Amicable Numbers

Date: 11/12/98 at 19:08:07
From: amanda sparks
Subject: Amicable numbers

What is the definition of amicable numbers? Please give two examples if 
possible.

Thank you for your help.

Date: 11/12/98 at 20:05:58
From: Doctor White
Subject: Re: Amicable numbers

Amanda:

Amicable numbers are a pair of numbers such that the sum of their 
proper divisors (not including itself) equals the other number. 

Because amicable numbers must come in pairs, they are sometimes called 
amicable pairs or amicable partners. 
 
For example, show that 220 and 284 are amicable numbers: 

   First, find the proper divisors of 220:

      1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110

   If you add all these numbers together, you can see they sum to 284.

   Now find the proper divisors of 284:

      1, 2, 4, 71, 142

   These sum to 220.

Thus, 220 and 284 are a pair of amicable numbers. 

For another example, let's take the pair 8 and 9:

   The proper divisors of 9 are 1 and 3, which sum to 4.
   The proper divisors of 8 are 1, 2, and 4, which sum to 7.

So, 8 and 9 are not a pair of amicable numbers since 4 does not equal 
8 and 7 does not equal 9.

If finding all of the proper divisors of a number and adding them 
together seems hard, you can find some shortcuts in another Dr. Math 
archive on amicable numbers, at:

   Amicable Partners
   http://mathforum.org/dr.math/problems/keelan7.23.98.html   

Some other well-known amicable pairs are: 

   17296 and 18416 (Fermat's pair)
   9363584 and 9437056 (Descartes' pair) 

Finally, there have been some interesting discussions on the history 
of amicable numbers in the math history discussion list, archived at:

   http://mathforum.org/kb/forum.jspa?forumID=193   

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!

Antreas P. Hatzipolakis quotes Martin Gardner:

   The Pythagorean brotherhood regarded 220 and 284 as symbols of 
   friendship. Biblical commentators spotted 220 in Genesis 32:14 as 
   the number of goats given Esau by Jacob. A wise choice, the 
   commentators said, because 220, being one of the amicable pair, 
   expressed Jacob's great love for Esau. During the Middle Ages this 
   pair of numbers played a role in horoscope casting, and talismans 
   inscribed with 220 and 284 were believed to promote love. One poor 
   Arab of the eleventh century recorded that he once tested erotic 
   effect *eating* something labeled with 284, at the same time having 
   someone else swallow 220, but he failed to add how the experiment 
   worked out. (p. 167)
   (Martin Gardner, Mathematical Magic Show. Viking, London 1984)

Hope this has helped. Come back to see us soon.

- Doctors Teeple and White, The Math Forum
  http://mathforum.org/dr.math/