Definition. (suggested new terminology) If n is an even perfect number, then the statement that p is the generating prime for n means that p is the integer such that n = (2^(p – 1))*(2^p – 1).

The following theorem is posted on Mathematics Stack Exchange:

Theorem. If n is an even perfect number, then n has 2p – 1 proper divisors, and their product is n^(p-1), where p is the generating prime for n.

Corollary. If n is an even perfect number, then the product of the divisors of n is a power of n.

(However, the converse is not true. For example, the product of the divisors of 8 is 64, and 64 = 8^2, but 8 is not a perfect number.)

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[Number Theory]

[even perfect numbers]

[a necessary condition for an even number to be perfect]

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