Careful scrutiny of a recently proposed proof of Fermat's last theorem (SN: 3/19/88, p.180) has turned up several flaws that cast doubt on the proof's validity. Japanese mathematician Yoichi Miyaoka, who is presently working at the Max Planck Institute for Mathematics in Bonn, West Germany, last week admitted that his proof has a serious problem. He is now studying how to revise his proof.

"That doesn't mean it's irreparable," says Barry Mazur of the Institut des Hautes Etudes Scientifiques near Paris, who has been discussing the proof with Miyaoka. "But it certainly means there's more work to do. It's a rather complex proof. If you change some things in one part of the proof, then all the other parts may be subject to change."

Fermat's last theorem concerns equations of the form xn + yn = zn. More than 300 years ago, amateur mathematician Pierre de Fermat stated that such equations have no positive-integer solutions when n is greater than 2, but he left no proof of his theorem. Ever since, innumerable mathematicians have tried to prove this conjecture. Although these attempts proved fruitless, they sometimes led to important new mathematical techniques that could be applied to other problems. And gradually, the potential for solving Fermat's theorem became linked with other questions in mathematics.

To solve the Fermat problem, Miyaoka, a specialist in algebraic geometry, which concerns the relationship between geometric surfaces and solutions of equations, ventured into a relatively new field known as arithmetic algebraic geometry. In this discipline, mathematicians look at surfaces that result when only integer solutions of equations are considered. Miyaoka tried to show that an inequality, or bound, that applies in algebraic geometry also fits an analogous case for equations with integer solutions.

That Miyaoka's initial attempt failed is hardly surprising or unsual in mathematical research. Normally, mathematicians privately circulate proposed proofs and discuss possible errors or oversights for months before gaining enough confidence to announce a proof publicly. In Miyaoka's case, the fact that Fermat's last theorem was such a famous unsolved problem put him in a spotlight that he had not sought.

Mathematicians are quite confident that someone, if not Miyaoka, will eventually come up with a proof of Fermat's last theorem. It's a little like waiting for an earthquake, says mathematician Ronald L. Graham of AT&T Bell Laboratories in Murray Hill, N.J. "All you know is that the longer you wait, the sooner the next one is going to be." -- I. Peterson