[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.