On the diophantine equation $x^{2} - 2^{m} = \pm \, y^{n}$

One of the purposes of this note is to correct the proof of a recent result of Y. Guo & M. Le on the equation $x^{2} - 2^{m} = y^{n}$. Moreover, we prove that the diophantine equation $x^{2} - 2^{m} = \pm \, y^{n}$, $x$, $y$, $m$, $n \in \mathbf {N}$, gcd$(x, y) =1$, $y>1$, $n>2$ has only finitely many solutions, all of which satisfying $n \le 7.3 \, 10^{5}$.