Perfect Number Finder

About this calculator

A perfect number is a positive integer that equals the sum of its own proper divisors (all divisors except itself) — 6 is perfect because 1+2+3=6, and 28 is perfect because 1+2+4+7+14=28. These numbers are extremely rare: only four were known to the ancient Greeks (6, 28, 496, 8128), and even today only a few dozen are known in total.

Every known even perfect number follows the pattern discovered by Euclid and completed by Euler nearly two thousand years later — the Euclid–Euler theorem — which states that an even number is perfect if and only if it has the form 2^(p−1) × (2^p − 1), where 2^p − 1 is a Mersenne prime. Whether any odd perfect numbers exist at all is still an unsolved problem in number theory. This finder is a number-theory teaching tool: enter an upper limit and it lists every perfect number up to that bound by checking each candidate’s divisor sum.