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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