# Amicable Pair/Examples/220-284

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## 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.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $220$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**amicable numbers** - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $220$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**amicable numbers** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**amicable numbers**

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