Amicable Numbers

[Note: In another article we presented the concept of Perfect Numbers, and the related two classes of numbers, Deficient and Abundant. If you have not read that one, it would be a good idea to do so now before proceeding here. The topic before us now is a natural extension of that article.]


For centuries men have felt that numbers had powers over the natural affairs of their lives. One small piece of evidence of this concept is that of Amicable, or Friendly, Numbers. These are merely

pairs of numbers that are connected in a special way. The math behind the relationship is the same that was used to classify numbers as either perfect, deficient, or abundant: the sum of the proper divisors of the number, or numbers, being considered. An example will make this clear:

The number pair of 220 and 284 are considered amicable, because the sum of the proper divisors of 220 is 284, while the sum of the proper divisors of 284 is 220. Got that? Look at these factor charts:

	          220		  284
		---|----	---|----
		 1 | 		 1 |
		 2 | 110	 2 | 142
		 4 |  55	 4 |  71
	         5 |  44
	        10 |  22
	        11 |  20

[NOTE: In this sort of work, divisor and factor are synonyms.]
The sum of the divisors in the larger chart {1, 2, 4, 5, 10, … , 110}
is equal to the other number (284), whereas the sum of the divisors in
the smaller chart {1, 2, 4, 71, 142} is the first number (220).

Nice, huh!


I have read that in times long ago when romantic love was, well, more cerebral than material, amicable numbers were used by a young man to express his love to his beloved. He would prepare two metal charms with the numbers 220 and 284 engraved on them. These were then put on chains to place around their necks. Maybe we should do it these days, too.

Just as mathematicians have searched long and hard to find perfect numbers, they have done the same for amicable ones. And as
you might expect, the sizes of the numbers soon become very large. (Some are so large that if a lover used them to make a love token, the charm would be a “back breaking” burden for the wearer.)

About 1200 pairs are known by the last count that I have access to.* Here are the smaller members of two different pairs: 2620 and 5020. I leave it as “homework” for you to find the partners that apply
to each of them.

But there is one pair that mathematicians missed in their search down through the centuries which was found by an Italian schoolboy in the last century. One number of the pair is 1184. We leave it to you, dear reader, to find its “loving” companion. [Hint: look in the cartoon above.]


Reference

* Joseph S. Madachy, Mathematics on Vacation, 1966, p. 155.


Postscript: 7/5/99

Recently David Einstein informed me that the figure given by Madachy’s
book is now very outdated. There are now more than 450,000 pairs of
amicable numbers. For more information about this, go to this website:
http://www.vejlehs.dk/staff/jmp/aliquot/knwnap.htm.


Postscript: 2/2/02

Here are some more sites with additional information:


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2 thoughts on “Amicable Numbers

  1. Hi there,

    I (vaguely) remember visiting your site around ten years ago; glad to see that your site is still around (now at a different URL).

    I also wanted to say thanks for linking to my site at http://www.stormloader.com/ajy/amicable.html . I wanted to let you know that I’ve moved my site and the page is now at http://mathlair.allfunandgames.ca/amicable.php , so you may want to update your link at your convenience.

    While I was reading this page, I noticed that the first and last links in the 2/2/02 postscript don’t work anymore, so you might want to remove/update those too.

    Cheers and keep up the good work.

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