Can a Mersenne number be a power (with exponent > 1) of a prime? Let $n \geq 1$ and consider the (Mersenne) number $M_n = 2^n-1$. Is it possible that $M_n = p^k$ for some prime $p$ and some (necessarily odd) $k > 1$? Thanks in advance.

It is not possible. Catalan's conjecture (proved in 2002) states that the only solution of $x^a-y^b=1$ ($x$ and $y$ are positive integers and $a$ and $b$ are integers greater than $1$) is $3^2-2^3=1$.
If a Mersenne's number were the power of a prime, we'd have
$$2^n-1=p^k$$
or
$$2^n-p^k=1$$
contradicting Catalan's conjecture.
