# Tagged: perfect number

## 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 $2^n-1$.

## Problem 74

(a) Show that if a group $G$ has the following order, then it is not simple.

1. $28$
2. $496$
3. $8128$

(b) Show that if the order of a group $G$ is equal to an even perfect number then the group is not simple.