed,395 reduced,235 Selberg,A.,73,81 Index Self-conjugate partition,279 Spread of a splicing scheme,411 Sequence,10 Square, aliquot,268 Euler-Mullin,78 Fibonacci,30 diabolic,187 magic,186 Square pseudorandom number generator, 397-399 formula for terms,10 integer,11 Square root,modular,423-424 probing,206,429 Square-free integer,120 Sidon,53 Square-free part,561 spectrum,14 Squaring an integer with final digit 5,60 super-increasing,332 Squares, sums of, 542-548, 599-602 Stark,Harold,260 Series, Farey,100 Strauss,E.,29 harmonic,27 Step, Set, basis,23 countable, 11,478 inductive,23 ordered,606 Stream cipher, 310-311 uncountable,11,478 Stridmo,Odd M.,264 well-ordered,6 Shadows,342 Shamir,Adi,323,324,340,463 Strip,Mobius,271 Strong pseudoprime, 229, 373-376, 454 Sharing,secret,342-343 Strongly multiplicative function, 247 Shift transformation,294 Subexponential time,128 Shifting,57 Substitution cipher,293 Shuffling cards,224 Subtraction,algorithm for,56 Sidon, Simon,53 Subtraction,complexity of,54 Sidon sequence,53 Sum, telescoping,18 Sierpinski,Wadaw,384 Sum of divisors function,249,634 Sierpinski number,384 Sieve, multiplicativity of,251 Summation, of Eratosthenes,71-72 index of,16 number field,129 notation, 16 quadratic,129 terms of a geometric series,18 Signature, digital, 339-340, 344-345, 405-407 Signed message,339 Simple continued fraction, 482 Summations, properties of,17 Summatory function,243 of Mobius function,270-271 Shafer,Michael,263 Sums of cubes,549-550 Sinning,301 Sums of squares,542-548,599-602 Skewes, S.,84 Super-increasing sequence,332 Skewes' constant,84 Superperfect integer,268 Sloane,Neil,11 SWAC,262 Slowinski,D., 262 Sylvester,James Joseph,96,266,280 Smith,Edson,264 Symbol, Sneakers, 324 Jacobi,443 Solovay-Strassen probabilistic primality test, Kronecker,451 460 Solving Legendre,417 Symmetric cipher,321 linear congruences,157-160 Symmetric property,147 linear diophantine equations,137-141 System,index,377 polynomial congruences,171-177 System of congruences,178-185 Splicing of telephone cables,411-412 749 System of linear congruences,174 181 750 Index System of residues, Thue, Axel, 551 complete, 148 Thue's lemma, 551 reduced, 235 Tijdeman, R., 537 Tournament, round-robin, 202-203 Tower of Hanoi, 28,259 Table, factor, 627-633 Transcendental number, 7,452, 476 478 of arithmetic functions, 634 Transformation, affine, 294, 316 of continued fractions, 640 Transformation, shift, 294 of indices, 636 639 Transitive property, 147 of primitive roots, 635 Transposition cipher, 316 Tao, Terrence, 87 Trial division, 71,127 t-congruent number, 574 Triangle, Heron, 574 Team, away, 203 Pascal's, 609 610 home, 203 Pythagorean, 522 Telephone cables, 411 413 Telescoping sum, 18 right, integer, 560 right, rational, 560 Ten most wanted integers, 133 Triangular number, 19,20 Term, initial, of a geometric progression, 10 Trichotomy law, 606 Terminate, 472 Trivial zeros, 83 Terminating base b expansion, 472 Tuberculosis, 62,232, 254,434 Test, Tunnell, J., 571-572 divisibility, 191-194 Tuckerman, Bryant, 262 Kasiski, 302 Twin prime conjecture, 86 Lucas-Lehmer, 260 Twin primes, 86 Miller's, 228-229 Pepin's, 438 439 primality, 71-72, 74-75, 228-230, 378-383,460 asymptotic formula conjecture, 92 application to hashing, 206 Two squares, sums of, 542-545, 601-602 Two's complement representation, 51 probabilistic primality, 228-230, 460 Tetrahedral number, 21 Ujjain, astronomical observatory at, 555 Theorem, Ulam, S M., 15 Bezout's, 95 Ulam number, 15 binomial, 610 611 Uncountable set, 12,15,478 479 Chinese remainder, 162-163 Unique factorization, 112-114 Dirichlet's, 9, 73, 118, 497 Euler parity, 283 of Gaussian integers, 592-594 Unique factorization, failure of, 114, 121, Euler's, 234 598 Euler's pentagonal number, 284 Unit, in the Gaussian integers, 581 Fermat's last, 530-536 Unit circle, Fermat's little, 219-220 rational points on, 526,527 fundamental, of arithmetic, 112 Unit fraction, 29 Gauss's generalization of Wilson's, 224 Unity Green-Tao, 87 primitive root of, 276, 441 Lagrange's (on continued fractions), root of, 441 506-507 Lagrange's (on polynomial congruences), 355 Lame's, 105-106 Universal exponent, 386 Universal product code, 213 Unrestricted partitions, 278 Uzbekistan, 57 prime number, 81 Wilson's, 217 Threshold scheme, 342-343,359-360 Valle-Poussin, C de la, 79, 81 van der Corput, Johannes, 87 Index Variable, dummy, 16, 20 Wilson, John, 217 Vega, Jurij, 79 Wilson prime, 224 Vegitarianism, 254 Wilson's theorem, 217-218 Verifier, in a zero-knowledge proof, 462 Gauss' generalization of, 224 Vemam, Gilbert, 311 Gaussian integers, analogue for, 604 Vemam cipher, 311 Winning move in game of Euclid, 111 Vigenere, Blaise de 300, 301, 312 Winning position in nim, 52 Vigenere cipher, 300-301 Wisdom, House of, 57 cryptanalysis of, 302-305 Wolfskehl prize, 534 von Humboldt, Alexander, 434 Woltman, George, 262 von Neumann, John, 394 Word size, 54 World, end of, 28 Wagstaff, Samuel, 133, 532 Wallis, John, 554-555 Year end day, 201 Waring, Edward, 217, 549 Year, leap, 197-198 Waring's problem, 549 Yildrim, Cem, 86 Web, Number Theory, 625 Web sites, 624-625 Zeckendorf representation, 34 Wedeniwski, Sebastian, 83 Zeller, Christian Julius, 200 Weights, 50, 169 Zero-knowledge proof, 461-462 Well-ordered set, 6, 606 Zeros, trivial, 83 Well-ordering property, 6, 606 Zeta function, Riemann, 81, 83 Welsh, Luke, 262 ZetaGrid, 83 Wiener, M., 328, 500 501 Ziegler's Giant Bar, 63 Wiles, Andrew, 533-534 751 Photo Credits Courtesy of The MacTutor History of Mathematics Archive, University of St Andrews, Scotland: Stanislaw M Ulam, Fibonacci, Francois-Edouard-Anatole Lucas, Paul Gustav Heinrich Bachmann, Edmund Landau, Pafnuty Lvovich Chebyshev, Jacques Hadamard, Alte Selberg, Joseph Louis Fran9ois Bertrand, G Lejeune Dirichlet, Gabriel Lame, David Hilbert, Karl Friedrich Gauss, Kurt Hensel, Joseph Louis LaGrange, Georg Friedrich Bernhard Reimann, Leonhard Euler, Joesph Liouville, Srinivasa Ramanujan, Marin Mersenne, August Ferdinand Mobius, Adrien-Marie Legendre, Ferdinand Gotthold Max Eisenstein, Carl Gustav Jacob Jacobi, Leopold Kroneker, Georg Cantor, Pythagoras, Sophie Germain, Ernst Eduard Kummer, Andrew Wiles, Claude Bachet, Edward Waring, Axel Thue, and Blaise Pascal; Eratosthenes©Culver Pictures, Inc.; Paul Erdos©1985 W lodzimierz Kuperberg; Euclid©The Granger Collection; Pierre de Fermat©Giraudon/Art Resource, N.Y.; Derrick H Lehmer© 1993 the American Mathematical Society; Gilbert S Vernam©Worcester Polytechnic Institute, Class of 1914; Adi Shamir©the Weizmann Institute of Science, Israel; Ronald Rivest©1999 Ronald Rivest; Leonard Adleman©1999 Eric Mankin, University of Southern California; John Von Neumann© Corbis; Emil Artin ©The Hall of Great Mathematicians, Southern Illinois University, Edwardsville; Eugene Catalan ©Collections artistiques de l'Universite de Liege; Terrence Tao©Reed Hutchinson 2009/UCLA; Norman MacLeod Ferrers by permission of the Master and Fellows of Gonville and Caius College, Cambridge; Etienne Bezout, St Andrew's University; Clifford Cocks, Simon Singh; Wadaw Sierpinski, St Andrew's University; Robert Daniel Carmichael, Mathematical Association of America Records, 1916-present, Archives of American Mathematics, Dolph Briscoe Center for American History, University of Texas at Austin; 752 List of Symbols [x] Greatest integer, I: Summation, 16 TI Product, 19 n! Factorial, 20 In Fibonacci number, 30 alb Divides, 37 aJb Does not divide, 37 (a, b) (akak-1 Greatest common divisor, 39 · · · aiaoh Base b expansion, 48 O(f) Big-0 notation, 61 n(x) Number of primes, 72 f(x) rv g(x) (a1a2, ' an) Asymptotic to, 82 Greatest common divisor (of n integers), 98 :fn Farey series of order n, 100 min(x , y) Minimum, 115 y) Maximum, 116 max(x, [a, b] Pa II n Least common multiple, 116 Exactly divides, 121 [ai a2, , an] Least common multiple (of n integers), 123 Fn Fermat number, 131 a=b(modm) Congruent, 145 a¢ b (modm) Incongruent, 145 a Inverse, 159 A= B (modm) Congruent (matrices), 180 I Identity matrix, 182 A Inverse (of matrix), 182 adj(A) Adjoint, 183 h(k) Hashing function, 204