Newsgroups: sci.math
Subject: Re: Impossibility proof X^5 + Y^5 = Z^2
From: Rusin
Date: [never mailed I guess] ~ Oct 7 1996 [from datestamp]

In article <267@stt.win-uk.net>, Iain Davidson <iain@stt.win-uk.net> wrote:
>
>I was wondering if anyone has a proof of impossibility
>for X^5 + X^5 = Z^2, X,Y,Z relatively prime integers XYZ <> 0.

I looked through Dickson's old number theory compendium. It would seem
that the question must have occured to other people, but there was
no reference to it. Euler addressed the corresponding problem with
cubes. An interesting formula with three fifth powers was found by
Bunyakowsky ca 1835 by integrating

	integral  (x+a)^4 - (x-a)^4  dx

first with the power rule and then by expanding the polynomial;
but I wouldn't expect a 2-power formula.

Given that you are trying to take norms (in  Q[1^(1/5)] ) of algebraic
integers in just a 2-dimensional subspace of this 4-dimensional
extension ring, I would expect it to happen only very infrequently that
this really is a square. But of course that's nothing like a proof.

dave