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.


First it is established (by means of an online big integer calculator and integer factorisation calculator):

$2 \times 1291 \times 5281^{19} - 1$ is prime
$2^3 \times 3^3 \times 5^2 \times 1291 \times 5281^{19} - 1$ is prime


Thus from Sigma Function of Integer: Corollary:

\(\displaystyle \) \(\) \(\displaystyle \map \sigma {3^4 \times 5 \times 11 \times 5281^{19} \times 29 \times 89 \times \paren {2 \times 1291 \times 5281^{19} - 1} }\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {3^5 - 1} {3 - 1} \times \paren {5 + 1} \times \paren {11 + 1} \times \paren {29 + 1} \times \paren {89 + 1} \times \dfrac {5281^{20} - 1} {5281 - 1} \times \paren {2 \times 1291 \times 5281^{19} }\)
\(\displaystyle \) \(=\) \(\displaystyle 121 \times 6 \times 12 \times 30 \times 90 \times 53912494548111776964581871379407163295641963924618078664526561134059220 \times \paren {2 \times 1291 \times 5281^{19} }\)
\(\displaystyle \) \(=\) \(\displaystyle 11^2 \times \paren {2 \times 3} \times \paren {2^2 \times 3} \times \paren {2 \times 3 \times 5} \times \paren {2 \times 3^2 \times 5} \times \paren {2^2 \times 5 \times 19 \times 41 \times 139 \times 311 \times 6661 \times 33331 \times 13 944481 \times 75 019421 \times 24027 536081 \times 92 192755 565941 \times 155 588291 031361} \times 2 \times 1291 \times 5281^{19}\)
\(\displaystyle \) \(=\) \(\displaystyle 2^8 \times 3^5 \times 5^3 \times 11^2 \times 19 \times 41 \times 139 \times 311 \times 1291 \times 5281^{19} \times 6661 \times 33331 \times 13 944481 \times 75 019421 \times 24027 536081 \times 92 192755 565941 \times 155 588291 031361\)


\(\displaystyle \) \(\) \(\displaystyle \map \sigma {3^4 \times 5 \times 11 \times 5281^{19} \times \paren {2^3 \times 3^3 \times 5^2 \times 1291 \times 5281^{19} - 1} }\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {3^5 - 1} {3 - 1} \times \paren {5 + 1} \times \paren {11 + 1} \times \dfrac {5281^{20} - 1} {5281 - 1} \times \paren {2^3 \times 3^3 \times 5^2 \times 1291 \times 5281^{19} }\)
\(\displaystyle \) \(=\) \(\displaystyle 121 \times 6 \times 12 \times 53912494548111776964581871379407163295641963924618078664526561134059220 \times \paren {2^3 \times 3^3 \times 5^2 \times 1291 \times 5281^{19} }\)
\(\displaystyle \) \(=\) \(\displaystyle 11^2 \times \paren {2 \times 3} \times \times \paren {2^2 \times 3} \times \paren {2^2 \times 5 \times 19 \times 41 \times 139 \times 311 \times 6661 \times 33331 \times 13 944481 \times 75 019421 \times 24027 536081 \times 92 192755 565941 \times 155 588291 031361} \times \paren {2^3 \times 3^3 \times 5^2 \times 1291 \times 5281^{19} }\)
\(\displaystyle \) \(=\) \(\displaystyle 2^8 \times 3^5 \times 5^3 \times 11^2 \times 19 \times 41 \times 139 \times 311 \times 1291 \times 5281^{19} \times 6661 \times 33331 \times 13 944481 \times 75 019421 \times 24027 536081 \times 92 192755 565941 \times 155 588291 031361\)


Then we calculate the sum, by means of the same online big integer calculator and integer factorisation calculator:

\(\displaystyle \) \(\) \(\displaystyle \paren {3^4 \times 5 \times 11 \times 5281^{19} \times 29 \times 89 \times \paren {2 \times 1291 \times 5281^{19} - 1} } + \paren {3^4 \times 5 \times 11 \times 5281^{19} \times \paren {2^3 \times 3^3 \times 5^2 \times 1291 \times 5281^{19} - 1} }\)
\(\displaystyle \) \(=\) \(\displaystyle 3^4 \times 5 \times 11 \times 5281^{19} \times \paren {29 \times 89 \times \paren {2 \times 1291 \times 5281^{19} - 1} + \paren {2^3 \times 3^3 \times 5^2 \times 1291 \times 5281^{19} - 1} }\)
\(\displaystyle \) \(=\) \(\displaystyle 3^4 \times 5 \times 11 \times 5281^{19}\)
\(\displaystyle \) \(\) \(\, \displaystyle \times \, \) \(\displaystyle (29 \times 89 \times 139175701888775976308855532899186267927088632551744230583288018723382689621\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle 375774395099695136033909938827802923403139307889709422574877650553133261979399)\)
\(\displaystyle \) \(=\) \(\displaystyle 3^4 \times 5 \times 11 \times 5281^{19}\)
\(\displaystyle \) \(\) \(\, \displaystyle \times \, \) \(\displaystyle (359212486574930794853156130412799757519815760616051859135466376325050721911801\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle 375774395099695136033909938827802923403139307889709422574877650553133261979399)\)
\(\displaystyle \) \(=\) \(\displaystyle 3^4 \times 5 \times 11 \times 5281^{19}\)
\(\displaystyle \) \(\) \(\, \displaystyle \times \, \) \(\displaystyle 734986881674625930887066069240602680922955068505761281710344026878183983891200\)
\(\displaystyle \) \(=\) \(\displaystyle 3^4 \times 5 \times 11 \times 5281^{19}\)
\(\displaystyle \) \(\) \(\, \displaystyle \times \, \) \(\displaystyle 2^8 \times 3 \times 5^2 \times 11 \times 19 \times 41 \times 139 \times 311 \times 1291 \times 661 \times 33331 \times 13 944481 \times 75 019421 \times 24027 536081 \times 92 192755 565941 \times 155 588291 031361\)
\(\displaystyle \) \(=\) \(\displaystyle 2^8 \times 3^5 \times 5^3 \times 11^2 \times 19 \times 41 \times 139 \times 311 \times 1291 \times 5281^{19} \times 6661 \times 33331 \times 13 944481 \times 75 019421 \times 24027 536081 \times 92 192755 565941 \times 155 588291 031361\)

$\blacksquare$


Sources