Amicable (which means friendly) numbers were discovered by Pythagoreans a really, really, really long time ago (see, even old guys liked finding “cutesy” math things). It is said that their properties stem from Pythagoras’ response to the question, “What is a friend?” He responded, “A friend is one who is the other I.” Now most of us here in the 21st century hear that response and say, “Huh?,” but when we dissect two amicable numbers, it becomes much clearer. There are many ways to define or describe Amicable numbers, but the clearest definition I’ve found so far is this: Two numbers are called Amicable (or friendly) if each equals to the sum of the proper divisors of the other (proper divisors are all the divisors excluding the number itself).
The first pair of amicable numbers ever discovered are 220 and 284. Let’s take a look:
220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220
284: 1, 2, 4, 71, 142, 284
I put the final factor of each number in italics to show that these are not proper factors of each number, so when summing the factors we will not be adding these. If you plug the factors 1-110 into your calculator and add, you should get 284 as the answer. Similarly, if you plug the factors 1-142 into your calculator and add, you should get 220 as the answer. And there we have it, Amicable numbers.
Thabit ibn Qurra, braniac he was, decided to take this a step further–he proved that numbers of the form T(n)=3*(2^n)-1 (now called 321 numbers) are also very special. If the 321 number for n and n-1 is prime and 9*2^(2n-1)-1, then the numbers (2^n)*T(n)*T(n-1) and (2^n)*(9*2^(2n-1)-1) are amicable. Don’t ask me how he managed to figure this all out–he must have had a lot (and I mean a lot) of time on his hands! And, even after all this work that Thabit did, the only known examples of this are for n=2, 4 and 7. That’s it. Check ’em out and see for yourself! (Hint: n=2 is our 220 and 284 example :))
For many years, this was the only known pair (220 and 284) of amicable numbers. It wasn’t until 1636 that Pierre de Fermat discovered another: 17296 and 18416. Descartes contributed the third pair: 9363584 and 9437056, and from there Euler took over, finding 63 pairs. Today, roughly 11994387 pairs of amicable numbers are known.