Local solubility of generalised Fermat equations

Abstract

For every $n\ge 2$ 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 $a{x}^n+b{y}^n+c{z}^n=0$ 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.