What is a perfect number? Are all perfect numbers even?
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Definition: A Perfect Number N is defined as any positive integer where the sum of its divisors minus the number itself equals the number. The first few of these, already known to the ancient Greeks, are 6, 28, 496, and 8128.
A Perfect Number “n”, is a positive integer which is equal to the sum of its factors, excluding “n” itself.
Euclid, over two thousand years ago, showed that all even perfect numbers can be represented by,
N = 2p-1(2p -1) where p is a prime for which 2p -1 is a Mersenne prime.
That is, we have an even Perfect Number of the form N whenever the Mersenne Number 2p -1 is a prime number. Undoubtedly Mersenne was familiar with Euclid’s book in coming up with his primes.
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