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