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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:

  • 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.

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