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

## 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.