Primalities of the Prime Numbers

Prime Number, are one of the most complex pattern known to Mathematicians. That is because, Mathematicians have not yet figured out a pattern to them, but deep down...we know it is there.

Natural Numbers can be divided into 3 categories - One, Composite Numbers, and Prime Numbers.

If you want to understand this Blog, the definition of these 3 categories should be known.

Composite Numbers - Divisible by at least one natural number other than 1 and the number itself. (More than two factors)

Prime Numbers - Divisible by only 1 and the number itself. (Exactly two factors)

1 - 1 is a special number because it has only one factor, 1, i.e. the number itself.

Now, back to Prime Numbers.

One fact about Prime Numbers that most people know is that they are never ending, and there is quite an elegant proof to show that.

Assume That all the Prime Numbers are contained in the Set P =  

{}.  Now, according to the Fundamental Theorem of Arithmetics, any natural number can be expressed as a product of primes. Let S =  , the product of all the primes. Then, S + 1 will also be prime, as it does not have any prime factor other than itself. This can go on forever....and thus the number of primes in the number system is never ending.

One simple test for prime numbers is the square root method:

A number is prime is it is not divisible by any natural number lesser than or equal to its square root.

It is quite an easy thing to understand. If we divide a number into two distinct factors, we can see that one factor is more, and the other less than the square root of the number. They are, in some ways symmetrical.

Ex :- Let N be the number in question, and x and y positive numbers, such that its two factors -   are both positive Natural Numbers. We, then see that -

Equality only holds when x and y are both 0, and the factors is the square root of N.

Therefor, either x or y have to be negative and the other positive. Now, if we have the value of one factor, we have the value of the other. So, we only need to look at the numbers before the root of N.

Cool Terms related to Prime Numbers -

1) Twin Prime - Twin Primes are the Prime Numbers which differ by 2. Eg:- {3, 5}, {5, 7}, {4193801, 4193803}, etc. It is cool because of the fact that, like primes, these numbers are also found high up in the number line, but there is no proof whether they are finite, or never ending. The proof is worth A million Dollars.

2) Emirp - Emirps are the reverse of Primes. These are numbers, which when reversed, behold another prime number.

3) Mersenne Primes - Mersenne Primes are primes of the form . These ar pretty useful in finding Perfect Numbers.

4) Perfect Numbers - Perfect Numbers are the numbers which are equal to the sum of their proper factors(excluding the number, itself). Ex :- 6 -> 6 = 1 + 2 + 3

                                                                     28 -> 1 + 2 + 4 + 7 + 14

Actually, Mersenne Primes and Perfect Numbers are pretty close related. 

If  is a prime number, then ()() is a perfect number.

For 6, n = 2 and for 28, n = 3.

One formula to know:

In the above formula, p, q, r, ..., z are prime factors of N.

Prime Number Theorem - The Prime Number Theorem is a pretty useful tool, which helps in calculating the number of prime numbers in a region. The number of prime numbers from 1 to n is given by :-

This is also consistent with the fact that there are infinite number of primes, as :-

Bertrand's Postulate -  Bertrand's Postulate states that for every n ≥ 2,

              

where p is a prime number. This postulate has been tested for all n in range [2, 3 * ].

Fermat's Little Theorem - If p be a prime number, and n be an integer such that HCF(n, p) = 1, then,

It can be proven in many ways, but I like Induction best. First, we can can see it is true for (a, p) = (7, 2), then, if we assume it is true for (a, p), we need to prove it is true for (a -1, p) :

Hence, proved.

And now, a prime quote to append the topic - “The best number is 73. Why? 73 is the 21st prime number. Its mirror, 37, is the 12th, and its mirror, 21, is the product of multiplying seven and three ... and in binary, 73 is a palindrome, 1001001, which backwards is 1001001."