Fermat couldn't possibly have had this proof.
proof
But what has made this problem special for amateurs is that there's a tiny possibility that there does exist an elegant 17th-century proof.
A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories.
Then you start another book and suddenly the galley proofs of the last one come in and you have to wrench your attention away from what you're writing and try to remember what you were thinking when you wrote the previous one.
An administration without a police executive is powerless and there were many proofs of this.
Thus, in a sense, mathematics has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs.
There are proofs that date back to the Greeks that are still valid today.
Those who have racked their brains to discover new proofs have perhaps been induced to do so by a compulsion they could not quite explain to themselves. Instead of giving us their new proofs they should have explained to us the motivation that constrained them to search for them.
proof motivation
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