Scientists Buzzing--Fermat’s Last Theorem May Have Been Proved

Mathematicians around the world were buzzing with excitement Monday over the prospect that Fermat’s Last Theorem, one of the oldest and most famous conjectures in mathematics, has finally been proved.

A Japanese mathematician working in Bonn, West Germany, has proposed what may be a proof of the conjecture, which has stumped the greatest minds since the French mathematician Pierre de Fermat proposed it in 1637. Experts who have examined the work of Yoichi Miyaoka at the Max Planck Institute in Bonn say they find no mistakes in it, but they are not yet saying for sure that the conjecture has been proved.

“It’s still not definite, but it looks fine,” Don Zagier, an American mathematician at the institute, said Monday by telephone from Bonn. “There’s no specific reason to say that it’s wrong, but there’s still a possibility that there’s a mistake somewhere.”

Fermat’s Last Theorem says that equations of the form “Xn plus Yn equals Zn “ have no solutions when n is a positive integer greater than 2.

When n is equal to 2, there are infinitely many solutions. An example is the famous Pythagorean Theorem from high school geometry: The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides: 32plus 42equals 52; 52plus 122equals 132; and so on.

Fermat wrote in the margin of one of his books that he had a proof that there are no solutions to this type of equation when n is greater than 2 but that the margin was too small to contain it. From that day to this, no one has found either Fermat’s proof or anyone else’s, but many, many people have tried and failed.

Many mathematicians regularly receive supposed (and incorrect) proofs from amateurs. The great number theorist Edmund Landau received so many such proofs that he had postcards printed that read, “Dear Sir or Madam: Your attempted proof of Fermat’s Theorem has been received and is herewith returned. The first mistake is on page --, line -- .”

So many efforts to prove Fermat’s Last Theorem have failed over the years that there has been speculation recently that it was independent of the basic assumptions that mathematicians make about numbers. As a result, some mathematicians thought, the theorem could neither be proved true nor could any counterexample be found to show that it was false.

But a week ago Friday, Miyaoka gave a talk to 30-odd mathematicians in Bonn in which he outlined an attack on Fermat’s Last Theorem based on proving a theorem in algebraic geometry. It was recently shown that that theorem would imply Fermat’s Last Theorem. If Miyaoka’s proof is correct, Fermat’s Last Theorem would follow as a corollary.

Zagier took extensive notes on Miyaoka’s talk, and these notes have now been circulated to experts in Paris, who have found no mistakes. “Some people felt there was a very good chance that it was going to work out,” Zagier said. Others raised questions, he said, but in each case Miyaoka has convinced them that his method is sound.

As word of Miyaoka’s talk has spread, dozens of mathematicians from around the world have been phoning Bonn to get direct word of this result. Miyaoka is expected to have a full manuscript available in the next day or two, which will enable precise checking of every step in his argument.

Final word will take at least a few more days.

Miyaoka’s approach through algebraic geometry has been successfully used before. Five years ago, Gerd Faltings proved the so-called Mordell Conjecture in the same way, a result that had significance for Fermat’s Last Theorem. By proving a theorem in algebraic geometry, Faltings was able to show that if there were any solutions to Fermat-type equations, there could not be infinitely many of them.

The successful use of this approach in the past lends credence to Miyaoka’s use of it now.

FERMAT’S LAST THEOREM Fermat’s Last Theorem says that equations of the form:

Xn + Yn = Zn

have no solutions when n is a positive integer greater than 2.