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WILES, Andrew. Modular Elliptic Curves and Fermat's Last Theorem. WITH: WILES, Andrew and TAYLOR, Richard. Ring-Theoretic Properties of Certain Hecke Algebras. IN: Annals of Mathematics, Second Series, Vol. 141, No. 3, pp. 443-572. (Princeton: Princeton University Press), 1995. Octavo, original printed paper wrappers. Housed in a custom three-quarter morocco clamshell box. $8800.First edition, in the journal Annals of Mathematics where it originally appeared, of Wiles' famous proof of Fermat's Last Theorem, which had confounded mathematicians for centuries.In a marginal note in the section of his copy of Diophantus' Arithmetica (1621) dealing with Pythagorean triples (positive whole numbers x, y, z satisfying x^2 + y^2 = z^2 Ð of which an infinite number exist), Fermat stated that the equation x^n + y^n = z^n, where n is any whole number greater than 2, has no solution in which x, y, z are positive whole numbers. Tantalizingly, Fermat wrote that he had found a wonderful proof but the margin of the book was too small to contain it. For 350 years, no mathematician succeeded in finding a proofÑthough many tried. Soon after the Second World War computers helped to prove the theorem for all values of n up to 500, then 1000, and then 10,000. In the 1980's Samuel S. Wagstaff of the University of Illinois raised the limit to 25,000 and more recently mathematicians could claim that Fermat's Last Theorem was true for all values of n up to 4,000,000. But no general proof was found until Andrew Wiles announced his proof, using the most advanced tools of modern mathematics, at a 1993 conference at the Isaac Newton Institute in Cambridge, England. (His former student Richard Taylor then helped him fix a flaw discovered in his original proof; both papers are present in this single journal issue.)Not wanting to be distractedÑor beaten to the punchÑWiles worked on his proof in secrecy for seven years. "To prove that something is true for an infinite number of cases required Wiles to pull together some of the most recent breakthroughs in number theory, and in addition invent new techniques of his ownÉ At each stage Wiles could never be sure that he could complete his proof. He realized that even if he did have the correct strategy, the mathematical techniques required might not yet existÑhe might be on the right track, but living in the wrong century. Eventually, in 1993, Wiles felt confident that his proof was reaching completion. The opportunity arose to announce his proof of a major section of the Shimura-Taniyama conjecture, and hence Fermat's Last Theorem, at a special conference to be held at the Isaac Newton Institute in Cambridge, England. Because this was his home town, where he had encountered the Last Theorem as a child, he decided to make a concerted effort to complete the proof in time for the conference. On June 23rd he announced his seven-year calculation to a stunned audience. His secret research program had apparently been a success, and the mathematical community and the worldÕs press rejoiced. The front page of the New York Times exclaimed 'At Last, Shout of "Eureka!" in Age-Old Math Mystery,' and Wiles appeared on television stations around the world" (Simon Singh, Fermat's Enigma).When peer review revealed a fatal flaw in Wiles' initial proof, he demanded the opportunity to correct the problem himself. After months of frustration, he took his former student Richard Taylor into his confidence. Returning to an approach Wiles had discarded early in his process, Wiles and Taylor discovered the solution, and his proof was indeed confirmed by the mathematical community shortly thereafter. "I haven't let go of this problem for nearly seven years," Wiles told the New York Times in 1993. "I've almost forgotten the experience of getting up and thinking about something elseÉ For many of us, [Fermat's] problem drew us in and we always considered it something you dream about but never actually doÉ There is a sense of loss, actually." Fi.
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