Number theory through inquiry

151 Pages·2007·0.724 MB·English

by  Marshall, David C., Odell, Edward, Starbird, Michael P

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Number Theory
Through Inquiry
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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
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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
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Council on Publications
James Daniel, Chair
MAA Textbooks EditorialBoard
Zaven A. Karian, Editor
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Wayne Roberts
StanleyE. Seltzer
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HollyS. Zullo
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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
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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
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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
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