What was Fermat's Next-To-Last Theorem?

Title pretty much says it all, I think.

Can’t really be determined.

Fermat’s “Last” Theorem wasn’t even his last theorem, chronologically. So, it’s a bit of a misnomer. He lived (and worked) 40 years after his infamous margin note of a marvelous proof.

Fermat wasn’t technically a “professional” mathematician, so his work was never published during his lifetime. That makes any attempt at organizing his work nigh impossible. We only know about much of his output because his son Fermat’s work posthumously and it’s also known through examination of his correspondence.

Fermat had a habit of claiming proofs that he never disclosed, even to his colleagues or even in his own notes. Many of these were subsequently proved by other mathematicians. It is likely he didn’t actually have proofs to many of his results. And it is also likely he had proofs to many results that we will never find (hence the continuing hunt by some people for an “elementary” proof to FLT).

Like Great Antibob wrote above, Fermat was an amateur mathematician in the 17th century; mathematicians weren’t as prolific or specialized as they are now. Given the rest of his work, his penultimate theorem was probably something in number theory, or low-dimensional Diophantine equations in particular. As far of results of his that are still remembered today, there’s Fermat’s little theorem, Fermat’s theorem about stationary points (which I hadn’t realized was due to, or even named for, him— it’s the result that extrema of a differentiable function must have f’ vanish), and the method of infinite descent. The first two are one-line proofs, but the latter is nontrivial. It’s still occasionally useful for specific Diophantine equations, but it’s also the basis (in a more sophisticated form) of the height machine.

Well, the thread title gave me my daily chuckle. Very interesting question, and unfortunately it doesn’t seem to have an actual answer.

Makes one wonder: What was Custer’s next-to-last-Stand?

I concur on the ingenuity of the question! My request: Second-Last Tango in Paris!

And what is the penultimate answer to life, the universe, and everything? 41?

The Penultimate Supper.

Which he used to prove the n=3 and n=4 cases of his more famous theorem. It’s probable that when he discovered the method, he thought that it would generalize to all n, and so wrote his little margin note. When he tried it and discovered that it didn’t, meanwhile, there was no reason for him to retract his marginal claim, since it was never published.

Fermat’s Last Theorem is so-called because it was the last remaining unproved one. So, presumably, one could find a list of Theorems that Fermat is credited with having claimed a solution to (but not providing the proof), then look up when each of them was later proved, and the one with the latest date prior to 1995 would be Fermat’s Penultimate Theorem. So, the question almost certainly has an answer, but it might be one that’s hard to find without some serious digging into mathematical history. Which I’m not willing to do. But I will nose around Wikipedia for a few minutes.

Wikipedia lists 6 Theorems ascribed to Fermat. Of those, two don’t provide a proof date. Of those that provide a proof date, the chronologically penultimate one is Fermat’s Polygonal Number Theorem, proved by Cauchy in 1813