Counting primitive integral solutions to spherical generalized Fermat equations

A solution (x, y, z) ∈ Z 3 -{(0, 0, 0)} to a generalized Fermat equation ( 1)By work of Beukers [Beu98], we know that in the spherical regime (that is, when the Euler characteristic χ = 1 a + 1 b + 1 c -1 is positive), if Equation (1) 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 [PSS07], to refine Beukers' result to an asymptotic count of the number of primitive integral solutions of bounded height.MSC class: 11D41 Contents 1. Introduction 1 2. Belyi maps and triangle groups 8 3. Counting rational points in the image of a rational function 10 4. Proof of main results 15 References 19 1 This is a ``preproof'' accepted article for Canadian Mathematical Bulletin This version may be subject to change during the production process.