Fermat's Last Theorem

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Against the Gods: The Remarkable Story of Risk by Peter L. Bernstein

Alan Greenspan, Albert Einstein, Alvin Roth, Andrew Wiles, Antoine Gombaud: Chevalier de Méré, Bayesian statistics, behavioural economics, Big bang: deregulation of the City of London, Bretton Woods, business cycle, buttonwood tree, buy and hold, capital asset pricing model, cognitive dissonance, computerized trading, 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 engineering, financial innovation, full employment, Great Leap Forward, index fund, invention of movable type, Isaac Newton, John Nash: game theory, John von Neumann, Kenneth Arrow, linear programming, loss aversion, Louis Bachelier, mental accounting, moral hazard, Myron Scholes, Nash equilibrium, Norman Macrae, Paul Samuelson, Philip Mirowski, Post-Keynesian economics, probability theory / Blaise Pascal / Pierre de Fermat, prudent man rule, random walk, Richard Thaler, Robert Shiller, Robert Solow, spectrum auction, statistical model, stocks for the long run, The Bell Curve by Richard Herrnstein and Charles Murray, The Wealth of Nations by Adam Smith, Thomas Bayes, trade route, transaction costs, tulip mania, Vanguard fund, zero-sum game

At one point, Fermat believed he might have discovered a formula that would always produce a prime number as its solution, but he warned that he could not prove theoretically that the formula would always do so. His formula produced 5, then 17, then 257, and finally 65,537, all of which were prime numbers; the next number to result from his formula was 4,294,967,297. Fermat is perhaps most famous for propounding what has come to be known as "Fermat's Last Theorem," a note that he scribbled in the margin of his copy of Diophantus's book Arithmetic. The notion is simple to describe despite the complexity of its proof. The Greek mathematician Pythagoras first demonstrated that the square of the longest side of a right triangle, the hypotenuse, is equal to the sum of the squares of the other two sides.

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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. The solution to the problem of the points begins with the recognition that the player who is ahead when the game stops would have the greater probability of winning if the game were to continue.

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Eric Temple Bell, one of Gauss's biographers, believes that mathematics might have been fifty years further along if Gauss had been more forthcoming; "Things buried for years or decades in [his] diary would have made half a dozen great reputations had they been published promptly."' Fame and secretiveness combined to make Gauss an incurable intellectual snob. Although his primary achievement was in the theory of numbers, the same area that had fascinated Fermat, he had little use for Fermat's pioneering work. He brushed off Fermat's Last Theorem, which had stood as a fascinating challenge to mathematicians for over a hundred years, as "An isolated proposition with very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of."9 This was not an empty boast.

Structure and interpretation of computer programs by Harold Abelson, Gerald Jay Sussman, Julie Sussman

Andrew Wiles, conceptual framework, Donald Knuth, Douglas Hofstadter, Eratosthenes, Fermat's Last Theorem, functional programming, Gödel, Escher, Bach, higher-order functions, industrial robot, information retrieval, iterative process, Ivan Sutherland, Johannes Kepler, loose coupling, machine translation, Multics, probability theory / Blaise Pascal / Pierre de Fermat, Richard Stallman, Turing machine

The first published proof was given by Euler in 1736 (and an earlier, identical proof was discovered in the unpublished manuscripts of Leibniz). 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.

Diaspora by Greg Egan

conceptual framework, cosmic abundance, cosmic microwave background, Fermat's Last Theorem, gravity well, Jacquard loom, stem cell, telepresence, telepresence robot, Turing machine

"Exponential growth is a curse in all its forms. You know it almost wiped out the fleshers? If we were insane enough, we could try turning the whole planet-or the whole galaxy-into some kind of machine able to exert the necessary brute computational force . . . but even then, I doubt we'd reach Fermat's Last Theorem before the end of the universe." Yatima had persisted. "You could make the programs more sophisticated. More discriminating. Let them generalize from examples, form conjectures ... aim for proofs." Radiya had conceded, "Perhaps it could be done. Some fleshers tried that approach before the Introdus-- and if you're short-lived, slow, and easily distracted, it almost makes sense to let unthinking software find the lodes you'd never hit before you died.

Adapt: Why Success Always Starts With Failure by Tim Harford

An Inconvenient Truth, Andrew Wiles, banking crisis, Basel III, behavioural economics, Berlin Wall, Bernie Madoff, Black Swan, Boeing 747, business logic, car-free, carbon footprint, carbon tax, Cass Sunstein, charter city, Clayton Christensen, clean water, cloud computing, cognitive dissonance, complexity theory, corporate governance, correlation does not imply causation, creative destruction, credit crunch, Credit Default Swap, crowdsourcing, cuban missile crisis, Daniel Kahneman / Amos Tversky, Dava Sobel, Deep Water Horizon, Deng Xiaoping, disruptive innovation, double entry bookkeeping, Edmond Halley, en.wikipedia.org, Erik Brynjolfsson, experimental subject, Fall of the Berlin Wall, Fermat's Last Theorem, financial engineering, Firefox, food miles, Gerolamo Cardano, global supply chain, Great Leap Forward, Herman Kahn, Intergovernmental Panel on Climate Change (IPCC), Isaac Newton, Jane Jacobs, Jarndyce and Jarndyce, Jarndyce and Jarndyce, John Harrison: Longitude, knowledge worker, loose coupling, Martin Wolf, mass immigration, 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, SpaceShipOne, special economic zone, spectrum auction, Steve Jobs, supply-chain management, tacit knowledge, the market place, The Wisdom of Crowds, too big to fail, trade route, Tyler Cowen, Tyler Cowen: Great Stagnation, Virgin Galactic, web application, X Prize, zero-sum game

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”?

From Bacteria to Bach and Back: The Evolution of Minds by Daniel C. Dennett

Ada Lovelace, adjacent possible, Alan Turing: On Computable Numbers, with an Application to the Entscheidungsproblem, AlphaGo, Andrew Wiles, Bayesian statistics, bioinformatics, bitcoin, Bletchley Park, Build a better mousetrap, Claude Shannon: information theory, computer age, computer vision, Computing Machinery and Intelligence, CRISPR, deep learning, disinformation, double entry bookkeeping, double helix, Douglas Hofstadter, Elon Musk, epigenetics, experimental subject, Fermat's Last Theorem, Gödel, Escher, Bach, Higgs boson, information asymmetry, information retrieval, invention of writing, Isaac Newton, iterative process, John von Neumann, language acquisition, megaproject, Menlo Park, Murray Gell-Mann, Necker cube, Norbert Wiener, pattern recognition, phenotype, Richard Feynman, Rodney Brooks, self-driving car, social intelligence, sorting algorithm, speech recognition, Stephen Hawking, Steven Pinker, strong AI, Stuart Kauffman, TED Talk, The Wealth of Nations by Adam Smith, theory of mind, Thomas Bayes, trickle-down economics, Turing machine, Turing test, Watson beat the top human players on Jeopardy!, Y2K

There is not just coevolution between memes and genes; there is codependence between our minds’ top-down reasoning abilities and the bottom-up uncomprehending talents of our animal brains. And if our future follows the trajectory of our past—something that is partly in our control—our artificial intelligences will continue to be dependent on us even as we become more warily dependent on them. 101Simon Singh, “The Whole Story,” http://simonsingh.net/books/fermats -last-theorem/the-whole-story/, is by far the most accessible account I have found, an edited version of an essay published in Prometheus magazine. 102Watson, notably, does have some specialized self-monitoring layers: it must assess its “confidence” in each candidate answer and can also adjust its threshold of confidence, taking more or less risk in answering.

Antarctica by Kim Stanley Robinson

company town, Exxon Valdez, Fermat's Last Theorem, gravity well, hiring and firing, Kim Stanley Robinson, late capitalism, lateral thinking, no-fly zone, Occam's razor, Turing test, Zeno's paradox

The home-conducting subculture was of course mostly uninterested in these overplayed warhorses of the concert hall, exactly the material from which computer conducting had freed them; and indeed Wade was coming to this piece from several years' work on unrecorded music in the opuses of D'Indy, Poulenc, and Martinu; but recently there had been movements, obviously related, to explore both "sensuous surface" and the neglected warhorses, and Wade felt that this performance he had put together was perhaps good enough to justify sending it out, at least to the Tchaikovsky crowd, the finale of the Fifth being somewhat in the nature of Fermat's last theorem, to which his version might be the Wiles proof. But there was the phone beeping. And of course it was Phil, Senator Philip Krishna Chase of California, his boss and friend, calling from the other side of the world and uncertain of the time difference as always. Phil had two years before lost the chair of the Senate Foreign Relations Committee, and was often spoken of as passe following the Republican retaking of both Senate and House that year; he had seldom been in Washington since (not that he had been here all that often before), and he appeared intent on proving that as the great telesenator he could do his job while on a more-or-less permanent junket/global pilgrimage (depending on who was characterizing it).

Coders at Work by Peter Seibel

Ada Lovelace, Bill Atkinson, bioinformatics, Bletchley Park, Charles Babbage, cloud computing, Compatible Time-Sharing System, Conway's Game of Life, Dennis Ritchie, domain-specific language, don't repeat yourself, Donald Knuth, fallacies of distributed computing, fault tolerance, Fermat's Last Theorem, Firefox, Free Software Foundation, functional programming, George Gilder, glass ceiling, Guido van Rossum, history of Unix, HyperCard, industrial research laboratory, information retrieval, Ken Thompson, L Peter Deutsch, Larry Wall, loose coupling, Marc Andreessen, Menlo Park, Metcalfe's law, Multics, no silver bullet, Perl 6, premature optimization, publish or perish, random walk, revision control, Richard Stallman, rolodex, Ruby on Rails, Saturday Night Live, side project, slashdot, speech recognition, systems thinking, the scientific method, Therac-25, Turing complete, Turing machine, Turing test, type inference, Valgrind, web application

Seibel: I haven't really done a study, but it seems like the vast majority of mathematical and scientific papers are typeset with TeX these days. There must have been things you've seen typeset in TeX that made you think, “Wow, my program played a part in this.” Knuth: Well, the proof of Fermat's Last Theorem was one of those. It's one of the most famous mathematical papers. And it happens all the time that I see books that I know wouldn't have been written if the authors had had to go through channels the way they used to. It's again a little bit of the black-box thing. It used to be, you would have to type something up and that would go into a compositor and it would come back in galley proofs, and so on.

The Art of Computer Programming: Fundamental Algorithms by Donald E. Knuth

Charles Babbage, discrete time, distributed generation, Donald Knuth, fear of failure, Fermat's Last Theorem, G4S, Gerard Salton, Isaac Newton, Ivan Sutherland, Jacquard loom, Johannes Kepler, John von Neumann, linear programming, linked data, Menlo Park, probability theory / Blaise Pascal / Pierre de Fermat, sorting algorithm, stochastic process, Turing machine

. — SHERLOCK HOLMES, in The Valley of Fear A888) ANSWERS TO EXERCISES / am not bound to please thee with my answers. Shylock, in The Merchant of Venice (Act IV, Scene 1, Line 65) NOTES ON THE EXERCISES 1. An average problem for a mathematically inclined reader. 4. See W. J. LeVeque, Topics in Number Theory 2 (Reading, Mass.: Addison-Wesley, 1956), Chapter 3; P. Ribenboim, 13 Lectures on Fermat's Last Theorem (New York: Springer-Verlag, 1979); A. Wiles, Annals of Mathematics 141 A995), 443-551. SECTION 1.1 1. t -f- a, a -f- b, b <— c, c <— d, d -f- t. 2. After the first time, the values of the variables m and n are the previous values of n and r, respectively; and n > r. 3. Algorithm F (Euclid's algorithm).

The Art of Computer Programming: Sorting and Searching by Donald Ervin Knuth

card file, Charles Babbage, Claude Shannon: information theory, complexity theory, correlation coefficient, Donald Knuth, double entry bookkeeping, Eratosthenes, Fermat's Last Theorem, G4S, information retrieval, iterative process, John von Neumann, linked data, locality of reference, Menlo Park, Norbert Wiener, NP-complete, p-value, Paul Erdős, RAND corporation, refrigerator car, sorting algorithm, Vilfredo Pareto, Yogi Berra, Zipf's Law

Be off, or I'll kick you down stairs!" LEWIS CARROLL, Alice's Adventures Under Ground A862) NOTES ON THE EXERCISES 1. An average problem for a mathematically inclined reader. 3. See W. J. LeVeque, Topics in Number Theory 2 (Reading, Mass.: Addison-Wesley, 1956), Chapter 3; P. Ribenboim, 13 Lectures on Fermat's Last Theorem (New York: Springer-Verlag, 1979); A. Wiles, Annals of Mathematics 141 A995), 443-551. SECTION 5 1. Let p(l). . . p(n) and q(l)... q(n) be different permutations satisfying the condi- conditions, and let i be minimal with p(i) ^ q(i). Then p{i) = q(j) for some j > i, and q(i) = p(k) for some k > i.