From: Hull Loss Incident
Subject: Re: Solutions of generalized FLT
Date: Tue, 02 Jan 2001 20:07:52 -0500
Newsgroups: sci.math
William Elliot wrote:
> > [discussing the Diophantine equation x^a + y^b = z^c]
> >Granville's joint work with Darmon proved finiteness of primitive
> >solutions for x^a + y^b = z^c with a,b,c fixed.
> __
> What are the limitations on a,b,c? All greater than 2?
When 1/a + 1/b + 1/c < 1, number of primitive solutions is
finite. Non-primitive solutions come in infinite families
when one of a,b,c is relatively prime to the others. When
each pair of exponents have a common factor, such as
x^p + y^q = z^pq (or x^n + y^n = z^n !), solutions
correspond to rational points on curves of genus > 1 so
are finite in number up to rescaling.
> What are the solutions? In particular what's the situation with
> x^2 + y^p = z^2p odd prime p
No primitive integer solutions, by Darmon's work of 1993 (in Intl Math Res
Notices, see reference in "Beal conjecture" thread). Non-primitive
solutions give points on hyperelliptic curve A^2 = B^p + 1, so are
finite in number up to rescaling (x,y,z) --> (xT^p, yT^2,zT).
> x^2 + y^p = z^p p > 2
No primitive integer solutions for p>3 (theorem of Darmon 1993).
Nonprimitive integer solutions, and rational solutions, come in
infinite families with (x^2, y^p, z^p) proportional
to any given triple (C^p - B^p, B^p, C^p).
When p=3 primitive integer solutions exist and come in a finite
number of parametrized polynomial families (plus nonprimitive
or rational rescalings of these).
> 1 + x^p = y^q p,q > 1
For p,q > 3, finite number of rational points by Mordell conjecture
(Faltings theorem).