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/
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