perfect number

Some numbers are perfectly balanced against their own parts. Add up all the divisors of 6 that are smaller than 6 — that is 1, 2, and 3 — and you get back exactly 6. A number with this rare self-equilibrium is called a perfect number.

Formally, a positive integer is perfect if it equals the sum of its proper divisors, meaning all its positive divisors except the number itself. The first few are 6, 28, 496, and 8128. Euclid found that whenever 2^p − 1 is prime (such primes are called Mersenne primes), the number 2^(p−1) times (2^p − 1) is perfect, and Euler later proved every even perfect number has exactly this shape.

Much remains genuinely unknown, which is part of the charm. No one has ever found an odd perfect number, yet no one has proved that none exists — it is one of the oldest open problems in mathematics. It is also unknown whether there are infinitely many perfect numbers at all, since that hinges on whether there are infinitely many Mersenne primes.