Fermat's last theorem: a promising approach.

Fermat's last theorem: A promising approach

The end of a centuries-long search for a proof of Fermat's last theorem, one of the most famous unsolved problems in mathematics, may at last be in sight. A Japanese mathematician, Yoichi Miyaoka of the Tokyo Metropolitan University, has proposed a proof for a key link in a chain of reasoning that establishes the theorem's truth. If Miyaoka's proof survives the mathematical community's intense scrutiny, then fermat's conjecture (as it ought to be called until a proof is firmly established) can truly be called a theorem.

Miyaoka's method builds on work done by several Russian mathematicians and links important ideas in three mathematical fields: number theory, algebra and geometry. Though highly technical, his argument fills fewer than a dozen manuscript pages -- short for such a significant mathematical proof. Miyaoka recently presented a sketch of his ideas at a seminar at the Max Planck Institute for Mathematics in Bonn, West Germany.

"It looks very nice," mathematician Don B. Zagier of the Max Planck Institute told Science News. "There are many nice ideas, but it's very subtle, and there could easily be a mistake. It'll certainly take days, if not weeks, until the proof's completely checked."

Fermat's conjecture is related to a statement by the ancient Greek mathematician Diophantus, who observed that there are positive integers, X, and y, and z, that satisfy the equation X.sup.2 + y.sup.2 = z.sup.2.. For example, if x = 3 and y = 4, then z = 5. In fact, this equation has an infinite number of such solutions.

In the 17th century, French amateur mathematician Pierre de fermat, while reading a book by Diophantus, scribbled a note in a margin proposing that there are no positive-integer solutions to the equation x.sup.n + y.sup.n = z.sup.n., when n is greater than 2. In other words, when n = 3, no set of positive integers satisfies the equation x.sup.3 + y.sup.3 = z.sup.3., and so on. Then, in a tantalizing sentence that was to haunt mathematicians for centuries to come, Fermat added that althouhg he had a wonderful proof for the theorem, he didn't have enough room to write it out.

Later mathematicians found proofs for a number of special cases, and a computer search performed a decade ago showed that Fermat's last theorem was true for all exponents less than 125,000. But despite the efforts of innumerable mathematicians, a proof for the general case remained elusive ('SN: 6/20/87, p.397).

In 1983, Gerd Faltings, now at Princeton (N.J.) University, opened up a new direction in the search for a proof. As one consequence of his proof of the Mordell conjecture (SN: 7/23/83, p.58), he showed that if there are any solutions to Fermat's equations, then there are only a finite number of them for each value of n. However, that was still far from the assertion that there are no such solutions.

Some of the key ideas for Falting's proof came from the work of Russian mathematician S. Arakelov, who was looking for connections between prime numbers, curves and geometrical surfaces. Both Arakelov and Faltings found that analogs of certain classical theorems already well established for geometrical surfaces could apply to curves and provide information about statements, such as Fermat's last theorem, that involve only integers.

About a year ago, A.N. Parshin of the Steklov Institute in Moscow, following Arakelov's lead, proved that if the arithmetical analog of an inequality, or bound, governing certain geometrical structures were true, then Fermat's last theorem would also be true. That inequality, in its original geometric form, had been discovered by Miyaoka and Shing-Tung Yau, now at Harvard University. By showing that the so-called Miyaoka-Yau inequality, in a modified form, also applies to the appropriate arithmetical structures, Miyaoka has apparently completed the chain of reasoning leading to a proof of Fermat's last theorem.

Miyaoka's results also demonstrate the increasing number of links being forged between diverse mathematical fields. If Miyaoka's proof turns out to be correct, then, according to some experts in arithmetical algebraic geometry (as this new field is called), similar methods may be useful for tackling a variety of tough mathematical problems.

"Fermat's last theorem is not important in mathematics directly," says Zagier. "It has no consequences." But the search for a proof has, over the years, prompted the development of much new mathematics. "It's a pity," he says, "that this goal may disappear."

Miyaoka is now busy carefully rechecking his proof and waiting for word from other expers who are studying his manuscript. "Things are looking good at the moment," says mathematician Lawrence C. Washington of the University of Maryland in College Park, who has been monitoring the situation. "But I don't think anyone wants to certify the proof yet." It's a time for both caution and excitement.