Math Girls2 Fermat’s Last Theorem a +b = c Hiroshi Yuki Translated by Tony Gonzalez http://bentobooks.com MATH GIRLS 2: FERMAT’S LAST THEOREM by Hiroshi Yuki. Copyright © 2008 Hiroshi Yuki. All rights reserved. Originally published in Japan by Softbank Creative Corp., Tokyo. English translation © 2012 Bento Books, Inc. Translated by Tony Gonzalez with Joseph Reeder. Additional editing by Alexander O. Smith and M. D. Hendon. No portion of this book in excess of fair use considerations may be reproduced or transmitted in any form or by any means without written permission from the copyright holders. Published by Bento Books, Inc. 217 Tierra Grande Court Austin, TX 78732-2458 USA bentobooks.com ISBN 978-0-9839513-3-9 (hardcover) ISBN 978-0-9839513-2-2 (trade paperback) Library of Congress Control Number: 2012918353 Printed in the U.S.A. First edition, December 2012 Math Girls2: Fermat’s Last Theorem Contents 1 Infinity in Your Hand5 1.1 The Milky Way5 1.2 Discoveries6 1.3 Odd One Out7 1.4 Clock Math9 1.5 Conditions for a Complete Cycle 15 1.6 Going Cycling 16 1.7 Beyond Human Limits 20 1.8 What Things Really Are 22 2 Pythagorean Triples 25 2.1 Rooftop Lunch 25 2.2 Rational Points 28 2.3 Yuri 30 2.4 The Pythagorean Juicer 31 2.5 Primitive Pythagorean Triples Revisited 32 2.5.1 Checking Parity 33 2.5.2 Trying Equations 34 2.5.3 As a Product 36 2.5.4 Relatively Prime 37 2.5.5 Prime Factorization 40 2.6 Finding the Trail 45 2.7 Squeaky 47 2.8 Rational Points on the Unit Circle 48 3 Relatively Prime 53 3.1 Hidden Depths 53 3.2 Fractions 54 3.3 GCDs and LCMs 56 3.4 Making Sure 60 3.5 The Pop-In 61 3.6 Sequences of Prime Exponents 62 3.6.1 An Example 62 3.6.2 Taking It Up a Notch 64 3.6.3 Multiplication 65 3.6.4 Greatest Common Divisors 66 3.6.5 On to Infinite Dimensions 67 3.7 In Awe of Miruka 68 4 Proof by Contradiction 69 4.1 Writing Proofs 69 4.1.1 The Definition 69 4.1.2 The Proposition 72 4.1.3 The Equation 73 4.1.4 Proofs are Eternal 79 4.2 Another Approach 81 4.2.1 Parity 81 4.2.2 Contradictions 84 5 Broken Primes 87 5.1 Imagining Numbers 87 5.1.1 A Quiz for Tetra 87 5.1.2 Defining Numbers 89 5.1.3 On to Quadratics 90 5.2 Sums and Products of Complex Numbers 92 5.2.1 Adding Complex Numbers 92 5.2.2 Multiplying Complex Numbers 93 5.2.3 ±i in the Complex Plane 97 5.3 Five Lattice Points 100 5.3.1 A New Card 100 5.3.2 Pigeonholing 102 5.4 Breakable Primes 106 6 Abelian Groups 119 6.1 The Accident 119 6.2 Day One 121 6.2.1 Introducing Operations into Sets 121 6.2.2 Operations 122 6.2.3 Associativity 124 6.2.4 The Identity Element 125 6.2.5 The Inverse Element 126 6.2.6 Defining Groups 127 6.2.7 Example Groups 127 6.2.8 The Smallest Group of All 130 6.2.9 A Group with Two Elements 131 6.2.10 Isomorphism 133 6.3 Day Two 134 6.3.1 The Commutative Property 134 6.3.2 Regular Polygons 136 6.3.3 Interpreting Mathematical Statements 139 6.3.4 Braided Axioms 140 6.4 True Forms 140 6.4.1 Essence and Abstraction 140 6.4.2 The Essence of a Heart 142 7 Modular Arithmetic 145 7.1 Clocks 145 7.1.1 Defining Remainders 145 7.1.2 What Clocks Really Tell Us 148 7.2 Congruence 149 7.2.1 Remainders of the Day 149 7.2.2 Congruence Relations 151 7.2.3 Different Congruences? 154 7.2.4 More or Less Equivalent 154 7.2.5 Equalities and Congruences 155 7.2.6 The Exception 156 7.2.7 Crutches 159 7.3 The Nature of Division 160 7.3.1 Math and Hot Chocolate 160 7.3.2 Studying the Operation Table 161 7.3.3 The Proof 166 7.4 Groups, Rings, and Fields 167 7.4.1 Reduced Residue Class Groups 167 7.4.2 From Groups to Rings 170 7.4.3 From Rings to Fields 174 7.5 Mod Hairstyles 179 8 Infinite Descent 181 8.1 FLT 181 8.2 Tetra’s Triangles 185 8.2.1 In the Zone 185 8.2.2 A Hint from Tetra 190 8.3 The Journey 191 8.3.1 Setting Out 191 8.3.2 Atoms and Elementary Particles 195 8.3.3 Into the Particles 196 8.3.4 The Shores of Quantum Foam 199 8.4 Yuri’s Awakening 201 8.4.1 Math, Interrupted 201 8.4.2 The Playground 202 8.4.3 Drinks with Pythagoras 203 8.5 Miruka’s Proof 209 8.5.1 Wax On, Wax Off 209 8.5.2 An Elementary Proof 210 8.5.3 Final Pieces 214 9 Euler’s Identity 217 9.1 The Most Beautiful Equation 217 9.1.1 Components of the Identity 217 9.1.2 Euler’s Formula 219 9.1.3 Exponent Rules 222 9.1.4 Negative and Rational Exponents 226 9.1.5 The Exponential Function 228 9.1.6 Staying True to the Equations 231 9.1.7 A Bridge to Trig 233 9.2 Party Preparations 238 9.2.1 Solve M for Mystery 238 9.2.2 Mom 239 10 Fermat’s Last Theorem 241 10.1 The Seminar 241 10.2 Some History 243 10.2.1 The Problem 243 10.2.2 The Era of Elementary Number Theory 244 10.2.3 The Era of Algebraic Number Theory 244 10.2.4 The Era of Arithmetic Geometry 245 10.3 How It Was Done 246 10.3.1 Back in Time 246 10.3.2 Getting to the Problem 248 10.3.3 Semistable Elliptic Curves 250 10.3.4 Whetting Our Appetite 251 10.4 The World of Elliptic Curves 252 10.4.1 Elliptic Curves 101 252 10.4.2 Rational Number Fields to Finite Fields 253 10.4.3 The Finite Field F2 255 10.4.4 The Finite Field F3 257 10.4.5 The Finite Field F5 259 10.4.6 How Many Points? 261 10.4.7 Prisms 261 10.5 Automorphic Forms 262 10.5.1 Staying True to Form 262 10.5.2 q-Expansions 264 10.5.3 From F(q) to a(k) 265 10.6 The Modularity Theorem 268 10.6.1 Two Worlds 268 10.6.2 Frey Curves and Semistability 270 10.7 Math Party 272 10.7.1 Everyone Arrives 272 10.7.2 A Zeta Variation 273 10.7.3 The Fruits of Solitude 275 10.7.4 Yuri’s Realization 276 10.7.5 No Coincidence 280 10.7.6 Silent Night 281 10.8 They’re Doing Math in Andromeda 281 Index 301 To my readers This book contains math problems covering a wide range of difficulty. Some will be approachable by middle school students, while others may prove challenging even at the college level. The characters often use words and diagrams to express their thoughts, but in some places equations tell the tale. If you find yourself faced with math you don’t understand, feel free to skip over it and continue on with the story. Tetra and Yuri will be there to keep you company. If you have some skill at mathematics, then please follow not only the story, but also the math. You might be surprised at what you discover. You may even learn something about the wonderful tale that you yourself are living. —Hiroshi Yuki Prologue God made integers. All else is the work of man. Leopold Kronecker We count in a world of integers. We count birds, stars, the number of days until the weekend. When we’re children, we count to see how long we can hold our breath. We draw in a world of figures. We use compasses to draw arcs, rulers to draw lines, and we are amazed at the constructions that result. We run through the schoolyard dragging an umbrella, and turn to see a long, winding line stretching toward the horizon. We live in a world of mathematics. God made the integers, Kro- necker said. But Pythagoras and Diophantus bound the integers in right triangles. And then came Fermat... Ah, Fermat, and his silly little note. A problem that anyone could understand, but no one could solve. History’s greatest puzzle—if it’s fair to call a problem that took mathematicians three centuries to solve a mere “puzzle.” But true forms are hidden. Things once lost are found again. That which has vanished reappears. Loss and rediscovery, death and rebirth. The joy of life and the burden of time. Consider the meaning of growth. Question the meaning of solitude. Know the meaning of words. 4 Fermat’s Last Theorem Memories are ghosts lost in the mist. I recall only fragments: the silver Milky Way, a warm hand, a trembling voice, chestnut hair. So that’s where I’ll begin, on a Saturday afternoon Chapter 1 Infinity in Your Hand Gauss’s path was the way of mathematics, a road paved by induction.