Comment by jerf
3 days ago
There is known to be a number of superficially compelling proofs of the theorem that are incorrect. It has been conjectured that the reason why we don't have Fermat's proof anywhere is that between him writing the margin note and some hypothetical later recording of the supposed proof, he realized his simple proof was incorrect. And of course, saw no reason to "correct the historical record" for a simple margin annotation. This seems especially likely to me in light of the fact he published a proof for the case where n = 4, which means he had time to chew on the matter.
Or, maybe he had a sense of humor, and made his margin annotation knowing full well that this would cause a lot of headscratching. It may well be the first recorded version of nerdsniping.
More likely he decided to leave it in as a nerdsnipe rather than he wrote it in the first place as a nerdsnipe (seems more likely he thought he had it?)
Or he forgot about it? Why should he have a margin note at the top of his mind?
I make notes all the time that I accidentally discover years later with some amusement.
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What are some believable but wrong proofs of FLT? Wikipedia also claims that there were historically a lot of them[1], but doesn't provide examples.
[1]: https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem#Prizes...
Among well-known mathematicians, Gabriel Lamé claimed a proof in 1847 that was assuming unique factorization in cyclotomic fields.
This was not obvious at the time, and in fact, Ernst Kummer had discovered the assumption to be false some years before (unbeknownst to Lamé) and laid down foundations of algebraic number theory to investigate the issue.