Blog Post # 85 - Interesting pattern of prefixes

Take 729 which is a perfect square (27 x 27)

Now prefix 1 - 1729 which is Ramanujam number and also 19 x 91 (19 and 91 are anagrams)
Now prefix 11 - 11729 which is 37 x 317 (317 has 37 as its subset)
Now prefix 111 - 111729 which is 3 x 37243 (Notice 37 again and 243 x 3 is 729 which is the original number!)
Now prefix 1111 - 1111729 which is 1019 x 1091 - Voila! 19 x 91 component is back again
Now prefix 11111 - 11111729 which is 37 x 300317! (37 and 317 are back!)
Now prefix 111111 - 111111729 which is 3 x 37037243!

To summarize

729 - 27 x 27
1729 - 19 x 91
11729 - 37 x 317
111729 - 3 x 37243
1111729 - 1019 x 1091
11111729 - 37 x 300317
111111729 - 3 x 37037243

Blog # 84 - Uniqueness of the square series

In one of my earlier post, I had posted a question about uniqueness of the squares between 1600 to 3600.. To recap, let's look at these squares

1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 2916, 3025, 3136, 3249, 3364, 3481

As you we could seem as the first two digits increase by 1, the last 2 digits are perfect squares in decreasing order till we hit 2500 and again goes in increasing order till we reach 3600!

Now the fun doesn't just stop there!

Let's look at the squares preceding these

Listing the ones before 1600

1521, 1444, 1369, 1296, 1225, 1156, 1089, 1024, 961, 900, 841, 784, 729, 676, 625, 576, 529, 484, 441, 400, 361, 324, 289, 256, 225, 196, 169, 144, 121, 100, 81, 64, 49, 36, 25, 16, 9, 4, 1

Listing the ones after 3600

3721, 3844, 3969, 4096, 4225, 4356, 4489, 4624, 4761, 4900, 5041, 5184, 5329, 5476, 5625, 5776, 5929, 6084, 6241, 6400, 6561, 6724, 6889, 7056, 7225, 7396, 7569, 7744, 7921, 8100, 8281, 8464, 8649, 8836, 9025, 9216,  9409, 9600, 9801

The pattern is very much clear..Isn't it!

a) Last 2 digits are same in both the series

b) Add the first 2 digits and you will see a pattern - 52 for the first few numbers and then 54 and the increases by 2 till 98 (1 can be written as 0001)

One of the reason behind this is the fact that these squares fit in the formulae a^2 - b ^2 if you closely observe it. Though it is not obvious the resulting patterns are indeed very interesting!

Blog #83 - Amazing 65536!

65536 is synonymous with computers as 64 KB translates to 65536 Bytes and it is a commonly used term in day to day life of a Computer personnel

65536 has also many factors of 2's multiple - 2 ^ 16, 4 ^ 8, 256 x 256, 16 ^ 4 and so on

But a lesser known fact of 65536 is what I am focussing on and is exactly the opposite of the above attributes. Check this out...

Most of the odd number subset of 65536 is either a prime or product of two primes

a) 65 - 13 x 5

b) 655 - 131 x 5

c) 6553 - Prime

d) 553 - 7 x 79

e) 55 - 11 x 5

f) 53 - Prime

g) 653 - Prime

Go one step further and add or subtract combination of digits which are a subset

h) 65 + 536 - 601 - Prime

i) 655 + 36 - 691 - Prime

j) 655- 36 - 619 - Prime

k) 6 + 553 - 559 - 13 x 43

l) 6553 + 6 - 6559 - 7 x 937

m) 6553 - 6 - 6547 - Prime

n) 553 - 66 - 487 - Prime

Wow... A number which has 2 and its multiples as factors exhibits a completely different behaviour when the individual digits combine

Blog #82 - The curious case of the numbers 7056 and 5776

I had posted earlier about some of the attributes of the number 7056 and 57767 and this is an extension of that

7056

a) 7056 is 84 x 84

b) Now check this sequence - 84, 70, 56 (This is the only 4 digit square which exhibits this sequence)

c) 70 ^ 2 - 56 ^ 2 = 1764 which is 42 x 42 and 42 is half of 84!

d) Also 42 extends the AP sequence highlighted in (b) = 84, 70, 56, 42!

e) Anagram of 42 is 24 and whose square is 576 (0576) which is anagram of 7056!

5776

a) 5776 is 76 x 76

b) Only 4 digit square (not ending with 25) whose last 2 digits are the original number itself

c) 57 - 19 x 3 and 76 - 19 x 4 and again one of the few squares which exhibit this sequence

d) Similar to 7056, 576 is again a subset of 5776 and a perfect square

e) Add 2000 to 5776 gives 7776 which is unique in its own sense - 3 ^ 5 x 2 ^ 5

Last but not the least - Prefix 1 to both 7056 and 5776

The resulting numbers are 17056 and 15776 and the similarities are striking

17056 - 2 ^ 5 x 13 x 41

15776 - 2 ^ 5 x 17 x 29

Both have 2 ^ 5 and a combination of 2 odd numbers as a factor

Post No. 81 - Ramanujam number and 2520!

This post is to showcase some of the interesting patterns generated when we combine two unique numbers

1729 - No introduction required... The Great Ramanujam number
2520 - Lowest number to have all factors from 1 to 10

Now for some interesting patterns

i) To start with 1729 + 2520 = 4249 which is 7 x 607.

ii) Now add 42 and 49 - we get 91 is a factor of 1729

iii) Add the digits of 607 - we get 13. Multiply by 7 and we get 91 again.

iv) Now add the digits of 4249 - we get 19!

v) 19 x 91 gives back 1729!

vi) Now add digits in pairs of 1729 and 2520 - 17 + 29 + 25 + 20 - Voila - it is 91 again!

vii) Now 2520 - 1729 = 791 which is 7 x 113 - Add the digits of 113 - we get 5. Multiply by 7 and we get 35. 35 is the odd factor component of 2520 (35 x 72)

We could go on creating patterns of this nature and there is no end at all but it is beautiful to see how some of the special numbers create further magic when they come together

Post No.78 - Dedicated to India vs Aus T20 - 9 is the glue!

Scenario 1:

24-03-2011 - India vs Aus QF and India won

Now for some Maths - 2+4+3 = 9, 2+0+1+1 = 4; 9 x 4 = 36

Fast Forward to 27-03-2016 - India vs Aus (virtual QF again)

Maths again - 2+7+ 3 = 12, 2+0+1+6 = 9; 12 x 9 = 108

The end result has gone up 3 times but 9 remains the common factor and so the match result is obvious!

Scenario 2:

24-03-2011
 Multiply all the non-zero digits - 48

Reverse of 48 is 84 and their difference is 36 which is factor of 9

27-03-2016
Multiply all the non-zeros - 504

504 is a factor of 9 again and the largest 3 digit number to have all factors from 1 to 9

9 remains the glue again!

Scenario 3:

24-03-2011
Add all the non-zero digits - 13

13 is 9 + 4 (compare with scenario 1 where 36 was 9 x 4)

27-03-2016

Add all the non-zero digits - 21

21 is 9 + 12 (compare with scenario 1 where 108 was 9 x 12)

9 remains the glue again!

Blog Post 77: Ramanujan's number and BITS 91 batch reunion

22nd December was the birth anniversary of the Mathematical genius Ramanujan and one of his well known all time favourite number is 1729 which is a product of 19 and 91... 

Coincidentally, four days after the birthday of the brilliant mathematician, on 26th Dec, the two magical numbers 19 and 91 once again come together in the form of 1991 batch of Pilani to meet at the "Prime" location - 333031! 

BITS 1991 batch will always remain special and happy to be part of Sir Ramanujan's history as well!