Communicating Math: Amicable Numbers

Amicable numbers are two numbers whose factors, not including the number itself, add up to the same value. For instance, since the time of Pythagoras, 220 and 284 is a common example of a pair of numbers that are amicable. If we add up the factors of 220 and remember not to include 220 we get

1+2+4+5+10+11+20+22+44+55+110 = 284.

Similarly, if we add up the factors, expect itself, of 284 we yield

1+2+4+71+142 = 220.

To find other pairs of amicable numbers, Thabit ibn Qurra came up with some general formulas to do so. If

T(n)=3*(2^n) - 1 (also known as the 321 number),

T(n - 1)=3*(2^(n - 1)) - 1,

and

9*(2^(2n - 1)) - 1

are prime numbers, then the numbers

(2^n)*T(n)*T(n - 1)

and

(2^n)*(9*(2^(2n - 1)) - 1)

are amicable. 

As of today, the only values of n that give us amicable numbers following Thabit's stipulations are 2, 4, and 7. On the other hand, these formula's don't work for the thousands of amicable pairs that have been found in recent years. One example is the pair of 1184 and 1210. We are not able to get these two numbers from an integer value n by following Thabit's rules, yet they are still amicable. 

It is interesting to think about why someone would first think to add up factors of a number and try to match the sum of factors for another number. Hundreds of thousands of pairs have been found but will there ever be a sure fire formula that provides two amicable numbers? Are there infinitely many pairs of amicable numbers out there? This concept of amicable numbers is kind of an ongoing mystery that keeps mathematicians curious and searching for more answers. 

Reference:
http://mathlair.allfunandgames.ca/amicable.php