Perfect number/Advanced

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An advanced level version of Perfect number.

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:

  • 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.

  1. From Hardy and Wright, Introduction to the Theory of Numbers