Amicable Pair/Examples/3^4 x 5 x 11 x 5281^19 x 29 x 89 (2 x 1291 x 5281^19 - 1)-3^4 x 5 x 11 x 5281^19 (2^3 x 3^3 x 5^2 x 1291 x 5281^19 - 1)

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Example of Amicable Pair

These integers:

$3^4 \times 5 \times 11 \times 5281^{19} \times 29 \times 89 \paren {2 \times 1291 \times 5281^{19} - 1}$
$3^4 \times 5 \times 11 \times 5281^{19} \paren {2^3 \times 3^3 \times 5^2 \times 1291 \times 5281^{19} - 1}$

form an amicable pair.


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:

\(\displaystyle \map \sigma {3^4 \times 5 \times 11 \times 5281^{19} \times 29 \times 89 \paren {2 \times 1291 \times 5281^{19} - 1} }\) \(=\) \(\displaystyle \)


\(\displaystyle \map \sigma {3^4 \times 5 \times 11 \times 5281^{19} \paren {2^3 \times 3^3 \times 5^2 \times 1291 \times 5281^{19} - 1} }\) \(=\) \(\displaystyle \)




Historical Note

David Wells reports in his Curious and Interesting Numbers of $1986$ that this amicable pair was discovered by Hermanus Johannes Joseph te Riele.

He also states that each of the pair has $152$ digits.

However, the most recent (at the time) paper published by te Riele does not mention this pair, and indeed states that the largest pair found was of $38$ digits.

Hence this result needs to be corroborated.

Calculation of this supposed amicable pair is under way.


Sources