After 2,000 years, nobody knows if an odd perfect number exists
Add up the divisors of 6, leaving out 6 itself, and you get 1, 2 and 3, which total exactly 6. Numbers with that neat property have fascinated thinkers since Euclid, yet only 52 are known, every one of them even, and two basic questions about them remain wide open.
Euclid defined perfect numbers in the Elements and gave a recipe for making them: whenever 2 to the power p, minus 1, is prime, multiplying it by 2 to the power p minus 1 yields a perfect number. With p set to 2, 3, 5 and 7 that recipe produces 6, 28, 496 and 8128, the only four the early Greeks knew. Primes of that special shape now carry the name of Marin Mersenne, a seventeenth-century monk who studied them. Around AD 1000 Ibn al-Haytham suggested, without proof, that the recipe caught every even perfect number, and in the 18th century Leonhard Euler proved it. The pairing of even perfect numbers with Mersenne primes is called the Euclid-Euler theorem.
Ancient writers read cosmic meaning into them. Philo of Alexandria argued that the world took 6 days to make and the moon 28 days to circle because both numbers are perfect, and Augustine repeated the point about creation in The City of God. Nicomachus, around AD 100, knew 8128 but wrongly claimed that the final digits alternate between 6 and 8; the sixth perfect number ends in 6, just like the fifth. The Egyptian scholar Ismail ibn Fallus listed the next three in the 13th century, and in 1588 Pietro Cataldi pinned down the sixth and seventh.
Today the hunt belongs to computers. The GIMPS distributed project has searched exhaustively through the first 51 even perfect numbers and found one more beyond them. By October 2024, 52 Mersenne primes were known, and the largest resulting perfect number runs to 82,048,640 digits. Mersenne primes are rare, since p must itself be prime and even then the result often factors; 2 to the 11th minus 1 equals 23 times 89.
Even perfect numbers hide other patterns. Written in binary, each is a run of ones followed by a run of zeros, and repeatedly summing the digits of any except 6 always lands on 1. Whether an odd one exists is unresolved: Euler called it a most difficult question, while Carl Pomerance has offered a heuristic case that none should. Any odd example would need at least 101 prime factors. Nobody knows whether perfect numbers go on forever, either.
Source: Perfect number
