Perfect Numbers
Mersenne primes Mp are interesting to many for their connection to perfect numbers. In the 4th century BCE, Euclid demonstrated that, whenever 2p−1 is prime, 2p−1(2p−1) is an even perfect number. This number, also expressible as Mp(Mp+1)/2, is the Mpth triangular number and the 2p−1th hexagonal number. In the 18th century, Leonhard Euler proved that, conversely, all even perfect numbers have this form. It is unknown whether there are any odd perfect numbers, but it appears unlikely.
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