Re: A question related to a Thue equation

From: Randall (someone.not_at_aol.com)
Date: 02/23/05 

Date: Wed, 23 Feb 2005 16:30:04 +0000 (UTC)

> Given d a positive integer, we are considerig the Thue equation x^2 -
> d^2 x y + y^2 = 1.
> What is known about such equations when we in addition have that x^2 +
> y ^2 = z^2 for z an integer?
> It is easy to see that for d = 1 and d = 2 there are no non-trivial
> solutions.
> Do there exist any positive integer d such that we have non-trivial
> solutions?

Kent:

It appears not likely. Ralph Buchholz and I have been discussing this
equation, it is genus 3. I did a rather large computer search initially,
finding d^2 as a rational polynomial in x,y (setting them to the
Pythagorean form) with empty results for d in Q, not to mention in Z.

Randall