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A Year Later, Snag Persists In Math Proof

A Year Later, Snag Persists In Math Proof
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June 28, 1994, Section C, Page 1Buy Reprints
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ONE year ago, a shy and somewhat secretivemathematician stunned the world by announcing that he had proved Fermat's last theorem, the most famous unsolved problem in mathematics. Yet a year later, he still has not published his proof. Was the claim premature?

In short, it is probably too early to say. A subtle gap has been found in the manuscript of the proof. Its author, Dr. Andrew Wiles of Princeton University, is working in seclusion to close the gap. A tense quietus has settled over the community of mathematicians, a few predicting failure, others expressing confidence based on the fact that Dr. Wiles's proof is already agreed to have conquered part of another major mathematical peak known as the Taniyama conjecture.

It is routine for long mathematical works to circulate before publication and for reviewers to find flaws that the author can often fix. The ground broken by Dr. Wiles's work is so novel that it is hard to gauge the seriousness of the gap that has come to light.

Was the claim to have solved Fermat's last theorem premature, or will Dr. Wiles make good on his claim to have scaled a pinnacle of intellectual achievement? Dr. Wiles himself will not talk about his work on the proof. He did not answer telephone messages left at his office or a letter hand-delivered to his home in Princeton. His friends and colleagues at Princeton University say he seems to be in good spirits, but he does not offer progress reports nor, out of courtesy, do they ask how his work is going.

Dr. Peter C. Sarnak, a Princeton mathematician, says he sees Dr. Wiles every day at the community swimming pool. Dr. Wiles, he says, understands that he has made a major breakthrough. "Things can work out," he said in an interview. "If there is a new idea needed, then there's a new idea needed. But it's not like everything collapsed and there's nothing there."

And because the prize is so great, Dr. Wiles wants to solve the problem unaided. He has chosen to work alone, without asking others for help or even giving colleagues enough information that might risk their chiming in with unsolicited advice.

Although some mathematicians are getting restive, others are optimistic that the proof will be completed because, they say, the road map now in place is so compelling. On Climbing Everest

Dr. Andre Weil (pronounced VAY), a distinguished elder mathematician at the Institute for Advanced Study in Princeton, is a doubter. "I am willing to believe he has had some good ideas in trying to construct the proof, but the proof is not there," he said in an interview in Scientific American. "Also, to some extent, proving Fermat's theorem is like climbing Everest. If a man wants to climb Everest and falls short of it by 100 yards, he has not climbed Everest."

But Dr. Enrico Bombieri, a distinguished colleague of Dr. Weil at the Institute for Advanced Study in Princeton, said: "I'm rooting for him. I think he has a chance."

Dr. Kenneth Ribet, a mathematician at the University of California at Berkeley whose work was highly influential in showing the way for Dr. Wiles, said: "My best guess is that it will be proved and it will be proved by a method that is not too far from the method of Wiles. All the strategy is in place for the final proof."

The theorem was stated 357 years ago by a French mathematician and physicist, Pierre de Fermat. In his copy of Diophantus's equations, he wrote that he had a marvelous proof of the theorem but could not fit it into the margin. Mathematicians have been trying to supply the missing proof ever since.

The theorem is simple to state: The equation x-squared + y-squared = z-squared is true when the exponent is 2, or squared, but for no higher whole number.

Dr. Wiles's proposed proof was indirect. He tried to prove a much more sweeping theorem, called the Taniyama conjecture after the Japanese mathematician Yutaka Taniyama. The conjecture, which is central to much of modern mathematics, concerns elliptic curves, which are mathematical equations that give rise to objects that look like the surface of a doughnut. What Taniyama Said

The Taniyama conjecture says that all elliptic curves are modular, a mathematical description that has a peculiar geometric meaning, said Dr. Karl Rubin of Ohio State University. It means that every elliptic curve can be covered in a special way by a surface made up of the upper complex half plane. This is the plane of the complex numbers, those that consist of ordinary numbers appended to multiples of the square root of minus one. The Taniyama conjecture says that every elliptic curve can be enveloped by this infinite plane wrapping around it over and over again.

In 1987, Dr. Ribet, following a suggestion made in the mid-1980's by Dr. Gerhard Frey of the University of the Saarland in Germany, proved that if the Taniyama conjecture was true, so was Fermat's last theorem. It turned out that if there were solutions to the equations of Fermat's theorem, those equations would give rise to certain elliptic curves. But those elliptic curves, according to the Taniyama conjecture, could not exist -- they would not be modular. So if those curves could not exist, neither could the solutions to Fermat's equations. That would mean that the theorem saying such solutions were impossible was true.

"I knew when I heard about Frey and Ribet's result that the landscape had changed," Dr. Wiles said in an interview a year ago. "There is no question that what they did changed the problem for me psychologically."

After working quietly for seven years in an attic office of his Tudor house in Princeton, Dr. Wiles finally got to a point where he was ready to announce his results to the world. Using methods so new and so complex, Dr. Bombieri said, that they are on the very frontier of mathematics, Dr. Wiles devised a proof that could be understood by just a small fraction of mathematicians.

Last year, at a conference in Cambridge, England, attended by the few mathematicians who knew the difficult mathematics needed for his proof, Dr. Wiles convinced his audience that he had proved the Taniyama conjecture and, as a consequence, Fermat's last theorem. Discovery of Gap

Dr. Wiles submitted his 200-page manuscript to the journal Inventiones Mathematicae, edited by Dr. Barry Mazur of Harvard University, and awaited comments from reviewers. The gap in the proof emerged after Dr. Mazur sent out different parts of the paper to different experts.

To fill the gap requires what is turning out to be an unexpectedly difficult calculation of the precise upper limit of a mathematical object called the Selmer group in the semi stable case.

"He reduced the Taniyama conjecture to the calculation of some cohomology group," said Dr. Nicholas Katz of Princeton University. "The problem is in estimating the size of the cohomology group. In some incredibly technical sense, the mistake that hasn't been repaired occurs on page something, on line something else, in a 70- or 80-page proof of the estimate," Dr. Katz said.

It is impossible to say how hard it will be to fill the gap, Dr. Katz added. "After it either turns out to be O.K. or in some worst case scenario turned out false, then we can say in retrospect, 'Oh yes, it is relatively easy,' or 'No wonder it took awhile because it involves some incredibly new ideas.' If we can't prove it, if it's actually false or it can never be done, then we haven't advanced."

Dr. Katz said: "A year ago, everyone thought that his new ideas, which are very logically compelling, would certainly work. The only thing that might make one less optimistic now is the fact that time is passing. But the internal coherence of the whole wonderful idea of the thing is just as good as it was a year ago."

In any case, mathematicians say, the material preceding the gap is in itself a stunning achievement, and it is intact. Dr. Wiles has lectured on it at Princeton and has circulated his manuscript. It has been checked and re-checked by other experts and is, mathematicians say, certifiably correct.

"What has been done so far is a spectacular achievement," Dr. Rubin said.

Dr. Bombieri explained that this first part of the proof showed that the Taniyama conjecture was true for a particular infinite set of equations. "This is the first proof of the Taniyama conjecture for infinite cases," Dr. Bombieri said. "This is the first general result and it is a great achievement."

"This is a major, major breakthrough, no question about it," Dr. Sarnak said.

In an odd way, mathematicians say, the fact that Fermat's theorem is a consequence of the Taniyama conjecture has distorted the work on the conjecture. The Taniyama conjecture is central to much of mathematics. Fermat's last theorem has no consequences by itself, but it is so famous that it is the problem that the world wants to see solved.

Dr. John Conway, a Princeton University mathematician, explained: "There is this absolutely amazing achievement that Wiles has solved so many cases of the Taniyama conjecture and set up a new approach. Of course, what the public is interested in is not the Taniyama conjecture that they never heard of but Fermat. So you say, here's this fantastic thing that he's done, but there's an even more fantastic thing that he has not done."

A correction was made on 
July 20, 1994

An article in Science Times on June 28 about a snag in a previously an nounced proof of a famous math con jecture, Fermat's last theorem, mis stated the theorem. It states that the equation xn + yn = zn , where n is an integer greater than 2, can have no solution in which x, y and z are posi tive integers.

How we handle corrections

A version of this article appears in print on  , Section C, Page 1 of the National edition with the headline: A Year Later, Snag Persists In Math Proof. Order Reprints | Today’s Paper | Subscribe

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