Blog Post #143: Case of Amicable Squares!

In my last blog post, I had written about "Amicable numbers" which has been well documented and researched

Last week, we met for a Team lunch and when I received the bill, I noticed that it was a perfect square! (1764). I have written about 1764 in one of my earlier blog almost 5 to 6 years ago but given the recent topic on amicable numbers, I was wondering if some of the squares like 1764 have something in common and started exploring that further

Guess what - Numbers don't disappoint us any time and I did get to see a peer/counterpart for 1764 which had similar attributes and patterns if not a 100% match

Here we go and look at some interesting points about 1764 and 2916 which i would like to call as "Amicable Squares" for the purpose of this blog

1. To begin with 1764 and 2916 are perfect squares of 42 and 54

2. Both squares are formed by a combination of prime number and another perfect square - 17 and 64, 29 and 16

3. One of the anagram of 1764 is 4761 and similarly 9216 is an anagram of 2916 - Both turn out to be perfect squares again (of 69 and 96) and on top of that, as we could see 69 and 96 are reverse of each other!

4. Another anagram of 1764 turns out to be the famous "Kaprekar constant" - 6714 (More about it could be found through this link (https://en.wikipedia.org/wiki/6174_(number)#:~:text=6174%20is%20known%20as%20Kaprekar's,Kaprekar.)

5. On similar lines, 9261 is an anagram of 2916 and is a perfect cube while 1296 which is another anagram, is a perfect square of 36 which in turn is a perfect square of 6!

6. Replace 64 with 29 to get 1729 which is Ramanujan number again! Cannot leave this magic number out of any number game :)

7. Add the digits of 1764 and 2916 and we would get 18 in both cases!

1764 and 2916 are a great example of perfect squares which demonstrate similar behavior and it would be great fun to see if there are similar square pairs (formed by a combination of prime number and a square)

#NumbersAreFun #PerfectSquares

Blog Post #142: Numbers which complement each other as a friend

Thanks to the recent post by Dilip and my friend Sukumar for tagging me on that, which acted as the trigger for this next blog of mine. The numbers in limelight are 220 and 284 are part of what Mathematicians call as “Amicable Numbers” and these two are the smallest of that series

https://en.m.wikipedia.org/wiki/Amicable_numbers

Given the friendly nature of these amicable numbers, I was wondering if there are more attributes or qualities that these two numbers stand for and started exploring further this weekend. Here are few of my observations and findings

1. Add 220 and 284 and the resultant is 504 which is the smallest three digit number with factors from 1 to 9 except 5 ofcourse. When multiplied by 5 the resulting number is 2520 which is a special number - the lowest whole number which has factors from 1 to 10!

2. Subtract 220 from 284 and we get a perfect square

3. Both 220 and 284 are resultant of a perfect square minus 5. 220 is 225 - 5 while 284 is 289 - 5

4. Swap the last digit of these two numbers to form 224 and 280 which are successive multiples of 56! Truly amicable and friendly even when they swap a digit!

5. Let’s reverse both numbers and we get 022 and 482. Look at these closely and we realise that both are multiples of a prime number with 2! 11 x 2 and 241 x 2 !!

6. Suffix 9 to 220 and we get 2209 which is a perfect square of 47. So what’s big in that one may wonder…Replace the last digit 4 in 284 with 09 and we get 2809 which is a perfect square of 53. Guess what 53 and 47 in turn adds up to 100!!!

7. I am always fascinated by 1729 (Ramanujan number) and strongly believe 220 and 284 should have some relation with 1729. Guess what… 220 + 284 is 504 as we saw earlier and 504 + a perfect square (1225) yields 1729. Voila! That’s the brilliance of Ramanujan number

No wonder 220 and 284 are part of the Amicable or Friendly number list given they share so many common attributes as seen above. Draw a parallel to real life and/or corporate world, these are the qualities that we would look up for amongst friends or colleagues when they work together as a team - flexibility, fungibility, work towards a common goal, adapt to any ambiguous situation and produce results.

Coincidentally this is my 142nd blog which happens to be half of 284 :)