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PerfectNumbers

A perfect number is a number equal to the sum of its divisors (except n itself).

For Example:
n = 6 is perfect, because 1 + 2 + 3 = 6

The Evens

It has been shown that all even perfect numbers have the form 2n-1(2n-1), where (2ⁿ-1) is prime. In fact, the converse holds as well -- all numbers of that form are perfect.

MersenneNumbers? are numbers of the form (2ⁿ-1). Therefore, the study and discovery of even perfect numbers is closely tied to the study of MersennePrimes?.

There are currently about 30 known mersenne primes and corresponding perfect numbers.

But what about the Odds ?

We don't know! really!

Currently we are not sure if there exist odd perfect numbers, but we do know that if there exists such a number, it must be greater than 10³⁰⁰. See the OddPerfectNumbers page for more information.

Why the big fuss about perfect numbers?

A number of different societies and cultures have appreciated the properties of perfect numbers, especially 6 and 28. Of course, had civilizations in the past developed advanced computing equipment, they might have revered 230(231-1)

RobinsonTryon - 14 Dec 2002