Latest Proof Is Questioned : Fermat’s Last Theorem Remains a Math Mystery

Despite a flurry of excitement around the world over the last few weeks, Fermat’s Last Theorem, the oldest and most famous unsolved conjecture in mathematics, remains unproved.

After lengthy examination of a proof proposed by Yoichi Miyaoka, a Japanese number theorist working in West Germany, the experts have concluded that the proof is not complete.

“The answer seems to be no,” Don Zagier, an American mathematician and colleague of Miyaoka’s at the Max Planck Institute, said Tuesday by telephone from Bonn.

“It’s not capable of convincing the experts,” Zagier said of Miyaoka’s proposed solution. “The problem has not yet succumbed.”

Fermat’s Last Theorem, which was proposed by the French mathematician Pierre de Fermat around 1637, asserts that there is no combination of whole numbers x , y and z (other than 0) that satisfies the equation Xn + Yn = Zn , where n is a positive integer greater than 2.

Simply put, Fermat asserted that no perfect cube is the sum of two perfect cubes, no 4th power is the sum of two 4th powers, no 5th power is the sum of two 5th powers and so on ad infinitum. (However, as he knew, if n equals 2, there are infinitely many combinations of whole numbers that satisfy the equation. That is, there are infinitely many perfect squares th are the sum of two perfect squares. For example, 25 (52) is the sum of 9 (3c,um,52) and 16 (42).)

Fermat asserted his theorem in the margin of his copy of Bachet’s “Diophantus,” adding, “I have found a truly marvelous proof of this theorem, but this margin is too narrow to contain it.”

But Fermat’s proof (if he had one) has never been found, and neither has anyone else’s despite tremendous efforts by generations of professional and amateur mathematicians.

A month ago, Miyaoka, who is a prominent number theorist, touched off a wave of excitement when he gave a talk in Bonn in which he outlined a new attack on Fermat’s Last Theorem. Though would-be proofs of the theorem are regularly announced, this was the first one in a long time that mathematicians took seriously.

But close examination revealed problems in Miyaoka’s proof. About two weeks ago, he completed writing his paper giving full details of his proof, and he sent it off to experts in Europe and in the United States.

The experts found holes in the proof. “Serious questions have been raised,” Zagier said Tuesday. “I simply don’t know how deep the problems are. But it’s not looking as promising as it did a few weeks ago. There is still a possibility that after a certain amount of further work the basic ideas will still turn out to work. But there seems to be a lot more that has to be done.”