Perfect number

Thus, 6 = 1 + 2 + 3 is a perfect number, since 1, 2 and 3 are the numbers which divide 6 evenly. The next perfect number is 28, as 28 = 1 + 2 + 4 + 7 + 14.

Perfect numbers are related to Mersenne primes (prime numbers that are one less than a power of 2): if M is a Mersenne prime, then M×(M+1)/2 is a perfect number. (This was proved by Euclid in the 4th century BC.) Furthermore, all even perfect numbers are of this form (as proved by Euler in the 18th century). So we have a concrete one-to-one association between even perfect numbers and Mersenne primes.

Only finitely many Mersenne primes (hence even perfect numbers) are presently known. It is unknown whether there are infinitely many of them. See the entry on Mersenne prime for additional information concerning the search for these numbers.

It is unknown whether there are any odd perfect numbers. Various results have been obtained, but none that have helped to locate one or otherwise resolve the question of their existence. It is known that if an odd perfect number does exist, it must be greater than 10³⁰⁰. Also, it must have at least 8 distinct prime factors (and at least 11 if it is not divisible by 3), and it must have at least one prime factor greater than 10⁷, two prime factors greater than 10⁴, and three prime factors greater than 100.

Considering the sum of proper divisors gives various other kinds of numbers. Numbers where the sum is less than the number itself are called deficient, and where it is greater, abundant; these, together with perfect numbers, come from Greek numerology. A pair of numbers which are the sum of each other's proper divisors are called amicable, and larger cycles of numbers are called sociable[?]. A number which is amicable to itself is perfect.

Some other related information can be found at Amicable Numbers (http://xraysgi.ims.uconn.edu:8080/amicable)