Poor Fermat. Two of the most famous things named after him turned out to be very wrong. His conjecture that all Fermat numbers are prime is spectacularly wrong: in fact, none of them beyond F4 seem to be prime, and disproving the primality of, and factoring, F5 should have been well within the mathematical techniques he himself discovered.
It seems unlikely that he found an elementary proof of Fermat's Last Theorem, so whatever he thought he had was probably dreadfully wrong.
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