Blog Post #99: Ramanujan Birthday special!

22-12-2017 - Today is the 130th birth anniversary of the Mathematics Genius Shri. Ramanujan and this post is a humble tribute to the Man who knew Infinity! Number 13 seems to hog the action and it is fascinating to see the same

Here we go!

a) 2212 + 2017 = 4229 which is 65 ^2 + 2 ^ 2 (65 is a multiple of 13)

b) 4229 in turn is a prime number and it doesn't stop there - Let's look at the relationship between 4229 and 1729 which is Ramanujan number itself

4229 - 1729 = 2500 which is a perfect square! Wow - How do we explain that? Ramanujan number is always in action in some form or the other

c) 2212 - 2017 - 195 which is 65 ^ 3 (65 is a multiple of 13)

d) Now 2017 - 1887 (birth year of Ramanujan) is 130 which is again a multiple of 13

e) 22-12 = 10 and 20-17 = 3 and 10 + 3 gives 13 again!

f) Last but not the least the ever famous Ramanujan number 1729 by itself is a multiple of 13 and that explains why 22-12-2017 in conjunction with 22-12-1887 is so special

Blog Post # 98: The curious case of two six digit squares

Consider these two squares - 531441 and 254016

On the face of it, they appear like any other normal square. Let's look at some surprising facts

For starters 531441 is 729 * 729 and 254016 is 504 * 504 - Nothing unique as of now

a) Now look at the last 3 digits of both squares - 441 and 016 and both are perfect squares

b) Last digit of the square roots (729 and 504) - 9 and 4 and both are perfect squares

c) Difference between the square roots (729 and 504) is a perfect square again

d) 729 by itself is a perfect square as well as a cube

e) 504 is a unique number as it is the lowest three digit to have all factors from 1 to 9 except 5 (Once we multiple this number by 5 we get 2520 which is the smallest 4 digit number to have factors from 1 to 10)

f) Now dissect the original squares as following - 53 + 14 + 41 which gives 108 while 25 +40 + 16 gives 81 (108 and 81 are successive multiples of 27 which in turn is square root of 729!)

g) Multiple the non zero digits of both squares - 5*3*1*4*4*1 = 240, 2*5*4*1*6 = 240 again!

h) Prefix 1 to the square root of 531441 (which is 729) and we get the famous Ramanujan Number 1729. Now subtract the other square root 504 from 1729 and you get a perfect square 1225!

i) Suffix 1 to the square root of 254016 (which is 504) and we get a perfect square 5041 again!

j) Add 531441 and 254016 and the resultant number has factors 3,3,3,3 and a prime number 9697

k) Add 729 and 504 and the resultant number has factors 3,3 and a prime number 137! (see the pattern)

l) Subtract 531441 and 254016 and the resultant number has factors 3,3,3,3,5,5 and 137 again!

Blog Post #97: Dissecting squares

Example 1: Consider 78 * 78

The product is 6084

Now look at the number 4964 - First two digits is 7*7 and last 2 digits is 8*8

4964 + 1120 would give us 6084 which is 78*78

Why 1120 - 1120 is nothing but 7*8 (of 78) multiplied by 20

Example 2: 63 * 63

The product is 3969

Now look at the number 3609 - First two digits is 6*6 and last 2 digits is 3*3

3609 + 360 would give us 3969 which is 63*63

Why 360 - 360 is nothing but 6*3 (of 63) multiplied by 20

Blog Post #96: The brilliance of Armstrong number 153

153 is an Armstrong number by definition and that's a given but there are some unique features that this number exhibits in and around its ecosystem

Here are few gems starting from some obvious ones to not so obvious

a) 153 - 9 is a perfect square (note 9 is also a perfect square)

b) 153 + 16 is a perfect square (note 16 is also a perfect square)

c) 153 - 72 is a perfect square (81)

d) 153 + 72 is a perfect square again (225)

e) 153 + 81 = 234 and 153-144 = 09 and when we concatenate 234 and 09 we get 23409 which is 153 * 153! (153 squared)

f) 153 is sum of two consecutive numbers 77 and 76 which is normal for any odd number but here is when it gets interesting - 7776 by itself is 6 ^ 5, 7776 + 8100 (which is a perfect square( is another perfect square (15876), 7776 + 7600 (00 concatenated with 76) gives 15376 which is not only a perfect square but it is our Armstrong number 153 combining with 76!

g) Continuing from the previous point, the beauty of numbers become evident when you try to combine 153 with 77 - 15377 is a prime number with no factors unlike 15376 which is a perfect square

h) 1 ^ 3 + 5 ^ 3 + 3 ^ 3 = 153 as we know. Now add 4 ^ 3 to that and we get 217. Now add 153 and 217 and we get 370 which by itself is another Armstrong number!

Blog Post No.95: Extending Pythagoras pairs

Pythagoras theorem doesn't need an introduction and has lot of relevance in day to day applications and not just a mere theorem

Just made an attempt to look at some of the existing Pythagoras pairs to see if they throw some interesting patterns when we bring a cube into picture

Here you go...

a) 12 ^ 2 + 5 ^ 2 = 169 which is 13 ^ 2. Now add 3 ^ 3 to 13 ^ 2 --> 13 ^ 2 + 3 ^ 3 = 196 which is 14 ^ 2

b) 24 ^ 2 + 7 ^ 2 = 625 which is 25 ^ 2. Now add 6 ^ 3 to 25 ^ 2 --> 25 ^ 2 + 6 ^ 3 = 841 which is 29 ^ 2

Can someone figure out more patterns of this nature?

Blog Post 94: Interesting patterns

These are few patterns which may be obvious for Maths enthusiasts and also there is a formulae based proof but neverthless pretty fascinating when we see how these play out

a) 111 x 111 = 12321

11 + 1 = 12 and 12x12 = 144 which is 123+21

b) 112 x 112 = 12544

11+2 = 13 and 13x13 = 169 which is 125+44. Not only that this number is more unique for the following reasons

11-2 = 9 and 9x9 = 81 which is 125-44! (we don't see that in the earlier example due to a very standard formulae based logic

c) 113x 113 = 12769

11+3 = 14 and 14x14 = 196 which is 127+69

d) 114x114 = 12996

11+4 = 15 and 15x15 = 225 which is 129+96

This pattern doesn't repeat when we go the series starting with 12 or 13. Can someone figure out why this is happening?

Blog Post #93: 3249 and its offshoots!

As I drove back home on Friday evening from office got a glimpse of 3249 at couple of places - Car registration number ahead of me, a prize tag on one of the online sites and that's when I decided this Blog is on 3249 and its offshoots!


a) For starters, 3249 is a perfect square and that's one reason this caught my attention - 57 x 57

b) 32 + 49 is 81 which is a perfect square of 9. Now if we add the first 3 digits we get 9 and multiply it with the last digit will give us 81 again

c) 324 (first 3 digits) is a perfect square (18 ^ 18) and 9 (last digit) is a perfect square)

d) 32 is 2 ^ 5 and 49 is 7 ^ 2. Now let's flip that - 5 ^ 2 (instead of 2 ^ 5) and 2 ^ 7 (instead of 7 ^ 2). Add the resultant - 25 + 128 = 153 which is Armstrong number!

e) 32 ^ 2 is 1024 and 49 ^ 2 is 2401 - 1024 and 2401 are both anagrams - Voila!

f) For all the factors, squares that 3249 exhibited till now, look at this - Prefix 1 to the number and you get 13249 which is a Prime. Suffix 1 and the resultant (32491) is a prime again!

Blog Post # 92: Ramanujan Number Neighbourhood

1729 has a special place in the Number world thanks to the genius of Ramanujan

Let's explore the neighbourhood of 1729 as I am sure some of those will exhibit special attributes as well being closer to GOD Number!

1728 is in Focus in this Blog

- For starters 1728 is multiple of two cubes - 3 ^ 3 x 4 ^ 3 (27 x 64)

- Now take the two factors (27 and 64) - Add both and you get 91. 91 multiplied with 19 (reverse) gives back Ramanujan Number!

- 1728 is also difference of two perfect squares - 1764 - 36. Now add the digits of 1764 in pairs (17 + 64 = 81) and subtract 36. The result is 45 which is sum of the digit pairs of 1728 (17 + 28)!

- One of the anagrams of 1728 is 2187 which 3 ^ 7 (refer to Blog No.91)

- Multiply 18 and 72 (digit pairs of 1728) and you get another perfect square - 1296

As always the best is reserved for the last

Sum of the cubes of its digits - 1 ^ 3 + 7 ^ 3 + 2 ^ 3 + 8 ^ 3 = 864 which is exactly half of 1728. Very few numbers exhibit this

The neighbourhood of Ramanujan number looks amazing and watch out this space for more!

Blog Post No.91 - The magical number 2187

Another number that provides some fascinating insights on how digits 2, 3, 7 combine and create magic

- 2187 is 3 ^ 7
- 3 x 7 is 21 which are the first two digits
- First digit (2) ^ 3 (sum of first 2 digits) is 3rd digit - 8
- First 2 digits (21) divided by sum of first 2 digits is 4th digit - 7

And for few more medium complex ones,

- 218 + 7 is 225
      - Insert 0 as the 2nd digit makes it 2025 which is 45 ^ 2 and 45 in turn is 27 + 18 or 28 + 17 (sum of digits)
      - Insert 0 as the 3rd digit makes it 2205 which is 441 x 5 (3 ^ 2 x 7 ^ 2 x (3+7)/2)

- 2187 is product of its own digits - 27 x 81

- 2187 can also be written as - ( (7 ^ 3) - (7+3) ^ 2 ) x 3 ^ 2

And best is reserved for the last

21 + 87 is 108 and 87 - 21 is 66. Multiplication of 108 x 66 gives 7128 which is an anagram of 2187