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The Perfect Numbers By Burl Collins of MSRD Inc A perfect number is a positive integer that is the sum of it's proper positive divisors excluding the number itself. It can also be stated that a perfect number is half the sum of all of its positive divisors including itself. 6 is the first perfect number since 1, 2, and 3 are its proper positive divisors excluding itself and 1 + 2 + 3 = 6 and 6 is equal to half the sum of all its positive divisors including itself (1 + 2 + 3 + 6) / 2 = 6. The next highest perfect number is 28 = 1 + 2 + 4 + 7 + 14. This next two perfect numbers are 496 and 8128. There is a one to one association between the even perfect numbers and the Mersenne primes. At present there are 44 known perfect numbers which are also the known 44 Mersenne primes. Leonhard Euler proved that the formula 2n-1(2n - 1) will yield all the even perfect numbers. It is unknown whether there are any odd perfect numbers.

Contributed by dragon on August 29, 2008, at 9:41 PM UTC.

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