Perfect Number Checker

Perfect numbers: definition and Euclid-Euler theorem

A perfect number is a positive integer n equal to the sum of its proper divisors (divisors excluding n itself). Formally, σ(n) − n = n, where σ is the sum-of-divisors function. The smallest examples: 6 = 1 + 2 + 3; 28 = 1 + 2 + 4 + 7 + 14; then 496, 8128. Euclid (Elements, Book IX) proved that if 2^p − 1 is a Mersenne prime, then (2^p − 1) · 2^(p−1) is an even perfect number. Euler proved the converse: every even perfect number has this form. Only 51 even perfect numbers are known as of 2024 (each tied to a GIMPS Mersenne prime). The existence of odd perfect numbers remains an open problem — if any exist, they must exceed 10^3000.

Applications and historical context

Perfect numbers are primarily of theoretical interest in number theory. They underlie work by Erdős on aliquot sequences and are deeply linked to Mersenne primes, the largest primes humans have ever found. They appear in recreational mathematics, mathematical olympiad problems, and number-mysticism traditions (Pythagoreans considered 6 and 28 spiritually significant).

FAQ

Does any odd perfect number exist? Unknown — one of the oldest open problems in mathematics. Computer searches up to 10^3000 found none.

Are there infinitely many perfect numbers? Unknown; depends on whether infinitely many Mersenne primes exist (also conjectured but unproven).

What is the next perfect number after 8128? 33,550,336 (p = 13).

Why are they called "perfect"? Ancient Greek tradition — Pythagoreans saw the equality between a number and the sum of its parts as a sign of harmony.