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Modular elliptic curves and Fermat's Last Theorem. [with] Ring-theoretic properties of certain Hecke algebras.

Princeton: Princeton University Press, 1995. First edition. First edition, journal issue, of his proof of Fermat's Last Theorem, which was perhaps the most celebrated open problem in mathematics. 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 x2 + y2 = z2 - of which an infinite number exist), Fermat stated that the equation xn + yn = zn, where n is any whole number greater than 2, has no solution in which x, y, z are positive whole numbers. Fermat followed this assertion with what is probably the most tantalising comment in the history of mathematics: 'I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain.' Fermat believed he could prove his theorem, but he never committed his proof to paper. After his death, mathematicians across Europe, from the enthusiastic amateur to the brilliant professional, tried to rediscover the proof of what became known as Fermat's Last Theorem, but for more than 350 years none succeeded, nor could anyone disprove the theorem by finding numbers x, y, z which did satisfy Fermat's equation. When the great German mathematician David Hilbert was asked why he never attempted a proof of Fermat's Last Theorem, he replied, "Before beginning I should have to put in three years of intensive study, and I haven't that much time to squander on a probable failure." Soon after the Second World War computers helped to prove the theorem for all values of n up to five hundred, then one thousand, and then ten

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**1995**