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THE PROOF OF FERMAT S LAST THEOREM. 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 thousand. 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 four million. But no general proof was found until 1995. "Between 1954 and 1986 a chain of events of occurred which brought Fermat s Last Theorem back into the mainstream. The incident which began everything happened in post-war Japan, when Yutaka Taniyama and Goro Shimura, two young academics, decided to collaborate on the study of elliptic curves and modular forms. These entities are from opposite ends of the mathematical spectrum, and had previously been studied in isolation. "Elliptic curves, which have been studied since the time of Diophantus, concern cubic equations of the form: y2 = (x + a).(x + b).(x + c), where a, b and c can be any whole number, except zero. The challenge is to identify and quantify the whole solutions to the equations, the solutions differing according to the values of a, b, and c. "Modular forms are a much more modern mathematical entity, born in the nineteenth century. They are functions, not so different to functions such as sine and cosine, but modular forms are exceptional because they exhibit a high degree of symmetry. For example, the sine function is slightly symmetrical because 2? can be added to any number, x, and yet the result of the function remains unchanged, i.e., sin x = sin (x + 2?). However, for modular forms the number x can be transformed in an infinite number of ways and yet the outcome of the function remains unchanged, hence they are said to be extraordinarily symmetric … "Despite belonging to a completely different area of the mathematics, Shimura and Taniyama began to suspect that the elliptic curves might be related to modular forms in a fundamental way. It seemed that the solutions for any one of the infinite number of elliptic curves could be derived from one of the infinite number of modular forms. Each elliptic curve seemed to be a modular form in disguise. This apparent unification became known as the Shimura-Taniyama conjecture, reflecting the fact that mathematicians were confident that it was true, but as yet were unable to prove it. The conjecture was considered important because if it were true problems about elliptic curves, which hitherto had been insoluble, could potentially be solved by using techniques develo.
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