Amicable Pair/Examples/17,296-18,416

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Example of Amicable Pair

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

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


Proof

By definition, $m$ and $n$ form an amicable pair if and only if:

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

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


Thus:

\(\displaystyle \map \sigma {17 \, 296}\) \(=\) \(\displaystyle 35 \, 712\) $\sigma$ of $17 \, 296$
\(\displaystyle \) \(=\) \(\displaystyle 17 \, 296 + 18 \, 416\)
\(\displaystyle \) \(=\) \(\displaystyle \map \sigma {18 \, 416}\) $\sigma$ of $18 \, 416$

$\blacksquare$


Historical Note

The amicable pair $17 \, 296$ and $18 \, 416$ were discovered by Pierre de Fermat in $1636$.

As such, it appears that he re-discovered what had previously been discovered by the medieval Arab school.

It was, however, the second amicable pair to be known of by the Western mathematical world after $220$ and $284$, known of old.


Sources