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Fermat put forth his 'last theorem' in 1637 which says that

No two positive integers greater than two satisfy the equation

$a^n + b^n = c^n$

But Fermat claims that the margin in his notebook was not big enough to accommodate the large proof. Which simply made it a conjecture that was never to become a theorem until after 358 years of mathematicians attempting to prove it.

The first successful proof was published in 1995 by Andrew Wiles who used modern methods such as the modularity theorem which is a thing of the 1950's and Ribet's theorem from 1985.

My question is is there any way by which Fermat could have possibly known the proof for his theorem? Although Andrew Wiles used modern methods, there are many ways to prove a theorem.

The answer probably is that Fermat could have never possibly proved it with the methods he had. (Euler could not provide a general proof either) Now Mathematics still was extremely advanced at times such as that of Fermat and Euler and the theorem even slipped through Gauss, Lagrange, Legendre, Riemann and Dirichlet (all leading proponents of Number Theory). What is it that was lacking that postponed the proof to after three centuries of masters?

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    $\begingroup$ Ken Ribet, who was part of the chain of contributors that preceded the actual proof, told us that there was a simple error that Fermat might have made that could have led him to think he had a proof. In modern terms, it amounts to unique factorization in rings where that did not actually hold. I don't know that Prof. Ribet ever published any expository article on this. $\endgroup$ – Will Jagy Nov 6 '16 at 3:19
  • $\begingroup$ It is possible Fermat actually have a proof for the $n=3$ and $4$ case using his method of infinite descent and at some point of time, honestly believe that can be extended to the general case. Up to what I heard, Fermat frequently mention the result for the $n = 3, 4$ case but never the general one in later part of his life. $\endgroup$ – achille hui Nov 6 '16 at 3:29
  • $\begingroup$ The most widespread idea about the “wonderful proof” that Fermat claimed to have was working on extensions of rational but ignoring that in most of these, the factorization is not unique. Mathematician and historian Catherine Goldstein writing about Wiles-Taylor and Company said that "the legend still stands" referring to the alleged elemental proof declared by Fermat. Personally, I think it will be very unlikely, but not impossible. $\endgroup$ – Piquito Nov 6 '16 at 3:39
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    $\begingroup$ @Will Jagy. This possible "error" of Fermat is well known and exposed e.g. in Borevitch & Shafarevitch's "Number Theory". See also www.math.uconn.edu/~kconrad/blurbs/gradnumthy/fltreg.pdf. For an odd prime p and a primitive p-th root u of 1, one can decompose x^p + y^p into a product of linear factors in the ring Z[u], and if Z[u] were a UFD, one can show by "elementary" factoriality arguments that the Fermat equation of degree p has no non trivial solution. This was even the "proof" presented by Lamé to the Académie des Sciences de Paris around 1947... $\endgroup$ – nguyen quang do Nov 6 '16 at 14:09
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    $\begingroup$ ... But at that time, Kummer had already known that Z[u] is not a UFD for p = 37 and, by developping a theory of factorization into "ideal numbers" (nowadays called ideals) in cyclotomic rings - a prelude to the theory of Dedekind rings - he had proved FLT for the so called regular primes p ( = those for which Z[u] is a UFD). This is generally considered as the birth certificate of algebraic number theory. [Edit: ... by elementary factoriality arguments... no non trivial solution if p does not divide xyz.] $\endgroup$ – nguyen quang do Nov 6 '16 at 14:18
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I think it's absolutely possible he wasn't lying (and it's possible he wasn't mistaken). The thread linked to below talks about pieces of mathematics that were known and lost, some of which took decades, centuries, and even millenia to rediscover, so I don't think it's entirely unrealistic to think that perhaps something Fermat claimed to discover is something that just has yet to be "rediscovered".

https://mathoverflow.net/questions/176425/rediscovery-of-lost-mathematics

It's also not uncommon for theorems to have multiple very different proofs, and for more simple and elegant proofs to be discovered long after the initial validated proof. Some of these are other number theoretic ideas - some are even other theorems proposed by Fermat. A classic example is the two square theorem, which was, after several decades, eventually proved with a surprising solution that is only one sentence long (the Numberphile videos below explain this, it's a fun one). So who's to say Fermat didn't have such a proof that differs completely in methods from the methods Wiles used?

Part 1 (Problem/History): https://www.youtube.com/watch?v=SyJlRUBoVp0
Part 2 (Proof): https://www.youtube.com/watch?v=yGsIw8LHXM8

I personally like to think that Fermat did have a proof for the last theorem, and that we may stumble upon something based on foundational concepts someday instead of using the heavy machinery of the modularity theorem. Perhaps we'll never stumble upon it. Only time will tell. Sadly, we may never know whether he truly had the deep insight into this theorem that he claimed, but the mystery surrounding the theorem and its proof is part of what makes it one of the most fascinating parts of math history for me - it just kinda eats at me to wonder what he may have known or seen that we haven't.

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  • $\begingroup$ @bof My bad, I wasn't paying enough attention. The post has been edited. Thanks! $\endgroup$ – MightyTyGuy Nov 6 '16 at 17:59
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The most detailed proof Fermat offered of any result in mathematics was the $n=4$ case of Fermat's last theorem. For almost a century, it was the only case anyone had proven (except for $n\in4\mathbb{N}$, of course). But it is odd that Fermat would have published a special case if he had a full proof. It is more likely that he had a flawed proof early, later realised his mistake, and published the one proof he could manage.

To prove Fermat's last theorem, one need only check the cases where $n$ is $4$ or an odd prime. Efforts over the centuries to adapt Fermat's techniques to odd primes had limited success. At one point it looked promising, but it was discovered a particular technique assumed a unique factorisation result that only applies for small $n$. An interesting contribution by Sophie Germain notwithstanding, it looked unlikely all odd primes could be addressed by traditional techniques.

Wiles's proof never really built on the earlier results. Instead it proved a result in the theory of modular forms and elliptic curves, which was known since 1987 to imply Fermat's last theorem. Techniques that only use number theory and complex numbers could only go so far for technical reasons. It is unlikely there is some unknown way to prove FLT using these methods.

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  • $\begingroup$ Ordinarily I wouldn't quibble about typos (or I would just correct them myself; gotta love edit privileges!), but I'm not certain what "n if $4$ of an odd prime" was supposed to be. Perhaps "n is $4$ or an odd prime"? $\endgroup$ – Ian Nov 10 '16 at 14:46
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Maybe he indeed took the Number-Theoretic Path or other path, like a Synthetic one...

Fermat's Theorem – a Geometrical View

3d Pythagoras theorem

...but, so far, those manuscripts have not been found.

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