## Even Perfect Numbers and Mersenne Prime Numbers

## Problem 496

Prove that if $2^n-1$ is a Mersenne prime number, then

\[N=2^{n-1}(2^n-1)\]
is a perfect number.

On the other hand, prove that every even perfect number $N$ can be written as $N=2^{n-1}(2^n-1)$ for some Mersenne prime number $2^n-1$.

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