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Fermat’s Last Theorem by Simon Singh
Albert Einstein, Andrew Wiles, Antoine Gombaud: Chevalier de Méré, Arthur Eddington, Augustin-Louis Cauchy, Fellow of the Royal Society, Georg Cantor, Henri Poincaré, Isaac Newton, John Conway, John von Neumann, kremlinology, probability theory / Blaise Pascal / Pierre de Fermat, RAND corporation, Simon Singh, Wolfskehl Prize
SIMON SINGH Fermat’s Last Theorem THE STORY OF A RIDDLE THAT CONFOUNDED THE WORLD’S GREATEST MINDS FOR 358 YEARS In memory of Pakhar Singh Birring CONTENTS Cover Title Page Dedication Foreword 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 Appendices Suggestions for Further Reading Index About the Author Also by the Author Copyright About the Publisher Foreword We finally met across a room, not crowded, but large enough to hold the entire Mathematics Department at Princeton on their occasions of great celebration. On that particular afternoon, there were not so very many people around, but enough for me to be uncertain as to which one was Andrew Wiles. After a few moments I picked out a shy-looking man, listening to the conversation around him, sipping tea, and indulging in the ritual gathering of minds that mathematicians the world over engage in at around four o’clock in the afternoon. He simply guessed who I was. 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.
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 of just such a world. So in proving Fermat, Andrew Wiles had cemented some of the most important number theory of the post-war 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 maths was in its infancy, had been waiting for this moment. The story of Fermat had ended in the most spectacular fashion. For Andrew Wiles, it meant the end of professional isolation of a kind almost alien to maths, which is usually a collaborative activity. Ritual afternoon tea in mathematics institutes the world over is a time when ideas come together, and sharing insight before publication is the norm.
In Chapter 2 we shall find out more about the mysterious Pierre de Fermat and how his theorem came to be lost, but for the time being it is enough to know that Fermat’s Last Theorem, a problem that had captivated mathematicians for centuries, had captured the imagination of the young Andrew Wiles. Sat in Milton Road Library was a ten-year-old boy staring at the most infamous problem in mathematics. Usually half the difficulty in a mathematics problem is understanding the question, but in this case it was simple – prove that xn + yn = zn has no whole number solutions for n greater than 2. Andrew was not daunted by the knowledge that the most brilliant minds on the planet had failed to rediscover the proof. He immediately set to work using all his textbook techniques to try and recreate the proof. Perhaps he could find something that everyone else, except Fermat, had overlooked. He dreamed he could shock the world. Thirty years later Andrew Wiles was ready. Standing in the auditorium of the Isaac Newton Institute, he scribbled on the board and then, struggling to contain his glee, stared at his audience.
The Music of the Primes by Marcus Du Sautoy
Ada Lovelace, Andrew Wiles, Arthur Eddington, Augustin-Louis Cauchy, computer age, Dava Sobel, Dmitri Mendeleev, Eratosthenes, Erdős number, four colour theorem, Georg Cantor, German hyperinflation, global village, Henri Poincaré, Isaac Newton, Jacquard loom, Jacquard loom, music of the spheres, New Journalism, Paul Erdős, Richard Feynman, Richard Feynman, Search for Extraterrestrial Intelligence, Simon Singh, Solar eclipse in 1919, Stephen Hawking, Turing machine, William of Occam, Wolfskehl Prize, Y2K
Much of mathematics had found itself entangled with physics over the past few decades. Despite being a problem with its heart in the theory of numbers, the Riemann Hypothesis had for some years been showing unexpected resonances with problems in particle physics. Mathematicians were changing their travel plans to fly in to Princeton to share the moment. Memories were still fresh with the excitement of a few years earlier when an English mathematician, Andrew Wiles, had announced a proof of Fermat’s Last Theorem in a lecture delivered in Cambridge in June 1993. Wiles had proved that Fermat had been right in his claim that the equation xn + yn = zn has no solutions when n is bigger than 2. As Wiles laid down his chalk at the end of the lecture, the champagne bottles started popping and the cameras began flashing. Mathematicians knew, however, that proving the Riemann Hypothesis would be of far greater significance for the future of mathematics than knowing that Fermat’s equation has no solutions.
Fundamental particles have been given some crazy names – gluons, cascade hyperons, charmed mesons, quarks, the last of these courtesy of James Joyce’s Finnegans Wake. But ‘morons’? Surely not! Bombieri has an unrivalled reputation for appreciating the ins and outs of the Riemann Hypothesis, but those who know him personally are also aware of his wicked sense of humour. Fermat’s Last Theorem had fallen foul of an April Fool prank that emerged just after a gap had appeared in the first proof that Andrew Wiles had proposed in Cambridge. With Bombieri’s email, the mathematical community had been duped again. Eager to relive the buzz of seeing Fermat proved, they had grabbed the bait that Bombieri had thrown at them. And the delights of forwarding email meant that the first of April had disappeared from the original source as it rapidly disseminated. This, combined with the fact that the email was read in countries with no concept of April Fool’s Day, made the prank far more successful than Bombieri could have imagined.
But it was striking how, as the century drew to a close, more and more mathematicians were prepared to talk about attacking it. The proof of Fermat’s Last Theorem only helped to fuel the expectation that great problems could be solved. Mathematicians had enjoyed the attention that Wiles’s solution to Fermat had brought them as mathematicians. This feeling undoubtedly contributed to the desire to believe Bombieri. Suddenly, Andrew Wiles was being asked to model chinos for Gap. It felt good. It felt almost sexy to be a mathematician. Mathematicians spend so much time in a world that fills them with excitement and pleasure. Yet it is a pleasure they rarely have the opportunity to share with the rest of the world. Here was a chance to flaunt a trophy, to show off the treasures that their long, lonely journeys had uncovered. A proof of the Riemann Hypothesis would have been a fitting mathematical climax to the twentieth century.
Infinite Ascent: A Short History of Mathematics by David Berlinski
Alan Turing: On Computable Numbers, with an Application to the Entscheidungsproblem, Albert Einstein, Andrew Wiles, Benoit Mandelbrot, Douglas Hofstadter, Eratosthenes, four colour theorem, Georg Cantor, Gödel, Escher, Bach, Henri Poincaré, Isaac Newton, John von Neumann, Murray Gell-Mann, Stephen Hawking, Turing machine, William of Occam
Fermat believed that he had discovered a marvelous proof of his own conjecture, and within the margins of his own paper noted sadly that the margins were too small to contain it. Very good mathematicians were intrigued and often obsessed. Amateurs and cranks, all of them curiously aware of my e-mail address, busied themselves with crackpot proofs, some of them fiendishly ingenious. For more than three centuries the conjecture remained unyielding. And then in 1993, the English mathematician Andrew Wiles announced a proof, one retrospectively validating Fermat. The old boy had been right after all. Wiles’ proof ran to more than two hundred pages and it made use of an immense body of modern mathematics. A first version, announced in a very dramatic setting at Oxford University, contained an error. The proof required revision. But then everything came right. Although his paper addresses an old problem, it is completely an exercise in the most modern mathematics.
The coordination between certain equations and certain structures in space proceeds directly up the dimensional chain. The fourth-order equation V = x2y2 + y4/4, for example, describes an undulating surface in a three-dimensional space. Analytical geometry may be conducted in four dimensions, if need be, and although the results cannot easily be seen—let us be honest: They cannot be seen at all—the analysis is much the same. When Andrew Wiles offered his proof of Fermat’s conjecture, he used an immense array of tools, but at the very center of his proof a tingling trail led backward to Descartes, for what he had succeeded in proving was the Taniyama-Shimura conjecture, a thesis about elliptical equations and modular forms, one that in the complexity of its formulation hid that old, shrub-covered trail between the form of the discriminant and various curves in the plane.
Ten years later, algebraic geometry seemed to mathematicians simply to scintillate, Alexander Grothendieck dominating the field with what René Thom once described as crushing technical superiority. Grothendieck has since given up mathematics and is said to be resident in a cave somewhere in the south of France, where he is occupied by various ecological issues. The classification of the finite simple groups, I suppose, is next, and after that the Taniyama-Shimura conjecture and the proof of Fermat’s famous theorem, the work collectively of Andrew Wiles, Ken Ribet, Barry Mazur, and Gerhard Frey. But this list, resembling as it does various trite accounts of what is in and what is out, or what is hot and what is not, could easily be rewritten in a dozen different ways, evidence that mathematics no longer has what for so long it had, and that is a stable center. If lacking a center, the modern era in mathematics nonetheless displays certain identifiable but inconsistent tendencies, almost as if a river were suddenly to separate itself into a number of hissing streams.
The Simpsons and Their Mathematical Secrets by Simon Singh
Albert Einstein, Andrew Wiles, Benoit Mandelbrot, cognitive dissonance, Erdős number, Georg Cantor, Grace Hopper, Isaac Newton, John Nash: game theory, mandelbrot fractal, Menlo Park, Norbert Wiener, P = NP, Paul Erdős, probability theory / Blaise Pascal / Pierre de Fermat, Richard Feynman, Richard Feynman, Schrödinger's Cat, Simon Singh, Stephen Hawking, Wolfskehl Prize, women in the workforce
Fermat’s last theorem has also appeared in novels (The Girl Who Played with Fire by Stieg Larsson), in films (Bedazzled with Brendan Fraser and Elizabeth Hurley), and plays (Arcadia by Tom Stoppard). Perhaps the theorem’s most famous cameo is in a 1989 episode of Star Trek: The Next Generation titled “The Royale,” in which Captain Jean-Luc Picard describes Fermat’s last theorem as “a puzzle we may never solve.” However, Captain Picard was wrong and out of date, because the episode was set in the twenty-fourth century and the theorem was actually proven in 1995 by Andrew Wiles at Princeton University.5 Wiles had dreamed about tackling Fermat’s challenge ever since he was ten years old. The problem then obsessed him for three decades, which culminated in seven years of working in complete secrecy. Eventually, he delivered a proof that the equation xn + yn = zn (n > 2) has no solutions. When his proof was published, it ran to 130 dense pages of mathematics. This is interesting partly because it indicates the mammoth scale of Wiles’s achievement, and partly because his chain of logic is far too sophisticated to have been discovered in the seventeenth century.
In other words, the Doctor is tacitly acknowledging that Wiles’s proof exists, but he rightly does not accept that it is Fermat’s proof, which he considers to be the “real one.” Perhaps the Doctor went back to the seventeenth century and obtained the proof directly from Fermat. So, to summarize, in the seventeenth century, Pierre de Fermat states that he can prove that the equation xn + yn = zn (n > 2) has no whole number solutions. In 1995, Andrew Wiles discovers a new proof that verifies Fermat’s statement. In 2010, the Doctor reveals Fermat’s original proof. Everyone agrees that the equation has no solutions. Thus, in “The Wizard of Evergreen Terrace,” Homer appears to have defied the greatest minds across almost four centuries. Fermat, Wiles, and even the Doctor state that Fermat’s equation has no solutions, yet Homer’s blackboard jottings present us with a solution: 3,98712 + 4,36512 = 4,47212 You can check it yourself with a calculator.
By the time this episode aired in 1998, Wiles’s proof had been published for three years, so Cohen was well aware that Fermat’s last theorem had been conquered. He even had a personal link to the proof, because he had attended some lectures by Ken Ribet while he was a graduate student at the University of California, Berkeley, and Ribet had provided Wiles with a pivotal stepping-stone in his proof of Fermat’s last theorem. Cohen obviously knew that Fermat’s equation had no solutions, but he wanted to pay homage to Pierre de Fermat and Andrew Wiles by creating a solution that was so close to being correct that it would apparently pass the test if checked with only a simple calculator. In order to find his pseudosolution, he wrote a computer program that would scan through values of x, y, z, and n until it found numbers that almost balanced. Cohen finally settled on 3,98712 + 4,36512 = 4,47212 because the resulting margin of error is minuscule—the left side of the equation is only 0.000000002 percent larger than the right side.
The Golden Ticket: P, NP, and the Search for the Impossible by Lance Fortnow
Alan Turing: On Computable Numbers, with an Application to the Entscheidungsproblem, Albert Einstein, Andrew Wiles, Claude Shannon: information theory, cloud computing, complexity theory, Erdős number, four colour theorem, Gerolamo Cardano, Isaac Newton, John von Neumann, linear programming, new economy, NP-complete, Occam's razor, P = NP, Paul Erdős, Richard Feynman, Richard Feynman, smart grid, Stephen Hawking, traveling salesman, Turing machine, Turing test, Watson beat the top human players on Jeopardy!, William of Occam
After Karp, computer scientists started to realize the incredible importance of the P versus NP problem, and it dramatically changed the direction of computer science research. Today, P versus NP has become a critical question not just in computer science but in many other fields, including biology, medicine, economics, and physics. The P versus NP problem has achieved the status of one of the great open problems in all of mathematics. Following the excitement of Andrew Wiles’s 1994 proof of Fermat’s Last Theorem, the Clay Mathematics Institute decided to run a contest for solutions to the most important unsolved mathematical problems. In 2000, the Clay Institute listed seven Millennium Problems and offered a $1 million bounty for each of them. 1. Birch and Swinnerton-Dyer conjecture 2. Hodge conjecture 3. Navier-Stokes equations 4. P versus NP 5. Poincaré conjecture 6.
In other words, there are no natural numbers a, b, and c all greater than zero and n greater than 2 such that an + bn = cn. Fermat never mentioned this proof again, so it is likely he never had a true solution. The problem gained great notoriety as it became the classic unsolvable math puzzle. Kids like me dreamed of being the first person to solve this famous problem. One of those kids grew up and did just that. In 1994 the Princeton mathematician Andrew Wiles, building on a long series of papers in number theory, developed a proof of Fermat’s claim and became an instant celebrity, at least as much of a celebrity as a mathematician could be. This chapter won’t show how to solve the P versus NP problem, or this would have been a very different book. Instead we explore a few of the ideas that people have tried to resolve the P versus NP problem. Alas, these ideas have not panned out to anything close to a solution to the problem.
Adapt: Why Success Always Starts With Failure by Tim Harford
Andrew Wiles, banking crisis, Basel III, Berlin Wall, Bernie Madoff, Black Swan, car-free, carbon footprint, Cass Sunstein, charter city, Clayton Christensen, clean water, cloud computing, cognitive dissonance, complexity theory, corporate governance, correlation does not imply causation, credit crunch, Credit Default Swap, crowdsourcing, cuban missile crisis, Daniel Kahneman / Amos Tversky, Dava Sobel, Deep Water Horizon, Deng Xiaoping, double entry bookkeeping, Edmond Halley, en.wikipedia.org, Erik Brynjolfsson, experimental subject, Fall of the Berlin Wall, Fermat's Last Theorem, Firefox, food miles, Gerolamo Cardano, global supply chain, Isaac Newton, Jane Jacobs, Jarndyce and Jarndyce, Jarndyce and Jarndyce, John Harrison: Longitude, knowledge worker, loose coupling, Martin Wolf, Menlo Park, Mikhail Gorbachev, mutually assured destruction, Netflix Prize, New Urbanism, Nick Leeson, PageRank, Piper Alpha, profit motive, Richard Florida, Richard Thaler, rolodex, Shenzhen was a fishing village, Silicon Valley, Silicon Valley startup, South China Sea, special economic zone, spectrum auction, Steve Jobs, supply-chain management, the market place, The Wisdom of Crowds, too big to fail, trade route, Tyler Cowen: Great Stagnation, web application, X Prize
Christensen, The Innovator’s Solution (Harvard Business School Press, 2003), p. 198. 243 Whole structure of Virgin Group has always been: Richard Branson, Business Stripped Bare (Virgin Books, 2008), pp. 169–214. 244 ‘I’ll be damned if I permit’: anonymous officer quoted in John Nagl, Learning to Eat Soup with a Knife (University of Chicago Press, 2005), p. 172. 8 Adapting and you 247 ‘He was not a very careful person’: Shimura is quoted in ‘Andrew Wiles and Fermat’s Last Theorem’, MarginalRevolution.com, 29 August 2010, http://www.marginalrevolution.com/marginalrevolution/2010/08/andrew-wiles-and-fermats-last-theorem.html 247 ‘Let us try for once not to be right’: Tristan Tzara214. , 1918. 247 ‘Stupefyingly clichéd and almost embarrassingly naïve’: Hedy Weiss, ‘Good music, flashy moves can’t fill emotional void’, Chicago Sun- Times, 21 July 2002. 247 ‘Crazily uneven’: Michael Phillips, ‘“Movin’ Out”? Maybe not; Broadway-bound Tharp-Joel show has to get acts together’, Chicago Tribune, 22 July 2002. 248 ‘If you stand in Twyla’s way’: Cathleen McGuigan, ‘Movin’ to Broadway: Twyla Tharp heads uptown with Billy Joel’, Newsweek,28 October 2002. 248 Oklahoma!
A Beautiful Mind by Sylvia Nasar
Al Roth, Albert Einstein, Andrew Wiles, Brownian motion, cognitive dissonance, Columbine, experimental economics, fear of failure, Henri Poincaré, invisible hand, Isaac Newton, John Conway, John Nash: game theory, John von Neumann, Kenneth Rogoff, linear programming, lone genius, market design, medical residency, Nash equilibrium, Norbert Wiener, Paul Erdős, prisoner's dilemma, RAND corporation, Ronald Coase, second-price auction, Silicon Valley, Simon Singh, spectrum auction, The Wealth of Nations by Adam Smith, Thorstein Veblen, upwardly mobile
And he was looking forward to hearing Busemann’s seminar on the state of Soviet mathematics because everyone knew that the Russians were doing great things, but the authorities were no longer allowing even abstracts of their mathematics articles to be translated into English. The signal event of the summer institute turned out to be the surprise announcement, within a day or two of the start of the meetings, of Milnor’s proof of the existence of exotic spheres.2 For the mathematicians gathered there, it had the same electrifying effect as the announcement of a solution of Fermat’s Last Theorem by Andrew Wiles of Princeton University four decades later. It stole Nash’s thunder. Nash reacted to the news of Milnor’s triumph with a display of adolescent petulance.3 The mathematicians were all camping out in a student dormitory and eating their meals in the cafeteria. Nash protested by grabbing gigantic portions. Once he demolished a pile of bread. Another time, he threw a glass of milk at a cashier.
“Yes,” he says, “my son is in a mental hospital in Arkansas but he got a job offer!” He is laughing at the absurdity of this juxtaposition. This is too much for Alicia. “You have to be fair to Johnny,” she returns. Nash says nothing. But later in the evening he goes to some lengths to make amends. He brings an offering, maps of Mexico, that he found in books on the Borels’ shelves, to Alicia. He takes the opportunity — during a conversation about Andrew Wiles’s successful proof of Fermat’s Last Theorem — to point out that Johnny had done some “classical” number theory in graduate school. Johnny had published “one correct result, one incorrect, but the correct one was a breakthrough of sorts,” he tells the other guests. Alicia responds by paying attention, by taking in what he means. Much of the renewal of their marriage has taken place since the Nobel.
Cohen, interview, 1.5.96. 4. Stanislaw Ulam, “John von Neumann, 1903–1957,” op. cit., p. 5. 5. Hardy, op. cit. 6. Felix Browder, interview, 11.10.95. 7. Harold Kuhn, interview, 7.95. 8. Ibid. 9. John Nash, plenary lecture, op. cit. 10. Elias Stein, interview, 12.28.95. 11. Cohen, interview. 12. E. T. Bell, Men of Mathematics, op. cit. 13. Enrico Bombieri, interview, 12.6.95. 14. Bell, op. cit. 15. Andrew Wiles, professor of mathematics, Princeton University, personal communication, 6.97. 16. Lars Hörmander, interview, 2.13.97. 17. F. Browder, interview. 18. John Forbes Nash, Jr., Les Prix Nobel 1994, op. cit. 19. Bell, op. cit. 20. Ibid. 21. Ibid. 22. Jacob Schwartz, professor of computer science, Courant Institute, interview, 1.29.96. 23. Jerome Neuwirth, interview, 5.27.97. 24. Stein, interview. 25.
The Infinity Puzzle by Frank Close
Albert Einstein, Andrew Wiles, Arthur Eddington, dark matter, El Camino Real, en.wikipedia.org, Ernest Rutherford, Isaac Newton, Murray Gell-Mann, Richard Feynman, Richard Feynman, Ronald Reagan, Simon Singh
At that time, it never occurred to me as suspicious that a child of ten had seen what generations of the ablest minds had overlooked. When the editor told me of the typo, I was astonished; not at my naïveté (I was still too naive) but that rotating a multiplication sign through 45 degrees to turn it into “+” had such profound consequences. That defeat convinced me there was no point in continuing with Fermat’s puzzle. A few years later, Andrew Wiles came across Fermat’s theorem in a book at his local library. Like me, he was then ten years old; like me, he decided to solve it; unlike me, he succeeded—though it took him more than a decade of dedicated work some thirty years later. ’t Hooft had the perseverance of Wiles, or at least more than most. Fermat’s Last Theorem had tantalized mathematicians for three centuries, whereas the problem of building a viable theory of the weak interaction had existed for only three decades.
The remarkable emergence of ’t Hooft, a mere student solving the puzzle that had defeated great masters, gained its own life. Over the years, Veltman’s role tended to be overlooked. What ’t Hooft had achieved was indeed remarkable, but no one, least of all himself, would claim that he did it alone. My comparison with Fermat’s Last Theorem is an example of the media oversimpliﬁcation, not least because Andrew Wiles solved Fermat on his own; ’t Hooft’s triumph, by contrast, was the culmination of a vast effort. In our analogy of climbing Everest, one could say that Veltman had made a map of the route, prepared the equipment, and almost reached the summit before ’t Hooft started. When Veltman, nearing the summit, had been faced with an impassable crevasse, ’t Hooft found a route, but even then it was Veltman’s tools that established this to be the way and got them successfully through.
American Society of Civil Engineers: Report Card, Andrew Wiles, Bernie Madoff, Black Swan, call centre, correlation does not imply causation, cross-subsidies, Daniel Kahneman / Amos Tversky, edge city, Emanuel Derman, facts on the ground, Gary Taubes, John Snow's cholera map, moral hazard, p-value, pattern recognition, profit motive, Report Card for America’s Infrastructure, statistical model, the scientific method, traveling salesman
Whereas the pure scientist is chiefly concerned with “what’s new,” applied work must deal with “how high,” as in “how high would profits go?” or “how high would the polls go?” In addition to purely technical yardsticks, applied scientists have goals that are societal, as with the Minnesota highway engineers; or psychological, as with the Disney queue managers; or financial, as with hurricane insurers and loan officers. The pursuit of pure science is rarely limited by time; as an extreme example, mathematician Andrew Wiles meticulously constructed his proof of Fermat’s last theorem over seven years. Such luxury is not afforded the applied scientist, who must deliver a best effort within a finite time limit, typically in the order of weeks or months. External factors, even the life cycle of green produce or the pipeline of drug innovations, may dictate the constraint on time. What use would it be to discover the cause of an E. coli outbreak the day after the outbreak dies down?
Andrew Wiles, Asian financial crisis, Berlin Wall, bonus culture, British Empire, business process, Cass Sunstein, computer age, credit crunch, Daniel Kahneman / Amos Tversky, discounted cash flows, discovery of penicillin, diversification, Donald Trump, Fall of the Berlin Wall, financial innovation, Gordon Gekko, greed is good, invention of the telephone, invisible hand, Jane Jacobs, Long Term Capital Management, Louis Pasteur, market fundamentalism, Nash equilibrium, pattern recognition, purchasing power parity, RAND corporation, regulatory arbitrage, shareholder value, Simon Singh, Steve Jobs, The Death and Life of Great American Cities, The Predators' Ball, The Wealth of Nations by Adam Smith, ultimatum game, urban planning, value at risk
The oblique solution complicates the problem to simplify it: The direct solution is inefficient, the oblique more direct. Invented puzzles frequently have this paradoxical character. They are a response to the everyday pleasure we take in obliquity. Most people regard arithmetic as a boring task—they do not enjoy long division or calculating the square root of a large number—but many like mathematical puzzles. The most famous such problem—Fermat’s Last Theorem—continues to intrigue. The 1994 proof by Andrew Wiles demands powerful computational tools.4 But Fermat hinted at a simpler solution that has not yet been rediscovered. Perhaps there is an oblique approach that, like Brunelleschi’s egg, or the presentation of perspective, or the Japanese assault on Singapore, is direct once thought of. If we sometimes recast problems before we begin, more often we revise our specification in the process of actually tackling them.
Turing's Vision: The Birth of Computer Science by Chris Bernhardt
Ada Lovelace, Alan Turing: On Computable Numbers, with an Application to the Entscheidungsproblem, Albert Einstein, Andrew Wiles, British Empire, cellular automata, Claude Shannon: information theory, complexity theory, Conway's Game of Life, discrete time, Douglas Hofstadter, Georg Cantor, Gödel, Escher, Bach, Henri Poincaré, Internet Archive, Jacquard loom, Jacquard loom, John Conway, John von Neumann, Joseph-Marie Jacquard, Norbert Wiener, Paul Erdős, Turing complete, Turing machine, Turing test, Von Neumann architecture
Post died of a heart attack shortly after receiving his last electroshock at the age of fifty seven. 2. In fact, there are no positive integer solutions to xn + yn = zn when n is larger than 2. This is the famous Fermat’s Last Theorem. Fermat wrote the statement in the margin of a book with the comment that he had a marvelous proof, but that the margin wasn’t large enough for him to include it. In 1994, 357 years later, Andrew Wiles finally proved it. Most mathematicians don’t believe that Fermat had a legitimate proof. The result should be more properly called Fermat’s conjecture or Wiles’s theorem. 3. All three of the problems we looked at have the property that there was an algorithm that works in the case when the answer of the decision problem was yes, but not in the case when the answer was no. Problems like this, where there is an algorithm for the Yes case, but not for the No case are sometimes called partially decidable, but not decidable (or recursively enumerable, but not recursive).
The Nature of Technology by W. Brian Arthur
Andrew Wiles, business process, cognitive dissonance, computer age, double helix, Geoffrey West, Santa Fe Institute, haute cuisine, James Watt: steam engine, joint-stock company, Joseph Schumpeter, Kevin Kelly, knowledge economy, locking in a profit, Mars Rover, means of production, railway mania, Silicon Valley, Simon Singh, sorting algorithm, speech recognition, technological singularity, The Wealth of Nations by Adam Smith, Thomas Kuhn: the structure of scientific revolutions
It is valid if it can be constructed under accepted logical rules from other valid components of mathematics—other theorems, definitions, and lemmas that form the available parts and assemblies in mathematics. Typically the mathematician “sees” or struggles to see one or two overarching principles: conceptual ideas that if provable provide the overall route to a solution. To be proved, these must be constructed from other accepted subprinciples or theorems. Each part moves the argument part of the way. Andrew Wiles’s proof of Fermat’s theorem uses as its base principle a conjecture by the Japanese mathematicians Taniyama and Shimura that connects two main structures he needs, modular forms and elliptic equations. To prove this conjecture and link the components of the argument, Wiles uses many subprinciples. “You turn to a page and there’s a brief appearance of some fundamental theorem by Deligne,” says mathematician Kenneth Ribet, “and then you turn to another page and in some incidental way there’s a theorem by Hellegouarch—all of these things are just called into play and used for a moment before going on to the next idea.”
The Half-Life of Facts: Why Everything We Know Has an Expiration Date by Samuel Arbesman
Albert Einstein, Alfred Russel Wallace, Amazon Mechanical Turk, Andrew Wiles, bioinformatics, British Empire, Chelsea Manning, Clayton Christensen, cognitive bias, cognitive dissonance, conceptual framework, David Brooks, demographic transition, double entry bookkeeping, double helix, Galaxy Zoo, guest worker program, Gödel, Escher, Bach, Ignaz Semmelweis: hand washing, index fund, invention of movable type, Isaac Newton, John Harrison: Longitude, Kevin Kelly, life extension, meta analysis, meta-analysis, Milgram experiment, Nicholas Carr, p-value, Paul Erdős, Pluto: dwarf planet, randomized controlled trial, Richard Feynman, Richard Feynman, Rodney Brooks, social graph, social web, text mining, the scientific method, Thomas Kuhn: the structure of scientific revolutions, Thomas Malthus, Tyler Cowen: Great Stagnation
Instead, in one of the most maddening episodes in math history, he scribbled this idea in the margins of a book and wrote that he had a brilliant proof, but, alas, the margin was too small to contain it. We now think he might have been mistaken. But no one had found any numbers greater than 2 that fit the equation since he wrote this statement. So it was assumed to be true, but no one could prove it. This elegant problem in number theory had gone unproven since the seventeenth century, until Andrew Wiles completed a proof in 1995, using pages and pages of very complex math, which would most certainly not have fit in Fermat’s margin. But, crucially, along the way, mathematicians proved other, smaller, proofs in their quest to crack Fermat’s Last Theorem. When finally solved, whole new pieces of math were involved in the construction of the proof. I decided to tackle predicting the proof of one of the most famous unsolved problems, something known as P versus NP.
The Infinite Book: A Short Guide to the Boundless, Timeless and Endless by John D. Barrow
Albert Einstein, Andrew Wiles, anthropic principle, Arthur Eddington, cosmological principle, dark matter, Edmond Halley, Fellow of the Royal Society, Georg Cantor, Henri Poincaré, Isaac Newton, mutually assured destruction, Olbers’ paradox, prisoner's dilemma, Ray Kurzweil, short selling, Stephen Hawking, Turing machine
If they could search systematically through all possibilities in a finite amount of our time, then they could print out ‘true’ or ‘false’ and stop. This is not as exciting to mathematicians as it might sound. Mathematicians are not only interested in whether conjectures like Goldbach’s are true or false, they are interested in the forms of reasoning needed to prove it. They want to see new types of argument. A classic example was the proof of Fermat’s Last Theorem by Andrew Wiles and Richard Taylor.18 The truth of Fermat’s conjecture emerged as a particular case of a much more general result that opened up types of proof and alternative formulations of old questions. A ‘proof ’ by direct search would provide no new insights of that sort. It would, in effect, be like looking up the answer in the back of the book. In fact, if a conjecture like Goldbach’s were shown to be true by an infinity machine, then we would feel aggrieved at being denied the insight provided by a proof.
Alex's Adventures in Numberland by Alex Bellos
Andrew Wiles, Antoine Gombaud: Chevalier de Méré, Black Swan, Black-Scholes formula, Claude Shannon: information theory, computer age, Daniel Kahneman / Amos Tversky, family office, forensic accounting, game design, Georg Cantor, Henri Poincaré, Isaac Newton, pattern recognition, Paul Erdős, probability theory / Blaise Pascal / Pierre de Fermat, random walk, Richard Feynman, Richard Feynman, SETI@home, Steve Jobs, The Bell Curve by Richard Herrnstein and Charles Murray, traveling salesman
Fermat never produced a proof – marvellous or otherwise – of his proposition even when unconstrained by narrow margins. His jottings in Arithmetica may have been an indication that he had a proof, or he may have believed he had a proof, or he may have been trying to be provocative. In any case, his cheeky sentence was fantastic bait to generations of mathematicians. The proposition became known as Fermat’s Last Theorem and was the most famous unsolved problem in maths until the Briton Andrew Wiles cracked it in 1995. Algebra can be very humbling in this way – ease in stating a problem has no correlation with ease in solving it. Wiles’s proof is so complicated that it is probably understood by no more than a couple of hundred people. Improvements in mathematical notation enabled the discovery of new concepts. The logarithm was a massively important invention in the early seventeenth century, thought up by the Scottish mathematician John Napier, the Laird of Merchiston, who was, in fact, much more famous in his lifetime for his work on theology.
Against the Gods: The Remarkable Story of Risk by Peter L. Bernstein
Albert Einstein, Andrew Wiles, Antoine Gombaud: Chevalier de Méré, Big bang: deregulation of the City of London, Bretton Woods, buttonwood tree, capital asset pricing model, cognitive dissonance, Daniel Kahneman / Amos Tversky, diversified portfolio, double entry bookkeeping, Edmond Halley, Edward Lloyd's coffeehouse, endowment effect, experimental economics, fear of failure, Fellow of the Royal Society, Fermat's Last Theorem, financial deregulation, financial innovation, full employment, index fund, invention of movable type, Isaac Newton, John Nash: game theory, John von Neumann, linear programming, loss aversion, Louis Bachelier, mental accounting, moral hazard, Nash equilibrium, probability theory / Blaise Pascal / Pierre de Fermat, random walk, Richard Thaler, Robert Shiller, Robert Shiller, spectrum auction, statistical model, The Bell Curve by Richard Herrnstein and Charles Murray, The Wealth of Nations by Adam Smith, trade route, transaction costs, tulip mania, Vanguard fund
The problem is, in fact impossible, as by my method I am able to prove with all rigor."11 Fermat observes that Pythagoras was correct that a2 + b2 = c2, but a3 + b3 would not be equal to c3, nor would any integer higher than 2 fit the bill: the Pythagorean theorem works only for squaring. And then Fermat wrote: "I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain."12 With this simple comment he left mathematicians dumbfounded for over 350 years as they struggled to find a theoretical justification for what a great deal of empirical experimentation proved to be true. In 1993, an English mathematician named Andrew Wiles claimed that he had solved this puzzle after seven years of work in a Princeton attic. Wiles's results were published in the Annals of Mathematics in May 1995, but the mathematicians have continued to squabble over exactly what he had achieved. Fermat's Last Theorem is more of a curiosity than an insight into how the world works. But the solution that Fermat and Pascal worked out to the problem of the points has long since been paying social dividends as the cornerstone of modem insurance and other forms of risk management.
Pathfinders: The Golden Age of Arabic Science by Jim Al-Khalili
agricultural Revolution, Albert Einstein, Andrew Wiles, Book of Ingenious Devices, colonial rule, Commentariolus, Dmitri Mendeleev, Eratosthenes, Henri Poincaré, invention of the printing press, invention of the telescope, invention of the wheel, Isaac Newton, Islamic Golden Age, Joseph Schumpeter, retrograde motion, Silicon Valley, Simon Singh, stem cell, Stephen Hawking, the scientific method, Thomas Malthus, trade route, William of Occam
In fact, for three and a half centuries it should have more correctly been called a ‘conjecture’ rather than a theorem. Stated mathematically, it says that there are no whole number values for x, y and z such that xn + yn = zn, when n is greater than 2. For instance, there are no whole numbers for which the sum of the cubes of two integers equals the cube of another (unless they are all equal to zero, of course). A proof was finally found by the British mathematician Andrew Wiles in 1995, and I for one have no intention of checking his method, since it runs to more than a hundred pages and took him seven years to complete. None of this should be credited to Diophantus, of course, but what it is meant to show is that his interest, like Fermat’s, was more in the properties of numbers than in the algebraic manipulation of symbols. In the seventh century the great Hindu mathematician Brahmagupta took up the challenge of tackling another Diophantine equation – what is known today as the Pell equation, which has the general form x2 − ay2 = 1.
How I Became a Quant: Insights From 25 of Wall Street's Elite by Richard R. Lindsey, Barry Schachter
Albert Einstein, algorithmic trading, Andrew Wiles, Antoine Gombaud: Chevalier de Méré, asset allocation, asset-backed security, backtesting, bank run, banking crisis, Black-Scholes formula, Bonfire of the Vanities, Bretton Woods, Brownian motion, business process, buy low sell high, capital asset pricing model, centre right, collateralized debt obligation, corporate governance, correlation coefficient, Credit Default Swap, credit default swaps / collateralized debt obligations, currency manipulation / currency intervention, discounted cash flows, disintermediation, diversification, Emanuel Derman, en.wikipedia.org, Eugene Fama: efficient market hypothesis, financial innovation, fixed income, full employment, George Akerlof, Gordon Gekko, hiring and firing, implied volatility, index fund, interest rate derivative, interest rate swap, John von Neumann, linear programming, Loma Prieta earthquake, Long Term Capital Management, margin call, market friction, market microstructure, martingale, merger arbitrage, Nick Leeson, P = NP, pattern recognition, pensions crisis, performance metric, prediction markets, profit maximization, purchasing power parity, quantitative trading / quantitative ﬁnance, QWERTY keyboard, RAND corporation, random walk, Ray Kurzweil, Richard Feynman, Richard Feynman, Richard Stallman, risk-adjusted returns, risk/return, shareholder value, Sharpe ratio, short selling, Silicon Valley, six sigma, sorting algorithm, statistical arbitrage, statistical model, stem cell, Steven Levy, stochastic process, systematic trading, technology bubble, The Great Moderation, the scientific method, too big to fail, trade route, transaction costs, transfer pricing, value at risk, volatility smile, Wiener process, yield curve, young professional
Some problems are harder than others, and to assess how hard a problem is we look at how long it has remained unsolved and how many great mathematicians have failed to solve it. To my mind, the greatest mathematicians are like explorers—they search for solutions to problems that no one knows how to solve and they risk achieving nothing. A terrific example of this is the famous Fermat’s Last Theorem.3 Fermat stated his famous theorem around 1637 and great mathematicians of every generation tried to solve it until Andrew Wiles finally cracked it in the early 1990s after famously spending seven years working on the problem in secret and alone. Why did he do it? Why does any mathematician work on hard problems? Consider the stakes. In the case of Fermat’s Last Theorem, no one knew for sure whether we could solve the problem at all. The fundamental question—“Can it be solved?”—had not been resolved. Yet Wiles risked seven years of his career on a hunch that he could solve it.
Big Bang by Simon Singh
Albert Einstein, Albert Michelson, All science is either physics or stamp collecting, Andrew Wiles, anthropic principle, Arthur Eddington, Astronomia nova, Brownian motion, carbon-based life, Cepheid variable, Chance favours the prepared mind, Commentariolus, Copley Medal, cosmic abundance, cosmic microwave background, cosmological constant, cosmological principle, dark matter, Dava Sobel, Defenestration of Prague, discovery of penicillin, Dmitri Mendeleev, Edmond Halley, Edward Charles Pickering, Eratosthenes, Ernest Rutherford, Erwin Freundlich, Fellow of the Royal Society, fudge factor, Hans Lippershey, Harlow Shapley and Heber Curtis, Harvard Computers: women astronomers, Henri Poincaré, horn antenna, if you see hoof prints, think horses—not zebras, Index librorum prohibitorum, invention of the telescope, Isaac Newton, John von Neumann, Karl Jansky, Louis Daguerre, Louis Pasteur, luminiferous ether, Magellanic Cloud, Murray Gell-Mann, music of the spheres, Olbers’ paradox, On the Revolutions of the Heavenly Spheres, Paul Erdős, retrograde motion, Richard Feynman, Richard Feynman, scientific mainstream, Simon Singh, Solar eclipse in 1919, Stephen Hawking, the scientific method, Thomas Kuhn: the structure of scientific revolutions, unbiased observer, V2 rocket, Wilhelm Olbers, William of Occam
I am just grateful that this postscript essay has at least allowed me the opportunity to get these three curiosities out of my system. Read on Have You Read? Other titles by Simon Singh Fermat’s Last Theorem In 1963 a schoolboy browsing in his local library stumbled across the world’s greatest mathematical problem: Fermat’s Last Theorem, a puzzle that has baffled mathematicians for over 300 years. Aged just ten, Andrew Wiles dreamed that he would crack it. Wiles’s lifelong obsession with a seemingly simple challenge set by a long-dead Frenchman is an emotional tale of sacrifice and extraordinary determination. In the end Wiles was forced to work in secrecy and isolation for seven years, harnessing all the power of modern maths to achieve his childhood dream. Many before him had tried and failed, including an eighteenth-century philanderer who was killed in a duel.
Structure and interpretation of computer programs by Harold Abelson, Gerald Jay Sussman, Julie Sussman
Andrew Wiles, conceptual framework, Douglas Hofstadter, Eratosthenes, Fermat's Last Theorem, Gödel, Escher, Bach, industrial robot, information retrieval, iterative process, loose coupling, probability theory / Blaise Pascal / Pierre de Fermat, Richard Stallman, Turing machine
The most famous of Fermat's results – known as Fermat's Last Theorem – was jotted down in 1637 in his copy of the book Arithmetic (by the third-century Greek mathematician Diophantus) with the remark “I have discovered a truly remarkable proof, but this margin is too small to contain it.” Finding a proof of Fermat's Last Theorem became one of the most famous challenges in number theory. A complete solution was finally given in 1995 by Andrew Wiles of Princeton University. 46 The reduction steps in the cases where the exponent e is greater than 1 are based on the fact that, for any integers x, y, and m, we can find the remainder of x times y modulo m by computing separately the remainders of x modulo m and y modulo m, multiplying these, and then taking the remainder of the result modulo m. For instance, in the case where e is even, we compute the remainder of be/2 modulo m, square this, and take the remainder modulo m.
I Am a Strange Loop by Douglas R. Hofstadter
Albert Einstein, Andrew Wiles, Benoit Mandelbrot, Brownian motion, double helix, Douglas Hofstadter, Georg Cantor, Gödel, Escher, Bach, Isaac Newton, James Watt: steam engine, John Conway, John von Neumann, mandelbrot fractal, pattern recognition, Paul Erdős, place-making, probability theory / Blaise Pascal / Pierre de Fermat, publish or perish, random walk, Ronald Reagan, self-driving car, Silicon Valley, telepresence, Turing machine
And then along came a vast team of mathematicians who had set their collective bead on the “big game” of Fermat’s Last Theorem (the notorious claim, originally made by Pierre de Fermat in the middle of the seventeenth century, that no positive integers a, b, c exist such that an + bn equals c n, with the exponent n being an integer greater than 2). This great international relay team, whose final victorious lap was magnificently sprinted by Andrew Wiles (his sprint took him about eight years), was at last able to prove Fermat’s centuries-old claim by using amazing techniques that combined ideas from all over the vast map of contemporary mathematics. In the wake of this team’s revolutionary work, new paths were opened up that seemed to leave cracks in many famous old doors, including the tightly-closed door of the small but alluring Fibonacci power mystery.
Structure and Interpretation of Computer Programs, Second Edition by Harold Abelson, Gerald Jay Sussman, Julie Sussman
Andrew Wiles, conceptual framework, Douglas Hofstadter, Eratosthenes, Gödel, Escher, Bach, industrial robot, information retrieval, iterative process, loose coupling, probability theory / Blaise Pascal / Pierre de Fermat, Richard Stallman, Turing machine, wikimedia commons
The most famous of Fermat’s results—known as Fermat’s Last Theorem—was jotted down in 1637 in his copy of the book Arithmetic (by the third-century Greek mathematician Diophantus) with the remark “I have discovered a truly remarkable proof, but this margin is too small to contain it.” Finding a proof of Fermat’s Last Theorem became one of the most famous challenges in number theory. A complete solution was finally given in 1995 by Andrew Wiles of Princeton University. 46 The reduction steps in the cases where the exponent is greater than 1 are based on the fact that, for any integers , , and , we can find the remainder of times modulo by computing separately the remainders of modulo and modulo , multiplying these, and then taking the remainder of the result modulo . For instance, in the case where is even, we compute the remainder of modulo , square this, and take the remainder modulo .
Albert Einstein, Andrew Wiles, asset allocation, availability heuristic, backtesting, Black Swan, capital asset pricing model, cognitive dissonance, compound rate of return, Daniel Kahneman / Amos Tversky, distributed generation, Elliott wave, en.wikipedia.org, feminist movement, hindsight bias, index fund, invention of the telescope, invisible hand, Long Term Capital Management, mental accounting, meta analysis, meta-analysis, p-value, pattern recognition, Ponzi scheme, price anchoring, price stability, quantitative trading / quantitative ﬁnance, Ralph Nelson Elliott, random walk, retrograde motion, revision control, risk tolerance, risk-adjusted returns, riskless arbitrage, Robert Shiller, Robert Shiller, Sharpe ratio, short selling, statistical model, systematic trading, the scientific method, transfer pricing, unbiased observer, yield curve, Yogi Berra
It is a dog. Therefore, has 4 legs. If a dog, then has 4 legs. It is not a dog. Therefore, not 4 legs. Denying The Consequent Fallacy: Affirming Consequent If a dog, then has 4 legs. Legs not equal to 4. Therefore not a dog. If a dog, then has 4 legs. Has 4 legs. Therefore dog. FIGURE 3.5 Conditional syllogisms: example. hinted at in the margin of a book in 1665 but not proven until 1994, by Andrew Wiles and Richard Taylor. Inductive Logic Induction is the logic of discovery. It aims to reveal new knowledge about the world by reaching beyond the knowledge contained in the premises of an inductive argument. However, this new knowledge comes with a price—uncertainty. Conclusions reached by induction are inherently uncertain. That is, they can only be true with some degree of probability. Thus, the notion of probability is intimately connected with induction.