Blog Post #61 - Five digit Number fun

Sunday musings!

Find this five digit number - ABCDE (Digits can repeat and not necessarily distinct)

- It can be expressed as (a ^ b) x (c ^ b) where a, b, c are integers
- (b ^ a) x (b ^ c) is a special number in its own sense
- Remove the 2nd digit (B) and it is a perfect square
- Add 1 to B and swap with A - (BACDE) and it is still a perfect square!

Blog #60 - X ^ Y - Y ^ X - An observation

I was randomly thinking through some of the scenarios for the formulae (X ^ Y) - (Y ^ X) where one is odd and other is even. For eg. ( 4 ^ 7) - (7 ^ 4)

Surprisingly, most of the resultant answer is either prime or a number whose factors are predominantly prime numbers and odd as well

3 ^ 4 - 4 ^ 3= 17
2 ^ 7 - 7 ^ 2 = 79
2 ^ 9 - 9 ^ 2= 431
4 ^ 5 - 5 ^ 4 = 399 (19 x 7 x 3)

Is there any pattern others see? And any justification using a mathematical equation (I did try logarithms but don't think it helps)

Blog #59 - Find the 4 digit number

- It is a 4 digit number (call it as ABCD)
- (AB + CD) ^ 2 is one of it anagrams
- One of its three digit anagram is also a perfect square
- ABCD - 1 results in a number which has 11 as one of its factor
- ABCD + 1 results in a number which is prime
- ABCD + 9 ^ 2 is again a perfect square
- Last not but not the least this number itself is pretty unique and famous for a reason (Clue: Uniqueness in terms of its factors)

Blog Post #58 - Armstrong number and their patterns

The well known Armstrong numbers are 153, 370, 371, 407 (apart from 0 and 1). Here are few more patterns exhibited by these numbers around squares, cubes and prime numbers

While most of the natural numbers will exhibit pattern of these sorts, it is interesting to see how fe number pairs play a larger role for these 4 numbers

a)  153

12 ^ 2 + 3 ^ 2
13 ^ 2 - 4 ^ 2

One of the anagram of 153 is 513 is 8 ^ 3 + 1 ^ 3

b) 370

7 ^ 3 + 3 ^ 3

19 ^ 2+ 3 ^ 2

One of the anagrams of 370 is 703 (which is not an Armstrong number but check the pattern)

703 = 19 x 37 (19 is there in earlier pattern exhibited by 370 and 37 is a subset of 370!)

c) 371

7 ^ 3 + 3 ^ 3 + 1 ^ 3 (sub of 3 different cubes)

One of the anagram of 371 is 731 which is formed by the digits 7, 3, 1 again

d) 407

7 ^ 3 + 4 ^ 3 or 7 ^ 3 + 8 ^ 2

Its anagram 704 is 7 ^ 3 + 19 ^ 2 (again notice the role number 19 plays in some of these numbers)

Last but not the least the Armstrong numbers show intricate relationship pattern among themselves

407 in turn is 370 + 37 (effectively 37 * 10, 37 * 11)
407 is also 371 + 6 ^ 2

Blog Post #57 - Prime numbers around 1729

This is a recap of one of my earlier post but with few more details around on prime factorization for one of the most famous number - "RAMANUJAM NUMBER - 1729"

Here are more amazing facts about this number that's not very obvious...

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

1729 in turn is 13 x 7 x 19 (13, 7, 19 are all prime numbers!)

Now look at the odd number anagrams on 1729...

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

Note that odd anagrams are either prime or have factors which are prime and cannot be broken down further

Dig deeper... (Some of these in turn are sum of squares +/- prime number)

1279 is (36 ^ 2) - 17
1296 is (36 ^ 2) + 1
1927 is (44 ^ 2) + 17
9127 is (96 ^ 2) - 89
9217 is (96 ^ 2) + 1

Hats off to the Number 1729 again!

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