On the equation $x^2+dy^6=z^p$ for square-free $1\le d\le 20$

TL;DR

This paper employs the modular method and innovative techniques like symplectic arguments and multi-Frey approaches to prove the non-existence of solutions for the equation x^2+dy^6=z^p with square-free d between 1 and 20.

Contribution

It introduces new techniques such as symplectic arguments over ramified extensions and a multi-Frey approach to eliminate solutions, advancing the application of the modular method.

Findings

No primitive non-trivial solutions exist for the equation with given parameters.

The methods successfully extend previous approaches to a broader class of equations.

The approach can be adapted to similar exponential Diophantine equations.

Abstract

The purpose of the present article is to show how the modular method together with different techniques can be used to prove non-existence of primitive non-trivial solutions of the equation $x^2+dy^6=z^p$ for square-free values $1 \le d \le 20$ following the approach of [PT]. The main innovation is to make use of the symplectic argument over ramified extensions to discard solutions, together with a multi-Frey approach to deduce large image of Galois representations.