Amicable Pair/Examples/59,554, 936,495, 441,481, 044,788, 091,271, 148,664, 944,796, 300,859, 243,635, 311,219, 048,448 - 59,554, 936,495, 441,891, 385,123, 332,422, 108,719, 776,971, 992,921, 810,832, 072,976, 105,472

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Example of Amicable Pair

$59 \, 554 \, 936 \, 495 \, 441 \, 481 \, 044 \, 788 \, 091 \, 271 \, 148 \, 664 \, 944 \, 796 \, 300 \, 859 \, 243 \, 635 \, 311 \, 219 \, 048 \, 448$

and

$59 \, 554 \, 936 \, 495 \, 441 \, 891 \, 385 \, 123 \, 332 \, 422 \, 108 \, 719 \, 776 \, 971 \, 992 \, 921 \, 810 \, 832 \, 072 \, 976 \, 105 \, 472$

are an amicable pair:

\(\ds \) \(\) \(\ds \map {\sigma_1} {59 \, 554 \, 936 \, 495 \, 441 \, 481 \, 044 \, 788 \, 091 \, 271 \, 148 \, 664 \, 944 \, 796 \, 300 \, 859 \, 243 \, 635 \, 311 \, 219 \, 048 \, 448}\)
\(\ds \) \(=\) \(\ds \map {\sigma_1} {59 \, 554 \, 936 \, 495 \, 441 \, 891 \, 385 \, 123 \, 332 \, 422 \, 108 \, 719 \, 776 \, 971 \, 992 \, 921 \, 810 \, 832 \, 072 \, 976 \, 105 \, 472}\)
\(\ds \) \(=\) \(\ds 119 \, 109 \, 872 \, 990 \, 883 \, 372 \, 429 \, 911 \, 423 \, 693 \, 257 \, 384 \, 721 \, 768 \, 293 \, 781 \, 054 \, 467 \, 384 \, 195 \, 153 \, 920\)
\(\ds \) \(=\) \(\ds 59 \, 554 \, 936 \, 495 \, 441 \, 481 \, 044 \, 788 \, 091 \, 271 \, 148 \, 664 \, 944 \, 796 \, 300 \, 859 \, 243 \, 635 \, 311 \, 219 \, 048 \, 448\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds 59 \, 554 \, 936 \, 495 \, 441 \, 891 \, 385 \, 123 \, 332 \, 422 \, 108 \, 719 \, 776 \, 971 \, 992 \, 921 \, 810 \, 832 \, 072 \, 976 \, 105 \, 472\)


Proof

By definition, $m$ and $n$ form an amicable pair if and only if:

$\map {\sigma_1} m = \map {\sigma_1} n = m + n$

where $\map {\sigma_1} n$ denotes the divisor sum function.


From $\sigma_1$ of $59 \, 554 \, 936 \, 495 \, 441 \, 481 \, 044 \, 788 \, 091 \, 271 \, 148 \, 664 \, 944 \, 796 \, 300 \, 859 \, 243 \, 635 \, 311 \, 219 \, 048 \, 448$:

\(\ds \) \(\) \(\ds \map {\sigma_1} {59 \, 554 \, 936 \, 495 \, 441 \, 481 \, 044 \, 788 \, 091 \, 271 \, 148 \, 664 \, 944 \, 796 \, 300 \, 859 \, 243 \, 635 \, 311 \, 219 \, 048 \, 448}\)
\(\ds \) \(=\) \(\ds 119 \, 109 \, 872 \, 990 \, 883 \, 372 \, 429 \, 911 \, 423 \, 693 \, 257 \, 384 \, 721 \, 768 \, 293 \, 781 \, 054 \, 467 \, 384 \, 195 \, 153 \, 920\)


From $\sigma_1$ of $59 \, 554 \, 936 \, 495 \, 441 \, 891 \, 385 \, 123 \, 332 \, 422 \, 108 \, 719 \, 776 \, 971 \, 992 \, 921 \, 810 \, 832 \, 072 \, 976 \, 105 \, 472$:

\(\ds \) \(\) \(\ds \map {\sigma_1} {59 \, 554 \, 936 \, 495 \, 441 \, 891 \, 385 \, 123 \, 332 \, 422 \, 108 \, 719 \, 776 \, 971 \, 992 \, 921 \, 810 \, 832 \, 072 \, 976 \, 105 \, 472}\)
\(\ds \) \(=\) \(\ds 119 \, 109 \, 872 \, 990 \, 883 \, 372 \, 429 \, 911 \, 423 \, 693 \, 257 \, 384 \, 721 \, 768 \, 293 \, 781 \, 054 \, 467 \, 384 \, 195 \, 153 \, 920\)


while:

\(\ds \) \(=\) \(\ds 119 \, 109 \, 872 \, 990 \, 883 \, 372 \, 429 \, 911 \, 423 \, 693 \, 257 \, 384 \, 721 \, 768 \, 293 \, 781 \, 054 \, 467 \, 384 \, 195 \, 153 \, 920\)
\(\ds \) \(=\) \(\ds 59 \, 554 \, 936 \, 495 \, 441 \, 481 \, 044 \, 788 \, 091 \, 271 \, 148 \, 664 \, 944 \, 796 \, 300 \, 859 \, 243 \, 635 \, 311 \, 219 \, 048 \, 448\)
\(\ds \) \(\) \(\, \ds + \, \) \(\ds 59 \, 554 \, 936 \, 495 \, 441 \, 891 \, 385 \, 123 \, 332 \, 422 \, 108 \, 719 \, 776 \, 971 \, 992 \, 921 \, 810 \, 832 \, 072 \, 976 \, 105 \, 472\)

$\blacksquare$


Sources

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