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Odd Perfect Numbers

Problem

The divisors of a positive integer, excluding the number itself, are called the proper divisors . If the sum of proper divisors is equal to the number we call the number perfect. For example, the divisors of 28 are 1, 2, 4, 7, 14, and 28, so the sum of proper divisors is 1 + 2 + 4 + 7 + 14 = 28.

The first eight perfect numbers are 6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128.

It is well known that P is an even perfect number iff it is of the form $2^{n-1}(2^n - 1)$ where $2^n - 1$ is prime.

No one has yet discovered an odd perfect number, and the existence of them is in doubt.

However, if $Z$ is an odd perfect, prove that it must be of the form $c^2 q^{4k+1}$ where $q \equiv 1 \mod 4$ is prime.

Problem ID: 328 (05 Jul 2007)     Difficulty: 4 Star

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