A perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. For example, 1 + 2 + 3 = 6 implies 6 is a perfect number.
Unsolved Problem: Are there any odd perfect numbers?
The belief, by some, that there are none goes back over 500 years (wikipedia).
If you want to check out some recent research into this problem, see oddperfect.org.
(Another unsolved problem: Are there an infinite number of even perfect numbers?)