# 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

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $1184$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $1184$

- Weisstein, Eric W. "Amicable Pair." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/AmicablePair.html