Equivalence of Definitions of Perfect Number

Theorem

The following definitions of the concept of Perfect Number are equivalent:

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.

Proof

Consider a strictly positive integer $n$.

By Definition 1, $n$ is a perfect number if and only if $n$ equals the sum of its aliquot parts.

By definition of divisor sum function, $\map {\sigma_1} n$ equals the sum of all the divisors of $n$.

Thus $\map {\sigma_1} n - n$ equals the aliquot sum of $n$.

So by Definition 2, $n$ is a perfect number if and only if $\map {\sigma_1} n - n = n$.

Hence the definitions are equivalent.

The equivalence of definition 4 to definition 2 follows directly.

The equivalence of definition 3 to definition 2 follows from the definition of abundance.

$\blacksquare$

Sources

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.2$: More about Numbers: Irrationals, Perfect Numbers and Mersenne Primes