# Aperiodvent, Day 2: The Euclid-Euler Theorem

Today’s entry is a Theorem of the Day:

The Euclid-Euler Theorem:

An even positive integer is a perfect number, that is, equals the sum of its proper divisors, if and only if it has the form $2^{n−1}(2^n − 1)$, for some n such that $2^n − 1$ is prime.

This theorem describes the relationship between perfect numbers and Mersenne primes. For more information, see the full listing at Theorem of the Day: the Euclid-Euler Theorem.

This is part of the Aperiodical Advent Calendar. We’ll be posting a new surprise for you each morning until Christmas!