Odd Amicable Pair/Examples/12,285-14,595

From ProofWiki
Jump to navigation Jump to search

Example of Odd Amicable Pair

$12 \, 285$ and $14 \, 595$ are the $7$th amicable pair and the smallest odd amicable pair:

$\map {\sigma_1} {12 \, 285} = \map {\sigma_1} {14 \, 595} = 26 \, 880 = 12 \, 285 + 14 \, 595$


Proof

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

$\map {\sigma_1} m = \map {\sigma_1} n = m + n$

where $\map {\sigma_1} n$ denotes the divisor sum function.


Thus:

\(\ds \map {\sigma_1} {12 \, 285}\) \(=\) \(\ds 26 \, 880\) $\sigma_1$ of $12 \, 285$
\(\ds \) \(=\) \(\ds 12 \, 285 + 14 \, 595\)
\(\ds \) \(=\) \(\ds \map {\sigma_1} {14 \, 595}\) $\sigma_1$ of $14 \, 595$


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

$\blacksquare$


Historical Note

The odd amicable pair $12 \, 285$ and $14 \, 595$ was discovered by B.H. Brown, who reported on them in the American Mathematical Monthly in $1939$.


Sources