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

TL;DR

This paper completely characterizes the integer solutions to the equation x^2 - 2^m = y^n for m in integers and n ≥ 3, using modularity methods to resolve previously unsolved cases.

Contribution

It proves that the only solutions to x^2 - 2 = y^n for n ≥ 2 are (±1, -1) when n is odd, completing the classification for this equation.

Findings

Only solutions are (±1, -1) for n odd.

Uses modularity method to solve unsolved cases.

Completes the classification of solutions for the equation.

Abstract

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.