> 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!
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!