Fermat's Last Theorem for n = 3 and 4

Date: 12/16/2004 at 04:14:02
From: Lucy
Subject: Fermat's last theorem

I understand that Fermat's solution (if one did exist) to his famous 
last problem is unknown, but what was his solution for a^n + b^n = 
c^n  for the cases, n = 3 and 4?  I know he published a solution for 
n=4 and that if he did not solve for n=3 then Euler came up with a 
solution that was later corrected.  I can not find these proofs 
anywhere on the net, please help me or guide me in the right 
direction!

Date: 12/16/2004 at 14:35:06
From: Doctor Vogler
Subject: Re: Fermat's last theorem

Hi Lucy,

Thanks for writing to Dr. Math.  That's a good question.  I first saw
Fermat's proof for the n=4 case in a book called "Elliptic Curves" by
Anthony W. Knapp (pg 81).  It is a proof by descent that

  x^4 + y^4 = z^2

has no solutions in positive integers.  Proof by descent was Fermat's
new idea, and it is a technique similar to induction whereby you show
that there are no positive integer solutions by assuming you have one
and showing that you can use this solution to get a smaller solution.
Since positive integers can't keep getting infinitely smaller, this
means there can't be any positive integer solutions.

Not wanting to copy the whole proof out of that book, I did a Google
search for

  "Fermat's Last Theorem" Euler proof n=3

and found a few interesting sites, including one at

    http://www.mathreference.com/num-zext,flt.html 

that you might find interesting.  However, they proved the n=4 case in
a rather different way by analyzing numerous related equations.  So I
did a Google search for

  "Fermat's Last Theorem" descent n=4

and found some better sites describing Fermat's proof by descent, such as

    http://www.math.toronto.edu/mathnet/questionCorner/fermat4.html 

and

    http://homepages.cwi.nl/~dik/english/mathematics/flt.html 

and the latter also has Euler's proof for the n=3 case.

So it seems you just needed to do more searching; they're on the net
after all!

If you have any questions about this or need more help, please write
back and show me what you have been able to do, and I will try to
offer further suggestions.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/