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

Blog #135: Decoding the magic of 998001

I was inspired to write about this 6 digit number based on a Youtube link shared by my ex-colleague and friend Karthik Narayan and I decided to unbundle few more amazing characteristics of this numbers

1. For starters, 998001 is a multiple of 9 as evident from a simple divisibility test. It doesn't stop there though

2. 998001 is the largest 6 digit perfect square and its square root is 999

3. 998001/9 = 110889. If you add the first 3 digits and the last 3 digits of the dividend as well as quotient you get 999, which turns out to be the square root as in #2!

4. If we multiply 9 with the reverse of 998001 (100899), the resulting answer is 908091 which is an anagram of 998001

5. Now let's look at some subsets of 998001 - 900, 9801 are perfect squares themselves and if you look at the reverse, 100899 then we get 1089, 100 as the perfect squares!

6. And finally, here's the trigger point for this blog post of mine - the Youtube link that was shared by Karthik (https://www.youtube.com/watch?v=daro6K6mym8).

When we divide 1 by 998001, we get all 3 digit numbers from 000 to 999 in order except "998"! 

Wow... Maths and Numbers in particular is always fascinating and 998001 is sheer magic

Blog #134: Welcoming 2022 with more hope and optimism!

2022 is here and along with it more hope and optimism as we navigate our way out of the pandemic that changed the world order and gave us a new perspective of how to approach life in general

A short mathematical note to welcome 2022 with Hope and Optimism as the "Prime" and "Unique" reason as we look forward to getting back to normalcy and have more in-person interactions with friends and family members without getting anxious about what's around!

a) 2022 + 2021 (the year before) gives 4043 which is a product of two "primes" - 13 and 311

b) 2022 + 2023 (the year after) results in 4045 which is again a product of two "primes" - 5 and 809

c) Now let's look at the factors of 2022 - It can be written as 337 x 6. When you look closely at 337, it is a "prime" number and more interestingly, the anagrams of 337 which are 373 and 733 are also "prime"!

d) 2022 is very interestingly placed between two square pairs. Take a look

The two even squares greater than 2022 are 2116 (46x46) and 2304 (48x48). Now difference between 2116 and 2022 is 94 while between 2304 and 2022 it is 282. 282 is 3 times 94 and also if you add the square roots (46 and 48), we get 94 again!

Now let's take a look at the even squares lesser than 2022 - 1936 (44x44) and 1764 (42x42). We would observe a similar pattern as above. The difference between 2022 and 1936 is 86 and between 2022 and 1764 is 258. 258 is 3 times 86 and if you add the square roots (42 and 44), we get 86 again!

e) Add 20 and 22 and we get 42. Reverse of 42 is 24 and when we multiply both we get the number 1008 which has a unique place in the field of astrology, spirituality, astronomy among many other fields

As we could see, 2022 has its own prime and unique properties and that personally gives me more hope as we look forward to this New Year

Wishing everyone a very Happy, Prosperous and Healthy 2022

#Hope #Optimism #2022

Blog #133: 2021 Recap - Teamwork and Collaboration at its best

If I look back at 2021, it was filled with full of challenges as well as opportunities for many of us both on the personal and professional front. On the professional side, what stood out to me within Mr. Cooper Group was the seamless collaboration, teamwork and commitment exhibited across teams spread across multiple locations, where employees came together, partnered with each other and delivered towards a common cause. 

I constantly saw people from Technology and Business functions working on new automation initiatives and ideas as well as employees from Business enabler functions like HR, Facilities, Finance partnering with their Business teams and innovate at every possible opportunity - Extremely proud of this work culture and credit goes to each one of the Coopers and Xomies

In the mathematics world, Numbers show a striking similarity and behavior - Whether it is an odd or even or prime number or a perfect square or cube, they display amazing patterns when they combine together and create magic under different circumstances

Here are few of my favorites and this is dedicated to all my team members across India and US.

1) 17 and 29 are two prime numbers on the face of it and so unique in that aspect. 

    a) Both are odd numbers but also when you look deeper, 17 and 29 are sum of two perfect squares - 17 is 16 + 1 while 29 is 25 + 4!

   b) Now when they combine together they become more special..1729 is Ramanujan number which doesn't require any introduction!

   c) Add the individual digits 1+7+2+9 = 19. Reverse of 19 is 91 and when you multiply both we get 1729 again!

2) Next in line 16 and 81. Both are perfect squares to begin of two other perfect squares but one of them is even and other is odd. That's how unique they are to begin with

Now let's see what happens when they combine together in various forms and shape!

     a) 1681 - Yet another perfect square (41 ^ 41)

16 x 81 - 1296 which is a perfect square again but it doesn't stop there. Anagrams of 1296 are 9216 (perfect square), 2916 (perfect square again!)

     b) Now add all the digits of 16 and 81, one by one..1 + 6 + 8 + 1 = 16! One of the original number and also a perfect square

     c) Add the digits in pairs - 16 + 81 = 97 which is the largest 2 digit prime

     d) Finally 168 + 1 is 169 whose square root is 13 and if you add the square roots of 16 and 81 you get 13 again!

3) Borrowing from one of my earlier blog posts, I have randomly selected two prime numbers 29 and 59 for this exercise

     a) Next form a 4 digit number combining 29 and 59 which gives us 2959. 2959 is not a prime but again a product of two primes! - 11 and 269

     b) Let's go a step further and form a 5 digit number with 11 and 269 which gives us 11269. What's interesting is that 11269 follows the same pattern - It is not a prime but again it is a product of two primes - 59 and 191 (As a side trivia, check how 59 appears across multiple numbers here)

    c) Next in line - the 5 digit number 59191 (59 and 191 together). Voila - 59191 is also a product of two primes 11 and 5381!!

   Tempted to go one step further?

    d) 115381 - this is a product of 7 and 16483 while 16483 in turn is a product of 53 and 311

        Though in some sense the sequence is "broken" as 115381 is a product of three prime numbers instead of two but we could see how these numbers come together and deliver something unique

If we look at the above 3 examples, we notice that numbers with unique attributes and background, when combined together through various Mathematical operators, deliver something very unique. Drawing a parallel, it is up to each one of us to identify and nurture the talent in every individual within the team and enable them with all the support required to collaborate under different circumstances to deliver towards a common goal. Believe me, each of those teams would innovate as they move along and create an ecosystem which is best in class

Thanks again to all my team members, colleagues and business partners for a wonderful 2021 and look forward to 2022

#WeAreMrCooper #WeAreXome #NumbersAreFun #Teamwork #Collaboration #ThankYou

Blog #132 - Srinivasa Ramanujan Birthday special

As we take a step closer to the beginning of a New Year, here's a short blog post on the occasion of the birth anniversary of the great mathematician Srinivasa Ramanujan - The Man Who Knew Infinity

Dec 22nd 2021 marks the 134th birth anniversary of the Mathematical genius and here's a humble tribute from yours truly to undoubtedly the greatest Mathematician of the last century

Let's look at few unique number patterns

a) 1887 + 2021 is 3908 which is a sum of two perfect squares 3844 and 64

b) 1887 can be represented as 1936 - 49 (44 ^ 2 - 7 ^ 2) while 2021 can be expressed as 2025 - 4 (45 ^ 2 - 2 ^ 2). Difference of two perfect squares

c) Now for the Ramanujan number itself to do some magic - 1729

1887 - 1729 = 158

Add 2021 and 158 and we get 2179

Now check this out

2021 - 1729 = 292

Add 1887 and 292 and bingo we get 2179 again!!

And do note that 2179 is the anagram of 1729!

Ramanujan Number at its best

d) In the previous step let's do subtraction instead of addition and see what happens

2021 - 158 = 1863 which is 1729 + 134 (Ramanujan number + 134th anniversary)!

1887 - 292 = 1595 which is 1729 - 134

(Ramanujan number - 134th anniversary)!

I am sure we could find more such interesting patterns especially when it is related to the Mathematical genius

#MathematicalGenius #srinivasaramanujan #TheManWhoKnewInfinity

#BirthAnniversary #NationalMathematicsDay