Asymptotic Fermat's Last Theorem for a family of equations of signature $(2, 2n, n)$

Abstract

In this paper, we study the integer solutions of a family of Fermat-type equations of signature $(2, 2n, n)$, $Cx^2 + q^ky^{2n} = z^n$. We provide an algorithmically testable set of conditions which, if satisfied, imply the existence of a constant $B_{C, q}$ such that if $n > B_{C,q}$, there are no solutions $(x, y, z, n)$ of the equation. Our methods use the modular method for Diophantine equations, along with level lowering and Galois theory.