Amicable Numbers

Let me start with the basic definition of Amicable Numbers.

Definition : If the sum of divisors of N1 is N2 and the sum of divisors of N2 is N1 , then N1 and N2 are called as amicable numbers.

          For example: Smallest Known Amicable Number pair are 220 and  284.

The proper divisors of 220( N1)  are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284( N2) ; and the proper divisors of 284( N2)  are 1, 2, 4, 71 and 142, of which the sum is 220(N1) .

Origin of concept of amicable numbers dates back to time of Pythagoras.Credit of the properties of these numbers goes to Pythagoras. 

Some of the Arabian Mathematician who studied  Al-Majriti , Al-Baghdadi and Al-Farisi.

Fermat and Descartes also rediscovered pairs of amicable numbers known to Arab mathematicians” 

Iraqi Mathematician Thabit ibn Qurra gave the first general method of generation of Amicable Numbers.

Thābit ibn Qurra theorem

The Thābit ibn Qurra theorem a method for discovering amicable numbers.

Statement: If  p = 3 × 2n − 1 − 1, q = 3 × 2n − 1, r = 9 × 22n − 1 − 1, where n > 1 is an integer and p, q, and r are prime numbers, then 2n×p×q and 2n×r are a pair of amicable numbers.


Euler’s rule

Euler’s rule is a generalization of the Thâbit ibn Qurra theorem. It states that if

p = (2(n – m)+1) × 2m − 1,
q = (2(n – m)+1) × 2n − 1,
r = (2(n – m)+1)2 × 2m + n − 1,

where n > m > 0 are integers and p, q, and r are prime numbers, then 2n×p×q and 2n×r are a pair of amicable numbers. Thābit ibn Qurra ‘s theorem corresponds to the case m=n-1. Euler’s rule creates additional amicable pairs for (m,n)=(1,8), (29,40) with no others being known.




About Praveenkumar P Kalikeri

ABOUT US Hey,Thanks for dropping by .First I would like to Appreciate for your love towards the Mathematics Subject. My name is Praveenkumar Kalikeri. I'm an Engineering Student from Karnataka , India. My Passion for Mathematics , encouraged me to start this blog . I Started loving the Mathematics subject from my schooling days.I was greatly motivated by teachers, parents who always helped me .        Numbers have Beauty !! They fascinated me .. As we all know best method to try out something new is to study old theories , concepts , its drawbacks, etc. I ,myself Started learning Vedic Mathematics at age of 15. Here , I will share vedic maths tricks learned and modified by me in simple manner which will be useful in various competitive exams. "If You Believe , You Can Do".
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