Friend Or Amicable Numbers

                                         Amicable /Friend Numbers

Two numbers are called Amicable if each of which is equal to the sum of all the exact divisors of the other except the number itself. 220 and 284 are the first pair of amicable numbers.

220 has the exact divisors 1,2,4,5,10,11,20,22,44,55 and 110 whose sum is 284 

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

and 284 has the exact divisors 1, 2, 4, 71, and 142 whose sum is 220. 

1 + 2 + 4 + 71 + 142 = 220

Another pair of amicable number is 1184 and 1210.

The set of 220 and 284 was the first known set of amicable numbers. Pythagoras discovered the relationship and coined the term Amicable because he considered the numbers to be a symbol of friendship.

There is an unauthenticated story related to Amicable numbers that a prince whose name from the stand point of numerology was equivalent to 284 sought a bride whose name would represent 220 believing the fact that such combination would guarantee for a happy marriage throughout the life.

Here is the list of first ten Amicable Pairs:-


1 ---  (220 ,284)
2 --- (1184 , 1210)
3 --- (2620 , 2924)
4 ---  (5020 ,5564 )
5 ---  (6232 ,6368)
6 ---  (10744 ,10856)
7 -- (12285, 14595)
8 -- (17296 , 18416)
9 -- (63020 , 76048)
10 -- (66928 , 66992)

An Arab Mathematician tried to generalize a rule to find the Amicable Number but he could not succed completely. The Thābit ibn Qurra theorem is a method for discovering amicable numbers invented in the ninth century by the Arab mathematician Thābit ibn Qurra.

It states that if

where n > 1 is an integer and p, q, and r are prime numbers, then 2^n×p×q and 2^n ×r are a pair of amicable numbers.

Dr Rajesh Kumar Thakur