Can a Mersenne Number $> 1$ ever be a perfect power?

Question

Can anyone prove the possibility of a Mersenne Number $2^p-1$ that is a perfect power (i.e. $2^p-1 = a^k$ for integers $(p, a, k) > 1$? There are none known with the conditions mentioned. Thanks for help!

Answer 1

Note that $2^p - a^k = 1$ implies that $2^p$ and $a^k$ are consecutive powers. By Mihailescu's Theorem, the only two consecutive powers of natural numbers are

$$3^2 - 2^3 = 1$$

so such $(p, a, k) > 1$ would not exist.

Answer 2

For an elementary solution, see this question. Some related questions around consecutive powers (provided with elementary solutions) are discussed in this conference paper.