Perfect Number Generator

A number is perfect when its proper divisors sum to itself. 6 is the first: 1 plus 2 plus 3. Next comes 28, then 496, then 8128 — and then the jump is brutal, because the fifth is 33,550,336. That rarity is what makes the subject interesting: only 51 perfect numbers are known, all found through a formula two thousand years old.

Euclid showed that when two to the p minus 1 is prime, then two to the p minus 1, multiplied by that prime, is perfect. Euler proved, two millennia later, that every even perfect number has exactly that shape. That is why the page generates the list from the known exponents: there is no other way to find them, and brute-force searching would be hopeless.

Primes of the form two to the p minus 1 are the Mersenne primes, and each one discovered yields a new perfect number. The search is carried out today by a distributed computing project, and the largest known has tens of millions of digits. The simplest question of all remains open: is there an odd perfect number? Nobody has found one, nobody has proved none exists, and it is known that if one exists it exceeds 1500 digits.