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.
\(\ds \) | \(\) | \(\ds \map {\sigma_1} {59 \, 554 \, 936 \, 495 \, 441 \, 481 \, 044 \, 788 \, 091 \, 271 \, 148 \, 664 \, 944 \, 796 \, 300 \, 859 \, 243 \, 635 \, 311 \, 219 \, 048 \, 448}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 119 \, 109 \, 872 \, 990 \, 883 \, 372 \, 429 \, 911 \, 423 \, 693 \, 257 \, 384 \, 721 \, 768 \, 293 \, 781 \, 054 \, 467 \, 384 \, 195 \, 153 \, 920\) |
\(\ds \) | \(\) | \(\ds \map {\sigma_1} {59 \, 554 \, 936 \, 495 \, 441 \, 891 \, 385 \, 123 \, 332 \, 422 \, 108 \, 719 \, 776 \, 971 \, 992 \, 921 \, 810 \, 832 \, 072 \, 976 \, 105 \, 472}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 119 \, 109 \, 872 \, 990 \, 883 \, 372 \, 429 \, 911 \, 423 \, 693 \, 257 \, 384 \, 721 \, 768 \, 293 \, 781 \, 054 \, 467 \, 384 \, 195 \, 153 \, 920\) |
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
- 1994: S.Y. Yan and T.H. Jackson: A New Large Amicable Pair (Computers & Mathematics with Applications Vol. 27, no. 6: pp. 1 – 3)
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $2^4 \times 7 \times 9,288,811,670,405,087 \times 145,135,534,866,431 \times 313,887,523,966,328,699,903$