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arXiv · 2109.07291

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

Franco Golfieri·Ariel Pacetti·Lucas Villagra Torcomian
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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.

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Franco Golfieri, Ariel Pacetti, Lucas Villagra Torcomian. 2021-09-15. On the equation $x^2+dy^6=z^p$ for square-free $1\le d\le 20$. https://arxiv.org/abs/2109.07291

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