Odd Amicable Pair/Examples/1,175,265-1,438,983

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Example of Odd Amicable Pair

$1 \, 175 \, 265$ and $1 \, 438 \, 983$ are the $9$th odd amicable pair:

$\map \sigma {1 \, 175 \, 265} = \map \sigma {1 \, 438 \, 983} = 2 \, 614 \, 240 = 1 \, 175 \, 265 + 1 \, 438 \, 983$


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:

\(\displaystyle \map \sigma {1 \, 175 \, 265}\) \(=\) \(\displaystyle 2 \, 614 \, 240\) $\sigma$ of $1 \, 175 \, 265$
\(\displaystyle \) \(=\) \(\displaystyle 1 \, 175 \, 265 + 1 \, 438 \, 983\)
\(\displaystyle \) \(=\) \(\displaystyle \map \sigma {1 \, 438 \, 983}\) $\sigma$ of $1 \, 438 \, 983$

$\blacksquare$


Historical Note

The odd amicable pair $1 \, 175 \, 265$ and $1 \, 438 \, 983$ was discovered by G.W. Kraft in the $17$th century.

It was the $1$st odd amicable pair to be discovered.


Sources