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 :)

Blog Post 141: Twin Primes - Part 2

In an earlier blog post (#137), I had documented about one of the properties of Twin Primes - If we multiple any Twin prime (except 3, 5) and add the digits of the resulting number, the answer is always 8

Here we look at another simple property of Twin Primes, which is very well documented by eminent Mathematicians but not very obvious to everyone

Again the Twin prime of (3, 5) is an outlier and doesn't directly satisfy this condition

Few examples for illustration

{11, 13} - If we add these two numbers then the resulting number is 24 which is a multiple of 6

{179, 181} - 360 is the resulting number and again a multiple of 6

{333029, 333031} - 666060 is the resulting number and a multiple of 6

Taking it a step further - Let's add two successive Twin Prime pairs {11, 13} and {17, 19}. The resulting answer is 60 which is the sum of {29, 31} which is the next Twin Prime pair in that sequence. {41, 43} is the next Twin prime pair which adds up to 84 and that is in turn the sum of {11, 13} and {29, 31}. Can we look at similar patterns as we go further?

Interestingly if we look at {3,5} which remains as the exception, their sum is 8 while their product is 15 and if we add the digits of 15, the resultant is 6! It is the complete opposite of what we observe with other twin primes

Drawing an analogy to the corporate world as I always do - Team members working together whether as part of a large or small group, produce consistent results as long as they have unique attributes and working towards a common goal.

#MathsIsFun #AnandMathemagic

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

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

Here are few patterns associated with his birthdate and magic numbers, which many mathematicians and number enthusiasts like me have always looked at in awe. Some of these are well documented but when we dig deeper, we get to unearth very interesting patterns and it is great fun to do that

a) 22121887 happens to be a Prime number! Yes - That happens to be his Birth date and the 8 digit number is a Prime!!!

b) 2212 + 1887 is 4099 which is again a prime!

c) The digits of 22-12-1887, when added in pairs, give 139 (22+12+18+87) which is a prime number. Go one step deeper - Take the first and third digit of 139 (19) and multiply that by its reverse (91) - We get 1729 which is Ramanujan number! 

d) When we add each digit of 22-12-1887 (2+2+1+2+1+8+8+7) we get 31 which is a prime and guess what - If we add each digit of 22-12-2022 (2+2+1+2+2+0+2+2) we get another prime - 13, which also happens to be the reverse of 31! The magic doesn't stop there though - 31 ^ 2 is 961 and 13 ^ 2 is 169 which are also mirror images of each other!

e) This is the 135th birth anniversary of Dr. Ramanujan and definitely the number 135 would show some interesting combinations and permutations given its association with the Man himself!

------->   1) For starters, the digits of 135 add up to a perfect square - 9

------->   2) The square of the digits add upto 35 which is a subset of 135

------->   3) The cube of the digits results in 153 which is the one of the 4 three digit Armstrong numbers (370, 371, 407 being the other three)

------->   4) 135 itself could be written as 61 + 74 and 61 and 74 combined would give us the famous Kaprekar constant - 6174!

f) Finally none of these blog posts around Ramanujan can be complete without writing about 1729

1729 = 12 ^ 3 + 1

9271 is one of the anagrams of 1729 

9271 = 21 ^ 3 + 10

21 is reverse of 12 and 10 is reverse of 01!

1729 + 9271 gives 10000 which is a perfect square (100 ^ 2)

#MathematicalGenius #srinivasaramanujan #TheManWhoKnewInfinity

 #NationalMathematicsDay

Blog #137: Twin Primes and Paired programming!

Thanks to my cousin Rajaram V for triggering a conversation around Prime number distribution this week which resulted in this short blog of mine. 

As per the standard mathematical definition, Twin primes are those two numbers which are prime in nature and the difference between them is 2

For e.g (3, 5), (5,7), (41, 43), (101, 103) are all examples of twin primes.

Taking this one step further let's look at the following twin primes

(11, 13) - Multiply 11 and 13 and we get 143. Add the digits of 143 till we get a single digit number. The resultant is 8

(17, 19) - Multipl1 17 and 19 and we get 323. Add the digits and the resultant is 8

(41, 43) - Repeat the above process. 1763 is the product and adding the digits will result in 8 again (17 followed by 8)

(137, 139) – The product is 19043 and adding the digits will result in 8 again!

Now for my favourite prime number – 333031, which is the Zip code of my Alma mater BITS, Pilani. Coincidentally it has a twin prime which is 333029! Multiple 333029 and 333031 and the resultant product is 110,908,980,899. Add the digits and we get 8 yet again!!!

Strangely if you apply this for the first twin prime (3,5) it results in 6 and not 8 and that seems to be the outlier

A simple analogy with the corporate world

Benefits of Paired Programming - Two individuals having overlapping as well as unique skillsets (odd numbers but with different properties of their own), pair up with each other towards a common cause or goal (Resultant 8 above) irrespective of the changing environment or dynamics around (whether it is a 2 digit or 6 digit prime number)

Does the sequence break somewhere or is there anything else unique about Twin primes? Would explore further in my next blog

Blog #136: Mathematical squares - What message do they convey based on how they are formed?

Let's look at 3 or 4 digit squares today and we would start forming them using pairs, till we reach 00 or if there are no more squares that can be formed

Series 1:

I would start with 41..Now if we have to form a 4 digit square with 41 there is no suitable pair as 4096 and 4225 are consecutive squares and there is no perfect square starting with 41. So technically the series ends with 41!

41

Series 2:

Now let's choose 37 next

a) 37 along with 21 gives us a perfect square 3721

b) With 21 as the starting number, we could combine it with 16 to get 2116 which is a perfect square - 37 21 16

c) With 16, there are 2 options, 00 and 81 (1600 and 1681 are perfect squares)

37 21 16 00 (Series would come to a stop since technically it becomes a 2 digit square again)

37 21 16 81

d) With 81, we could combine with 00 to get 8100 and the series stops there

37 21 16 81 00

Series 3:

We would go with 70 next. Following the same approach as above, we will get

70 56 25 00

Series 4:

Lets begin with 27 and follow the same logic

27 04 84 64 00

27 04 41

27 04 00

Lets look at two more examples where we start with smaller number like 03 as the starting digits

Series 5:

Beginning number - say 03

03 24 01 21 16 81

03 24 01 21 16 00

03 24 01 44 89

03 24 01 69

03 24 01 96 04 00

03 24 01 96 04 41

03 24 01 96 04 84 64 00

03 61

If we observe, the smaller the starting number, the more combinations we get to see till we reach the end

Drawing a simple analogy to the corporate world...

A bottoms-up approach, where we get all the team members right from the junior most person in the organization, to actively contribute towards the broader vision, would bring in more number of ideas and foster innovation rather than a top down mandate