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Q: Fermat's Last theorem ( Answered,   4 Comments )
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:


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:

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. 


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.
Subject: Re: Fermat's Last theorem
From: mplungjan-ga on 23 Apr 2002 01:14 PDT's_last_theorem.html:

Fermat's Last Theorem states that 

xn + yn = zn 
has no non-zero 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 



"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: yggdrassil-ga 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: petebevin-ga on 23 Apr 2002 07:09 PDT
Elliptic Curves and Modular Forms.

Dr Math has a good article explaining the concepts.

If you're interested, I suggest you read the book Fermat's Enigma by Simon 

Also, the sci.math FAQ has some basic information:
Subject: Re: Fermat's Last theorem
From: zhiwenchong-ga on 26 Jun 2002 21:59 PDT
The major bridge towards the proof of the FLT was the Taniyama-Shimura
conjecture. I'm surprised nobody mentioned it.

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