Perfect numbers are numbers that are equal to the sum of their factors: 6 is perfect because its factors are 1, 2, and 3, and 1 + 2 + 3 = 6. Likewise 28 = 1 + 2 + 4 + 7 + 14; and so on. So far, 44 perfect numbers are known.
Puzzle: Can you prove that if 2n - 1 is prime, then 2n - 1(2n - 1) is perfect?
planx_constant mentioned that little theorem to me over vacation. It was first proved by Euclid. Millenia later, Euler proved that all even perfect numbers are produced by this formula. But it is not known whether there are any odd perfect numbers. Most mathematicians seem to think there are none. Here's James Joseph Sylvester, writing in 1888:
...a prolonged meditation on the subject has satisfied me that the existence of any one such—its escape, so to say, from the complex web of conditions which hem it in on all sides—would be little short of a miracle.