Blog Post 103: 13 and its anagram 31 - Awesome magical pair!

13 is not seen as a lucky number by many and there are numerous reasons, examples quoted for the same. But 13 as a number along with its anagram pair 31 displays some incredible patterns and attributes

Here's why I think this number is special

a) 13 squared is 169 and it is one of the first set of Pythagoras family member - 13 ^ 2 = 12 ^ 2 + 5 ^ 2

b) Reverse of 169 is 961 which is 31 ^ 2 and 31 is reverse or anagram of 13

So far so good and most of you must be saying that there's nothing new. Now things get a bit interesting

c) 13 and 31 together is 1331 which is 11 * 11 * 11 and notice the first and last digit and two digits in the middle - 11 and 33 = 11 * 3 = 33 while the number itself is 11 ^ 3! And ofcourse to top it all 1331 is a palindrome!

d) Concatenate 169 and 961 and we get 169961...Without a doubt this a perfect palindrome!! There are only few squares like these which are perfect palindromes - 144 and 441 being the other (12 ^ 2 and 21 ^ 2)

e) Now 169961 is a product of two prime numbers - 11 and 15451... And if we notice 15451 is also a perfect palindrome and a prime number!

f) 13 ^ 3 is 2197 and 31 ^ 3 is 29791 and as we could notice 2197 is a subset of 29791!

Can you find more patterns of this nature? I bet there are few more if we continue to deep dive!

Blog Post #102: The curious case of 91

While I was chatting with one of my colleage over coffee, we discussed about 1729, Ramanujan number and how special the number was. I told my friend to look closely at the digits in pairs - 19 and 72, add both and you get 91 and now multiple 19 and 91 (which happen to be reverse of each other) - The answer is 1729!

Wow... There is something interesting about 91 he exclaimed and wanted to see if we can dig deeper.

The conversation started with the obvious - 91 is a product of two prime numbers 13 and 7. Well the next question was does 91 or its prime factors influence the number world in some way...

That discussion led to my next blog dedicated to 91! Here is a pattern that it influences right off the bat...

13 x 11 x 7 = 1001 (The first and last digit of the product is the number in the middle - 11)
13 x 12 x 7 = 1092 (first and last digit together is 12)
13 x 13 x 7 = 1183 (first and last digit together is 13)
13 x 14 x 7 = 1274 (first and last digit together is 14)
13 x 15 x 7 = 1365 (first and last digit together is 15)
13 x 16 x 7 = 1456 (first and last digit together is 16)
13 x 17 x 7 = 1547 (first and last digit together is 17)
13 x 18 x 7 = 1638 (first and last digit together is 18)
13 x 19 x 7 = 1729 (first and last digit together is 19 and ofcourse Ramanujan number!)
13 x 20 x 7 = 1820 (does not exhibit the same pattern but if you look closer 20 has sneaked in)
13 x 21 x 7 = 1911 (outlier again like the earlier one - Guess why?)

Now if you look at the first 9 output, look closely at the two digits in the middle - Yes you got it - They are multiples of 9 from 0 to 72 (9 * 0 till 9 * 8)

Now try to check the pattern from 22 onwards

13 x 22 x 7 = 2002 (the pattern has resumed!)
13 x 23 x 7 = 2093 (And it continues...)

Try to work on this sequence further and see where the pattern breaks again or rather where it becomes encapsulated as part of the output

Needless to say this behavioural pattern occurs due to a basic Mathematics principle and I am sure some of you have already figured that out!

Blog Post No: 101 - Dedicated to Dr.Kaprekar Constant

This blog post is dedicated to Dr. Kaprekar constant which is 6174 - More details on that is available at this link - https://en.wikipedia.org/wiki/6174_(number)

Let's look at some more unique characteristics of this number

a) 61 + 74 is 135 and 13x5 is 65. Now 65 - 61 and 74 - 65 are both perfect squares (4 and 9) and 4 + 9 is 13 which is the first 2 digits of 135 and 9 - 4 is 5 which is the last digit!

b) 6174 can be written as (7 ^ 3) * (3 ^ 2) * (2 ^ 1) * (1 ^ 0) - Note that starting from left to right the exponents and indices alternate from 3 to 1

c) Two of the anagrams of 6174 are perfect squares - 4761 and 1764

d) The following takes the cake!

    61 + 74 = 135. Add a perfect square to it - 135 + 9 and you get another perfect square - 144.

    617 + 4 = 621. Add a perfect square to it - 621 + 4 and you get another perfect square - 625

    6 + 174 = 180. Add a perfect square to it - 180 + 16 and you get another perfect square - 196

Blog Post #100: The sheer brilliance of Ramanujan number - Special Blog

This is my 100th Blog post and what could be better than dedicate this to my all time favourite (Ramanujan Number - 1729)

1) Let's start with the Year 2018 - 2018 is sum of 1729 and 289 which a perfect square of 17 (first 2 digits of 1729 as well)!

2) 172 * 9 is 1548 and 172 + 9 is 181 - The difference between 1729 and 1548 is 181 as well!

3) Sum of the digits of 1729 is 19 (1+7+2+9) and when you multiply 19 by its reverse (91) you get 1729 again!

4) Let's now look at some of the anagrams of 1729 and see the properties they exhibit as sum of two perfect squares with one of them being 1 (similar to 1729 which is sum of 12 ^ 3 + 1 ^ 3)

- 1297 is 36 ^ 2 + 1 ^ 2

- 9217 is 96^ 2 + 1 ^ 2

- 2917 is 54 ^ 2 + 1 ^ 2


5) Square of 17 is 289 and square of 289 is 841. Now look at square 1729 which turns out to be 29,89,441. 289 and 841 are subsets of 29,89,441 and there is a visible pattern seen

6) 1729 is 12 ^ 3 + 1 ^ 3 as we are all aware. Now (12+1) ^ 3 is 2197 which is an anagram of 1729!

7) Multiple the 2nd and 3rd digits with 1st and 4th digits (72 x 19) and we get 1368. Now 1729 - 1368 = 361 which is again 19 ^ 2! (square of the 1st and 4th digit)

8) Ramanujan lived between 1887 and 1920... Now look at how these two years combine with his magic number and exhibit an unique property

1729 + 1887 -----> 3616 (6 ^ 2 and 4 ^ 2)

1729 + 1920 -----> 3649 (6 ^ 2 and 7 ^ 2)

9) 1729 is a multiple of 3 distinct prime numbers namely 7, 13, 19. Now let's look at the number 17299271 is formed by appending 9271 (exact reverse of 1729) to the number itself. 17299271 is also a multiple of 4 distinct prime numbers 11, 31, 97, 523!

There are many more such interesting patterns which we could continue to unearth and this just reflects the sheer brilliance of Shri Ramanujan