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As an exercise in Python programming I've written a script that finds amicable pairs, and I've had fun doing so. But I'm wondering if amicable pairs have a use, in mathematics or elsewhere?

As a reminder:
Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. (A proper divisor of a number is a positive integer divisor other than the number itself. For example, the proper divisors of 6 are 1, 2, and 3.)

For example, the smallest pair of amicable numbers is (220, 284); for the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71, and 142, of which the sum is 220. (from the Wikipedia article)

The first few amicable pairs, according to Math World, are: (220, 284), (1184, 1210), (2620, 2924) (5020, 5564), (6232, 6368), (10744, 10856), (12285, 14595), (17296, 18416), (63020, 76084)

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up vote 10 down vote accepted

Apparently, John Conway says no.

For example, John Conway says they are not very useful mathematically:

The only application or use for these numbers is the original one - you insert a pair of amicable numbers into a pair of amulets, of which you wear one yourself and give the other to your beloved!

From Dr. Math.

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Amicable numbers have a long history in magic and astrology and were used for preparing horoscopes, making love potions and talismans. If you find that useful. For me, more interesting is that none of them are square numbers. Read more about amicable numbers in my blog:

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