Fermat famously claimed to have a proof of his last theorem that he didn’t have room to write down. Mathematicians have speculated ever since what this proof must have been, though everyone is convinced the proof must have been wrong.
The usual argument for Fermat being wrong is that since it took over 350 years, and some very sophisticated mathematics, to prove the theorem, it’s highly unlikely that Fermat had a simple proof. That’s a reasonable argument, but somewhat unsatisfying because it’s risky business to speculate on what a proof must require. Who knows how complex the proof of FLT in The Book is?
André Weil offers what I find to be a more satisfying argument that Fermat did not have a proof, based on our knowledge of Fermat himself. Dana Mackinzie summarizes Weil’s argument as follows.
Fermat repeatedly bragged about the n = 3 and n = 4 cases and posed them as challenges to other mathematicians … But the never mentioned the general case, n = 5 and higher, in any of his letters. Why such restraint? Most likely, Weil argues, because Fermat had realized that his “truly wonderful proof” did not work in those cases.
Every mathematician has had days like this. You think you have a great insight, but then you go out for a walk, or you come back to the problem the next day, and you realize that your great idea has a flaw. Sometimes you can go back and fix it. And sometimes you can’t.
The quotes above come from The Universe in Zero Words. I met Dana Mackinzie in Heidelberg a few weeks ago, and when I came home I looked for this book and his book on the formation of the moon, The Big Splat.