Fermat's Last Theorem - Did Fermat Possess A General Proof?

Did Fermat Possess A General Proof?

The mathematical techniques used in Fermat's "marvelous" proof are unknown. Only one detailed proof of Fermat has survived, the above proof that no three coprime integers (x, y, z) satisfy the equation x4 − y4 = z2.

Taylor and Wiles's proof relies on mathematical techniques developed in the twentieth century, which would be alien to mathematicians who had worked on Fermat's Last Theorem even a century earlier. Fermat's alleged "marvellous proof", by comparison, would have had to be elementary, given mathematical knowledge of the time, and so could not have been the same as Wiles' proof. Most mathematicians and science historians doubt that Fermat had a valid proof of his theorem for all exponents n.

Harvey Friedman's grand conjecture implies that Fermat's last theorem can be proved in elementary arithmetic, a rather weak form of arithmetic with addition, multiplication, exponentiation, and a limited form of induction for formulas with bounded quantifiers. Any such proof would be elementary but possibly too long to write down.