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### 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
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

"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!

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/
```
Associated Topics:
College Number Theory

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