Re: Could Fermat's Last Thereom be prooved using older mathematics?

In the summer of '93, the news raced around the world via email. Andrew Wiles had done what mathematicians had been trying to accomplish for centuries, he had finally proved Fermat's Last Theorem! The proof turned out to be incomplete, but in the next two years Wiles patched up the gap and the problem is now believed to have been rigorously proven.

However, the proof that Wiles gave was obviously not the one Fermat was thinking of when he wrote "I have discovered a truly remarkable proof which this margin is too small to contain." Wiles' proved the theorem as a corollary of a conjecture in the theory of elliptic curves, an area that wasn't even invented until well after Fermat's death in 1665. So did Fermat have a different, short and elegant proof in mind using only the number theory of the time when he was scribbling away?

The short answer is no, there is virtually no chance that Fermat himself proved Fermat's Last Theorem. The long answer is a bit more complex.

One way of to prove in a mathematical sense that Fermat did not have a proof would be to exhaustively list all number theoretical proofs below a certain size, a task impossible to complete in our lifetimes. So we are left to weigh the historical evidence surrounding Fermat to see what he had in mind when writing in the margin.

First, keep in mind that Fermat was not a mathematician by trade, he was in fact a lawyer. This fact meant that he published almost none of his results. On the other hand, he was certainly on of the most brilliant amateur mathematicians that ever lived. One of his hobbies was proving difficult theorems on his own, and then challenging other mathematicians to find proofs.

So Fermat never published any of his proofs for the problems he came up with, but all of his other challenge problems were proven true by others. But here's the kicker, what we call Fermat's Last Theorem (FLT) was never sent out as a challenge problem. Now, as every mathematician whos every tried to prove it know, FLT states that xⁿ + yⁿ = zⁿ has no nonzero integer solutions for x, y, and z when n is an integer greater than 2.

Fermat did send out as challenge problems FLT restricted to the cases when n = 3 and n = 4, and he clearly knew how to prove the result for these values of n. However, given that he never put the full version of FLT to the test of others, it is unlikely that he ever really had a proof himself.

So why write those tantalizing words that have inspired so many(including great mathematicians such as Euler, Cauchy, Legendre and Dirichlet) to search for a short proof? Probably Fermat just made a simple mistake in his proof that led him to believe that he had a proof for all n when in fact he had only proved it for a few values of n. The fact that all the combined brain power of three centuries of mathematicians have been brought to bear on the problem is further evidence that an elementary proof is unlikely to exist.

That marginal inscription only came to light when Fermat's son Samuel began to collect and publish his father's work. It is fortunate that it did, for work on FLT has led to some great mathematics, including commutative ring theory.

For information on the more than three centuries of history of Fermat's Last Theorem, I'd recommend this site as a good starting point.