[Ritjert]

Then look up Wiles' proof of the Taniyama-Shimura conjecture.

The math required is so advanced, and there have been numerous people who taught they solved it but turned out to be wrong.

It's extremely unlikely that Fermat had solved it.

PS. He wrote that note when he was young (and working on special cases, small n and such) and probably realized later on that he didn't have a proof for the general case.

Edit: There was no letter to a friend. It was in the margin of a book he was reading. More like a little reminder to himself to remember to jot down the proof.