# 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.