In the range $$1\le n\le 10^5$$ the only perfect powers of the form $$2^n\pm n$$ are $$2^5-5=3^3$$ and $$2^7-7=11^2$$
How can I prove that there are no more perfect powers of this form ?
The case of even $n$ allows at least to find the possible perfect squares, but I do not have an idea for a complete solution.