The divisors of these even perfect numbers can be listed in two columns, illustrated here for the divisors of 496.
1 2 4 8 16 |
31 62 124 248 (496) |
The first column lists powers of 2 from 2^{0} up through 2^{4}. The sum of these powers of 2 is 31, which is one less than 2^{5}. That number 31 appears at the top of the second column, and its repeated doubles up through 496 appear on the second column. In such a tableau, the sum of all the numbers, except the last, will equal the last.
The question of odd perfect numbers was not solved by Euclid. Probably the oldest open conjecture in mathematics is that there are no odd perfect numbers. There is no proof yet, but it is known that if there is an odd perfect number, then it has to be immensely huge.