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 $(2,2n,n)$, $C{x}^2+q^k{y}^{2n}=z^n$. 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.

20 pages

References in corpus (2)

• A brief survey on the generalized Lebesgue-Ramanujan-Nagell equation
• Differences between perfect powers : the Lebesgue-Nagell Equation

Cited by in corpus (2)

• Asymptotic Fermat equation of signature $(r,r,p)$ over totally real fields
• Generalised Fermat equation: a survey of solved cases