An application of the modular method and the symplectic argument to a Lebesgue-Nagell equation

In this paper, we study the generalized Lebesgue-Nagell equation [ x^2+7^{2k+1}=y^n. ] This is the last case of equations of the form $x^2+q^{2k+1}=y^n$ with $kgeq0$ and $q>0$ where $mathbb{Q}(sqrt{-q})$ has class number one. Our proof is based on the modular method and the symplectic argument.