# Perfect numbers

The sigma function is multiplicative.

If \(2^k - 1\) is prime, then \(2^{k - 1}\left(2^k - 1\right)\) is perfect.

If \(n\) is an even perfect number, then
$$n = 2^{p - 1}\left(2^p - 1\right)$$
for some prime \(p,\) and \(2^p - 1\) is also prime.

Page 59 of Elementary Number Theory by Dudley, 2nd ed.