Blog Post #108 - Year end tribute to Srinivasa Ramanujan!

As we near the end of the 2nd decade of the 21st century, here's a short blog post a week after the birthday of an eminent personality whom I admire the most - Mathematician Srinivasa Ramanujan

Dec 22nd 2019 marked the 132nd birth anniversary of the Mathematical genius and needless to say it is all about numbers when we talk about Ramanujan!

Let's look at few patterns

a) 1887 + 2019 is 3906 which is the product of two consecutive numbers 62 * 63 and in turn their sum (62+63) is a perfect cube (125)

b) 2019 - 1887 is 132 which is the product of two consecutive numbers again! 12 * 11

c) Now add the digits of 1887 and 2019 in pairs - 18 + 87 + 20 + 19 which gives us 144 and a perfect square

d) Now add the individual digits - 1 + 8 + 8 + 7 + 2 + 0 + 1 + 9 which gives us 36 and a perfect square again (36 is one fourth of 144)

Now for the Final Home run which is completely out of the Blue!

Extract the 1st and last digits of both 1887 and 2019 - 17 and 29. 17 and 29 together form 1729 which is the famous Ramanujan Number! Isn't it awesome that 2019, which turns out to be the last year of this decade along with the birth year of Ramanujan produces the magic number that he came up with

That's the sheer beauty of Ramanujan and Numbers - Cannot separate both!

It does not stop there.. Hold on! The remaining two digits are 88 and 01 after extracting the other two and the resulting number is 8801. So what? It is yet another 4 digit number - Yes but like 1729 it is also a multiple of 13! 1729 is 13 and 133 while 8801 is 13 * 677 (Notice two 3s and two 7s within the factors which is unique again!)

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 :)