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

In this paper, we study the integer solutions of a family of Fermat‐type equations of signature , . We provide an algorithmically testable set of conditions which, if satisfied, imply the existence of a constant such that if , there are no solutions of the equation. Our methods use the modular method for Diophantine equations, along with level lowering and Galois theory.