Amicable Numbers: A Brief Introduction And Its Fascinating Properties

There are plenty of examples of numbers with some interesting and magical properties in the history of theory of numbers. The numbers with some unique and fascinating properties have always received a great attention in the world of mathematics. There are so many kinds of numbers like perfect numbers, amicable numbers, fermats numbers, Fibonacci numbers, etc. which shows some special characteristics. It was well said by great mathematician Kronecker that ‘‘God created the natural numbers, and all the rest is the work of man.’’

One of the most special types of numbers with some fascinating properties is amicable number. Two integers are said to be amicable if the sum of the proper positive divisors of one of the integers is equal to the other integer and vice- versa. More generally we can defined that amicable numbers are a pair of numbers with the property that the sum of all the proper divisors of the first number (not including itself) is exactly equal to the second number while the sum of all the proper divisors of the second number (not including itself) likewise equal to the first number.

The first pair of amicable numbers is (220, 284) which was discovered by Pythagoreans. Let us check the two numbers 220 and 284.

Proper divisors of 220 are – 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110.

Adding all the divisors of 220 we get,

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

So, sum of the divisors of 220 is 284.

Proper divisors of 284 are – 1, 2, 4, 71, 142

Adding all the divisors of 284 we get,

1+2+4+71+142 = 220

So, sum of the divisors of 284 is 220.

Hence, the numbers 220 and 284 are amicable.

There are some amicable pairs (a,b), in which the sum of digits of a and b is equal.

For example, we consider the pair, (69615, 87633)

Sum of digits of 69615 is (6+9+6+1+5) = 27

Sum of digits of 87633 is (8+7+6+3+3) = 27

Hence, in the above amicable pair the sum of digits of both the number are same.

Some another pairs of amicable numbers are (1184, 1210), (2620, 2924), (5020, 5564) etc. As we have seen that amicable numbers have the property that one number represents the other number. This symbolizes friendship, harmony and love. The Greeks believed that these numbers had a special influence in establishing friendship between peoples. Amicable numbers have a huge application in astrology (in casting horoscopes) and magic.

After discovery of first amicable pair, Arabian mathematician Thabit described an explicit rule for finding certain types of amicable pairs. According to him ‘‘ If three numbers p= 3. 2n-1 -1, q= 3. 2n -1, and r = 9. 22n-1 -1 are all prime and n ≥ 2, then 2n pq and 2nr are amicable numbers.’’ Later another two mathematician Fermat and Descartes announced two pairs of amicable numbers (17296, 18416) and (9363584, 9437056). Swiss Mathematician Leonard Euler gave a list of total 64 pairs of amicable numbers .

After advancement of Computer and modern technology more and more amicable numbers of very big digits have generated in recent times. As of January 2018, there are over 1,221,159,849 known amicable pairs. But still searches are going on to find proper rule for finding all pairs of amicable numbers. Even the query has not yet established whether the number of amicable pairs are finite or infinite. So, the search is still going on….Let us conclude with a famous quotes of Paul Erdos.

‘‘Why are numbers beautiful? It’s like asking why is Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is.’’

References:

1. D. M. Burton, Elementary Number Theory, McGraw Hill Education (India) Private Limited, New Delhi, 2014.
2. Amicable numbers, S.S Gupta.
3. Wikipedia.

(Published in: Good Morning Science)