Counting primitive integral solutions to spherical generalized Fermat equations

TL;DR

This paper refines previous results on the number of primitive solutions to spherical generalized Fermat equations, providing an asymptotic count using Fermat descent methods.

Contribution

It introduces an asymptotic counting approach for primitive solutions in the spherical regime, extending Beukers' qualitative results.

Findings

Asymptotic count of primitive solutions with bounded height

Refinement of Beukers' result on solution infinitude

Application of Fermat descent method to generalized equations

Abstract

A solution $(x,y,z) \in \mathbb{Z}^3-\{(0,0,0)\}$ to a generalized Fermat equation \[ Ax^a + By^b + Cz^c = 0, \] is called \emph{primitive} if $\gcd(x,y,z) = 1$. By work of Beukers, we know that in the \emph{spherical} regime (that is, when the Euler characteristic $\chi = \tfrac{1}{a} + \tfrac{1}{b} + \tfrac{1}{c} - 1$ is positive), if the equation has one primitive solution, then it has infinitely many. In this work, we use the method of \emph{Fermat descent}, as employed by Poonen--Schaefer--Stoll, to refine Beukers' result to an asymptotic count of the number of primitive integral solutions of bounded height.