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Perfect numbers cannot be prime numbers. Here's why:

A number N is perfect if σ(N) = 2N (σ is the sum of divisors function). If there is a prime p that is a perfect number, then σ(p) = 2p. However, the only factors of p are 1 and p, so σ(p) is also equal to p+1. If 2p = p+1, then p=1, which is not prime, and 1 is defined to have only one factor, 1.

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Q: Do perfect numbers have to be prime numbers?
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They really are not


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Prime numbers have two factors. The sum of their proper divisors is always 1.


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