This implies that both $m-1$ and $m+1$ are both powers of $2$. However, there are no distinct powers of 2 that are a distance of 2 apart, so there are no solutions over the natural numbers.

This implies that both $m-1$ and $m+1$ are both powers of $2$. However, there are no distinct powers of 2 that are a distance of 2 apart, so there are no solutions over the natural numbers.