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perfect numbers
A perfect number is a positive initeger n such that the sum of all of its positive divisors is 2n. Euclid proved that if 2p-1 is prime then 2p-1(2p-1) is an even perfect number. Euler proved that the Euclid's example is the complete list of even perfect numbers.
Existence of odd perfect numbers is one of the most famous unsolved problems in number theory.