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Mathematicians / folks knowledgeable on the subject - do you think Fermat had an error in his (lost) proof? Or that there exists a solution that perhaps is more straightforward than Wiles's?
My understanding is that Wiles had to use a ton of math (and invent some new math?) that had not yet been invented in Fermat's time.
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One of the arguments I have heard that he did not have a proof is the following.
He wrote his note in his copy of Arithmetica around 1637.
He most likely wrote his proof for the case of n=4 in the 1640s.
He sent letters to other mathematicians in 1640, 1657 where he talks about the case of n=3 but writes in such a way that it seems like he does not have the answer.
Why would he write n=4 after? If he had a generalized proof?
Why would he tease other mathematicians with a special case in 1657 if he already had a generalized proof?
I don't know that much about math but I've read that one possibility is he had found a valid proof for n=4 and assumed that it generalized. Hopefully someone else who knows more about the subject chimes in!
Consensus is that there's no way he had a correct, general proof.
My personal guess is he figured out later that his proof was broken or limited and never got around to fixing it.