# Definition:Perfect Number

## Contents

## 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:

- $\sigma \left({n}\right) = 2 n$

where $\sigma: \Z_{>0} \to \Z_{>0}$ is the sigma 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:

- $\dfrac {\sigma \left({n}\right)} n = 2$

where $\sigma: \Z_{>0} \to \Z_{>0}$ is the sigma function.

## Sequence of Perfect Numbers

The sequence of perfect numbers begins:

\(\displaystyle 6\) | \(=\) | \(\displaystyle 2^{2 - 1} \times 2^2 - 1\) | |||||||||||

\(\displaystyle 28\) | \(=\) | \(\displaystyle 2^{3 - 1} \times 2^3 - 1\) | |||||||||||

\(\displaystyle 496\) | \(=\) | \(\displaystyle 2^{5 - 1} \times 2^5 - 1\) | |||||||||||

\(\displaystyle 8128\) | \(=\) | \(\displaystyle 2^{7 - 1} \times 2^7 - 1\) | |||||||||||

\(\displaystyle 33 \, 550 \, 336\) | \(=\) | \(\displaystyle 2^{13 - 1} \times 2^{13} - 1\) | |||||||||||

\(\displaystyle 8 \, 589 \, 869 \, 056\) | \(=\) | \(\displaystyle 2^{17 - 1} \times 2^{17} - 1\) |

## 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.

(*The Elements*: Book $\text{VII}$: Definition $22$)

## 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

- Theorem of Even Perfect Numbers: An even perfect number is of the form $2^{n - 1} \paren {2^n - 1}$, where $2^n - 1$ is prime.

- 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.

Nicomachus made the following conjectures:

- One Perfect Number for Each Number of Digits
- Last Digit of Perfect Numbers Alternates between $6$ and $8$

both of which are seen to be incorrect from the next few instances in the sequence:

- $6, 28, 496, 8128, 33 \, 550 \, 336, 8 \, 589 \, 869 \, 056, \ldots$

A manuscript of $1456$ correctly gives the $5$th perfect number as $33 \, 550 \, 536$.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $6$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**Euclid numbers** - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $6$