# Perfect number

This article defines a property that can be evaluated for a natural number, i.e., every natural number either satisfies the property or does not satisfy the property.
View a complete list of properties of natural numbers

## Definition

A natural number  is termed a perfect number if , where  denotes the divisor sum function: the sum of all the positive divisors of . In particular,  equals the sum of all its proper positive divisors.

## Facts

• If  (the  Mersenne number) is a prime number (and hence, a Mersenne prime), then  is a perfect number.
• Every even perfect number arises in the above fashion.
• The existence of odd perfect numbers is an open problem.