Blog Post No.88: The magic of 4096

4096 is well known in the computer world as 2 ^ 12 (64 x 64) and that's pretty straight-forward

Anagram of 4096 is 9604 which is also a perfect square - 98 x 98

The above are fairly straight-forward and simple but check out this one

4096 x 4096 = 16777216

Now carefully dissect the 8 digit number - 16 and 16 are perfect squares of 4 which is the first digit of 4096

7772 is 4 x 67 x 29 - Look at this again - 4 is the first digit of 4096 and the sum of other two factors (67 + 29) is 96 which are the last two digits of 4096

Not many squares exhibit this kind of a pattern and if there are few more would love to see that listed down

Blog Post #87: Attributes of 1312

During one of our conversations one of my pretty close school mate, Ram, said that his favourite number is 1312 and wanted to check if there are any cool attributes associated with this number

Here are few to start with

a) 1312 is sum of two perfect squares - 1296 + 16

b) 13 + 12 and 13-12 are perfect squares again

c) 1312 can be written as 1000 + 156 + 156 which is 10 ^ 3 + 13 x 12 + 13 x 12 - not many numbers exhibit this property where it is a sum of perfect cube (1000) added with a number (156) whose factors (13, 12 here) are sub-digits of the result! (1312)

d) Now this gets interesting - Look at the odd anagrams of 1312

1123 - Prime
1213 - Prime
1321 - Prime
2113 - Prime
2131 - Prime
2311 - Prime
3121 - Prime
3211 - Only exception but the factors are 13 and 19 which are prime numbers again and 13 is the first two digits of 1312!

e) 2131 is the mirror image of 1312 - 21 x 31 = 651 which is the mirror image of 156 (13 x 12)

Ram - over to you to jot down few more :)

Blog Post #86: Cool attributes of the number 12544

To start with 12544 is a perfect square and I always find perfect squares having lot of unique attributes

a) 12544 - 112 x 112
b) 125 - 44 = 81 which is 9 x 9
c) 125 + 44 = 169 which 13 x 13
d) Now 9 + 13 = 22 and 13 - 9 = 4; Now look at the square root which is 112 (11 + 11 = 22 and 2 + 2 = 4)

How many squares exhibit this kind of pattern? Can we explore further?

Few more interesting attributes

e) Go back to 12544 and add some of the digits: 144 + 25 = 169 which is again 13 x 13 and 144 is 12 ^ 2 and 25 is 5 ^ 2 (12 and 5 are the first 3 digits of 12544)
f) Now look at 112 again - 11 x 11 = 121, 2 x 2 = 4. 121 + 4 = 125 (which is the first 3 digits of 12544 again)

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