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So: as I understand it, Fermet claimed there was an elegant proof. The proof we've found later is very complex. Is the consensus that he never had the proof (h
by timmg 1y ago
So: as I understand it, Fermet claimed there was an elegant proof. The proof we've found later is very complex.
Is the consensus that he never had the proof (he was wrong or was joking) -- or that it's possible we just never found the one he had?
- wk_end 1y agoIt's possible we never found the one he had, but it's pretty unlikely given how many brilliant people have beaten their head against this. "Wrong or joking" is much more likely.
- School-Cotton 1y agoThe 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).
- Someone 1y agoUnlikely, 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_two_squares 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-actually-exists/answer/Senia-Sheydvasser 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)
- prerok 1y agoWell, 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.
- DoctorOetker 1y agowhy does that make it more likely?
- tslater2006 1y agoBecause 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
- DoctorOetker 1y agoIt is simply an obvious fault line in the nature of the problem statement: you can crack the problem in 2 parts: the x^4+y^4=z^4 part, and the part that claims x^p+y^p=z^p with p a prime. Suppose Fermat solved the proof by using this natural fault line -its just how this cookie crumbles- solved the n=4 case, and then smashed his head a thousand times against the problem and finally found the prime n proof. He challenges the community, and since they don't take up the challenge, "encourages" them in a manner that may be described as trollish, by showing how to do the n=4 case. (knowing full well the prime power case proof looks totally different)
- deleted 1y ago[deleted]
- prerok 1y agoThat's an interesting take but I think it's unlikely for two reasons: 1. In any case you view it, it's not trivial, which was the statement in the note. If it were, the effort to publish just for n=4 would be silly, because it would take equal effort to just publish for general case. That he withheld the proof just to mess with people is highly unlikely. 2. I definitely do not make private notes in my books just so that maybe somebody later on would pick up that book and wonder whether I had indeed discovered the secrets of the universe. I definitely do not write "challenges to the community" there.
- 1y ago
- ape4 1y agoIts a "dog ate my homework" situation
- jfengel 1y agoFermat lived for nearly three decades after writing that note about the marvelous proof. It's not as if he never got a chance to write it down. So it sure wasn't his "last theorem" -- later ones include proving the specific case of n=4. There are many invalid proofs of the theorem, some of whose flaws are not at all obvious. It is practically certain that Fermat had one of those in mind when he scrawled his note. He realized that and abandoned it, never mentioning it again (or correcting the note he scrawled in the margin).
- griffzhowl 1y agoIt was called Fermat's last theorem because it was the only one of the theorems stated by Fermat that remained to be proved at the time
- mdiesel 1y agoHe probably did know it, it's remarkably simple. I can't remember how to format maths in a HN comment though to put it here.
- prerok 1y agoYeah, I just figured out how to simply reconcile general relativity and quantum mechanics, but I am writing on my phone and it's too tedius to write here.
- thiht 1y agoThe proof is pretty trivial anyway so it’s left as an exercise to the reader
- throw46395 1y agoimport FLT theorem PNat.pow_add_pow_ne_pow (x y z : ℕ+) (n : ℕ) (hn : n > 2) : x^n + y^n ≠ z^n := PNat.pow_add_pow_ne_pow_of_FermatLastTheorem FLT.Wiles_Taylor_Wiles x y z n hn
- jerf 1y agoThere is known to be a number of superficially compelling proofs of the theorem that are incorrect. It has been conjectured that the reason why we don't have Fermat's proof anywhere is that between him writing the margin note and some hypothetical later recording of the supposed proof, he realized his simple proof was incorrect. And of course, saw no reason to "correct the historical record" for a simple margin annotation. This seems especially likely to me in light of the fact he published a proof for the case where n = 4, which means he had time to chew on the matter.
- jacquesm 1y agoOr, maybe he had a sense of humor, and made his margin annotation knowing full well that this would cause a lot of headscratching. It may well be the first recorded version of nerdsniping.
- ljlolel 1y agoMore likely he decided to leave it in as a nerdsnipe rather than he wrote it in the first place as a nerdsnipe (seems more likely he thought he had it?)
- lo_zamoyski 1y agoOr he forgot about it? Why should he have a margin note at the top of his mind? I make notes all the time that I accidentally discover years later with some amusement.
- ljlolel 1y agoYea I’m partly saying he came across it or remembered it much later and was amused to not correct it
- RainyDayTmrw 1y agoWhat are some believable but wrong proofs of FLT? Wikipedia also claims that there were historically a lot of them[1], but doesn't provide examples. [1]: https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem#Prizes_and_incorrect_proofs https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem#Prizes...
- hokkos 1y agoI also have an elegant proof, but it does't quite fit in a HN comment.
- glitchc 1y agoNo support for symbols, amirite?
- Xcelerate 1y agoI feel like there’s an interesting follow-up problem which is: what’s the shortest possible proof of FLT in ZFC (or perhaps even a weaker theory like PA or EFA since it’s a Π^0_1 sentence)? Would love to know whether (in principle obviously) the shortest proof of FLT actually could fit in a notebook margin. Since we have an upper bound, only a finite number of proof candidates to check to find the lower bound :)
- tliltocatl 1y ago> we have an upper bound Is Wiles' proof even in ZFC?
- openasocket 1y agoI would be surprised if it wasn’t. Maybe some part of depends on the continuum hypothesis, but ZFC is pretty powerful
- jibal 1y agoYour question is explored in https://www.cs.umd.edu/users/gasarch/BLOGPAPERS/fltlargecard.pdf https://www.cs.umd.edu/users/gasarch/BLOGPAPERS/fltlargecard...
- tliltocatl 1y agoThanks. So if I read this correctly - there is consensus that Wiles' proof can be reduced to ZFC and PA (and maybe even much weaker theories). But as presented Wiles proof relies on Grothendieck's works and Grothendieck could not care less about foundationalism, so no such reduction is known and we don't really have a lower bound even for ZFC.
- bubblyworld 1y agoEven super simple results like uniqueness of prime factorisation can pages of foundational mathematics to formalise rigorously. The Principia Mathematica famously takes entire chapters to talk about natural numbers (although it's not ZFC, to be fair). For that reason I think it's pretty unlikely.
- zarzavat 1y agoFermat was alive in the 1600s, long before the advent of mathematical rigour. What counted as a proof in those days was really more of a vibe check.
- DoctorOetker 1y agoIt is possible to discover mathematical relation haphazardly, in the style of a numerologist, just by noticing patterns, there are gradations of rigor. One could argue, being a lawyer put Fermat in the more rigorous bracket of contemporary mathematicians at least.
- ants_everywhere 1y agoThis is actually way false. Rigorous mathematical proof goes back to at least 300 BCE with Euclid's elements. Fermat lived before the synthesis of calculus. People often talk about the period between the initial synthesis of calculus (around the time Fermat died) and the arrival of epsilon-delta proofs (around 200 years later) as being a kind of rigor gap in calculus. But the infinitesimal methods used before epsilon-delta have been redeemed by the work on nonstandard analysis. And you occasionally hear other stories that can often be attributed to older mathematicians using a different definition of limit or integral etc than we typically use. There were some periods and schools where rigor was taken more seriously than others, but the 1600s definitely do not predate the existence of mathematical rigor.
- unexpectedtrap 1y agoEuclid’s Elements “rigorous proof” is not the same thing as the modern rigorous proof at all. >But the infinitesimal methods used before epsilon-delta have been redeemed by the work on nonstandard analysis. This doesn’t mean that these infinitesimal methods were used in a rigorous way.
- monkeyelite 1y agoNot true. Even if it’s more strict it’s just a matter of inserting more care and steps, not changing the original idea.
- DoctorOetker 1y agoThe consensus is that there is no consensus yet. I possess a very simple proof of FLT, and indeed it does not fit in a margin. I don't ask you to believe me, I just ask you to be patient.
- jibal 1y agoNo, the consensus among mathematicians is that Fermat did not have a proof.
- DoctorOetker 1y agoDon't confuse majority for consensus, soon the majority will flip, but the consensus will stay the same: that there is no consensus.
- deleted 1y ago[deleted]
- jibal 1y agoP.S. This is utter nonsense: "Don't confuse majority for consensus, soon the majority will flip, but the consensus will stay the same: that there is no consensus."
- Lucasoato 1y ago> In the words of mathematical historian Howard Eves, "Fermat's Last Theorem has the peculiar distinction of being the mathematical problem for which the greatest number of incorrect proofs have been published."