Definition 22

A perfect number is that which is equal to the sum of its own parts.

Guide

For example, the number 28 is perfect because its parts (that is, proper divisors) 1, 2, 4, 7, and 14 sum to 28. The four smallest perfect numbers were known to the ancient Greek mathematicians. They are 6, 28, 496, and 8128. In proposition IX.36 Euclid gives a construction of even perfect numbers. For more discussion on perfect numbers, see the Guide for the that proposition.

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 2⁰ up through 2⁴. The sum of these powers of 2 is 31, which is one less than 2⁵. 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.