[mellosouls]

Despite Fermat's brilliance, it's barely feasible that such a proof exists after 300 years of attempts by professionals and amateurs alike.

Ironically though, the tantalising idea of the existence of a simple proof - motivated by that infamous quote - contributed massively (along with the simplicity of the conjecture itself) to the theorem's notoriety and the myriad attempts to solve it.

[ubercow13]

Fermat was a brilliant social engineer as well as mathematician.

[rrmm]

It's the predecessor of the dictum about putting a wrong answer on the internet to get the right answer. Sorta, and with a few hundred years involved.

[QuesnayJr]

He wrote the note to himself, a long time before he died. He then publicly announced a proof for the n=4 case, and nothing about the general case. It strongly suggests that he realized his proof was wrong.