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PERFECT NUMBERS

An integer is called a perfect number if it is equal to the sum of all of its positive factors, excluding itself.  The first perfect number is 6 because 6=1+2+3, where 1, 2 and 3 are the factors of 6.  The next is 28 because 28=1+2+4+7+14.  The next two are 496 and 8128.

One theorem in Euclid's book Elements deals with perfect numbers.  The theorem states that:

If 2ⁿ - 1 is a prime number, then

      (2^n-1)(2ⁿ - 1) is a perfect number.

Until present, no one has found an odd perfect number.  No one has shown that odd perfect number does not exist either.

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