(2) Amicable Numbers

(This blog is For Students of IMS Indore, my Past Students, family, friends and colleagues.)

By Pankaj Khanna, Faculty Maths , IMS Indore

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(2) Amicable Numbers:

Posted on 5/9/2022

There are certain numbers which are known as Amicable Numbers or Friendly Numbers.The smallest pair of amicable numbers is (220, 284). 

Amicable numbers are two different natural numbers related in such a way that the sum of the proper divisors of each is equal to the other number. 

Proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, and their sum is 284. And the proper divisors of 284 are 1, 2, 4, 71,142 and their sum is 220. (A proper divisor of a number is a positive factor of that number other than the number itself. For example, the proper divisors of 6 are 1, 2, and 3.)

The first ten amicable pairs are: (220, 284), (1184, 1210), (2620, 2924), (5020, 5564), (6232, 6368), (10744, 10856), (12285, 14595), (17296, 18416), (63020, 76084), and (66928, 66992).

Now coming back to the philosophical part of the above numbers. An ideal friendship or relationship should be like these amicable numbers; to complement each other- even when one is absent- the other should represent the friend! If you have friends like such amicable numbers; you are truly blessed!

Now let us discus about the root of the word amicable.The word amicable is derived  from the Latin root amicus meaning ‘friend’.

There are many words originated from the root amicus. All of the them mean either friendship or love with different shades of meaning. Do read dictionaries before using such amorous words! A few of them are listed below:

Amiable

Amicable 

Amicus Curiae

Amorous

Amoratory

Amour

Amity

Enamour

Let us also look into the formula for generation of amicable numbers also:

p = (2^n − m + 1)×2^m − 1,

q = (2^n − m + 1)×2ⁿ − 1,

r = (2^n − m + 1)²×2^m + 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. 

You may wish to try generate a few of Amicable Numbers from the above formula.

On the occasion of Teachers Day, I wish to thank  all my students,  who  have been my  teachers also for last fifteen years. I am particularly thankful to all the Faculty and Students of IMS Indore who are now affectionately encouraging me to be more productive and proactive; after returning from the retirement!👍🙏

Pankaj Khanna

Indore (9424810575)

B.E. (Mech.) SGSITS Indore 1983 Batch

-Worked with HPCL from 1984 to 2007 as Mgr.

-Faculty Maths/DI-LR - T.I.M.E. Indore 2007-20

-Faculty Maths/DI-LR -IMS Indore 2022 Onwards.

_____________________________________________

List of  Articles of 'Epeolatry': 

1    Epeolatry by a Philomath.  

2    Amicable Numbers

3    Retrouvaille

4    Aeipathy

5    Ambigram

6    Gaslighting

7    Netiquettes

8    Helios

9    Sol

10  Nostalgia and Solastalgia

11  Lalochezia

12  Farrago

13  Muliebrity

14  Rodomontade

15. Kakistocracy

16. Breviloquance

You may also wish to  see my blog : Love thy Numbers. It may be helpful for CAT Preparation in Quant Section to the Serious Students who are willing to accept additional inputs.

(और अगर आप 'भविष्य की भाषा हिंदी' में वर्तमान में कुछ अच्छा, हल्का-फुल्का हास, परिहास और इतिहास  पढ़ना चाहते हैं तो आप तवा संगीत के आलेख भी पढ़ सकते हैं। ये हिंदी वाला ब्लॉग CAT में कुछ काम नहीं आएगा पर मजा आ सकता है:)