On the equation x2 + dy6 = zp for square-free 1 ≤ d ≤ 20

Abstract

The purpose of this paper 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 x2 + dy6 = zp for square-free values 1 ≤ d ≤ 20. The key ingredients are: the approach presented in [A. Pacetti and L. V. Torcomian, Q-curves, Hecke characters and some Diophantine equations, Math. Comp. 91(338) (2022) 2817–2865] (in particular its recipe for the space of modular forms to be computed) together with the use of the symplectic method (as developed in [E. Halberstadt and A. Kraus, Courbes de Fermat: Résultats et problèmes, J. Reine Angew. Math. 548 (2002) 167–234], although we give a variant over ramified extensions needed in our applications) to discard solutions and the use of a second Frey curve, aiming to prove large image of residual Galois representations.