# Perfect number/Advanced

## Definition in terms of the sum-of-divisors function

Perfect numbers can be succinctly defined using the sum-of-divisors function . If is a counting number, then is the sum of the divisors of . A number is perfect precisely when

- .

## Proof of the classification of even perfect numbers

Euclid showed that every number of the form

where is a Mersenne prime number is perfect. A short proof that every even perfect number must have this form can be given using elementary number theory.

The main prerequisite results from elementary number theory, besides a general familiarity with divisibility, are the following:

- is a multiplicative function. In other words, if and are relatively prime positive integers, then .

- If is a power of a prime number, then

### The proof^{[1]}

Let be an even perfect number, and write where and is odd. As is multiplicative,

- .

Since is perfect,

- ,

and so

- .

The fraction on the right side is in lowest terms, and therefore there is an integer so that

- .

If , then has at least the divisors , , and 1, so that

- ,

a contradiction. Hence, , , and

If is not prime, then it has divisors other than itself and 1, and

- .

Hence, is prime, and the theorem is proved.

- ↑ From Hardy and Wright, Introduction to the Theory of Numbers