# Perfect number

A perfect number is a number for which the sum of its proper factors is equal to the number itself. The proper factors of a number are all the positive factors of the number, excluding the number. For example, the proper factors of 12 are 1, 2, 3, 4, and 6. However, 12 is not a perfect number.

Examples

Determine whether the following numbers are perfect numbers.

1. 12:

As mentioned above, the proper factors of 12 are 1, 2, 3, 4, and 6.

1 + 2 + 3 + 4 + 6 = 16 ≠ 12

12 is not a perfect number.

2. 6:

The proper factors of 6 are 1, 2, and 3.

1 + 2 + 3 = 6

6 is a perfect number.

3. 28:

The proper factors of 28 are 1, 2, 4, 7, and 14.

1 + 2 + 4 + 7 + 14 = 28

28 is a perfect number.

Perfect numbers that are even can be found using Euclid's rule. Euclid's rule states that the following expression,

yields an even perfect number given that is prime.

Example

If p = 2, 2^{2} - 1 = 4 - 1 = 3. 3 is a prime number, so:

6, as we confirmed above, is a perfect number.

There are a total of 51 known perfect numbers, all of which are even. There are currently no known odd perfect numbers, and it is believed that none exist. However, this has yet to be proven. Below are the first 5 perfect numbers:

6, 28, 496, 8128, 33550336

The remaining perfect numbers quickly get much larger, with the current 51st known perfect number having 49,724,095 digits.

### Did you know?

Perfect numbers have been studied since ancient times. The early Egyptians are believed to have studied perfect numbers. Pythagoras, a famous Greek mathematician and philosopher, studied them, perhaps from a metaphysical (nonphysical) as well as a mathematical perspective. There is still much to learn about perfect numbers, and mathematicians still study them today.