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