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Wiles, Sir Andrew John

theorem proof fermat’s conjecture

(1953– ) British mathematician: proved Fermat’s Last Theorem.

Wiles was born and in part educated in Cambridge. When only ten years old, he became fascinated by Fermat’s Last Theorem, which states that there are no solutions for x n + y n = z n where n > 2. This theorem (more accurately a conjecture) had first been proposed by Pierre de sometime around 1637. Typically of Fermat, he had not written down his proof, but had scribbled the equation in the margin of a book together with the note ‘I have discovered a truly marvellous demonstration of this proposition, which this margin is too narrow to contain.’ Although a small number of specific cases were soon proved (for n = 3 and 4, etc.), and later for all values of n up to 125 000, the combined efforts of generations of mathematicians, over the following three and a half centuries, failed to provide a general proof, and many believed that Fermat must have been mistaken, and that the theorem was inherently unprovable.

After completing a first degree at Oxford, and a Cambridge doctorate on the subject of elliptic curves, Wiles moved to Princeton in the 1980s. Late in 1986 he decided to focus on the proof of Fermat’s Last Theorem, a topic he worked on single-mindedly and in secret until 1993, when he announced his proof. Unfortunately, a flaw was soon found in his logic, but a year and a half later he was able to resolve this satisfactorily, and the final proof was published in 1995.

Wiles’s proof, which runs to over 100 pages, can probably only be fully understood by a small group of fellow number theorists, but it builds on work by a German mathematician, Gerhard Frey, who had shown that that if Fermat’s Last Theorem is true, then the Taniyama–Shimura Conjecture (which proposes that every elliptic equation has an equivalent modular form) must also be true. Wiles succeeded in proving the Taniyama–Shimura Conjecture to be true, and therefore the Last Theorem also.

Whilst proving the Last Theorem may appear to be of little practical purpose, the series of new mathematical techniques developed by Wiles in the process of his proof, together with the proof of the Taniyama–Shimura Conjecture, and his resulting linkage between elliptic equations and modular forms, has revolutionized mathematics, and enabled others to attack a wide range of other problems.

Wiles was knighted in 2000. He held a professor-ship of mathematics at Princeton from 1982–8 and from 1990 onwards.

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