  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 $k\geq0$ and $q>0$ where $\mathbb{Q}(\sqrt{-q})$ has
class number one. Our proof is based on the modular method and the symplectic
argument.
