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

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.