Timeline for Theorems which say "such and such method cannot possibly prove FLT"
Current License: CC BY-SA 2.5
22 events
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| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Sep 27, 2012 at 15:19 | vote | accept | Craig Feinstein | ||
| Jan 2, 2012 at 0:47 | comment | added | Rob Harron | In a similar vein, like how the Main Conjecture of Iwasawa Theory was first proved by Mazur–Wiles using a modular approach, but then Rubin found a proof using a cyclotomic approach. | |
| Jul 15, 2010 at 3:18 | comment | added | KConrad | David, I don't think there is an argument for why a proof of FLT by cyclotomic methods was not found. For a comparison, consider Catalan's conjecture, another famous Diophantine equation involving variable powers. As with Fermat, one can reduce to the case of prime exponents and relations with class numbers of cyclotomic fields and Weiferich primes were found. If Mihailescu had not solved that problem in his way, perhaps someday someone would have found a non-cyclotomic approach and then there would be a question on MO asking why cyclotomic methods were inadequate for Catalan's conjecture... | |
| Jul 14, 2010 at 22:25 | answer | added | T.. | timeline score: 8 | |
| Jul 14, 2010 at 5:53 | comment | added | Theo Johnson-Freyd | Oh, why the vote to close? | |
| Jul 14, 2010 at 5:53 | comment | added | Theo Johnson-Freyd | Although OP's motivation for asking the question is to understand why there is not currently an "elementary" proof of FLT, the question asks something different, namely, are there any nontrivial results in the direction "such method cannot prove FLT" (as there are for P=NP). Most of the answers address the motivation and not the question, it seems to me. | |
| Jul 13, 2010 at 22:49 | answer | added | Richard Borcherds | timeline score: 19 | |
| Jul 13, 2010 at 21:43 | comment | added | Andy Putman | @David : I think that's a really great question. It's a bit different from the OP's question, which seemed to want to rule out "elementary approaches". Maybe you should ask it separately? | |
| Jul 13, 2010 at 19:34 | comment | added | David E Speyer | Here's an issue I'd be interested to hear experts comment on: There are roughly 100 years of attempts to prove FLT by working in the class group of $\mathbb{Q}(\zeta_p)$, but the final proof required working with the $p$-torsion points of an elliptic curve instead. Is there some fundamental reason why looking at cyclotomic fields was inadequate? | |
| Jul 13, 2010 at 17:47 | comment | added | Emerton | But it doesn't seem impossible that the proof secretly just works in PA, or some easily described extension thereof, in which case (as Charles Matthews points out in his answer below) one can write out a (very complicated) elementary proof. What seems to be being asked, though, is whether a simple proof can be found. And this is a question whose truth value will be unknown, until such time as one is found. But, in the meantime, I think that 350 years of experience is a reasonable guide. | |
| Jul 13, 2010 at 17:43 | comment | added | Emerton | An important technical remark is that it's not clear what the precise proof-theoretic level of the Wiles/Taylor-Wiles proof of FLT is. (In particular, I'm not sure whether it's clear that it is proved in Peano Arithmetic, or only in some strengthening thereof. The reason for this (at least to me) lack of clarity is that the arguments as written really apply to the standard natural numbers, and use many second order constructions, and as far as I know, no-one has gone through the argument and tried to strip it down to its logical essentials.) (To be cont'd ... 0 | |
| Jul 13, 2010 at 11:33 | comment | added | Thomas Bloom | Remember that elementary methods may not be the easiest methods or the methods which show why the theorem is true - the classic example is the elementary proof of the Prime Number Theorem. | |
| Jul 13, 2010 at 11:04 | answer | added | Charles Matthews | timeline score: 13 | |
| Jul 13, 2010 at 6:58 | comment | added | KConrad | That reminds me of the joke about two people who go into a store and someone comes over to them and says "Ladies, if you need any help my name is George". "Oh," one of them says, "and if we don't need any help who are you then?" | |
| Jul 13, 2010 at 6:44 | answer | added | T.. | timeline score: 10 | |
| Jul 13, 2010 at 4:52 | comment | added | Will Jagy | Andy, what about being on the phone with someone else's wife? | |
| Jul 13, 2010 at 4:44 | comment | added | Andy Putman | Actually, in trying to make my answer elementary I screwed up (ironic, isn't it?). What actually holds is that $x^p+y^p=z^p$ has a nontrivial soln in the $q$-adic integers for all primes $q$. In particular, for $n$ large there is a nontrivial soln to $x^p+y^p=z^p$ in $\mathbb{Z}/q^n \mathbb{Z}$. However, $x$ or $y$ might be divisible by $q^{n-1}$, so this soln might be trivial mod $q^{n-1}$. The moral still stands, however : congruence conditions themselves are not enough. Another moral to draw from this is that one shouldn't answer MO questions while on the phone with one's wife... | |
| Jul 13, 2010 at 3:24 | answer | added | Noah Snyder | timeline score: 26 | |
| Jul 13, 2010 at 2:30 | answer | added | Bill Dubuque | timeline score: 5 | |
| Jul 13, 2010 at 1:22 | comment | added | Andy Putman | I am far from an expert, but I'll make one very shallow remark here. Lots of the elementary "proofs" I've seen spend a lot of time mucking around with congruence conditions. However, this alone is bound to fail because for an odd prime $p$, the equation $x^p+y^p=z^p$ has nontrivial solutions mod $n$ for every $n$ (this is not hard to prove for yourself, or see Prop 6.9.11 in Cohen's "Number Theory Vol I"). In fancier language, the "Hasse principle" fails for FLT. | |
| Jul 13, 2010 at 0:30 | history | asked | Craig Feinstein | CC BY-SA 2.5 |