---
tag: 0SB7Q8SYSF
kind: definition
name: Perfect number
msc: [11A25]
defines: 0MXGYCH4SY
uses: [009QCP5T3Z]
sources:
  - url: https://en.wikipedia.org/wiki/Perfect_number
    locator: lead section
---

## Definition

A positive integer $n$ is perfect if it is equal to the sum of its positive divisors other than $n$.
Equivalently, $\sigma(n) = 2n$, where $\sigma$ is the [[sum of divisors|009QCP5T3Z]].

## Remarks

The first perfect numbers are $6$, $28$, $496$ and $8128$. For example, $28 = 1 + 2 + 4 + 7 + 14$.
The [[Euclid–Euler theorem|0WETJCKNRG]] describes all even perfect numbers by [[Mersenne primes|06GPJF9V7M]].
No odd perfect number is known, and it is an open problem whether one exists.

## Explanations

### L1 human

A perfect number is the sum of its smaller divisors.
The divisors of $6$ other than $6$ are $1$, $2$ and $3$, and $1 + 2 + 3 = 6$.