# Definition:Perfect Number

## Definition

### Definition 1

A perfect number is a (strictly) positive integer equal to its aliquot sum.

### Definition 2

A perfect number $n$ is a (strictly) positive integer such that:

$\map {\sigma_1} n= 2 n$

where $\sigma_1: \Z_{>0} \to \Z_{>0}$ is the divisor sum function.

### Definition 3

Let $A \left({n}\right)$ denote the abundance of $n$.

$n$ is perfect if and only if $A \left({n}\right) = 0$.

### Definition 4

A perfect number $n$ is a (strictly) positive integer such that:

$\dfrac {\map {\sigma_1} n} n = 2$

where $\sigma_1: \Z_{>0} \to \Z_{>0}$ is the divisor sum function.

## Sequence of Perfect Numbers

The sequence of perfect numbers begins:

 $\ds 6$ $=$ $\ds 2^{2 - 1} \times \paren {2^2 - 1}$ $\ds 28$ $=$ $\ds 2^{3 - 1} \times \paren {2^3 - 1}$ $\ds 496$ $=$ $\ds 2^{5 - 1} \times \paren {2^5 - 1}$ $\ds 8128$ $=$ $\ds 2^{7 - 1} \times \paren {2^7 - 1}$ $\ds 33 \, 550 \, 336$ $=$ $\ds 2^{13 - 1} \times \paren {2^{13} - 1}$ $\ds 8 \, 589 \, 869 \, 056$ $=$ $\ds 2^{17 - 1} \times \paren {2^{17} - 1}$

## Examples of Perfect Numbers

### 6

$6$ is a perfect number:

$1 + 2 + 3 = 6$

### 28

$28$ is a perfect number:

$1 + 2 + 4 + 7 + 14 = 28$

### 496

$496$ is a perfect number:

$1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496$

### 8128

$8128$ is a perfect number:

$1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 = 8128$

## Euclid's Definition

In the words of Euclid:

A perfect number is that which is equal to its own parts.

## Flow Chart

The following flow chart can be used to define an algorithm (not particularly efficient) for finding all perfect numbers under $500$:

## Also known as

The even perfect numbers are also known as the Euclid numbers.

However, on $\mathsf{Pr} \infty \mathsf{fWiki}$ we already have a definition for Euclid number, as one more than a primorial.

As the only known perfect numbers are all even anyway, the distinction is rarely considered worthy of a separate definition.

## Also see

• Results about perfect numbers can be found here.

## Historical Note

The first $4$ perfect numbers:

$6, 18, 496, 8128$

were known to the ancient Greeks.

All were listed by both Nicomachus of Gerasa and Iamblichus Chalcidensis.

Last Digit of Perfect Numbers Alternates between $6$ and $8$
$6, 28, 496, 8128, 33 \, 550 \, 336, 8 \, 589 \, 869 \, 056, \ldots$
A manuscript of $1456$ correctly gives the $5$th perfect number as $33 \, 550 \, 536$.