Number theory through inquiry PDF
Preview Number theory through inquiry
(cid:2) (cid:2) “NumberTheory_bev” — 2007/10/15 — 12:46 — page i — #1 (cid:2) (cid:2) Number Theory Through Inquiry (cid:2) (cid:2) (cid:2) (cid:2) i i \NumberTheory_bev" | 2011/2/16 | 16:14 | page ii | #2 i i About the cover: The cover design suggests themeaning and proof of the ChineseRemainderTheoremfromChapter3.Picturedaresolidwheelswith 5,7,and11teethrollinginsideofgroovedwheels.Asthesmallwheelsroll around a large wheelwith5(cid:2)7(cid:2)11D385grooves, onlypart of whichis drawn, the highlightedteeth from each small wheel would all arrive at the same groove inthebigwheel.The intermediate35groovedwheelsuggests an inductiveproof of thistheorem. Cover image by Henry Segerman Cover design by Freedom by Design, Inc. (cid:13)c 2007by TheMathematicalAssociationof America(Incorporated) Libraryof CongressCatalogCardNumber2007938223 Print ISBN978-0-88385-751-9 Electronic edition ISBN978-0-88385-983-4 Printedin theUnitedStatesofAmerica Current Printing (last digit): 1098 7654321 i i i i (cid:2) (cid:2) “NumberTheory_bev” — 2007/10/15 — 12:46 — page iii — #3 (cid:2) (cid:2) Number Theory Through Inquiry David C. Marshall Monmouth University Edward Odell The University of Texas at Austin Michael Starbird The University of Texas at Austin ® Published and Distributedby The Mathematical Association of America (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) “NumberTheory_bev” — 2007/10/15 — 12:46 — page iv — #4 (cid:2) (cid:2) Council on Publications James Daniel, Chair MAA Textbooks EditorialBoard Zaven A. Karian, Editor William C. Bauldry Gerald M Bryce George Exner Charles R. Hadlock Douglas B. Meade Wayne Roberts StanleyE. Seltzer Shahriar Shahriari Kay B. Somers Susan G. Staples HollyS. Zullo (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) “NumberTheory_bev” — 2007/10/15 — 12:46 — page v — #5 (cid:2) (cid:2) MAA TEXTBOOKS Combinatorics:AProblemOrientedApproach,DanielA. Marcus ComplexNumbersandGeometry,Liang-shinHahn ACourseinMathematicalModeling,DouglasMooneyandRandallSwift CreativeMathematics,H. S. Wall CryptologicalMathematics,Robert EdwardLewand Differential Geometryandits Applications,JohnOprea ElementaryCryptanalysis,AbrahamSinkov ElementaryMathematicalModels,DanKalman EssentialsofMathematics,Margie Hale FourierSeries,RajendraBhatia GameTheoryandStrategy,PhilipD. Straffin GeometryRevisited,H. S. M. Coxeter andS.L. Greitzer KnotTheory,CharlesLivingston MathematicalConnections:ACompanionforTeachersandOthers,Al Cuoco MathematicalModelingintheEnvironment,Charles Hadlock MathematicsforBusinessDecisionsPart1:ProbabilityandSimulation(electronic textbook), RichardB. ThompsonandChristopher G.Lamoureux MathematicsforBusinessDecisionsPart2:CalculusandOptimization(electronic textbook), RichardB. ThompsonandChristopher G.Lamoureux TheMathematicsofGamesandGambling,EdwardPackel MathThroughtheAges,William Berlinghoff andFernandoGouvea NoncommutativeRings,I.N. Herstein Non-EuclideanGeometry,H.S. M. Coxeter NumberTheoryThroughInquiry,David C. Marshall, Edward Odell, and Michael Starbird APrimerofRealFunctions,RalphP.Boas ARadicalApproachto RealAnalysis,2ndedition, DavidM. Bressoud RealInfinite Series,Daniel D. BonarandMichael Khoury,Jr. TopologyNow!, Robert MesserandPhilipStraffin UnderstandingourQuantitativeWorld,JanetAndersenandTodd Swanson MAA ServiceCenter P.O.Box91112 Washington,DC 20090-1112 1-800-331-1MAA FAX: 1-301-206-9789 (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) “NumberTheory_bev” — 2007/10/15 — 12:46 — page vi — #6 (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) “NumberTheory_bev” — 2007/10/15 — 12:46 — page vii — #7 (cid:2) (cid:2) Contents 0 Introduction 1 NumberTheoryand Mathematical Thinking . . . . . . . . . . . 1 Note on the approach and organization . . . . . . . . . . . . . 2 Methods of thought . . . . . . . . . . . . . . . . . . . . . . . 3 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . 4 1 Divide and Conquer 7 Divisibilityin the Natural Numbers . . . . . . . . . . . . . . . . 7 Definitionsand examples . . . . . . . . . . . . . . . . . . . . 7 Divisibilityand congruence . . . . . . . . . . . . . . . . . . . 9 The DivisionAlgorithm . . . . . . . . . . . . . . . . . . . . . 14 Greatest common divisors and linear Diophantineequations . 16 Linear Equations Through the Ages . . . . . . . . . . . . . . . . 23 2 Prime Time 27 The Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . 27 Fundamental Theorem of Arithmetic . . . . . . . . . . . . . . 28 Applications of the Fundamental Theorem of Arithmetic . . . 32 The infinitudeof primes. . . . . . . . . . . . . . . . . . . . . 35 Primes of special form. . . . . . . . . . . . . . . . . . . . . . 37 The distributionof primes . . . . . . . . . . . . . . . . . . . . 38 From Antiquityto the Internet . . . . . . . . . . . . . . . . . . . 41 3 A Modular World 43 ThinkingCyclically . . . . . . . . . . . . . . . . . . . . . . . . 43 Powers and polynomialsmodulon . . . . . . . . . . . . . . . 43 Linear congruences . . . . . . . . . . . . . . . . . . . . . . . 48 vii (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) “NumberTheory_bev” — 2007/10/15 — 12:46 — page viii — #8 (cid:2) (cid:2) viii Number TheoryThroughInquiry Systems of linear congruences: the Chinese RemainderTheorem. . . . . . . . . . . . . . . . . . . 50 A Prince and a Master . . . . . . . . . . . . . . . . . . . . . . . 51 4 Fermat’s Little Theorem and Euler’s Theorem 53 Abstractingthe Ordinary . . . . . . . . . . . . . . . . . . . . . . 53 Orders of an integer modulon . . . . . . . . . . . . . . . . . 54 Fermat’s Little Theorem . . . . . . . . . . . . . . . . . . . . . 55 An alternativeroute to Fermat’s LittleTheorem . . . . . . . . 58 Euler’s Theorem and Wilson’s Theorem . . . . . . . . . . . . 59 Fermat, Wilson and ...Leibniz? . . . . . . . . . . . . . . . . . . 62 5 Public Key Cryptography 65 Public KeyCodes and RSA . . . . . . . . . . . . . . . . . . . . 65 Public keycodes . . . . . . . . . . . . . . . . . . . . . . . . . 65 Overview of RSA . . . . . . . . . . . . . . . . . . . . . . . . 65 Let’s decrypt . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 6 PolynomialCongruences and Primitive Roots 73 Higher Order Congruences . . . . . . . . . . . . . . . . . . . . . 73 Lagrange’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 73 Primitiveroots . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Euler’s (cid:2)-function and sums of divisors . . . . . . . . . . . . 77 Euler’s (cid:2)-function is multiplicative . . . . . . . . . . . . . . . 79 Roots moduloa number . . . . . . . . . . . . . . . . . . . . . 81 Sophie Germain is Germane, Part I . . . . . . . . . . . . . . . . 84 7 The Golden Rule: Quadratic Reciprocity 87 Quadratic Congruences. . . . . . . . . . . . . . . . . . . . . . . 87 Quadratic residues . . . . . . . . . . . . . . . . . . . . . . . . 87 Gauss’ Lemma and quadratic reciprocity . . . . . . . . . . . . 91 Sophie Germain is germane, Part II. . . . . . . . . . . . . . . 95 8 Pythagorean Triples, Sums of Squares, and Fermat’s Last Theorem 99 Congruences toEquations . . . . . . . . . . . . . . . . . . . . . 99 Pythagorean triples . . . . . . . . . . . . . . . . . . . . . . . 99 Sums of squares . . . . . . . . . . . . . . . . . . . . . . . . . 102 Pythagorean triplesrevisited. . . . . . . . . . . . . . . . . . . 104 Fermat’s Last Theorem . . . . . . . . . . . . . . . . . . . . . 104 Who’s Represented? . . . . . . . . . . . . . . . . . . . . . . . . 106 Sums of squares . . . . . . . . . . . . . . . . . . . . . . . . . 106 (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) “NumberTheory_bev” — 2007/10/15 — 12:46 — page ix — #9 (cid:2) (cid:2) Contents ix Sums of cubes, taxicabs, and Fermat’s Last Theorem . . . . . 107 9 RationalsClose to Irrationals and the Pell Equation 109 DiophantineApproximationand PellEquations . . . . . . . . . 109 A plunge intorational approximation . . . . . . . . . . . . . . 110 Out with the trivial . . . . . . . . . . . . . . . . . . . . . . . 114 New solutionsfrom old . . . . . . . . . . . . . . . . . . . . . 115 Securingthe elusive solution . . . . . . . . . . . . . . . . . . 116 The structureof the solutions tothe Pell equations . . . . . . 117 Bovine Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 10 The Search for Primes 123 PrimalityTesting . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Is it prime? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Fermat’s Little Theorem and probable primes . . . . . . . . . 124 AKS primality . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Record Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 A MathematicalInduction: The DominoEffect 129 The InfinitudeOf Facts . . . . . . . . . . . . . . . . . . . . . . 129 Gauss’ formula. . . . . . . . . . . . . . . . . . . . . . . . . . 129 Another formula . . . . . . . . . . . . . . . . . . . . . . . . . 131 On your own . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Strong induction . . . . . . . . . . . . . . . . . . . . . . . . . 133 On your own . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Index 135 About the Authors 139 (cid:2) (cid:2) (cid:2) (cid:2)
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