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# Longtime Math Mystery Reported Solved

June 24, 1993| From Associated Press

NEW YORK — A mathematician from Princeton University claims to have solved the most famous problem in mathematics with a dense, 200-page argument proving a centuries-old theorem, a colleague said Wednesday.

"When we heard it, people started walking on air," said Simon Kochen, chairman of the Princeton mathematics department. "It was an incredible feeling that this has been done after all this time."

The proof was presented Wednesday at a conference of mathematicians at Cambridge University in England by Andrew Wiles, a Princeton mathematics professor. Wiles couldn't be reached Wednesday night in Cambridge.

Now, for Wiles, the waiting begins. Experts will spend days poring over his argument, trying to find flaws.

"There have been proofs claimed in the past, and eventually somebody found an error. Until it's checked and actually published, it's hard to say," said Tom Apostol, a mathematician at Caltech, in Pasadena.

"A lot of us believe the theorem is true, and we think a proof is going to come up eventually, and maybe the time is right," said Apostol, when told of the claim. "I hope it's correct. It will be a very exciting chapter in the history of mathematics."

The theorem--actually a conjecture--was stated by the French mathematician Pierre de Fermat, who with Rene Descartes was considered one of the leading mathematicians of the 17th Century.

The problem is intriguingly simple: If "n" represents any whole number larger than two, there is no solution to the equation "x to the nth power plus y to the nth power equals z to the nth power."

There are solutions if "n" equals two--for example, three squared (9) plus four squared (16) equals five squared (25). But if "n" equals three or more, according to Fermat, there are no solutions.

"This is something you could state to a high school boy, but it's so, so difficult to prove," Kochen said.

Interest in the problem has been especially keen because of a mischievous note Fermat left in the margin of a book. He claimed he had found an "admirable proof of this theorem, but the margin is too narrow to contain it."

"Most people don't anymore believe he had a proof for the general case," Kochen said.

Fermat did prove the truth of the theorem in certain special cases, as have others.

In 1988, a Japanese mathematician, Yoichi Miyaoka, stunned the mathematical world when he claimed an overall proof, only to find the claim wither under scrutiny.

Wiles' proof relies on the work of scores of mathematicians over the centuries, but it adds a conceptual breakthrough, Kochen said.

"There are many others whose work Wiles had to use," Kochen said. "He was throwing the kitchen sink at it, using all kinds of techniques that had been developed in recent years." Wiles has been working on the problem for five years, Kochen said.