Anand's world of Numbers and Mathematics
This a blog where I plan to share interesting facts and figures about Mathematics and Numbers
Sunday 31 December 2023
Blog #145: Welcoming 2024
Friday 22 December 2023
Blog #144: Tribute to the Maths Genius on his 136th birth anniversary
Saturday 13 May 2023
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
Monday 1 May 2023
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 :)
Sunday 9 April 2023
Blog Post 141: Twin Primes - Part 2
Saturday 31 December 2022
Blog Post #140: Welcoming 2023!
Here's a short Mathematical note from yours truly on the occasion of New Year!
The number 2023 has lots to do with prime numbers and prime factors and exhibits some unique characteristics
a) To begin with, prime factors of 2023 are 17, 17, 7
b) Add 20 and 23 to get 43 which is prime and 4+3 is again a prime
c) Prefix 1 to 2023 and the resultant is 12023 which is a product of 11 and 1093 (two primes)
Suffix 1 to 2023 and we get 20231 which by itself a prime!
d) Now for some sheer prime magic around 2023!
Adding all the digits of 2023 gives 7
Adding squares of the digits gives 17 which is the other prime factor of 2023 apart from 7!
Adding cubes of the digits gives 43 which was sum of 20 and 23 as seen earlier!
Adding the digits to the power 4 results in 113 which is again a prime!
Adding the digits to the power 5 results in 307 which is still a prime!
The prime streak is "broken" when we add the digits to the power of 6 where the resulting number is 893 which is a product of two primes in any case! 19 and 47.
e)
Having seen a "Prime fest" all along, let's do a square of 2023 and the resulting number is 40,92, 529. This shows a nice pattern when you dissect the digits
20 + 20 = 40, 23 x 4 = 92, 23 x 23 = 529
If you add all the digits of 4092529 we get 31 which is a prime and if you add the digits 40 + 92 + 529, the answer is 661 which is a prime again!!
It has been raining Primes as far as 2023 is concerned and that's going to be one "Prime" reason for me to look at welcoming 2023 with lot of optimism and hope!
Wishing everyone a very Happy, Healthy and Prosperous 2023
Wednesday 21 December 2022
Blog Post #139: A tribute to the Man Who Knew Infinity
Here's a small tribute to Dr. Srinivasa Ramanujan, the Mathematics Genius on the occasion of his 135th Birth anniversary - 22-12-2022
Dr. Ramanujan was born on 22-12-1887 and many of us are familiar with his magic square which has 22, 12, 18, 87 as the topmost row that adds up to 139 and hence dedicating my 139th post to the eternal genius and one of the greatest mathematician the world has ever seen