Counting primitive integral solutions to spherical generalized Fermat equations

A solution (x,y,z) ∈ Z³-\(0,0,0)\ to a generalized Fermat equation \[ Ax^a + By^b + Cz^c = 0, \] is called primitive if (x,y,z) = 1. By work of Beukers, we know that in the spherical regime (that is, when the Euler characteristic χ= 1/a + 1/b + 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 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.