Magazine article Science News

Fermat's Famous Theorum: Proved at Last?

Magazine article Science News

Fermat's Famous Theorum: Proved at Last?

Article excerpt

A year ago, mathematician Andrew Wiles of Princeton University faced a troubling gap in the logic he had followed to prove Fermat's last theorem. Now, he has apparently found a way to bridge the gap and complete his proof of Fermat's famous conjecture.

Last week, in a move that caught the mathematical community by surprise, Wiles began distributing copies of two new manuscripts addressing the concerns that had been raised about his original argument. The first, lengthy paper announces the revised proof, still following quite closely the strategy Wiles had outlined in his lectures in June 1993 at the University of Cambridge in England (SN: 7/3/93, p.5).

The second, short paper, produced in collaboration with Cambridge mathematician Richard L. Taylor, contains mathematical reasoning justifying a key step in the main proof. Instead of solving the original problem, Wiles and Taylor avoided it by using a different approach to reach the same conclusion.

Both papers have been submitted for publication in the ANNALS OF MATHEMATICS.

"The proof looks really beautiful, but it's too soon to comment in detail [on its validity]," says Fernando Q. Gouvea of Colby College in Waterville, Maine. "Everybody's being very cautious." Gouvea has been attending a seminar at Harvard University, where Taylor has been discussing aspects of the proof.

Pierre de Fermat's claim, made more than 350 years ago, was that for each whole number greater than 2, the equation [x.sup.n] + [y.sup.n] = [z.sup.n] has no solutions that are positive whole numbers. Over the centuries that followed, many mathematicians tried to prove Fermat's conjecture but invariably failed.

The attack chosen by Wiles relied on recently discovered links between Fermat's conjecture and the theory of elliptic curves. By assuming that Fermat's last theorem is false, mathematicians could construct a "weird" elliptic curve that they believe, for other mathematical reasons, shouldn't exist. …

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