FWIW, it seems pretty clear that while Fermat briefly thought he had a proof, he fairly quickly realized it was flawed since he never mentioned this publicly and later went on to develop a proof just for the simple n=4 case.
In 1847 Gabriel Lami presented a claimed (simple) general proof to the French Academy of Sciences, only for the flawed assumption in it to be pointed out immediately at the end of his presentation! This may have been the same proof that Fermat had in mind.
Oh, so if he later published a proof for a special case (and presumably had more paper available than the book's margin :-) ), that's convinces me that he never had a correct proof. Also that he was smart enough to realize that but perhaps just forgot to or didn't erase his original margin note. :-)
* A list of daily tasks small and quick too many to remember
* Tasks to do or check every three or four days ie about twice a week
* Tasks to do about once a week eg hoovering.
* Tasks to do every three months like wash house windows etc
Most days I put my todo list into ChatGPT and get it to put my todo list in order from quickest to longest.
Sometimes I ask ChatGPT to order my todo list by importance.