Counting primitive integral solutions to spherical generalized Fermat 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 $χ= \tfrac{1}{a} + \tfrac{1}{b} + \tfrac{1}{c} - 1$ 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.