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Fermat's Enigma

Published by Anchor on 1998-09-08
Paperback: $16.95

xn + yn = zn, where n represents 3, 4, 5, …no solution

"I have discovered a truly marvelous demonstration of this proposition which this margin is too narrow to contain."

With these words, the seventeenth-century French mathematician Pierre de Fermat threw down the gauntlet to future generations.  What came to be known as Fermat’s Last Theorem looked simple; proving it, however, became the Holy Grail of mathematics, baffling its finest minds for more than 350 years.  In Fermat’s Enigma–based on the author’s award-winning documentary film, which aired on PBS’s "Nova"–Simon Singh tells the astonishingly entertaining story of the pursuit of that grail, and the lives that were devoted to, sacrificed for, and saved by it.  Here is a mesmerizing tale of heartbreak and mastery that will forever change your feelings about mathematics.

(Paperback (1st Anchor Books ed), 1998-09-08)
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ASIN: 0385493622
ISBN: 9780385493628
EAN: 9780385493628

A boast in the margin of a book is the starting point for a wonderful journey through the history of mathematics, number theory and logic

The note, scribbled in Latin, simply states: "I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain."

Tim Radford is the author of The Address Book: Our Place in the Scheme of Things (Fourth Estate)

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Simon Singh received his Ph.D. in physics from the University of Cambridge. A BBC producer, he directed and coproduced an award-winning documentary film on Fermat's Last Theorem that aired on PBS's "Nova" series. He lives in London, England, and is at work on his second book.

FERMAT'S ENIGMA The Epic Quest to Solve the World's Greatest Mathematical Problem P S I λl O N S I N G H Foreword byJohn Lynch ANCHOR BOOKS A DIVISION OF RANDOM HOUSE, INC. NEW YORK

In memory of Pakhar Singh FIRST ANCHOR BOOKS EDITION, OCTOBER I998 Copyright © 1997 by Simon Singh Foreword copyright © 1997 by John Lynch All rights reserved under International and Pan-American Copyright Conventions. Published in the United States by Anchor Books, a division of Random House, Inc., New York. Originally published in hardcover in the United States by Walker and Company. The Anchor Books edition is published by arrangement with Walker and Company. Anchor Books and colophon are registered trademarks of Random House, Inc. Library of Congress Cataloging-in-Publication Data Singh, Simon. Fermat's engima : the epic quest to solve the world's greatest mathematical problem / Simon Singh : foreword by John Lynch.—1st Anchor Books ed. p. cm. Originally published: New York : Walker, 1997. Includes bibliographical references and index. 1. Fermat's last theorem. I. Tide. [QA244.S55 1998] 512'.74—dc21 98-27262 CIP ISBN: 0-385-49362-2 Printed in the United States of America 16 18 20 19 17 15

Contents Foreword by John Lynch Preface 1. "I Think I'll Stop Here" 2. The Riddler 3. A Mathematical Disgrace 4. Into Abstraction 5. Proof by Contradiction 6. The Secret Calculation 7. A Slight Problem Epilogue: Grand Unified Mathematics Appendixes Suggestions for Further Reading Picture Credits Index vii XV 1 35 71 121 171 205 255 279 287 301 306 307

Foreword I finally met Andrew Wiles across a room, not crowded, but large enough to hold the entire mathematics department at Princeton. On that particular afternoon, there were not so very many people around, but enough for me to be uncertain as to which one was Wiles. After a few moments I introduced myself to the shy-looking Wiles, who had been listening to the conversation around him and sipping tea. It was the end of an extraordinary week. I had met some of the finest mathematicians alive, and begun to gain an insight into their world. But despite every attempt to pin down Andrew Wiles, to speak to him, and to convince him to take part in a BBC Horizon documentary film on his achievement, this was our first meeting. This was the man who had recently announced that he had found the holy grail of mathematics the man who claimed he had proved Fermat's Last Theorem. As we spoke, Wiles had a distracted and withdrawn air about him, and although he was polite and friendly, it was clear that he wished me as far away from him as possible. He explained very simply that he could not possibly focus on anything but his work, which was at a critical stage, but perhaps later, when the current pressures had been resolved, he would be pleased to take part. I knew, and he knew I knew, that he was facing the collapse of his life's ambition, and that the holy grail he had held was now being revealed as no more than a rather beautiful, valuable, but straightVll

viii FERMAT'S ENIGMA forward drinking vessel. He had found a flaw in his heralded proof. The story of Fermat's Last Theorem is unique. By the time I first met Andrew Wiles, I had come to realize that it is truly one of the greatest stories in the sphere of scientific or academic endeavor. I had seen the headlines in the summer of 1993, when the proof had put math on the front pages of national newspapers around the world. At that time I had only a vague recollection of what the Last Theorem was, but saw that it was obviously something very special, and something that had the smell of a Hoήzm film to it. I spent the next weeks talking to many mathematicians: those closely involved in the story, or close to Andrew, and those who simply shared the thrill of witnessing a great moment in their field. All generously shared their insights into mathematical history, and patiently talked me through what little understanding I could achieve of the concepts involved. Rapidly it became clear that this was subject matter that perhaps only half a dozen people in the world could fully grasp. For a while I wondered if I was insane to attempt to make a film. But from those mathematicians I also learned of the rich history, and the deeper significance of Fermat to mathematics and its practitioners, and that, I realized, was where the real story lay. I learned of the ancient Greek origins of the problem, and that Fermat's Last Theorem was the Himalayan peak of number theory. I was introduced to the aesthetic beauty of mathematics, and I began to appreciate what it is to describe mathematics as the language of nature. Through Wiles's contemporaries I grasped the Herculean nature of his work in pulling together all the most recent techniques of number theory to apply to his proof. From his friends at Princeton I heard of the intricate progress of Andrew's years of isolated study. I built up an extraordinary pic-

FOREWORD IX ture around Andrew Wiles, and the puzzle that dominated his life. Although the math involved in Wiles's proof is some of the toughest in the world, I found that the beauty of Fermat's Last Theorem lies in the fact that the problem itself is supremely simple to understand. It is a puzzle that is stated in terms familiar to every schoolchild. Pierre de Fermat was a man in the Renaissance tradition, who was at the center of the rediscovery of ancient Greek knowledge, but he asked a question that the Greeks would not have thought to ask, and in so doing produced what became the hardest problem on earth for others to solve. Tantalizingly, he left a note for posterity suggesting that he had an answer, but not what it was. That was the beginning of the chase that lasted three centuries. That time span underlies the significance of this puzzle. It is hard to conceive of any problem, in any discipline of science, so simply and clearly stated that could have withstood the test of advancing knowledge for so long. Consider the leaps in understanding in physics, chemistry, biology, medicine, and engineering that have occurred since the seventeenth century. We have progressed from "humors" in medicine to gene-splicing, we have identified the fundamental atomic particles, and we have placed men on the moon, but in number theory Fermat's Last Theorem remained inviolate. For some time in my research I looked for a reason why the Last Theorem mattered to anyone but a mathematician, and why it would be important to make a program about it. Mathematics has a multitude of practical applications, but in the case of number theory the most exciting uses that I was offered were in cryptography, in the design of acoustic baffling, and in communication from distant spacecraft. None of these seemed likely to draw in an audience. What was far more compelling were the

FERMAT'S ENIGMA mathematicians themselves, and the sense of passion that they all expressed when talking of Fermat. Mathematics is one of the purest forms of thought, and to outsiders mathematicians may seem almost otherworldly. The thing that struck me in all my discussions with them was the extraordinary precision of their conversation. A question was rarely answered immediately. Instead, I would often have to wait while the precise structure of the answer was resolved in the mind, but it would then emerge, as articulate and careful a statement as I could have wished for. When I tackled Andrew's friend Peter Sarnak on this, he explained that mathematicians simply hate to make a false statement. Of course they use intuition and inspiration, but formal statements have to be absolute. Proof is what lies at the heart of math, and is what marks it out from other sciences. Other sciences have hypotheses that are tested against experimental evidence until they fail, and are overtaken by new hypotheses. In math, absolute proof is the goal, and once something is proved, it is proved forever, with no room for change. In the Last Theorem, mathematicians had their greatest challenge of proof, and the person who found the answer would receive the adulation of the entire discipline. Prizes were offered, andrivalryflourished.The Last Theorem has a rich history that touches death and deception, and it has even spurred on the development of mathematics. As the Harvard mathematician Barry Mazur has put it, Fermat added a certain "animus" to those areas of mathematics that were associated with early attempts at the proof. Ironically, it turned out thatjust such an area of math was central to Wiles's final proof. Gradually picking up an understanding of this unfamiliar field, I came to appreciate Fermat's Last Theorem as central to, and even a parallel for, the development of mathematics itself.

FOREWORD XI Fermat was the father of modern number theory, and since his time mathematics had evolved, progressed, and diversified into many arcane areas, where new techniques had spawned new areas of mathematics, and become ends in themselves. As the centuries passed, the Last Theorem came to seem less and less relevant to the cutting edge of mathematical research, and more and more turned into a curiosity. But it is now clear that its centrality to math never diminished. Problems around numbers, such as the one Fermat posed, are like playground puzzles, and mathematicians like solving puzzles. To Andrew Wiles it was a very special puzzle, and nothing less than his life's ambition. When he first revealed a proof in that summer of 1993, it came at the end of seven years of dedicated work on the problem, a degree of focus and determination that is hard to imagine. Many of the techniques he used had not been created when he began. He also drew together the work of many fine mathematicians, linking ideas and creating concepts that others had feared to attempt. In a sense, reflected Barry Mazur, it turned out that everyone had been working on Fermat, but separately and without having it as a goal, for the proof had required all the power of modern mathematics to be brought to bear upon its solution. What Andrew had done was tie together once again areas of mathematics that had seemed far apart. His work therefore seemed to be a justification of all the diversification that mathematics had undergone since the problem had been stated. At the heart of his proof of Fermat, Andrew had proved an idea known as the Taniyama-Shimura conjecture, which created a new bridge between wildly different mathematical worlds. For many, the goal of one unified mathematics is supreme, and this was a glimpse ofjust such a world. So in proving Fermat, Andrew Wiles had cemented some of the most important number theory of the

Xll FERMAT S ENIGMA postwar period and had secured the base of a pyramid of conjectures that were built upon it. This was no longer simply solving the longest-standing mathematical puzzle, but was pushing the very boundaries of mathematics itself. It was as if Fermat's simple problem, born at a time when math was in its infancy, had been waiting for this moment. For mathematicians, the level of emotion was intense. All were reveling in the glorious moment. With all this in train, small wonder at the weight of responsibility that Wiles felt as theflawhad gradually emerged over the autumn of 1993. With the eyes of the world upon him, and his colleagues calling to have the proof made public, somehow, and only he knows how, he didn't crack. He had moved from doing math in privacy and at his own pace to suddenly working in public. Andrew is an intensely private man, who fought hard to keep his family sheltered from the storm that was breaking around him. Throughout that week while I was in Princeton, I called, I left notes at his office, on his doorstep, and with his friends I even provided a gift of English tea. But he resisted my overtures, until that chance meeting on the day of my departure. A quiet, intense conversation followed, which in the end lasted barely fifteen minutes. When we parted that afternoon there was an understanding between us. If he managed to repair the proof, then he would come to me to discuss a film I was prepared to wait. But as I flew home to London that night it seemed to me that the television program was dead. No one had ever repaired a hole in the many attempted proofs of Fermat in three centuries. History was littered with false claims, and much as I wished that he would be the exception, it was hard to imagine Wiles as anything but another headstone in that mathematical graveyard. A year later I received the call. After an extraordinary math-

FOREWORD ematical twist, and a flash of true insight and inspiration, Wiles had finally brought an end to Fermat. The following year we found the time for him to devote to filming. By this time I had invited Simon Singh tojoin me in making the film, and together we spent time with Andrew, learning from the man himself the full story of those seven years of isolated study, and his year of hell that followed. As we filmed, Andrew told us, as he had told no one before, of his innermost feelings about what he had done how for thirty years he had hung on to a childhood dream how so much of the math he had ever studied had been, without his really knowing it at the time, really a gathering of tools for the Fermat challenge that had dominated his career how nothing would ever be the same of his sense of loss for the problem that would no longer be his constant companion and of the uplifting sense of release that he now felt. For a field in which the subject matter is technically about as difficult for a lay audience to understand as can be imagined, the level of emotional charge in our conversations was greater than any I have experienced in a career in science film making. For Andrew Wiles it was the end of a chapter in his life. For me it was a privilege to be close to it. The film, Fermat's Last Theorem, was transmitted in Britain on BBC Television as part of the Horizon series, and then in the United States on PBS's Nova series. Simon Singh has now developed those insights and intimate conversations, together with the full richness of the Fermat story and the history and mathematics that have always hung around it, into this book, which is a complete and enlightening record of one of the greatest stories in human thinking. John Lynch Editor of BBC TV's Horizon series March 1997

Preface The story of Fermat's Last Theorem is inextricably linked with the history of mathematics, touching on all the major themes of number theory. It provides a unique insight into what drives mathematics and, perhaps more important, what inspires mathematicians. The Last Theorem is at the heart of an intriguing saga of courage, skulduggery, cunning, and tragedy, involving all the greatest heroes of mathematics. Fermat's Last Theorem has its origins in the mathematics of ancient Greece, two thousand years before Pierre de Fermat constructed the problem in the form we know it today. Hence, it links the foundations of mathematics created by Pythagoras to the most sophisticated ideas in modern mathematics. In writing this book I have chosen a largely chronological structure that begins by describing the revolutionary ethos of the Pythagorean Brotherhood, and ends with Andrew Wiles's personal story of his struggle to find a solution to Fermat's conundrum. Chapter 1 tells the story of Pythagoras, and describes how Pythagoras's theorem is the direct ancestor of the Last Theorem. This chapter also discusses some of the fundamental concepts of mathematics that will recur throughout the book. Chapter 2 takes the story from ancient Greece to seventeenthcentury France, where Pierre de Fermat created the most profound riddle in the history of mathematics. Fermat was an extraordinary character whose contribution to mathematics xv

XVI FERMAT S ENIGMA goes far beyond the Last Theorem. I have spent several pages describing his life and some of his other brilliant discoveries. Chapters 3 and 4 describe some of the attempts to prove Fermat's Last Theorem during the eighteenth, nineteenth, and early twentieth centuries. Although these efforts ended in failure they led to a marvelous arsenal of mathematical techniques and tools, some of which have been integral to the very latest attempts to prove the Last Theorem. In addition to describing the mathematics, I have devoted much of these chapters to the mathematicians who became obsessed by Fermat's legacy. Their stories show how mathematicians were prepared to sacrifice everything in the search for truth, and how mathematics has evolved through the centuries. The remaining chapters of the book chronicle the remarkable events of the last forty years that have revolutionized the study of Fermat's Last Theorem. In particular, Chapters 6 and 7 focus on the work of Andrew Wiles, whose breakthroughs in the last decade astonished the mathematical community. These later chapters are based on extensive interviews with Wiles. This was a unique opportunity for me to hear at first hand one of the most extraordinary intellectual journeys of the twentieth century, and I hope that I have been able to convey the creativity and heroism that was required during Wiles's ten-year ordeal. In telling the tale of Pierre de Fermat and his baffling riddle I have tried to describe the mathematical concepts without resorting to equations, but inevitably x, y, and z do occasionally rear their ugly heads. When equations do appear in the text I have endeavored to provide sufficient explanation such that even readers with no background in mathematics will be able to understand their significance. For those readers with a slightly deeper knowledge of the subject I have provided a series of ap-

PREFACE XV11 pendices that expand on the mathematical ideas contained in the main text. In addition, I have included a list of further reading, which is generally aimed at providing the layperson with more detail about particular areas of mathematics. This book would not have been possible without the help and involvement of many people. In particular I would like to thank Andrew Wiles, who went out of his way to give long and detailed interviews during a time of intense pressure. During my seven years as a science journalist I have never met anybody with a greater level of passion and commitment to their subject, and I am eternally grateful that Professor Wiles was prepared to share his story with me. I would also like to thank the other mathematicians who helped me in the writing of this book and who allowed me to interview them at length. Some of them have been deeply involved in tackling Fermat's Last Theorem, while others were witnesses to the historic events of the last forty years. The hours I spent quizzing and chatting with them were enormously enjoyable, and I appreciate their patience and enthusiasm while explaining so many beautiful mathematical concepts to me. In particular I would like to thankJohn Coates,John Conway, Nick Katz, Barry Mazur, Ken Ribet, Peter Sarnak, Goro Shimura, and Richard Taylor. I have tried to illustrate this book with as many portraits as possible to give the reader a better sense of the characters involved in the story of Fermat's Last Theorem. Various libraries and archives have gone out of their way to help me, and in particular I would like to thank Susan Oakes of the London Mathematical Society, Sandra Cumming of the Royal Society, and Ian Stewart of Warwick University. I am also grateful to Jacquelyn Savani of Princeton University, Duncan McAngus, Jeremy Gray, Paul

xviii FERMAT'S ENIGMA Balister, and the Isaac Newton Institute for their help in finding research material. Thanks also go to Dawn Dzedzy, Patrick Walsh, Christopher Potter, Bernadette Alves, Sanjida O'Connell, and my parents for their comments and support throughout the last year. Finally, many of the interviews quoted in this book were obtained while I was working on a television documentary on the subject of Fermat's Last Theorem. I would like to thank the BBC for allowing me to use this material, and in particular I owe a debt of gratitude to John Lynch, who worked with me on the documentary, and who helped to inspire my interest in the subject. Although Fermat's Last Theorem has been the world's hardest mathematical problem, I hope that I have succeeded in conveying an understanding of the mathematics involved in tackling it, and an insight into why mathematicians have been obsessed by it for more than three centuries. Mathematics is one of the purest and most profound of intellectual disciplines, and my intention has been to provide readers a glimpse into this fascinating world.

Index Page numbers in italic refer to illustrations. Abel, Niels Henrik, 3 absolute proof, 20-26,134 absurdities, mathematical, 131, 294 Academy of Sciences, French, 109, 218 prize for proving Fermat's Last Theorem, 111-18 ACE (Automatic Computing Engine), 156 Adleman, Leonard, 94 Adler, Alfred, 2 Agnesi, Maria, 99-100,101,109 Alexandria, 45-46, 50-53, 99 Alexandrian Library, 45-46,53-54 Algarotti, Francesco, 102 algorithms, 75 amicable numbers, 58-59 Anglin, W. S., 71 Annals of Mathematics, 279 April fool E-mail, 271-73 Arago, Franςois, 73 Arakelov, Professor S., 232 Archimedes, 45, 102 Aristotle, 54 arithmetic algebraic geometrists, 232-33 Aήthmetica (Diophantus), 50-51, 54, 56, 57, 58 Clement-Samuel Fermat's edition, 63, 64-65 and elliptic equations, 165 Fermat's marginal notes, 58, 61-62, 63, 80 Latin translation, 52, 56-57 and Pythagorean triples, 60 axioms, 20-21,135-36 of arithmetic, 297 consistency of, 139-41 Babylonians, 7, 19, 54 Bachet de Meziriac, Claude Gaspar, 56-57 Latin translation of Aήthmetica, 52, 56-57 ProbUmes plaisants et delectables, 57 weighing problem, 57, 292 Barnum, P. X, 126-27 Bell, Eric Temple, 5-6,29,32,37, 105 Bernoulli family, 73-74 birthdays, shared, probability of, 42-43 Bombelli, Rafaello, 83-84 Bonaparte, Napoleon, 107, 214 Bourg-la-Reine, 212, 214, 218 Brahmagupta, 55 bridges, mathematical, 190-91 Bulletin of the London Mathematical Society, 186 calculus, 43-44 Carroll, Lewis, 126 Cauchy, Augustin Louis, 111-19, 722,218,219 chessboard, mutilated, problem of, 23-25 Chevalier, Auguste, 225, 227 Chudnovsky brothers, 48 Churchill, Sir Winston Leonard Spencer, 155 307

308 FERMAT'S ENIGMA cicadas, life cycles, 96-97 Circle Limit IV (Escher), 180,180 City of God, The (St. Augustine), 11 Clarke, Arthur C, 23 clock arithmetic, 166-69 closed groups, 228-29 Coates,John, 161,163, 164,169, 189-90, 206, 209, 238, 244, 248, 262, 279-80 code breaking, 93-95,150-56 Cohen, Paul, 143-44 Colossus (computer), 156 commutative law of addition, 135 completeness, 82-3,136,138,141 complex numbers, 86,116 computers early, 156-58 unable to prove Fermat's Last Theorem, 158-61 unable to prove Taniyama-Shimura conjecture, 211 conjectures, 67 unifying, 282 Constantinople, 55-56 continuum hypothesis, 143-44 contradiction, proof by, 46-47,49 Conway, Professor John H., 268 Coolidge, Julian, 37 cossists, 38 counting numbers, 11 Cretan paradox, 141 Croton, Italy, 9, 27-28 cryptography, 93-95,150-56 cubic equations, 217 Cwriosa Mathematka (Dodgson), 126 Cylon, 26-28 d'Alembert, Jean Le Rond, 87 Daltonjohn, 22 Darmon, Henri, 271, 273 Deals with the Devil, 68 defective numbers, 11 slightly, 13 Descartes, Rene, 39, 40, 59, 227 "Devil and Simon Flagg, The," 35,68 dΉerbinville, Pescheux, 223,226 Diderot, Denis, 76-77 differential geometry, 232, 234 Diffie, Whitfΐeld, 94 Digby, Sir Kenelm, 36, 60 Diophantine problems, 51 Diophantus of Alexandria, 50,51 riddle of his age, 51,291 Diophantus' Aήthmetica Containing Observations by P. deFermat, 63, 64-65 Dirichlet, Johann Peter Gustav Lejeune, 106,117,168 disorder parameters, 128-30 Disquisitiones aήthmeticae (Gauss), 105 Dodgson, Reverend Charles, 126 domino effect, 212 du Motel, Stephanie-Felicie Poterine, 223,226 Dudeney, Henry, 126 Dumas, Alexandre, 221-22 E-series, 118-19, 183-84, 190, 230-31 Ecole Normale Superieure, 220 Ecole Polytechnique, 103-4, 216 economics, and calculus, 46 Eddington, Sir Arthur, 121 Egyptians, ancient, 7-8 Eichler, 175 Eiffel Tower, 109 Einstein, Albert, 17,100 electricity, and magnetism, 184 Elements (Euclid), 46, 48, 50,114 Elkies, Noam, 159-60, 271-72 elliptic curves, 163

INDEX 309 elliptic equations, 163, 165-69, 181-82 families of, 239, 243 Frey's elliptic equation, 195-99, 202 and modular forms, 181, 182- 85, 187-94, 282 Enigma code, 150-56 Epimenides, 141 Escher, Mauritz, 180 Euclid infinite number of Pythagorian triples proof, 61, 292 infinity of primes proof, 90-91 and perfect numbers, 13 proves that V2 is irrational, 49, 289-90 and reductio ad absurdum, 46- 47, 48-49 unique factorization proof, 114 Euler, Leonhard, 32, 59, 70 attempts to solve Fermat's Last Theorem, 79-81,87 blindness and death, 87-88 forsakes theology, 73-74 and Kόnigsberg bridge puzzle, 77-79 phases of the moon algorithm, 75-76, 88 proves existence of God, 76-77 solves prime number theorem, 63, 66 Euler's conjecture, 159-60 Evens, Leonard, 261 Eves, Howard W., 205 excessive numbers, 11 slightly, 13 factorization, unique, 114,116 Faltings, Gerd, 232-34, 235, 277 Fermat, Clement-Samuel, 62-63 Fermat, Pierre de, 34 amateur mathematician, 37 Aήthmetica, 56, 57, 60-62 calculus, 43-44 career in civil service, 35-37,56 death, 62 education, 35 and elliptic equations, 165 and Father Mersenne, 39-40 ill with plague, 36-37 observations and theorems, 63, 66-67 probability theory, 40-41, 43 reluctant to reveal proofs, 39 Fermat's Last Theorem challenge of, 67-69 computers unable to prove, 158- 59 Fermat's proof, 283-84 Miyaoka's 'proof, 232-35 partial proofs by computer, 158 Germain's method, 104-6 n=3 (Euler), 81,86,89-90 n=4 (Fermat), 80-81, 89-90 w=5 (Dirichlet and Legendre), 106 n=7 (Lame), 106 n=irregular prime (Kummer and Mirimanoff), 157-58 publication of, 62-63 and Pythagoras's equation, 29- 32, 60-62 skepticism as to existence of proof, 117-118 simplicity of statement, 6, 68 and Taniyama-Shimura conjecture, 196-99, 201, 244, 245 and undecidability, 143-44,147 why called'Last', 67 Wiles's proof see Wiles, Andrew

310 FERMAT S ENIGMA Fermatean triple, 61 Flach, Matheus, 238 four-dimensional shapes, 233-34 four-dimensional space, 179-81 Fourier,Jean BaptisteJoseph, 219 "14-15" puzzle, 127-31,199 fractions, 11,49,81-82 Frey, Gerhard, 195-99, 202 Frey's elliptic equation, 195-99, 201-2 friendly numbers, 58-59 fundamental particles of matter, 22-23 fundamental theorem of arithmetic, 114 fundamental truths, 134-35 Furtwangler, Professor P., 138-39 Galileo Galilei, 37 Galois, Evariste, 3, 213 birth, 212 duel with dΉerbinville, 223, 225-26 education, 214-16,220 final notes, 223, 224, 225-26, 227 funeral, 226 and group theory, 228-29, 230-31 and quintic equations, 218, 219-20, 225, 227-28 revolutionary career, 218-22 game theory, 147-48, 297 Gardner, Martin, 58, 133 Gauss, Carl Friedrich, 105,107-9, 160 geometry, 7-8 Gerbert of Aurillac, 55 Germain, Sophie, 97, 98,101-5 career as a physicist, 108-9 and Evariste Galois, 221 relationship with Gauss, 107-8, 109 strategy for Fermat's Last Theorem, 104-6 Gibbon, Edward, 99 Globe, Le, 219 Gόdel, Kurt, 133,138-39,140,141 undecidable statements, 142-43 Goldbach, Christian, 81 Gombaud, Antoine, 40-41 Gouvea, Fernando, 279 Government Code and Cypher School, 150-55 gravity, theories of, 22 group theory, 228-29 Guardian, 250 hammers, harmony of, 15 Hardy, G. H., 1, 2-4, 47,146,147, 160-61,171 Hecke algebras, 276-77 Hein, Piet, 255 Heisenberg, Werner, 142-43 Hellman, Martin, 94 Hermite, Charles, 3 hieroglyphics, 191 Hubert, David, 92-94, 133, 136, 137,138 and basic axioms, 135-36 and Fermat's Last Theorem, 206-7 23 problems, 136,144 Hubert's Hotel, 92-93 Hippasus, 50 History of Mathematics (Montucla) 102 Hypatia, 99, 101 hyperbolic space, 179-81 Iamblichus, 14-15

INDEX 311 Illusie, Luc, 256, 259 imaginary numbers, 81, 84-87, 114-15 induction, proof by, 211-12,297- 99 infinite descent, method of, 80-81 infinity, 55, 91-93,158-59 International Congress of Mathematicians Berkeley (1986), 199 Paris (1900), 136 intuition, and probability, 41 invariants, 130,131,199 Inventiones Mathematicae y 255 irrational numbers, 47,49,81-82 Iwasawa theory, 237,238,274,275 Journal de Mathematiques pares et appliquees, 227 Kanada, Yasumasa, 48 Katz, Nick, 240, 241-43, 256-58, 259 knot invariants, 130-31, 221 Kolyvagin-Flach method, 237-38, 241-43, 257-58, 259, 269, 274-75 Kόnigliche Gesellschaft der Wissenschafien, 123-25,255 Kόnigsberg bridge puzzle, 77-79 Kovalevsky, Sonya, 101 Kronecker, Leopold, 47-48 Kummer, Ernst Eduard, 113-14, 775,116-19 Zrseries, 168 Lagrange,Joseph-Louis, 87,104, 219 Lame, Gabriel, 106, 770, 111, 113-14,116-19 Landau, Edmund, 100,132 Langlands, Robert, 192-93,282- 283 Langlands program, 192-93,232, 282-83 Last Problem, The (Bell), 5-6, 29, 32,68 Le Blanc, Antoine-August, 104 see also Germain, Sophie Legendre, Adrien-Marie, 106 Leibniz, Gottfried, 84 liar's paradox, 141-42 Libri-Carrucci dalla Sommaja, Count Guglielmo, 103, 221 limericks, 279,283 Liouville, Joseph, 113, 227-28 Lipman, Joseph, 261 Littlewood,John Edensor, 160-61 logic, mathematical, 134-35 logicians, 134-35,143 loopiness, in rivers, 17-18 Loyd Sam, 126-31 Loyd's puzzle s^ "14-15" puzzle lyre, tuning strings on, 14-16 M-series, 181-85,190, 230-31 magnetism, and electricity, 184 Mathematical Magic Show (Gardner), 58 mathematical proof, 20-21,23-26 Mathematician's Apology, A (Hardy), 2-3, 47,146 mathematicians collaboration amongst, 4-5 and compulsion of curiosity 145- 47 in India and Arabia, 54-56 mathematical life, 2-4 require absolute proof, 134-36, 138

312 FERMAT S ENIGMA mathematicians continued secretive nature, 38-39 self-doubt of, 72-73 youthfulness, 2-3 mathematics foundation for science, 26 objective subject, 27-28 relationship with science, 17,18 in seventeenth century, 37-38 Mathematics of Great Amateurs (Coolidge), 37 Mathematische Annalen, 171 Mazur, Barry, 190-91, 199, 201, 243, 244-45, 248, 255 Mersenne, Father Marin, 38-39 Method, The (Heiberg), 45 Milo, 9,27 Mirimanoff, Dimitri, 158 Miyaoka, Yoichi, 232-35 Miyaoka inequality, 234 modular forms, 175,179-82 and elliptic equations, 181, 183-85,188-94, 282 Monde, he, 250 7 51 Montucla,Jean-Etienne, 102 moon, predicting phases of, 75- 76 Moore, Professor L. T., 44 Mozans, H. J., 109 musical harmony, principles of 14- 16 natural numbers, 81-82 negative numbers, 81-84 New York, subway graffiti, 235 New York Times, 232,251, 252,260, 279 Newton, Isaac, 44, 74, 75 Noether, Emmy, 100-101 nothingness, concept of, 54-55 number line, 83-85,166-67 numbers, relationships between 11 numerals, Indo-Arab, 54-55 Oberwolfach symposium (1984) 194-99 Olbers, Heinrich, 105 order and chaos, 17 overestimated prime conjecture 160 Paganini, Nicolό, 59 parallelism, philosophy of, 232, 235 parasites, life cycles, 96-97 particle physics, 22-23 Pascal, Blaise, 38, 40-41, 43-44 Penrose, Roger, 178 Penrose tilings, 178-79 People, 251,268 perfect numbers, 11-13 philosopher, word coined by Pythagoras, 9-10 pi(π), 17-18 Pillow Problems (Dodgson), 126 Pinch, Richard, 262-63 Plato, 99 points (dice game), 41 polynomials, 217-18 Porges, Arthur, 35, 68 prime numbers, 63, bϋ and Fermat's Last Theorem, 89-91 Germain primes, 105-6 infinity of, 90-93 irregular primes, 116-17,158 practical applications, 93-97 333,333,331 not prime, 159 probability, 40-44 counterintuitive, 42-43 Problemes plaisants et delectables (Bachet), 57 puzzles, compendiums of, 126

INDEX 313 Pythagoras abhors irrational numbers, 47, 49-50 at Croton, 9-10, 27-28 death, 28 and mathematical proof, 25- 26 and musical harmony, 14-16 and perfect numbers, 11-13 and study of numbers, 7 travels, 7-8 Pythagoras's equation, 27-29 "cubed" version, 30-32 and Fermat's Last Theorem 32, 60-62 whole number solutions, 28- 30 Pythagoras's theorem, 6-7, 19- 20, 25-26, 287-88 Pythagorean Brotherhood, 9-11, 13, 26-28, 46, 47, 98 Pythagorean triples, 26-29, 60, 292 quadratic equations, 216-17 quantum physics, 142-43 quartic equations, 217-18 quintic equations, 217-18, 219, 225, 227-29 Ramanujan, Srinivasa, 2-3 rational numbers, 10-11 rearrangement of equations 195- 96 recipes, mathematical, 7-8, 218 reductio ad absurdum, 46-47, 49- 50 reflectional symmetry, 176 Reidemeister, Kurt, 130 religion, and probability, 43 Ribenboim, Paulo, 133 Ribet, Ken, 199, 200,201-3, 245, 248-49, 254, 266-67, 282 Fermat Information Service, 260 and significance of TaniyamaShimura conjecture, 199, 201-3 river ratio, 17-18 Rivest, Ronald, 94 Rosetta stone, 191 rotational symmetry, 176-77 Rubin, Professor Karl, 246-47, 276-77 Russell, Bertrand, 21, 41,133 St. Augustine (of Hippo), 11 Sam Loyd and his Puzzles: An Autobiographical Review, 126-27 Samos, Greece, 8-9 Sarnak, Peter, 263, 269 Schlichting, Dr. R, 133, 294-95 scientific proof, 21-22 scientific theories, 22-23 scrambling and unscrambling messages, 93-95, 148,150-56 Selmer groups, 264 Shamir, Adi, 94 Shimura, Goro, 171,172,173-75, 179,182,183 relationship with Taniyama, 185-87, 188 and Taniyama-Shimura conjecture, 187-90, 251 Shimura-Taniyama conjecture see Taniyama-Shimura conjecture Silverman, Bob, 262 Isaac Newton Institute, Cambridge, 4-5, 244 Sir Isaac Newton s Philosophy Explain 'd for the Use of Ladies (Algarotti), 102 6, perfection of, 11-12

314 FERMAT'S ENIGMA Skewes, S., 160-61 Skewes's number, 160-61 sociable numbers, 59-60 Socrates, 110 Somerville, Mary, 103 square, symmetries of, 176-77 square-cube sandwiches, 59 square root of one, 83-84 square root of two, 48-49,82-83, 288-90 strings and particles, 23 vibrating, 14-15,16,17 Suzuki, Misako, 186-87 symmetry, 175-82 Taniyama, Yutaka, 170,171,173- 75,179,182 death, 185-87 influence of, 188 and Taniyama-Shimura conjecture, 182-85 Taniyama-Shimura conjecture, 184-85,187-94 and Fermat's Last Theorem 194- 99, 201-3 Wiles and, 194, 203, 205-12, 236-39, 241-43, 251, 280-81 Taniyama-Weil conjecture see Taniyama-Shimura conjecture Taylor, Richard, 262, 269, 270, 273, 276-77 Thales, 26 Theano, 9,97-98 theorems, 66-67 Theory of Games and Economic Behaviour, The (von Neumann), 147 Thomson,J.J., 22 three-body problem, 75 tiled surfaces, symmetry of, 177-79 Titchmarsh, E. C, 147 Tokyo, international symposium (1955), 182 translational symmetry, 176-77 trichotomy, law of, 134-35 truels, 148,297 Turing, Alan Mathison, 148-57 uncertainty principle, 142-43 undecidability theorems, 139, 141-44 von Neumann,John, 139,147-48 Wagstaff, Samuel S., 158 Wallisjohn, 36,40,60 weighing problem, 57, 292 Weil, Andre, 141,189 Weil conjecture see TaniyamaShimura conjecture Weyl, Hermann, 136 whole numbers, 11 Wiener Kreis (Viennese Circle), 139 Wiles, Andrew, xx, 162, 204, 254, 278 adolescence and Fermat's Last Theorem, 5-6, 32, 71-72 graduate student days, 161,163 tackles elliptic equations, 163, 165-66,169 and Taniyama-Shimura conjecture, 194,203,205-12,236- 39,241-43,251,280-82 uses Galois's groups, 229-31, 236, 273 announces proof ofFermat's Last Theorem, 1-2,5,33,244-47, 250 reaction of media, 250-53 mathematical celebrity, 251-53, 268

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