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What was Fermat's Last Theorem?


Date: 07/23/97 at 14:39:52
From: Steve Gardner
Subject: Fermat's last theorem

If it is not too much trouble, I wonder if you might take the time to 
explain Fermat's Last Theorem to me.  I am an undergraduate in
mathematics, early in my degree (I have little formal education in
number theory; however, I am really interested in it). I know this is
asking the impossible, but please keep the explanation at an easy 
level.  As I said, I am early in my degree, so a somewhat superficial, 
qualitative answer would be perfect.


Date: 07/23/97 at 15:48:58
From: Doctor Rob
Subject: Re: Fermat's last theorem

In the margin of his copy of a book by Diophantus, Pierre de Fermat
wrote that it is possible to have a square be the sum of two squares,
but that a cube can not be the sum of two cubes, nor a fourth power
be a sum of two fourth powers, and so on. Further, he wrote that he
had found a truly marvelous proof which the margin was too small to
contain.

In modern language, he was stating that the equation x^n + y^n = z^n
has no solution in integers x, y, z > 0 and n > 2.

Long after all the other statements made by Fermat had been either
proved or disproved, this remained; hence it is called Fermat's *Last*
Theorem (actually, Conjecture would be more accurate than Theorem).

This conjecture was worked on by many famous mathematicians. Fermat
himself proved this theorem for n = 4, and Leonhard Euler did n = 3.
Special cases were dispatched one after another. New theories were
developed to attack this problem, but all attempts at a general proof
failed. They failed, that is, until this decade, when, building on
work of many famous mathematicians, Prof. Andrew Wiles of Princeton
University finally proved it. His method could not have been known
to Fermat. Fermat's "truly marvelous proof" is now believed to have
been faulty.

Is this what you wanted?  If not, write back with a better description
of what you want, and we will try to comply.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
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