Local solubility of generalised Fermat equations

TL;DR

This paper derives an asymptotic formula for counting integer triples with bounded absolute value that satisfy a generalized Fermat equation locally everywhere, confirming related conjectures and computing the leading constant.

Contribution

It provides the first asymptotic count for solutions to generalized Fermat equations and verifies conjectures by Loughran, Smeets, and Rome--Sofos for these cases.

Findings

Derived the asymptotic formula for solutions

Computed the leading constant explicitly

Confirmed conjectures for these equations

Abstract

For every $n \geq 2$ we determine the asymptotic formula for the number of integer triples $(a,b,c)$ of bounded absolute value such that the generalised Fermat equation given by $ax^n+by^n+cz^n=0$ is everywhere locally soluble. We compute the leading constant, answering a question of Loughran--Rome--Sofos, and determine that the conjectures of Loughran--Smeets and Loughran--Rome--Sofos hold for such equations.