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There are failures too

An attempt to realize the hope created by Faltings' proof, by delimiting once more the scope of potential solutions to Fermat's equation, was made in 1988 by Yoichi Miyaoka of the Tokyo Metropolitan University. As a sequel to a proof published by the Soviet mathematician A.N. Parshin a year earlier, Miyaoka presented a proof, using a technique similar to that of Faltings' proof, showing that the number of powers which contain solutions to FLT is finite, namely: there is a number n, such that for each prime power larger than n FLT is true. If we are lucky enough and this n is smaller than the number for which FLT was already checked and found to be true, then FLT has been proven. The proof of Miyaoka gained large publicity, although it faced some skepticism. Here is how the Science magazine told his readers about the news:

"For the first time in memory, the mathematics community is optimistic that its most famous open problem - Fermat's Last Theorem - may finally have been proved. Although no one will be completely confident until all the details have been thoroughly checked, those involved feel that Miaoka's proof has the best chance yet of settling the centuries-old problem."

Well, after a few weeks a non-reparable flaw was found in the proof.

Mishaps like this are not rare in the history of FLT. Ferdinand Lindemann, who solved the ancient problem of squaring the circle, namely constructing a square with the same area as a given circle using ruler and compasses alone, and so put an end to a two-thousand-year-old problem, tried to solve FLT too. In 1901, he published a seventeen pages proof of FLT, but it was found to be wrong. Seven years he tried to find a correction to that proof, but when he published his new proof, extended to sixty-three pages, a flaw was found almost in the beginning. So FLT offers some condolences to those who have hard time with mathematics - great mathematician also make mistakes sometimes.

Not every flaw in a proof makes it obsolete. In Euler's proof for the validity of FLT for n=3, there was a flaw, but it was removed, using methods not unknown to Euler. About 160 years after Kummer displayed his proofs about irregular primes, Vandiver found that the proofs were incomplete, and some time later managed to complete them. Moreover, not every wrong proof is useless. As in sports, where the victory is the main target but sometimes participating in the competition is an achievement by itself, so is the case in our story. In an article named How Not to Prove Fermat's Last Theorem, the mathematician John McCleary describes his original, but failed, attempt to prove FLT. "I feel that the method is of considerable interest in spite of its lack of success", he says in the beginning of the article.

Should mathematicians be ashamed of having such mistakes? Here is the answer of Keith Devlin, a mathematician of Stanford University:

"Only if it is shameful to push the limits of human mental ability, producing arguments that are so intricate that it can take the world's best experts weeks or months to decide if they are correct or not. As they say in Silicon Valley, where I live, if you haven't failed recently, you're not trying hard enough. No, I am not ashamed. I'm proud to be part of a profession that does not hesitate to tackle the hardest challenges, and does not hesitate to applaud those brave individuals who strive to reach the highest peaks, who stumble just short of the summit, but perhaps discover an entire new mountain range in the process."

Keith Devlin, The Shame of It, Devlin's Angle, MAA Online, May 2003.

© David Shay, 2003



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