Number Theory Through Inquiry

AMS / MAA TEXTBOOKS VOL 9

David C. Marshall, Edward Odell, and Michael Starbird

“NumberTheory_bev” — 2007/10/15 — 12:46 — page i — #1�

�

�

�

�

�

�

�

Number Theory

Through Inquiry

\NumberTheory_bev" | 2011/2/16 | 16:14 | page ii | #2i

i

i

i

i

i

i

i

About the cover: The cover design suggests the meaning and proof of the

Chinese Remainder Theorem from Chapter 3. Pictured are solid wheels with

5, 7, and 11 teeth rolling inside of grooved wheels. As the small wheels roll

around a large wheel with 5 � 7 � 11 D 385 grooves, only part of which is

drawn, the highlighted teeth from each small wheel would all arrive at the

same groove in the big wheel. The intermediate 35 grooved wheel suggests

an inductive proof of this theorem.

Cover image by Henry Segerman

Cover design by Freedom by Design, Inc.

c 2007 byThe Mathematical Association of America (Incorporated)

Library of Congress Catalog Card Number 2007938223

Print ISBN 978-0-88385-751-9

Electronic edition ISBN 978-0-88385-983-4

Printed in the United States of America

Current Printing (last digit):

10 9 8 7 6 5 4 3 2 1

“NumberTheory_bev” — 2007/10/15 — 12:46 — page iii — #3�

�

�

�

�

�

�

�

Number Theory

Through Inquiry

David C. MarshallMonmouth University

Edward OdellThe University of Texas at Austin

Michael StarbirdThe University of Texas at Austin

®

Published and Distributed byThe Mathematical Association of America

10.1090/text/009

“NumberTheory_bev” — 2007/10/15 — 12:46 — page iv — #4�

�

�

�

�

�

�

�

Council on PublicationsJames Daniel, Chair

MAA Textbooks Editorial BoardZaven A. Karian, Editor

William C. BauldryGerald M BryceGeorge Exner

Charles R. HadlockDouglas B. MeadeWayne RobertsStanley E. SeltzerShahriar ShahriariKay B. SomersSusan G. StaplesHolly S. Zullo

“NumberTheory_bev” — 2007/10/15 — 12:46 — page v — #5�

�

�

�

�

�

�

�

MAA TEXTBOOKS

Combinatorics: A Problem Oriented Approach, Daniel A. MarcusComplex Numbers and Geometry, Liang-shin HahnA Course in Mathematical Modeling, Douglas Mooney and Randall SwiftCreative Mathematics, H. S. WallCryptological Mathematics, Robert Edward LewandDifferential Geometry and its Applications, John OpreaElementary Cryptanalysis, Abraham SinkovElementary Mathematical Models, Dan KalmanEssentials of Mathematics, Margie HaleFourier Series, Rajendra BhatiaGame Theory and Strategy, Philip D. StraffinGeometry Revisited, H. S. M. Coxeter and S. L. GreitzerKnot Theory, Charles LivingstonMathematical Connections: A Companion for Teachers and Others, Al CuocoMathematical Modeling in the Environment, Charles HadlockMathematics for Business Decisions Part 1: Probability and Simulation (electronictextbook), Richard B. Thompson and Christopher G. Lamoureux

Mathematics for Business Decisions Part 2: Calculus and Optimization (electronictextbook), Richard B. Thompson and Christopher G. Lamoureux

The Mathematics of Games and Gambling, Edward PackelMath Through the Ages,William Berlinghoff and Fernando GouveaNoncommutative Rings, I. N. HersteinNon-Euclidean Geometry, H. S. M. CoxeterNumber Theory Through Inquiry, David C. Marshall, Edward Odell, and MichaelStarbird

A Primer of Real Functions, Ralph P. BoasA Radical Approach to Real Analysis, 2nd edition, David M. BressoudReal Infinite Series, Daniel D. Bonar and Michael Khoury, Jr.Topology Now!, Robert Messer and Philip StraffinUnderstanding our Quantitative World, Janet Andersen and Todd Swanson

MAA Service CenterP.O. Box 91112

Washington, DC 20090-11121-800-331-1MAA FAX: 1-301-206-9789

“NumberTheory_bev” — 2007/10/15 — 12:46 — page vi — #6�

�

�

�

�

�

�

�

“NumberTheory_bev” — 2007/10/15 — 12:46 — page vii — #7�

�

�

�

�

�

�

�

Contents

0 Introduction 1Number Theory and Mathematical Thinking . . . . . . . . . . . 1Note on the approach and organization . . . . . . . . . . . . . 2Methods of thought . . . . . . . . . . . . . . . . . . . . . . . 3Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . 4

1 Divide and Conquer 7Divisibility in the Natural Numbers . . . . . . . . . . . . . . . . 7Definitions and examples . . . . . . . . . . . . . . . . . . . . 7Divisibility and congruence . . . . . . . . . . . . . . . . . . . 9The Division Algorithm . . . . . . . . . . . . . . . . . . . . . 14Greatest common divisors and linear Diophantine equations . 16

Linear Equations Through the Ages . . . . . . . . . . . . . . . . 23

2 Prime Time 27The Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . 27Fundamental Theorem of Arithmetic . . . . . . . . . . . . . . 28Applications of the Fundamental Theorem of Arithmetic . . . 32The infinitude of primes . . . . . . . . . . . . . . . . . . . . . 35Primes of special form . . . . . . . . . . . . . . . . . . . . . . 37The distribution of primes . . . . . . . . . . . . . . . . . . . . 38

From Antiquity to the Internet . . . . . . . . . . . . . . . . . . . 41

3 A Modular World 43Thinking Cyclically . . . . . . . . . . . . . . . . . . . . . . . . 43Powers and polynomials modulo n . . . . . . . . . . . . . . . 43Linear congruences . . . . . . . . . . . . . . . . . . . . . . . 48

vii

“NumberTheory_bev” — 2007/10/15 — 12:46 — page viii — #8�

�

�

�

�

�

�

�

viii Number Theory Through Inquiry

Systems of linear congruences: the ChineseRemainder Theorem . . . . . . . . . . . . . . . . . . . 50

A Prince and a Master . . . . . . . . . . . . . . . . . . . . . . . 51

4 Fermat’s Little Theorem and Euler’s Theorem 53Abstracting the Ordinary . . . . . . . . . . . . . . . . . . . . . . 53Orders of an integer modulo n . . . . . . . . . . . . . . . . . 54Fermat’s Little Theorem . . . . . . . . . . . . . . . . . . . . . 55An alternative route to Fermat’s Little Theorem . . . . . . . . 58Euler’s Theorem and Wilson’s Theorem . . . . . . . . . . . . 59

Fermat, Wilson and . . . Leibniz? . . . . . . . . . . . . . . . . . . 62

5 Public Key Cryptography 65Public Key Codes and RSA . . . . . . . . . . . . . . . . . . . . 65Public key codes . . . . . . . . . . . . . . . . . . . . . . . . . 65Overview of RSA . . . . . . . . . . . . . . . . . . . . . . . . 65Let’s decrypt . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6 Polynomial Congruences and Primitive Roots 73Higher Order Congruences . . . . . . . . . . . . . . . . . . . . . 73Lagrange’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 73Primitive roots . . . . . . . . . . . . . . . . . . . . . . . . . . 74Euler’s �-function and sums of divisors . . . . . . . . . . . . 77Euler’s �-function is multiplicative . . . . . . . . . . . . . . . 79Roots modulo a number . . . . . . . . . . . . . . . . . . . . . 81

Sophie Germain is Germane, Part I . . . . . . . . . . . . . . . . 84

7 The Golden Rule: Quadratic Reciprocity 87Quadratic Congruences . . . . . . . . . . . . . . . . . . . . . . . 87Quadratic residues . . . . . . . . . . . . . . . . . . . . . . . . 87Gauss’ Lemma and quadratic reciprocity . . . . . . . . . . . . 91Sophie Germain is germane, Part II . . . . . . . . . . . . . . . 95

8 Pythagorean Triples, Sums of Squares,and Fermat’s Last Theorem 99Congruences to Equations . . . . . . . . . . . . . . . . . . . . . 99Pythagorean triples . . . . . . . . . . . . . . . . . . . . . . . 99Sums of squares . . . . . . . . . . . . . . . . . . . . . . . . . 102Pythagorean triples revisited . . . . . . . . . . . . . . . . . . . 104Fermat’s Last Theorem . . . . . . . . . . . . . . . . . . . . . 104

Who’s Represented? . . . . . . . . . . . . . . . . . . . . . . . . 106Sums of squares . . . . . . . . . . . . . . . . . . . . . . . . . 106

“NumberTheory_bev” — 2007/10/15 — 12:46 — page ix — #9�

�

�

�

�

�

�

�

Contents ix

Sums of cubes, taxicabs, and Fermat’s Last Theorem . . . . . 107

9 Rationals Close to Irrationals and the Pell Equation 109Diophantine Approximation and Pell Equations . . . . . . . . . 109A plunge into rational approximation . . . . . . . . . . . . . . 110Out with the trivial . . . . . . . . . . . . . . . . . . . . . . . 114New solutions from old . . . . . . . . . . . . . . . . . . . . . 115Securing the elusive solution . . . . . . . . . . . . . . . . . . 116The structure of the solutions to the Pell equations . . . . . . 117

Bovine Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

10 The Search for Primes 123Primality Testing . . . . . . . . . . . . . . . . . . . . . . . . . . 123Is it prime? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123Fermat’s Little Theorem and probable primes . . . . . . . . . 124AKS primality . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Record Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

A Mathematical Induction: The Domino Effect 129The Infinitude Of Facts . . . . . . . . . . . . . . . . . . . . . . 129Gauss’ formula . . . . . . . . . . . . . . . . . . . . . . . . . . 129Another formula . . . . . . . . . . . . . . . . . . . . . . . . . 131On your own . . . . . . . . . . . . . . . . . . . . . . . . . . . 132Strong induction . . . . . . . . . . . . . . . . . . . . . . . . . 133On your own . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Index 135

About the Authors 139

“NumberTheory_bev” — 2007/10/15 — 12:46 — page x — #10�

�

�

�

�

�

�

�

“NumberTheory_bev” — 2007/10/15 — 12:46 — page 135 — #145�

�

�

�

�

�

�

�

Index

abstract algebra, 53, 118Adleman, Leonard, 65Agrawal, Manindra, 126AKS primality test, 126al-Haytham, Abu, 62Archimedes, 109, 119, 122arithmetical geometry, 108Artin’s Conjecture, 85Artin, Emil, 85Aryabhata, 24asymmetrical key, 69

Bachet, Claude, 24Bessy, Frenicle de, 62Bhaskara, 122Binomial Theorem, 58, 62Brahmagupta, 24, 50, 52, 122Brouncker, William, 110, 122

Chinese Remainder Theorem, 51–52common divisor, 16

greatest, 17complete residue system modulo n, 47,

48, 56, 80, 84, 88, 91canonical, 47, 54

composite number, 29, 57, 125, 126congruent modulo n, 8, 43

Descartes, Rene, 41, 106Diffie, Whitfield, 69Diophantine approximation, 109

Diophantine equation, 20, 86, 99, 102,109

Diophantus of Alexandria, 24Dirichlet’s Rational Approximation

Theorem, 112, 113, 116Dirichlet, Lejeune, 37, 106discrete logarithm modulo p, 70Disquisitiones Arithmeticae, 52divide, 8divisibility tests, 14, 45–46Division Algorithm, 15

elliptic curve, 70, 108equivalence class, 47equivalence relation, 12Eratosthenes, 30

Sieve of, 30Euclid, 41Euclidean Algorithm, 18, 49, 66Euler �-function, 59, 60, 76, 77, 79–81Euler’s Criterion, 90, 91, 95Euler’s Theorem, 60, 62, 66, 82, 91Euler, Leonhard, 21, 24, 40, 42, 52, 62,

63, 89, 106, 107, 110, 127exponential time algorithm, 124

Fermat number, 127Fermat prime, 38Fermat’s Last Theorem, 86, 99, 105,

107, 108exponent 4, 105

135

“NumberTheory_bev” — 2007/10/15 — 12:46 — page 136 — #146�

�

�

�

�

�

�

�

136 Index

Fermat’s Little Theorem, 55–60, 62, 63,66, 74, 79, 91, 125

Fermat, Pierre de, 41, 52, 62, 104, 106,122

Four Squares Theorem, 107Fundamental Theorem of Algebra, 73,

74Fundamental Theorem of Arithmetic, 30–

35applications, 32–35statement, 31

Gauss’ Lemma, 92Gauss, Carl Friedrich, 39, 51, 91, 129Germain, Sophie, 85, 86Girard, Albert, 106Goldbach Conjecture, 40Goldbach, Christian, 40, 106Great Internet Mersenne Prime Search,

42greatest common divisor, 17Green, Ben, 37

Hardy, G. H., 107Hellman, Martin, 69

integer, 8inverse modulo p, 61irrational number, 33, 109, 110, 118irreducible polynomial, 134Ivory, James, 62

Kayal, Neeraj, 126key exchange, 69, 84Koblitz, Neal, 70

Lagrange’s Theorem, 74Lagrange, Joseph, 25, 52, 63, 73, 85,

107Law of Quadratic Reciprocity, 94, 99least common multiple, 23Legendre symbol, 89Legendre, Adrien-Marie, 39, 52, 89, 106Lehmer, D. H., 128Leibniz, Gottfried Wilhelm, 63linear congruence, 48–50, 53

systems of, 50linear Diophantine equation, 16–22, 48,

49, 67, 83, 109, 114Lucas, Edouard, 127Lucas-Lehmer Test, 128

mathematical induction, 28, 130, 131strong, 133, 134

Mersenne prime, 38, 42, 127, 128Mersenne, Marin, 41, 106method of descent, 105method of successive squaring, 44–45Miller’s Theorem, 86, 96Miller, G. A., 86Miller, Victor, 70multiplicative function, 80, 81

natural number, 8

order of a modulo n, 55–57, 74, 75, 86

Pascal, Blaise, 106Pell equation, 109, 110, 113–115, 117–

119, 121, 122non-trivial solutions, 114–117nontrivial solutions, 119trivial solutions, 114, 119

Pell, John, 110Pepin’s Test, 128Pepin, Theophile, 127perfect number, 41polynomial time algorithm, 124–126polynomials modulo n, 45, 73, 74Poulet number, 126primality test, 123–127prime number, 29, 47, 56, 65–67, 70,

76, 77, 79, 123, 1274k C 1 primes, 904k C 3 primes, 36, 90Fermat prime, 38in arithmetic progressions, 36–37infinitude of, 36, 41, 90Mersenne prime, 38, 127, 128Sophie Germain prime, 86, 95, 96

Prime Number Theorem, 40primitive Pythagorean triple, 100, 103

“NumberTheory_bev” — 2007/10/15 — 12:46 — page 137 — #147�

�

�

�

�

�

�

�

Index 137

primitive root, 70, 75, 76, 79, 81, 84–86, 88, 95

probable prime test, 125, 126Proth’s Test, 128Proth, Francois, 128public key codes, 65, 66, 70Pythagorean Theorem, 99Pythagorean triple, 100, 104, 106, 109

infinitude of, 101primitive, 100, 103

Pythagorean Triple Theorem, 101

quadratic non-residue, 88Quadratic Reciprocity Theorem, 93quadratic residue, 88, 103

Ramanujan, Srinivasa, 107rational number, 33, 109reducible polynomial, 134relatively prime, 17, 57, 60, 74, 80Rivest, Ronald, 65root, 73roots modulo n, 81–83RSA encryption, 65–70, 123

Saxena, Nitin, 126Shamir, Adi, 65Shimura-Taniyama Conjecture, 108Sieve of Eratosthenes, 30, 39Sophie Germain prime, 86, 95, 96strong mathematical induction, 133, 134sums of squares, 102, 106

representing numbers, 104representing primes, 102

symmetrical key, 69, 70system of linear congruences, 50–51

Tao, Terence, 37Taylor, Richard, 108triangular number, 121Twin Prime Question, 39

Wallis, John, 122Waring, Edward, 63Well-Ordering Axiom, 14, 119Wiles, Andrew, 105, 108Wilson’s Theorem, 61–63, 124

converse of, 62Wilson, John, 63

“NumberTheory_bev” — 2007/10/15 — 12:46 — page 138 — #148�

�

�

�

�

�

�

�

“NumberTheory_bev” — 2007/10/15 — 12:46 — page 139 — #149�

�

�

�

�

�

�

�

About the Authors

David Marshall was born in Anaheim, California and spent most of hisearly life in and around Orange County. After receiving a bachelor’s de-gree in mathematics from California State University at Fullerton he leftthe Golden State to attend graduate school at the University of Arizona.David received his Ph.D. in mathematics in 2000, specializing in the fieldof algebraic number theory. He held postdoctoral positions at McMasterUniversity in Hamilton, Ontario and The University of Texas at Austinbefore becoming an Assistant Professor at Monmouth University in WestLong Branch, New Jersey. David has been an active member of the MAAand AMS for over 10 years and currently serves as the Program Editor forthe MAA’s New Jersey Section.

Edward Odell was born in White Plains, New York. He attended the StateUniversity of New York at Binghamton as an undergraduate and receivedhis Ph.D. from MIT in 1975. After teaching 2 years at Yale Universityhe joined the faculty of The University of Texas at Austin where he hasbeen since 1977, currently as the John T. Stuart III Centennial Professor ofmathematics. He is an internationally recognized researcher in his area, thegeometry of Banach spaces and is a much sought after speaker. Odell wasan invited speaker at the 1994 International Congress of Mathematiciansin Zurich. He has given series of lectures at various venues in Spain andrecently at the Chern Institute in Tianjin, China. He is the co-author ofAnalysis and Logic and the co-editor of two books in the Springer Lectureseries.

Michael Starbird received his B.A. degree from Pomona College and hisPh.D. from the University of Wisconsin, Madison. He is a Distinguished

139

“NumberTheory_bev” — 2007/10/15 — 12:46 — page 140 — #150�

�

�

�

�

�

�

�

140 Number Theory Through Inquiry

Teaching Professor of mathematics at The University of Texas at Austinand is a member of UT’s Academy of Distinguished Teachers. He haswon many teaching awards, including the 2007 Mathematical Associationof America Deborah and Franklin Tepper Haimo National Award for Dis-tinguished College or University Teaching of Mathematics. He has writtentwo books with co-author Edward B. Burger: The Heart of Mathematics:An invitation to effective thinking and Coincidences, Chaos, and All ThatMath Jazz: Making Light of Weighty Ideas. Starbird has produced four videocourses for The Teaching Company: Change and Motion: Calculus MadeClear; Meaning from Data: Statistics Made Clear; What are the Chances?Probability Made Clear; and, with collaborator Edward B. Burger, The Joyof Thinking: The Beauty and Power of Classical Mathematical Ideas.

Number Theory Through Inquiry is an innovative textbook that leads students

on a carefully guided discovery of introductory number theory. The book has

two equally significant goals. One goal is to help students develop mathemat-

ical thinking skills, particularly, theorem-proving skills. The other goal is to help

students understand some of the wonderfully rich ideas in the mathematical

study of numbers. This book is appropriate for a proof transitions course, for

an independent study experience, or for a course designed as an introduction

to abstract mathematics. Math or related majors, future teachers, and students

or adults interested in exploring mathematical ideas on their own will enjoy

Number Theory Through Inquiry.

Number theory is the perfect topic for an introduction-to-proofs course. Every

college student is familiar with basic properties of numbers, and yet the explo-

ration of those familiar numbers leads us to a rich landscape of ideas. Number Theory Through Inquiry contains a carefully arranged sequence of challenges that

lead students to discover ideas about numbers and to discover methods of proof

on their own. It is designed to be used with an instructional technique variously

called guided discovery or Modified Moore Method or Inquiry Based Learning

(IBL). Instructors’ materials explain the instructional method. This style of

instruction gives students a totally different experience compared to a standard

lecture course. Here is the effect of this experience: Students learn to think inde-

pendently: they learn to depend on their own reasoning to determine right from

wrong; and they develop the central, important ideas of introductory number

theory on their own. From that experience, they learn that they can personally

create important ideas, and they develop an attitude of personal reliance and a

sense that they can think effectively about difficult problems. These goals are

fundamental to the educational enterprise within and beyond mathematics.

AMS / MAA TEXTBOOKS