Comment by umanwizard
3 days ago
The former.
We can't be 100% certain that Fermat didn't have a proof, but it's very unlikely (someone else would almost surely have found it by now).
3 days ago
The former.
We can't be 100% certain that Fermat didn't have a proof, but it's very unlikely (someone else would almost surely have found it by now).
Unlikely, but not unheard of. Fermat's theorem on sums of two squares is from 1640. https://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_... says:
“Fermat usually did not write down proofs of his claims, and he did not provide a proof of this statement. The first proof was found by Euler after much effort and is based on infinite descent. He announced it in two letters to Goldbach, on May 6, 1747 and on April 12, 1749; he published the detailed proof in two articles (between 1752 and 1755)
[…]
Zagier presented a non-constructive one-sentence proof in 1990“
(https://www.quora.com/What-s-the-closest-thing-to-magic-that... shows that proof was a bit dense, but experts in the field will be able to fill in the details in that proof)
Well, true, we cannot be 100% certain, but if he published the proof to n=4, we can say it's most likely he did not find the general proof.
why does that make it more likely?
Because if he had the general proof he wouldn't need to go out of his way to prove n=4, since it would be covered already by the general proof
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