Counting primitive integral solutions to spherical generalized Fermat equations

2025/08/18 by Arango-Piñeros, Santiago
#11D41 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2508.13093

Abstract

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 gcd(x,y,z) = 1. By work of Beukers, we know that in the spherical regime (that is, when the Euler characteristic χ= \tfrac1a + \tfrac1b + \tfrac1c - 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.

Citations

• Fermat descent
• Algebraic curves uniformized by congruence subgroups of triangle groups

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