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

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