@Mictorrani
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Amicable & Social Numbers
**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”.) Copyright © 2007, 2020 Meleonymica/Mictorrani. All Rights Reserved. If you are interested in numbers, you find all my articles related to numbers here, and please also join my community Numbers (7b6a). https://read.cash/@Mictorrani/index-over-my-articles-at-read-cash-5f7c1e34#numbers https://read.cash/c/numbers-7b6a You find all my writings on Read.Cash, sorted by topic, here. https://read.cash/@Mictorrani/index-over-my-articles-at-read-cash-5f7c1e34
17 comments
I contunie to learn a lot of about numbers and maths with your posts! Like it solid contribution to Read Cash!
Thanks. My pleasure. I am glad if people feel that they learn something from my articles. That makes writing more meaningful.
very useful and interesting article thanks for posting keep more doing
Thanks. Glad you enjoyed it.
This article is useful, educative and interesting. I have learned some things new today.
My pleasure @Olad. One should always try to learn something new every day.
Yes, I learn everyday. Learners are leaders. By learning, one can easily grow in knowledge.
very interesting and useful article keep doing more like this
Will do.
very knowleadgable and thinkable article,thank man @mictorrani
You're welcome.
Nice dependencies between the numbers. Mathematicians used to have to rack their brains to find such numbers. Today you can write a computer program which finds out such numbers easily.
Yes, finding by trial and error, because computers are fast; it's another thing to find general formulas or prove things, for that computers are not very useful. Thanks for the donation.
Very nice article and more interesting. I am not a maths student you post is cool
You mean you understand it even you are not a maths student? If that is so, I am glad, because I strive to explain things so an interested layman can understand them and find them interesting.
very interesting article mictorrani and waiting for more like this keep doing more
It will come more, but I have several topics to keep up with so every post I make will not be dedicate to numbers. But now and then, yes.