# Perfect number

From 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

## Contents

## 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.

## Relation with other properties

### Weaker properties

### Variations

- Almost perfect number: This requires .
- Quasiperfect number: This requires .

### Opposite properties

- Abundant number: This requires .
- Deficient number: This requires .

## 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.