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 $33 \, 550 \, 536$.