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).