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Numbthy-2015-01-07.Pdf Discovering the Art of Mathematics Number Theory by Julian Fleron with Philip K. Hotchkiss, Volker Ecke and Christine von Renesse c 2009{2015 (Rev.: 2014-12-31) Working Draft: Can be copied and distributed for educational purposes only. Educational use requires notification of the authors. Not for any other quotation or distribution without written consent of the authors. For more information, please see http://www.artofmathematics.org/. DRAFT c 2015 Julian Fleron, Philip Hotchkiss, Volker Ecke, Christine von Renesse ii Acknowledgements This work has been supported by the National Science Foundation under awards NSF0836943 and NSF1229515. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. These materials are also based on work supported by Project PRIME which was made possible by a generous gift from Mr. Harry Lucas. These materials were originally developed for use in the course Mathematical Explorations at Westfield State College. They are now part of the curriculum library for Discovering the Art of Mathematics, all of whose volumes are freely available at http://artofmathematics.org . Many fine students and faculty at Westfield State College have provided insight, feedback, and inspiration for this work. In particular, I would like to thank the great students in MA110 - Mathematical Explorations and Students in MA116 - Introduction to Mathematical Systems who played an active role in the genesis of this material and patiently worked through the original drafts as well. As an undergraduate, now Ph.D. and an leading expert in partition theory, Brandt Kronholm sparked my interest in partitions. He is one of those great students who you learn from more than you teach. Senior mathematics majors Gina Kennedy and Sarah Lewison carefully helped me revise and edit the first major draft of this work. The Department of Mathematics at Westfield State has provided fertile ground and enthusias- tic support for pedagogical innovation, scholarship on teaching and learning in mathematics, and classroom experimentation. Under the direction of Chairs Karin Vorwerk and previously John Judge, the department continues to be deeply committed to student learning - including in its many required core courses. We are thankful to all departmental faculty, current and past, for their support of Discovering the Art of Mathematics. Colleagues that provided particular guidance, support and feedback of this volume include Marcus Jaiclin, Robert McGuigan, James Robertson, and Larry Griffith. We are grateful for the support of our administration for many years, includ- ing Dean Robert Martin, Vice President William Lopes, Sponsored Programs Director Louann D'Angelo, President Elizabeth Preston, and Dean Andrew Bonacci. iii DRAFT c 2015 Julian Fleron, Philip Hotchkiss, Volker Ecke, Christine von Renesse In different incarnations, this work was reviewed and/or beta tested by George Andrews, Un- derwood Dudley, David Farmer, Charles Rocca, and Ethan Berkove. The author is grateful for their many insights and detailed work. Any and all imperfections remain with the author. Of course, many other cherished people deserve thanks for helping me to develop into the mathematics teacher that I am. Many fine mathematics teachers and mathematicians enabled this work by helping inspire my own mathematical exploration. My parents Frederic and Lou Jean, step-parents Kim and Jack, my sister Ingeri, my niece K.C., and many other fine teachers helped nurture not only my intellectual curiosity, but my passion for sharing this curiosity through teaching. Lastly, I would like to thank my partner Kris Hedblom and my kids, Addie and Jacob, great teachers who constantly remind me of the potential of all learners and the playful forms of explo- ration in which the best learning naturally takes. c 2010{2015 by Julian Fleron, Phil Hotchkiss, Volker Ecke, Christine von Renesse. iv Contents Acknowledgements iii Preface: Notes to the Explorer 1 Navigating This Book 5 Introduction: Number Theory 7 0.1 $1 Million Dollar Problems . 7 0.2 High Drama . 8 0.3 What Good is This? . 9 0.4 Recent Breakthroughs . 10 0.5 Yeah, But Can We Do It? . 10 1 Fibonacci Numbers 13 1.1 A Remarkable Sequence of Numbers . 13 1.2 Fibonacci . 14 1.3 Fibonacci's Problem . 15 1.4 Fibonacci's Rabbits . 16 1.5 Fibonacci Spirals in Nature . 17 1.6 Honeybee Family Trees . 18 1.7 Plant Growth . 19 1.8 Two Fibonacci Identities . 20 1.9 The Mandelbrot Set . 21 1.10 Fibonacci Numbers Everywhere? . 23 1.11 Phyllotaxis . 24 1.12 Fibonacci Spirals from Optimal Packing . 27 2 The Golden Ratio 31 2.1 The Golden Ratio . 31 2.2 Division into Extreme and Mean Ratio . 32 2.3 The Golden Ratio Algebraically . 34 2.4 Nested Radicals . 35 2.5 Continued Fractions . 36 2.6 Powers of φ......................................... 38 v DRAFT c 2015 Julian Fleron, Philip Hotchkiss, Volker Ecke, Christine von Renesse 2.7 Golden Rectangles . 39 2.8 Star Pentagrams . 40 2.9 Magical Rectangles . 41 2.10 Perspectives on the Golden Ratio . 43 3 Primes and Congruences 45 3.1 Primes . 45 3.2 Twin Primes and Other Arithmetic Progressions of Primes . 46 3.3 Fermat Primes . 50 3.4 Mersenne Primes . 51 3.4.1 Modern Analysis of Mersenne Primes before Computers . 52 3.4.2 Modern Analysis of Mersenne Primes with Computers . 53 3.5 The Twins . 54 3.6 Congruences: aka Clock Arithmetic . 55 3.7 Application of Congruences . 55 3.8 Powers and Congruences . 56 3.9 The Chinese Remainder Theorem - Mathematical Magic . 58 3.10 Secret Codes, Ciphers and Cryptography . 60 3.11 Connections . 63 3.11.1 History: Alan Turing and World War II Codebreaking . 63 3.11.2 History: Polish Mathematicians and World War II Codebreaking . 63 3.11.3 History: Navajo Code Talkers and World War II Encryption . 64 3.11.4 Secret Codes and Statistics . 65 3.12 Further Investigations . 66 3.12.1 Dirichlet and Primes in Arithmetic Progressions . 66 3.12.2 The Euler-Fermat Theorem . 66 3.12.3 Example of RSA Implementation . 68 4 Class Numbers: A Bridge Between Two $1 Million Dollar Problems 69 4.1 Guiding Problems . 69 4.2 Distribution of the Primes . 70 4.3 Class Numbers . 73 4.4 A Class Number Sieve . 74 4.5 Gauss' Class Number Problem . 76 4.6 The Riemann Hypothesis . 79 4.7 Another $ 1Million Problem . 80 5 Partitions 83 5.1 The Births of Modern Number Theory . 83 5.2 The Development of Mathematics Illustrated by Number Theory . 84 5.3 Connections Between Fermat and Euler . 84 5.3.1 Fermat Primes . 84 5.3.2 The (Mathematical) Key to Modern Encryption . 85 5.3.3 Primes and Sums of Two Squares . 85 5.3.4 Sums of Squares . 86 5.3.5 Fermat's Last Theorem . 86 vi DRAFT c 2015 Julian Fleron, Philip Hotchkiss, Volker Ecke, Christine von Renesse 5.4 Partitions . 87 5.4.1 Enumerating Partitions . 87 5.4.2 Counting Strategies . 88 5.4.3 Patterns in the Partition Function . 89 5.4.4 Amazing New Discoveries . 90 6 Power Partitions 93 6.1 Another Story about Gauss . 93 6.2 Mathematics Manipulatives . 93 6.3 Proofs Without Words . 96 6.4 Interesting Numbers . 99 6.5 Square Partitions . 99 6.6 Cubical Partitions . 100 6.7 Minimal Square Partitions . 100 6.8 Waring's Problem . 102 6.9 Solving Waring's Problem . 103 6.10 Further Investigations . 104 7 The World's Greatest Mathematical Problem 105 7.1 The Pythagorean Theorem . 105 7.2 Hundreds of Proofs . 106 7.3 Pythagorean Triples . 107 7.3.1 Pythagorean Triples as Partitions . 107 7.3.2 Finding Pythagorean Triples . ..
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