Amicable numbers

The unravelling of the hidden properties of numbers never fails to fascinate math enthusiasts. Pythagoreans, known for the famous Pythagoras Theorem, saw a property for the pair of numbers 220 and 284 which they called Amicable. It started the hunt for other pairs of amicable numbers. What is that make pair of numbers amicable or friendly?

Find the proper divisors (all divisors except the number itself) of 220 and add them. Guess what? You will get 284. If you do the same with 284, you will end up with 220.

The sum of proper divisors of 220:

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

The sum of the proper divisors of 284:

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

Two positive integers are amicable if the sum of proper divisors of one integer equals the other.

After Greeks, Arab mathematicians found several amicable numbers. Euler has given a rule to produce amicable numbers. We will look at Euler’s rule. Take prime numbers p, q, r as given below with n and m as positive integers with n > m.


Then Image are an amicable pair of integers.

For example, take n=2 and m=1. Then Image, q= 11 and r = 71. It follows that 4*5*11= 220 and 4*71= 284, the same two numbers we explored at the beginning. Euler foundnearly 60 amicable pairs using this rule.

However, Euler’s rule doesn’t provide all possible amicable numbers. There are amicable numbers not satisfying this rule. The amicable pair (1184, 1210) found by a sixteen year old Italian boy Nicolo Paganini in 1866 doesn’t satisfy Euler’s rule. 

Please look at this nice video presented by Dr. James Grime on Amicable numbers.

Also look at for further reading.