Definition:Perfect Number

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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:

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

This sequence is A000396 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


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 $500$:


PerfectNumbers.png


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.


Nicomachus made the following conjectures:

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

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


Sources