# Definition:Amicable Pair

## Definition

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

### 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 $2$.

## Sequence

The sequence of amicable pairs begins:

- $\tuple {220, 284}, \tuple {1184, 1210}, \tuple {2620, 2924}, \tuple {5020, 5564}, \tuple {6232, 6368}, \tuple {10744, 10856}, \tuple {12285, 14595}, \tuple {17296, 18416}, \tuple {63020, 76084}$

## Examples

### $220$ and $284$

$220$ and $284$ are the smallest amicable pair:

- $\map \sigma {220} = \map \sigma {284} = 504 = 220 + 284$

### $1184$ and $1210$

$1184$ and $1210$ are the $2$nd amicable pair:

- $\map \sigma {1184} = \map \sigma {1210} = 2394 = 1184 + 1210$

### $2620$ and $2924$

$2620$ and $2924$ are the $3$rd amicable pair:

- $\map \sigma {2620} = \map \sigma {2924} = 5544 = 2620 + 2924$

### $5020$ and $5564$

$5020$ and $5564$ are the $4$th amicable pair:

- $\map \sigma {5020} = \map \sigma {5564} = 10 \, 584 = 5020 + 5564$

### $6232$ and $6368$

$6232$ and $6368$ are the $5$th amicable pair:

- $\map \sigma {6232} = \map \sigma {6368} = 12 \, 600 = 6232 + 6368$

### $10 \, 744$ and $10 \, 856$

$10 \, 744$ and $10 \, 856$ are the $6$th amicable pair:

- $\map \sigma {10 \, 744} = \map \sigma {10 \, 856} = 21 \, 600 = 10 \, 744 + 10 \, 856$

### $12 \, 285$ and $14 \, 595$

$12 \, 285$ and $14 \, 595$ are the $7$th amicable pair and the smallest odd amicable pair:

- $\map \sigma {12 \, 285} = \map \sigma {14 \, 595} = 26 \, 880 = 12 \, 285 + 14 \, 595$

### $17 \, 296$ and $18 \, 416$

$17 \, 296$ and $18 \, 416$ are the $8$th amicable pair:

- $\map \sigma {17 \, 296} = \map \sigma {18 \, 416} = 35 \, 712 = 17 \, 296 + 18 \, 416$

## Also known as

Some sources refer to them as **friendly pairs**, but this is deprecated as the term **friendly pair** is usually used for something different.

Many sources refer to the elements of an **amicable pair** as **amicable numbers**. While this usage is common and widespread, it is preferred not to be used on $\mathsf{Pr} \infty \mathsf{fWiki}$ in order that the distinction is clear between **amicable pairs**, **amicable triplets** and so on.

## Also see

- Results about
**amicable pairs**can be found here.

## Historical Note

Amicable pairs are on record as having been studied by the Pythagoreans.

The first mathematician to explore amicable pairs systematically was Leonhard Paul Euler.

He published a list of $64$ examples.

There are now over $40 \, 000$ amicable pairs now known, including all such pairs where the smaller pair is under $1 \, 000 \, 000$.

Techniques for generating new amicable pairs from existing ones strongly suggest that there is an infinite number of them, but this still has to be rigorously proven.

## Sources

- Weisstein, Eric W. "Amicable Pair." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/AmicablePair.html