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.
Read more about this topic: Mersenne Prime
Famous quotes containing the words perfect and/or numbers:
“Perfect soldier, perfect gentleman ... never gave offence to anyone, not even the enemy.”
—A.J.P. (Alan John Percivale)
“I had but three chairs in my house; one for solitude, two for friendship; three for society. When visitors came in larger and unexpected numbers there was but the third chair for them all, but they generally economized the room by standing up.”
—Henry David Thoreau (1817–1862)