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Q: Fermat's Last theorem ( Answered,   4 Comments )
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 Subject: Fermat's Last theorem Category: Science > Math Asked by: dimitrism-ga List Price: \$5.00 Posted: 23 Apr 2002 00:39 PDT Expires: 30 Apr 2002 00:39 PDT Question ID: 3054
 Which were the major mathematical techniques that helped prove Fermat's Last theorem ?
 Subject: Re: Fermat's Last theorem Answered By: drdavid-ga on 23 Apr 2002 14:36 PDT
 There is a lot that has been written about Fermat's Last Theorem from assorted history to much interesting related mathematics. You can turn up an extensive list with a Google search on: ://www.google.com/search?q=fermat%27s+last+theorem One of my favorite places to start for understanding the mathematics behind Andrew Wiles' solution is Charles Daney's article "The Mathematics of Fermat's Last Theorem," posted on his home page: http://www.mbay.net/~cgd/flt/flt01.htm As petebevin notes in his Comment, a key part of the solution has to do with the mathematics of elliptical curves and modular forms. To quote from near the beginning of Daney's article: "At the highest level, the proof is extremely simple to understand, since it follows from just 2 theorems: Theorem A: If there is a solution (x, y, z, n) to the Fermat equation, then the elliptic curve defined by the equation Y^2 = (X-x^n)(X+y^n) is semistable but not modular. And Theorem B: All semistable elliptic curves with rational coefficients are modular. However, both of these theorems are very difficult themselves, and both have been proven only in the last 10 years. But given that both are now known, it follows that, in order to avoid a contradiction, there cannot be any solution to the Fermat equation. … Theorem A is obviously rather special in that it applies only if the Fermat equation has a solution. (And since we now know this isn't the case, the theorem has no further use.) It was first conjectured around 1982 by Gerhard Frey, and finally proved in 1986 by Ken Ribet, with help along the way from Jean-Pierre Serre. Theorem B is even harder still, and it is the theorem of which Andrew Wiles first claimed a proof in 1993, thus proving FLT as well. Although problems were found in Wiles' original proof, he managed to nail it down a year later, with help from Richard Taylor." Daney goes on to discuss the key concepts that were used in proving these two theorems: elliptical curves, elliptical functions, modular functions, zeta and L-functions and Galois representations. He then follows with an outline of the complete proof.