A perfect number equals the sum of its proper divisors (positive divisors less than itself). The first four are 6, 28, 496, and 8128 — known to the ancient Greeks. Euclid showed every \(2^{p-1}(2^p - 1)\) where \(2^p - 1\) is a Mersenne prime is perfect; Euler later proved this gives all even perfect numbers.
Finding even perfect numbers therefore reduces to finding Mersenne primes. As of 2024, only 51 perfect numbers are known, the largest having tens of millions of digits.
Whether any odd perfect number exists is one of the oldest unsolved problems in mathematics — none has been found, and any that exists must be larger than \(10^{1500}\) with very specific structural properties.