Fermat wrote that in the margin of his book Arithmetica that a proof existed, but there wasn’t space in the margin to write it. It took Wiles 385 years to find a proof, and it won’t fit in a margin. https://en.m.wikipedia.org/wiki/Fermat%27s_Last_Theorem

That’s the allure of the theorem; that a simple unknown proof may exist.

Yes, I remember when news of his proof broke, being disappointed at how voluminous and obscure (to someone like me) it was. I'd been hoping for something I might be able to get my head around. (Hard as it was, I don't think Wiles spent 385 years coming up with it btw!)

Whether there is an existing and verified proof or not, there is still a great mystery to be solved by figuring out what Fermat actually meant by what he thought as an elegant solution, whether it is an actual solution or not.

Well, it could have been like Kempe's chains... they finally realized there is a problem, and then it took like 100 years before Appel and Haken made what is probably the first computer-aided proof. And who can say it's really a "proof" if it doesn't explain "why" it's true.

Phrased in a way that doesn't imply Wiles's extreme longevity: it took 385 years of advances in mathematics to invent the tools and frameworks that allowed Wiles to come up with the proof.

> That’s the allure of the theorem; that a simple unknown proof may exist.

Well, Fermat made lots of similar claims wrt. other propositions, and for most of them the proof was found easily, or perhaps they were refuted altogether and shown to be wrong. FLT gets its name because it was a very rare case of a claim that just couldn't be solved, one way or the other. In fact, it seems that Fermat himself may have realized at some point that what he thought of as a proof he had, was in fact wrong - and dropped his claim altogether as a consequence. Which would then explain why it was only found as a margin note in a textbook. It's fascinating because it's such a simple claim to state, and yet the proof is incredibly complex. To be sure, logicians can predict that such cases will occur, in the abstract; it's a bit like having hard-to-solve instances of the SAT. But it's still nice to have such a natural example!

Yeah that's the math lore what I was referencing when I put the word margin in quotes :)

I wasn't sure if the author was making an intentional callout to Fermat's lost marvelous proof he couldn't fit in the margin. But indeed, looks like he was, since he doesn't seem to have really mentioned the length of any other proofs.

There's some speculation that fermat made the same mistake as lamé, who thought he had a proof. However lamé incorrectly assumed unique factorisation of a general number field, which was an easy mistake to make at the time.