On the equation $x^2-2^m=y^n$

In this article we finish the study of solutions of the equation $x^2-2^m=y^n$ for $m\in\mathbb{Z}$ and $n\geq3$. This is achieved using the modularity method in unsolved cases, namely, we prove that the only integer solutions of $x^2-2=y^n$ for $n\geq2$ are $(\pm1,-1)$ when $n$ is odd.