A Mersenne prime is a prime number which can be expressed in the form $2^n-1$ for some $n\in \mathbb{N}$. For example, $7=2^3-1$ is a Mersenne prime.

A perfect number is a number which is equal to the sum of its proper divisors. For example, the proper divisors of $6$ are $1,2$ and $3$; since $6=1+2+3$, $6$ is a perfect number.

Prove that, for any Mersenne prime $M$,

is a perfect number.