

Subject:
Fermat's Last theorem
Category: Science > Math Asked by: dimitrismga 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: drdavidga 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 = (Xx^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 JeanPierre 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 Lfunctions and Galois representations. He then follows with an outline of the complete proof. 

Subject:
Re: Fermat's Last theorem
From: mplungjanga on 23 Apr 2002 01:14 PDT 
http://wwwgroups.dcs.st and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html: Fermat's Last Theorem states that xn + yn = zn has no nonzero integer solutions for x, y and z when n > 2. Fermat wrote I have discovered a truly remarkable proof which this margin is too small to contain.  http://www.wsws.org/articles/1999/jan1999/ferm93.shtml says "Both the ancient Greeks and Babylonians were aware that the equation x2 + y2 = z2 had whole number solutions. If x, y and z are 3, 4 and 5 respectively, then 32 = 9 plus 42 = 16 is equal to 52 = 25. Another such solution is if x, y and z are the numbers 5, 12 and 13. There are an infinite number of such whole number solutions, known as Pythagorean triples. While studying the work of the Greek mathematician Diophantus who lived around 250 AD in Alexandria, Fermat generalised the equation to consider numbers raised to any power. In a famous note in the margin of his copy of Diophantus' Arithmetica, Fermat wrote: "To resolve a cube into the sum of two cubes, a fourth power into two fourth powers, or in general any power higher than the second into two of the same kind, is impossible; of which fact I have found a remarkable proof. The margin is too small to contain it." Put more concisely, the equation xn + yn = zn has whole number solutions if n = 2 but for all larger values of n, Fermat asserted that there were no whole number solutions. Fermat never wrote down his "remarkable proof"and mathematicians since then have cast doubt on whether, given the mathematical techniques available in the seventeenth century, it ever existed." 
Subject:
Re: Fermat's Last theorem
From: yggdrassilga on 23 Apr 2002 01:30 PDT 
Farmat's last theorem was mainly solved using eleptic curve math. 
Subject:
Re: Fermat's Last theorem
From: petebevinga on 23 Apr 2002 07:09 PDT 
Elliptic Curves and Modular Forms. Dr Math has a good article explaining the concepts. http://mathforum.org/dr.math/problems/renjen11.2.97.html If you're interested, I suggest you read the book Fermat's Enigma by Simon Singh. http://www.amazon.com/exec/obidos/ASIN/0385493622 Also, the sci.math FAQ has some basic information: http://www.faqs.org/faqs/scimathfaq/FLT/Fermat/ 
Subject:
Re: Fermat's Last theorem
From: zhiwenchongga on 26 Jun 2002 21:59 PDT 
The major bridge towards the proof of the FLT was the TaniyamaShimura conjecture. I'm surprised nobody mentioned it. 
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