From: "Iain Davidson" <Sttscitrans@tesco.net>
Subject: Re: A Fermat Proof and Number Theory
Date: Sat, 11 Sep 1999 03:15:55 +0100
Newsgroups: sci.math
Keywords: x^2 + 2 = y^3

MAppell917 <mappell917@aol.com> wrote in message
news:19990910125950.01182.00000454@ng-bg1.aol.com...
> Fermat had a proof of the following number theory problem:
>
> Prove that 26 is the ONLY number sandwiched between a square and a cube. That
> is to say, 25 is 5^2 and 27 is 3^3. Fermat proved that 26 was the ONLY number
> in all of the integers that were sandwiched like this.  I'm wondering if anyone
> knows how to prove this or if there is a proof anywhere on the internet. I'd
> like to see it.

In Dickson's "Theory of numbers" it says that "Fermat stated that he could
give
a rigorous proof that 25 is the only integral square which is less than a
cube by
2" . So Fermat did not actually publish a proof.

x^2 +2 = y^3
x,y odd

s^2 + 2t^2 = (s + t*sqrt(-2))(s - t*sqrt(-2))

Euler argued that  s^2 + 2t^2 could only be a cube
if  both (s + t*sqrt(-2)) and (s - sqrt(-2)) were cubes.

(s + t*sqrt(-2) = (a + b*sqrt(-2))^3
                      = a^3 + 3a^2*b*sqrt(-2) + -6a*b^2  -2b^3*sqrt(-2)
                      = (a^3 - 6a*b^2) +(3a^2*b -2b^3)sqrt(-2)

s = x, t = 1

b(3a^2 -2b^2) = 1

b = 1, (3a^2 -2b^2) = 1
b = -1, (3a^2 -2b^2) = -1
etc.


Euler's assumption is not generally valid however.
It works in this case because Z(sqrt(-2)) has unique
factorisation, 1, -1 are the only units and the factors are
relatively prime.

x^2 + 2 = (x + sqrt(-2))(x - sqrt(-2)) = y^3

Assume there is prime p in Z[sqrt(-2)] that divides
both  (x + sqrt(-2)) and (x - sqrt(-2)).

p then divides 2sqrt(-2) = (sqrt(-2))^3
sqrt(-2) is prime, so only sqrt(-2) and
its associate -sqrt(-2) can be a common divisor.

So sqrt(-2) divides (x - sqrt(-2))
So sqrt(-2) divides x, but x is an integer and so
must be even. x and y cannot both be even so
(x + sqrt(-2)) and (x - sqrt(-2)) are relatively
prime.

As factorisation in Z[sqrt(-2)] is unique then
(x + sqrt(-2)) and (x - sqrt(-2)) can only
have the form uv^3, where u is a unit.
The only units are 1, -1.

So Euler's assumption is correct in this case