The generalized Fermat equation $Ax^2 + By^r = Cz^p$ and applications

TL;DR

This paper extends the modular method to solve the generalized Fermat equation involving hyperelliptic curves, applying it to a specific conjecture related to exponential Diophantine equations.

Contribution

It develops a new approach within Darmon's program using Frey hyperelliptic curves for the generalized Fermat equation.

Findings

Successfully applied the modular method to the generalized Fermat equation.

Provided insights into the conjecture of Laradji, Mignotte, and Tzanakis.

Extended the framework of Darmon's program to hyperelliptic curves.

Abstract

In this paper, we develop the modular method for the generalized Fermat equation appearing in the title, within the framework of Darmon's program and using Frey hyperelliptic curves. As an application, we study a conjecture of Laradji, Mignotte, and Tzanakis concerning the equation $5x^2+q^{2n}=y^5$.