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Thread: Fermat's theorem

  1. #1 Fermat's theorem 
    Forum Masters Degree thyristor's Avatar
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    Does anybody know where you can find the proof of Fermat's theorem on the internet.


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  3. #2  
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    Which one? There’s Fermat’s little theorem, which is very easy: http://en.wikipedia.org/wiki/Proofs_...little_theorem

    Well, I presume you mean Fermat’s last theorem:

    http://math.stanford.edu/~lekheng/flt/wiles.pdf

    Happy reading!


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  4. #3  
    Forum Professor serpicojr's Avatar
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    Let me advise you that Mr. Wiles' proof will be pretty inaccessible. Understanding what he's doing requires a working knowledge of research level number theory. In particular, if you've never seen any elliptic curves or modular forms, you'll have no idea what he's doing. Wiles actually proved a conglomerate of conjectures by Weil, Taniyama, and Shimura which, in one of its simpler forms, states:

    Let E be an elliptic curve defined over the rationals, let L(s,E) be its Hasse-Weil L-function, and let N = N<sub>E</sub> be its conductor. Then there exists a weight 2 newform f of level N such that L(s,f) = L(s,E).

    That's quite a mouthful. Frey and Ribet developed mathematics that showed that this conjecture implies Fermat's Last Theorem, and so Wiles proof kind of has nothing directly to do with Fermat's Last Theorem.
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  5. #4  
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    If anyone wants to read about Fermat’s last theorem, I would highly recommend Simon Singh’s Fermat’s Last Theorem (1997). It is an excellent introduction to the subject indeed. The only thing bad I can say about it is that there are a lot digressions – places where the author would be talking about something, then stop suddenly and go off at a tangent to talk about something else, before returning a few pages later to continue talking about the first subject. However, if you can keep from being distracted by these tangents, then Simon Singh’s book should prove to be a most readable one on how Prof Wiles went about proving Fermat’s last theorem. 8)

    Here’s a summary:

    First, there are these things called elliptic equations. These are equations in two variables x and y of the form y<sup>2</sup> = x<sup>3</sup> + ax<sup>2</sup> + bx + c, where a, b and c are constants. Then there are these things called modular forms – some kind of inordinately symmetric structures in 4-dimensional hyperbolic space. A result, which was known at the time as the Taniyama–Shimura conjecture, states that every elliptic equation is modular.

    In the 1980s, Gerhard Frey stated that if the Fermat equation had a nonzero solution, say x = A, y = B, z = C so that A<sup>N</sup> + B<sup>N</sup> = C<sup>N</sup> for some integer N > 2, then the elliptic equation y<sup>2</sup> = x<sup>3</sup> + (A<sup>N</sup>B<sup>N</sup>)x<sup>2</sup> − A<sup>N</sup>B<sup>N</sup> would not be modular – contradicting the Taniyama–Shimura conjecture. Frey’s hypothesis was proved in 1986 by Ken Ribet of the University of California, Berkeley.

    So now all Andrew Wiles had to do to prove Fermat’s last theorem was to prove the Taniyama–Shimura conjecture!

    The road was not all that smooth though. In June 1993, Wiles thought he had got it – only for a flaw to be discovered later. The basis of Wiles’ proof was mathematical induction. The first step involved showing that the first term in every L-series of an elliptic equation corresponded to the first term in some M-series of a modular form. This was relatively easily done using Galois representations. It was in the inductive step, where Wiles used something called the Kolyvagin–Flach method, that the proof went astray. It took some racking of brains before Wiles realized that what he had to do was to complement Kolyvagin–Flach with Iwasawa theory. It worked. And with that, in September 1994, the Taniyama–Shimura conjecture – and hence Fermat’s last theorem – was proved.
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  6. #5  
    Forum Professor serpicojr's Avatar
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    Let me clear up a few things. This is all very close to my research, so I've got a lot to say on the subject.

    The modular forms under consideration are on 2-dimensional hyperbolic space; typically, they're defined as functions on the upper half-plane. The fact that modular forms exist isn't super-surprising--their existence is guaranteed by the geometric structure of so-called modular surfaces.

    The Taniyama-Shimura Conjecture shouldn't really be thought of as a reduction of Fermat's Last Theorem; instead, Fermat should be thought of a corollary to the conjecture. The conjecture has a much wider scope than just this application--in some sense, it gives a parameterization of elliptic curves, i.e. smooth genus 1 algebraic curves with rational points. The parameterization of genus 0 curves is classical; Faltings showed that genus g ≥ 2 curves only have finitely many rational points. So the elliptic curves case is kind of the borderline, and most interesting, case. Additionally, the truth of the conjecture is good evidence for the so-called Langlands program, which basically says all Galois representations (some of which arise from elliptic curves) come from automorphic representations (of which modular forms are an example).
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