Equivalence of Definitions of Amicable Pair

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Theorem

Let $m \in \Z_{>0}$ and $n \in \Z_{>0}$ be (strictly) positive integers.

The following definitions of the concept of Amicable Pair are equivalent:

Definition 1

$m$ and $n$ are an amicable pair if and only if:

the aliquot sum of $m$ is equal to $n$

and:

the aliquot sum of $n$ is equal to $m$.

Definition 2

$m$ and $n$ are an amicable pair if and only if:

$\map \sigma m = \map \sigma n = m + n$

where $\map \sigma m$ denotes the $\sigma$ function.

Definition 3

$m$ and $n$ are an amicable pair if and only if they form a sociable chain of order $2$.


Proof

Definition 1 is equivalent to Definition 2

Let $s \left({n}\right)$ denote the aliquot sum of (strictly) positive integer $n$.

The sum of all the divisors of a (strictly) positive integer $n$ is $\sigma \left({n}\right)$, where $\sigma$ is the $\sigma$ function.

The aliquot sum of $n$ is the sum of the divisors of $n$ with $n$ excluded.

Thus:

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


Suppose:

$s \left({n}\right) = m$

and:

$s \left({m}\right) = n$


Then:

\(\displaystyle \sigma \left({n}\right) - n\) \(=\) \(\displaystyle m\) Definition of Proper Divisor of Integer
\(\displaystyle \leadsto \ \ \) \(\displaystyle \sigma \left({n}\right)\) \(=\) \(\displaystyle m + n\)


Similarly:

\(\displaystyle \sigma \left({m}\right) - m\) \(=\) \(\displaystyle n\) Definition of Proper Divisor of Integer
\(\displaystyle \leadsto \ \ \) \(\displaystyle \sigma \left({m}\right)\) \(=\) \(\displaystyle m + n\)

Thus:

$s \left({n}\right) = s \left({m}\right) = m + n$


The argument reverses.


$\Box$


Definition 1 is equivalent to Definition 3

From the definition of definition 1 of an amicable pair:

$m$ and $n$ are an amicable pair if and only if:

the aliquot sum of $m$ is equal to $n$

and:

the aliquot sum of $n$ is equal to $m$.


From the definition of a sociable chain:

Let $m$ be a positive integer.

Let $\map s m$ be the aliquot sum of $m$.


Define the sequence $\sequence {a_k}$ recursively as:

$a_{k + 1} = \begin{cases} m & : k = 0 \\ \map s {a_k} & : k > 0 \end{cases}$


A sociable chain is such a sequence $\sequence {a_k}$ where:

$a_r = a_0$

for some $r > 0$.


Here it is seen that setting $r = 2$ gives that:

$s \left({a_0}\right) = a_1$
$s \left({a_1}\right) = a_0$

and the equivalence follows.

$\blacksquare$