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