Prime due to a reason

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Before starting off with another beautiful topic of "Prime" numbers let me try and convince myself why does table of 9 exhibit symmetry.....

9 X 5 = 45   - 9 X 6 = 54

9 X 4 = 36  -  9 X 7 = 63

9 X 3 = 27  -  9 X 8 = 72 

9 X 2 = 18   - 9 X 9 = 81

9 X 1 = 09   - 9 X 10 = 90

a two digit number can be expressed as (10x + y). Since, 10x+y is divisible by 9 as in the case above, so is (10y+x).

Ex : (Representation of a 2 digit number xy as 1x + y)
xy = 10 * x +  y
45 = 10 * 4 + 5
36 = 10 * 3 + 6
72 = 10 * 7 + 2

In table of 9, 45 and 54 both are divisible by 9 . similarly 72 and 27 .... etc

Now, 10x + y, i.e 27, 45, 54 , 72 is divisible by 9.
Since it is symmetrical 10y + x is also divisible by 9.

Observe that the sum of the digits is always 9. (Proof of which will be given in subsequent posts)

=> x + y = 9

=> 10x + y = 9x + (x + y)
=> 9x + 9 = 9(x+1)

9(x+1) is obviously divisible by 9, which leaves quotient as x+1. This result is also very interesting. But, we will leave table of 9 here ............. Let move to the most interesting of all .... The prime numbers

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1. Introduction to prime numbers and very interesting fact about 11,13,17 and 19...

Prime numbers are some numbers which have factors as 1 and the number itself. (Text book definition)

I never quite understood the concept of factors. I looked at it this way .... Prime numbers are those annoying numbers which never get divided.

There are some interesting thing to observe about prime numbers. They always end with 1 or 3 or 7 or 9 .

it can't end with even numbers and 0 because it would certainly get divided by 2. It can't end with 5 because it would get divided by 5. Remaining are 3,5,7 and 9.

Proof you ask ??
Proof : 11,13,17 and 19. (Isn't it weird that these are the continuous primes with same digit in 10's place. Does any sequence exist with same 10's place yet there exist 4 primes.)

2. Why are they prime or at least called so? (If I were to name them ... I would have named annoying numbers or freaks )

Ans : Definition of prime number by Euclid (Tougher than text book ones)

or

Prôtos arithmos estin ho monadi monêi metroumenos.

or 

Prime numbers are that unit alone measured.

or

A prime number is that which is measured by an unit alone.

 (Souce : primes.utm.edu/notes/faq/WhyCalledPrimes.html)

well, it is pretty simple :

Every number always have least mulitples as prime numbers : 

Ex: 

1> 64 = 2*2*2*2*2*2 (Well, 2 is the only even prime number... you knew it already !!!)

2> 33 = 11 * 3 

3> 96 = 3*2*2*2*2*2

All numbers have least mulitples as prime. (They call these as factors. Least divisible factors are prime numbers. Hence it is prime.)

3.Story about prime numbers

There is one puzzle which many mathematicians are trying to break. It has so much significance that, any person who solves it can get 1 million dollars. It is P vs NP problem. The mathematicians all over the world are trying to prove whether 

P == NP (Is P equal to NP) or P != NP (P not equal to NP)

If N =1, then  P == NP !!! Right ? Where the hell is my 1 million dollar?? (?????????!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! I may get killed if any P vs NP enthusiast reads this)

Jokes apart ,

Actually it is too complex to even understand. P is one class of problems and NP is another class of problems. They are trying to see if they can actually be segregated that way. I will try to explain in common man's language (as much as I have understood ... I might be wrong .. Who cares !!!)

P Problems :: These are the problems which we can solve easily following some algorithm. 

Ex: go on a date, add two numbers, multiply two numbers, sing a song etc ...

NP problems :: These are the set of problems which are too difficult or nearly impossible to solve. But, given the solution it can be verified whether it is correct or not.
Ex:  will she accept my proposal?, find the factors of a number which is a product of two very huge prime numbers(one which matters here !!), did people like my song ? etc ..

They are trying to see whether P == NP or not. (Don't take the examples seriously .. It was just to give an idea)

Given 14160, it is easy to find the factors (since it is obvious that it gets divided by 2 and find the subsequent smaller numbers)

But, Is it easy to find factors of 14161 ? Difficult right ?

 Now i say :

It is actually 119 * 119 .

You can verify the result easily. 

A system similar to this is used in many bank security systems. This is thought to be a NP problem. If they prove that, it is infact P ... then I can follow a few steps of some algorithm and break all security systems in the world. There will be chaos.... Sadists are trying to prove that P != NP ... If P == NP, there will be so many new things ... (Recent proof by Vinay deolalikar of HP labs has given a proof that P != NP and mathematicians say that the proof is not too legitimate... I have no idea about their arguments .. So, lets just stick to prime numbers.)

4. Are there infinitely many prime numbers ?

The answer is yes. The solution is very simple. 

Assume that there are finite number of prime numbers.

Let me state 2 facts (That should automatically prove the silly question): 

1 - there are infinite numbers. 

2 - All numbers have the lowest factors as prime numbers. (Or all numbers are product of prime numbers and 1)

If there are only finite set of prime numbers, that would mean that the set of numbers is in fact finite, which is false.

Ex : Lets assume there are only 4 prime numbers i.e 2,3,5 and 7

The numbers 26, 91 wouldn't exist because 13 and 17 do not exist ... Get the logic ??

5. Is there a formula to check whether a number is prime or not ? 

 Ans : No

If there existed a formula, then scientists and maniacs wouldn't be using supercomputers to find new prime numbers everyday. They would simply use the formula to get new prime numbers. There would be no ATMs with ATM cards ... They exist .. so I can safely scream out that there is NO FORMULA.

I wrote this entry because I there is no good book or blog on mathematics without mentioning of prime numbers. Hope this blog becomes good now !!!! :-) (get the joke ??) :-P