a book, …st Theorem.” While 109 pages was clearly not something that could be briefed in a margin of a book, no one knows if Fermat actually knew the proof to it. In fact, people were able to find proofs to every other problem presented in the book published by…
I have to quibble here. While we of course can't know for sure, no one really believes he had a proof.
The techniques used to finally prove it took over 300 years to develop. Fermat couldn't possibly have thought they could fit into a margin. The idea that there's some shortcut proof seems so doubtful as to be impossible.
More, there were other false proofs of Fermat's Conjecture in those 300 years, some of them fairly elementary.
Almost certainly Fermat was wrong. However, a genius's errors are often more interesting than a regular person's successes.