Amicable Pair/Examples/1184-1210

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Example of Amicable Pair

$1184$ and $1210$ are the $2$nd amicable pair:

$\map \sigma {1184} = \map \sigma {1210} = 2394 = 1184 + 1210$


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:

\(\displaystyle \map s {1184}\) \(=\) \(\displaystyle \map \sigma {1184} - 1184\)
\(\displaystyle \) \(=\) \(\displaystyle 2394 - 1184\) $\sigma$ of $1184$
\(\displaystyle \) \(=\) \(\displaystyle 1210\)


\(\displaystyle \map s {1210}\) \(=\) \(\displaystyle \map \sigma {1210} - 1210\)
\(\displaystyle \) \(=\) \(\displaystyle 2394 - 1210\) $\sigma$ of $1210$
\(\displaystyle \) \(=\) \(\displaystyle 1184\)

$\blacksquare$


Historical Note

The amicable pair $1184$ and $1210$ was discovered by Nicolò Paganini in $1866$, at the age of $16$.

It is remarkable that it had until that time escaped being found, even by Leonhard Paul Euler's systematic exploration.


Sources