Amicable Numbers

According to an anecdote, Pythagoras once defined a friend as the other self, as 220 is to 284. Indeed, these numbers are related in a special way, and such a pair is called "amicable".

This relation was known before Pythagoras's time. In Genesis 32:14, one can learn that Jacob sent his brother Esau "two hundred she goats, and twenty he goats, two hundred ewes, and twenty rams." At this time he had a special reason to show his friendliness to Esau, and it certainly was no coincidence that he chose two times 220 animals.

Other stories involving the numbers 220 and 284 and their amicable symbolism (or magic) derive from Arabian tradition, where they are more related to love than to friendship.

But what is so special about 220 and 284?

The number 220 is divisible by the following integers: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110. The sum of these divisors is 284.

The number 284 is divisible by these integers: 1, 2, 4, 71, and 142. Their sum is 220.

Thus 220 and 284 are referred to as an "amicable pair of numbers". That is two numbers, each of which is equal to the sum of the divisors of the other. To be formally correct, one should say "aliquot divisors" here. In addition to that, the divisor must be an integer, and the division must result in another integer. We are dealing with whole numbers.

The Arabian mathematician Thabit ibn Qurrah (9th century) first developed a method to find amicable pairs. It was not perfect, but until then only 220 and 284 had been known. Other methods were developed later.

A large number of amicable pairs are known today.

The first 10 pairs are:

220 - 284

1184 - 1210

2620 - 2924

5020 - 5564

6232 - 6368

10744 - 10856

12285 - 14595

17296 - 18416

63020 - 76084

66928 - 66992

Other amicable pairs are, for example:

9363584 - 9437056

111448537712 - 118853793424

4522265534545208537974785 - 4539801326233928286140415

Sociable Numbers

We have just seen how amicable numbers are connected: an amicable pair consists of two numbers, each of which is equal to the sum of the proper divisors of the other.

A set (cycle) of social numbers consists of a sequence of numbers where the sum of the proper divisors of the first one equals the second number; the sum of the proper divisors of the second one equals the third, etc. After a certain amount of steps one returns to the first number again.

The number of steps is expressed as the "order" of the sequence. Amicable numbers are social of the order 2.

12496 is a social number of the 5th order. That means after 5 steps you get back to 12496 again. Interested readers can try to make the calculations and get the whole cycle.

(This material has previously been published in TMA/Meriondho Leo and in my e-book “Numericon”.)

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