Counting primitive integral solutions to spherical generalized Fermat equations

A solution $(x,y,z)\in {\mathbb{Z}}^3-\{(0,0,0)\}$ to a generalized Fermat equation $$A{x}^a+B{y}^b+C{z}^c=0,$$ 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 $\chi =\frac{1}{a}+\frac{1}{b}+\frac{1}{c}-1$ 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.