Definition:Perfect Number
This page is about Perfect in the context of Number Theory. For other uses, see Perfect.
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.
(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): Euclid numbers
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $6$