Blog Post # 107: The world of 5 digit squares

Let's take a peek into the world of 5 digit squares and you will be surprised on some of the characteristics and patterns that is in play

I have chosen two numbers 59049 and 65536 since they are multiples of the two smallest prime numbers - 2 and 3!

59049 is 3 ^ 10 and 65536 is 2 ^ 16

a) 59049

- 243 ^ 2 or 9 ^ 5
- One of the anagram of 59049 is the number itself
- Last 2 digits is a perfect square
- Digits 4 and 5 are perfect squares as well!
- 9049 (last 4 digits) is a prime number

Now for the fascinating part - Replace the first digit (5) with numbers from 1 to 9 and we would notice that none of them are prime and at the same time product of 2 or 3 prime numbers

- 19049 is 43 * 443
- 29049 is 3 * 23 * 421
- 39049 is 17 * 2297
- 49049 is 7^ 3 * 11 * 13
- 69049 is 29 * 2381
- 79049 is 137 * 577
- 89049 is 3 * 29683
- 99049 is 37 * 2677

b) 65536

- Well know number in computer parlance
- It has so many factors of multiples of 2 - 256 ^ 2 or 16 ^ 4 or 4 ^ 8 or 2 ^ 16
- Its immediate neighbour 65537 is a prime number though!
- Again one of the anagram of 65536 is the number itself
- Last 2 digits again is a perfect square


Contrast to 59049, this is an even square but look at the following combinations

- 655 + 36 is 691 which is a prime
- 65 + 536 is 601 which is again a prime
- 6553 + 6 is 6559 which is a product of two prime numbers 7 and 937


Further let's add 59049 and 65536 - the end result is 124585

- 124585 is product of two prime numbers - 5 and 24917
- 124 + 585 is a prime number
- 12 + 4585 is a prime number

Last but not the least the last two digits of both numbers are both perfect squares (49 ad 36) and their difference is 13 which is a prime number. The sum of the square roots is also 13!

Square root of 59049 is 243 and that of 65536 is 256 - Guess what their difference is also 13!

And we can go on and on!
  

Blog Post #106 - Similarities between 2601 and 7056

On the face of it, what's common between 2601 and 7056

- Both are perfect squares - 51 ^ 2 and 84 ^ 2

Beyond that there is something unique when you look closely at both the number and its square

Let's write it in this format

51 26 01 (51 ^ 2 => 2601)

84 70 56 (84 ^ 2 => 7056)

Now if you closely look at the number segment, two digits at a time, we can notice a pattern which is an Arithmetic progression

51-26=25
26-1 = 25

84-70= 14
70-56 = 14

Can you find any other square pair exhibiting a similar pattern?

Now here's some more work for the brain... The below number exhibits a slightly different pattern.
81 ^ 2 = 6561

81 65 61

81 - 65 = 16 => 4 ^ 2
65 - 61 = 4 =>   4 ^ 1

Does it have a pair which shows a similar trend?

Blog Post #105: Know more about 370 and 371

I am referring to whole numbers 370 and 371 and nothing else :) I case you like numbers continue to read on else if you were looking for something else on 370 and 371 you can go back to FB and Twitter feeds!!!

Here are some interesting facts about 370 and 371

a) Both are called Armstrong numbers

b) 3 ^ 3 + 7 ^ 3 + 0 ^ 3 = 370 and 3 ^ 3 + 7 ^ 3 + 1 ^ 3 = 371 (hence Armstrong numbers)

c) Adding a prime number (37) to 370 or another perfect square (36) to 371 gives 407 which is the last 3 digit Armstrong number!

d) 371 is a product of two prime number 53, 7. Prefix 1 before 53 and you get 153 which is the first Armstrong number!

e) 370 is also sum of two squares - 361 and 9 (19 and 3 which are prime numbers again)

f) 370 + 371 = 741 which is a product of three prime numbers 19, 3 and 13 (Yes 19 and 3 again!)

g) 370 in Reverse is 73 and 371 in Reverse is 173 and both are Prime numbers again!

h) 370 + 407 is 777 which a product of three prime numbers again - 37, 3, 7

We can go on and on and find more patterns about this Armstrong number pair..Amazing!

Blog Post #104: Welcoming 2019!

I happened to see few interesting facts through a WhatsApp forward about 2019 - it is the smallest 4 digit number which can be expressed as a sum of three prime squares in six different ways

That egged me to write a small blog for 2019 as we welcome the New Year!

Look at all perfect squares from 10 to 100 in steps of 10 - 100, 400, 900 and so on and add or subtract to arrive at 2019. Notice that the number that is added or subtracted is a product of two or three unique primes!

100 + 1919 ===> 1919 is a product of two prime numbers

400 + 1619 ===> 1619 is a prime number

900 + 1119 ===> 1119 is a product of two prime numbers again

1600 + 419 ===> 419 is a prime number

2500 - 381 ===> 381 is a product of two prime numbers

3600 - 1581 ===> 1581 is a product of three prime numbers

4900 - 2881 ===> 2881 is a product of two prime numbers

6400 - 4381 ===> 4381 is a product of two prime numbers

8100 - 6081 ===> 6081 is a product of two prime numbers again!

10000 - 7981 ===> 7981 is a product of two prime numbers!

Now if you look at 2019 - it is also a product of two unique primes! (673 and 3)

This is one prime reason why 2019 could be unique :)

Blog Post 103: 13 and its anagram 31 - Awesome magical pair!

13 is not seen as a lucky number by many and there are numerous reasons, examples quoted for the same. But 13 as a number along with its anagram pair 31 displays some incredible patterns and attributes

Here's why I think this number is special

a) 13 squared is 169 and it is one of the first set of Pythagoras family member - 13 ^ 2 = 12 ^ 2 + 5 ^ 2

b) Reverse of 169 is 961 which is 31 ^ 2 and 31 is reverse or anagram of 13

So far so good and most of you must be saying that there's nothing new. Now things get a bit interesting

c) 13 and 31 together is 1331 which is 11 * 11 * 11 and notice the first and last digit and two digits in the middle - 11 and 33 = 11 * 3 = 33 while the number itself is 11 ^ 3! And ofcourse to top it all 1331 is a palindrome!

d) Concatenate 169 and 961 and we get 169961...Without a doubt this a perfect palindrome!! There are only few squares like these which are perfect palindromes - 144 and 441 being the other (12 ^ 2 and 21 ^ 2)

e) Now 169961 is a product of two prime numbers - 11 and 15451... And if we notice 15451 is also a perfect palindrome and a prime number!

f) 13 ^ 3 is 2197 and 31 ^ 3 is 29791 and as we could notice 2197 is a subset of 29791!

Can you find more patterns of this nature? I bet there are few more if we continue to deep dive!

Blog Post #102: The curious case of 91

While I was chatting with one of my colleage over coffee, we discussed about 1729, Ramanujan number and how special the number was. I told my friend to look closely at the digits in pairs - 19 and 72, add both and you get 91 and now multiple 19 and 91 (which happen to be reverse of each other) - The answer is 1729!

Wow... There is something interesting about 91 he exclaimed and wanted to see if we can dig deeper.

The conversation started with the obvious - 91 is a product of two prime numbers 13 and 7. Well the next question was does 91 or its prime factors influence the number world in some way...

That discussion led to my next blog dedicated to 91! Here is a pattern that it influences right off the bat...

13 x 11 x 7 = 1001 (The first and last digit of the product is the number in the middle - 11)
13 x 12 x 7 = 1092 (first and last digit together is 12)
13 x 13 x 7 = 1183 (first and last digit together is 13)
13 x 14 x 7 = 1274 (first and last digit together is 14)
13 x 15 x 7 = 1365 (first and last digit together is 15)
13 x 16 x 7 = 1456 (first and last digit together is 16)
13 x 17 x 7 = 1547 (first and last digit together is 17)
13 x 18 x 7 = 1638 (first and last digit together is 18)
13 x 19 x 7 = 1729 (first and last digit together is 19 and ofcourse Ramanujan number!)
13 x 20 x 7 = 1820 (does not exhibit the same pattern but if you look closer 20 has sneaked in)
13 x 21 x 7 = 1911 (outlier again like the earlier one - Guess why?)

Now if you look at the first 9 output, look closely at the two digits in the middle - Yes you got it - They are multiples of 9 from 0 to 72 (9 * 0 till 9 * 8)

Now try to check the pattern from 22 onwards

13 x 22 x 7 = 2002 (the pattern has resumed!)
13 x 23 x 7 = 2093 (And it continues...)

Try to work on this sequence further and see where the pattern breaks again or rather where it becomes encapsulated as part of the output

Needless to say this behavioural pattern occurs due to a basic Mathematics principle and I am sure some of you have already figured that out!

Blog Post No: 101 - Dedicated to Dr.Kaprekar Constant

This blog post is dedicated to Dr. Kaprekar constant which is 6174 - More details on that is available at this link - https://en.wikipedia.org/wiki/6174_(number)

Let's look at some more unique characteristics of this number

a) 61 + 74 is 135 and 13x5 is 65. Now 65 - 61 and 74 - 65 are both perfect squares (4 and 9) and 4 + 9 is 13 which is the first 2 digits of 135 and 9 - 4 is 5 which is the last digit!

b) 6174 can be written as (7 ^ 3) * (3 ^ 2) * (2 ^ 1) * (1 ^ 0) - Note that starting from left to right the exponents and indices alternate from 3 to 1

c) Two of the anagrams of 6174 are perfect squares - 4761 and 1764

d) The following takes the cake!

    61 + 74 = 135. Add a perfect square to it - 135 + 9 and you get another perfect square - 144.

    617 + 4 = 621. Add a perfect square to it - 621 + 4 and you get another perfect square - 625

    6 + 174 = 180. Add a perfect square to it - 180 + 16 and you get another perfect square - 196