Blog Post #56 - The curious world of Prime numbers

After a small hiatus again here's a post on Prime numbers... No better place to start than my Alma Mater - Pilani Pin Code - 333031!!!

333031 is a prime number and I decided to look at its reverse 130333 which has 7, 43, 433 as factors
So I thought of looking at prime numbers around 333031

Listed below the prime numbers starting 333019 till 333049 as a sample

333019 – Prime, 910333 – 17, 53549

333023 – Prime, 320333 – 13, 41, 601

333029 – Prime, 920333 - Prime

333031 – Prime, 130333 – 7, 43, 433

333041 – Prime, 140333 - Prime

333049 – Prime, 940333 – 373, 2521

When I look at the factors of the numbers which are reverse of the original number, I see that either they are prime or have factors which are prime themselves!!!

Now that Pilani gave me some good for thought to start off on a Saturday morning, I started looking at the 4 digit numbers in random... Take a look below

1913 - Prime, 3191 - Prime

1747 - Prime, 7471 - 31, 241

3041 - Prime, 1403 - 23, 61

3371 - Prime, 1733 - Prime

3863 - Prime, 3683 - 29, 127

5981 - Prime, 1895 - 5, 379

7879 - Prime, 9787 - Prime

Isn't that wonderful to see? 

Needless to say if the original number starts with 2, 4, 6, 8 (or sometimes even 5), the number in reverse may not exhibit this pattern always

Puzzle #55 - Characteristics of 15625

What are some of the characteristics of 15625 which sets it apart from other numbers? Is there any other similar number which exhibits some of those characteristics?

There is no right or wrong answer for this question and so keep your creative part of the brain buzzing throughout till you find as many as possible!

Puzzle #54 - Find the odd man out

Among the following perfect squares which is the odd man out and why

121, 144, 169, 196, 225, 256, 289, 324

Interesting series - Puzzle # 53

Look at the series below

1, 16, 169, 1369, 13689,....

Do you see the pattern here?

If yes, can you find similar series of this nature?

Puzzle No. 52 - Amazing facts of 1729

We all know that 1729 is the ever famous Ramanujam number and even in the past, we have looked at some interesting patterns that emerge out of this number

Here are few more amazing facts about this number that's not very obvious... Focus on the odd number anagrams of 1729 in particular

1729 is 10 ^ 3 + 9 ^ 3 = 12 ^ 3 + 1 ^ 3 (This is the Ramanujam number as we all know)

Now look at the odd number anagrams...

1279 - Prime number
1297 - Prime number
1927 - 41 * 47 - Both are prime in turn
7129 - Prime number
7219 - Prime number
7291 - 23 * 317 - Both are prime in turn
7921 - 89 * 89 which in turn is a prime
9127 - Prime number
9217 - 13 * 709 - Both are prime in turn
9271 - 73 * 127 - Both are prime in turn
9721 - Prime number

It is beautiful to see that all the odd anagrams are either prime or have factors which are prime and cannot be broken down further

This number continues to amaze me everytime I think about the combinations and permutations!

Puzzle N0.51 - What's the next in this series and why

19, 36, 18, 49, 17, 64, 16, 81, 15, ?, ?, ?

Additional clue: the missing numbers are all 2 digits

Rare number pattern - sum of cubes and Armstrong number!

Check these patterns which are very unique!

a) 343 + 64 = 407

343 is 7 ^ 3 and also it exhibits this unique property (3+4) ^ 3 = 343

64 is 4 ^ 3

407 which is the resultant is the Armstrong number - 4 ^ 3 + 0 ^ 3 + 7 ^ 3

Now notice that numbers 4 and 7 and their cubes effectively combine to create this pattern

b) 343 + 27 = 370

343 is 7 ^ 3 and also it exhibits this unique property (3+4) ^ 3 = 343

27 is 3 ^ 3

370 which is the resultant is the Armstrong number again - 3 ^ 3 + 7 ^ 3 + 0 ^ 3

Again notice that numbers 3 and 7 and their cubes effectively combine to create this pattern like before!

c) 343 + 28 = 371

343 is 7 ^ 3 and also it exhibits this unique property (3+4) ^ 3 = 343

28 is 3 ^ 3 + 1 ^ 3 :)

371 which is the resultant is the Armstrong number again - 3 ^ 3 + 7 ^ 3 + 1 ^ 3

Yet again notice that the numbers 1, 3 and 7 and their cubes combine to create this pattern!!!

Couldn't resist adding one more pattern!

d) 125 + 28 = 153

125 is 5 ^ 3

28 is 3 ^ 3 + 1 ^ 3

153 which is the resultant is an Armstrong number - 1 ^ 3 + 5 ^ 3 + 3 ^ 3

Yet again numbers 1, 3 and 5 and their cubes combine to create this pattern!

No wonder numbers are beautiful and mesmerizing!