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