# Amicable Pair/Examples/220-284

## Example of Amicable Pair

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

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

## Proof

Let $\map s n$ denote the aliquot sum of $n$.

By definition:

$\map s n = \map \sigma n - n$

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

Thus:

 $\displaystyle \map s {220}$ $=$ $\displaystyle \map \sigma {220} - 220$ $\displaystyle$ $=$ $\displaystyle 504 - 220$ $\sigma$ of $220$ $\displaystyle$ $=$ $\displaystyle 284$

 $\displaystyle \map s {284}$ $=$ $\displaystyle \map \sigma {284} - 284$ $\displaystyle$ $=$ $\displaystyle 504 - 284$ $\sigma$ of $284$ $\displaystyle$ $=$ $\displaystyle 220$

It can be determined by inspection of the aliquot sums of all smaller integers that there is no smaller amicable pair.

$\blacksquare$

## Historical Note

The amicable pair $220$ and $284$ were, according to Iamblichus Chalcidensis, known to Pythagoras of Samos.

However, it is strongly supposed by some commentators that they were known even further back than that.