A positive integer is perfect if its aliquot sum equals \(n\). Equivalently, a positive integer \(n\) is perfect if \(\sigma(n)=2n\).
Q105002: Are there any odd perfect numbers?
Q105003: Are there infinitely many perfect numbers?
| Author | Date Published | |
| Nicomachus of Gerasa | ~AD 100 |
By the Euclid–Euler theorem, a number is perfect if and only if it is of the form \( 2^{p-1}(2^p-1) \) where \(2^p-1\) is a Mersenne prime. There is a one-to-one Correspondence between the even perfect numbers and the Mersenne primes.
The first four perfect numbers \(6, 28, 496, 8128\) were known to the ancient Greeks. There are 51 known perfect numbers, the largest being \( 2^{82,589,933-1}(2^{82,589,933}-1)\).
Hyperperfect numbers
Superperfect numbers
Quasiperfect numbers