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I was searching in SE and Google for elementary proofs of Fermat's Last Theorem (FLT) and I found a lot of false claims about an elementary proof.

I wonder: more than two decades after Andrew Wiles published his proof of FLT based on the modularity theorem, why are people still trying to find an elementary proof? What motivates these attempts?

Xander Henderson
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Mithridates the Great
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    I don't see any mystery...the problem looks as if it might possess a quick and easy solution. Certainly it looks like you can attack it with elementary techniques. And, even post Wiles, there'd be a lot of fame and glory for anyone who came up with an easy proof. – lulu Dec 12 '18 at 22:01
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    Why have people found literally hundreds of proofs of the Pythagorean Theorem? – Adrian Keister Dec 12 '18 at 22:13
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    There is motivation if it brings up new understanding of Mathematics. Look at the elementary proof of the prime number theorem. Some new mathematics had to be 'discovered' for the elementary proof. Also, mathematicians do more things than solving unsolved problems. Solving conjectures is a side effect of doing mathematics and not their main profession. Mathematicians are involved in all aspects of mathematics. Simplifying knowledge is another important aspect (though elementary proofs need not simplify things). – user1952500 Dec 12 '18 at 22:16
  • The "too large to fit in the margin" story caught people's attention. Certainly if Fermat had a proof, it would be an elementary one (as more advanced machinery was not available to him). The general consensus among experts is that whatever proof Fermat thought he had was wrong, but still... – Robert Israel Dec 12 '18 at 22:19
  • @RobertIsrael "The general consensus among experts is that whatever proof Fermat thought he had was wrong" it is just a conjecture by itself or mathematicians have some sort of proof for it? I mean is it possible to prove that there is no elementary proof of FLT and close these attempts forever? – Mithridates the Great Dec 12 '18 at 22:20
  • It depends on what you mean by elementary. See e.g. here – Robert Israel Dec 13 '18 at 00:57
  • Odds speak clearly against the possibility that Fermat had a proof. Of course, it cannot be rigorously decided whether he had. The best evidence that no elementary proof exists is the effort that was necessary to finally prove the theorem. – Peter Dec 27 '18 at 10:39
  • @AloneProgrammer I would be very surprised if the impossibility of an elementary proof could be proven. – Peter Dec 27 '18 at 10:42
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    @Peter: It depends on what is meant by "elementary". For example, see this post under the section "Fermat's last theorem in weak arithmetic theories", which explain that some weak theories of arithmetic cannot prove FLT, and that even the whole theory of naturals cannot prove FLT if the exponentiation involved is simply a binary function-symbol and we only know the basic exponentiation 'laws' for that function-symbol. – user21820 Jan 02 '19 at 12:10
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    In layman's terms, we know that FLT cannot be proven algebraically, and requires at least some induction, but Fermat did have his method of infinite descent (which corresponds to what is known today as the well-ordering principle, which is as strong as induction), and hence we likely can never convincingly determine that Fermat didn't have a proof using his type of tools. – user21820 Jan 02 '19 at 12:14

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Look at it this way, OP - Wiles' proof is long, and involved. It's the kind of stuff only professional mathematicians could probably understand. Yet the theorem itself? Exceptionally easy to digest, no?

Yes, Wiles' proof is valid, but at the same time it's not accessible to the layperson in this respect. Hell, I've nearly completed my undergrad in mathematics and probably couldn't understand it fully.

The simplicity of the theorem itself lends one to think that there is a proof with more elementary, simple techniques that a novice could understand. What complicates the matter further is that Fermat himself likely would not have come up with Wiles' proof (was some of the theory in said proof even around in Fermat's lifetime?). One could brush it off as "well, duh, then Fermat's proof was invalid" - but what if it wasn't? (Footnote: Apparently the general consensus is Fermat's proof would be invalid, but just roll with me here.) After all, having not seen the proof, it's not on us to judge: there need not be one specific proof for any given theorem. Aren't there books published literally full of nothing but dozens, hundreds of proofs of the Pythagorean theorem? Fermat definitely could've had a proof, however unlikely it might seem. And on top of that - one that could fit in the margin of a page in a book? That certainly seems exceptionally elementary and simple at first glance.

(Granted, if I remember correctly, you could also use the fact that $\zeta(2) = \pi^2/6$ and thus say $\pi$ is irrational. So length and simplicity aren't always hand in hand, but I do find that often short solutions, if not elementary or simple, are at least beautiful in their own way. And I imagine that the layperson would certainly think as much.)

And of course, the sheer magnitude of how much the theorem has ingrained itself in our culture is probably a big contributor. Of all places to find math humor, it's referenced repeatedly in The Simpsons with near-miss solutions (Mathologer on YouTube has done several videos on this very topic). Proving Fermat's Last Theorem was the kind of thing that made world news back in the 1990s, and resulted in Andrew Wiles being knighted by English royalty.

I, if cynically, don't think nearly the same would happen if someone proved the Riemann hypothesis or otherwise solved any of the other Millennium Prize Problems - and yes, I say that in spite of the the million-dollar prize on them. Then again, I don't know if the proof of the Poincaré conjecture made news; yet on the other hand, I only have heard other people talk about it at any length in a STEM magazine I read in 10th grade and on Numberphile.

In short:

  • The simplicity of the theorem. (Anyone with rudimentary knowledge of math could understand it!)
  • The complexity of Wiles' proof. (A 200-or-so page proof of high level math.)
  • The possibility of Fermat having a more elementary proof. (Why he never wrote it down, though, I don't know. Surely there were more than enough book margins to scribble it down.)
  • The penetration into our culture. (Literally in The Simpsons, it still blows my mind a little.)

All of these are big reasons why people want a more elementary proof: a lot of people know about it, but not many are able to understand Wiles' proof. Fermat himself claimed to have a proof, and (were it true) it likely would be way different from Wiles'.

People probably just want a proof that they can understand without devoting years of their life trying to understand Wiles'. Sure, you can always take the word of a mathematician and go "they say it's right," but on a personal level I feel that's different than understanding it for yourself, and it feels more fulfilling as well. (Even moreso with Fermat's Last Theorem - I mean, at first glance, it would almost seem to suggest that some solutions exist, right?)

Plus, as another personal anecdote - there's a beauty in simple/elementary proofs. So perhaps that might be why even professionals might want alternate proofs to some theorems (Fermat's Last Theorem included): they can illustrate the concepts in a totally different way, maybe even one that's easier to understand, or more intuitive (even if longer). Perhaps it could invoke new thoughts and ideas, shed light on unthought-of concepts. It's not a fruitless effort, in the end, even if it seems pointless at first glance.

PrincessEev
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    There's quite a lot of maths in The Simpsons. I think there may even be a Wikipedia article about it.. – timtfj Dec 12 '18 at 23:16
  • Well,a Guardian article. https://www.theguardian.com/tv-and-radio/2013/sep/22/the-simpsons-secret-formula-maths-simon-singh Also a book by Simon Singh. – timtfj Dec 12 '18 at 23:30