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Prime Number Theorem

When prime numbers are considered, we know that there are infinite number of them. To understand this concept of infinite number here, we use a theorem called the prime number theorem.
The prime number theorem defines about the density of prime numbers that is how prime numbers are distributed among all the positive integers.
 
Let assume $n$ is any positive real number and let $\pi(x)$ denotes the number of primes numbers less than or equal to x.
So  ${\pi (n)\to \propto }$,  as ${(n)\to \propto }$
For example $\pi(30)$ denotes  the number of prime numbers less than 30.
The prime numbers less than 30 are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.
So $\pi(30)$  =  10.
For small number, its possible to estimate all prime number, but as the numbers get larger, it can be seen that these prime numbers are irregularly distributed. In such situation prime number theorem helps a lot to get an approximate answer.
 
The prime number theorem implies that the probability that a random number n, which is selected from a set of positive number, will be prime, is about $\frac{\ (1)}{ln(x)}$.
This means in a group of n numbers, $1$ out of every $ln(n)$ will be a prime.
For example among the positive integers up to and including n = 153, about 1 in $ln(153)=5.03\sim 5$ will be a prime number.

So the prime number theorem states that, the limit of the quotient of two functions $\pi(n)$ and $\frac{\ (n)}{ln(n)}$ will be $1$ as n tends to infinity.
The asymptotic law of distribution of prime numbers, can be formulated as
$\lim_{x\to \propto}\frac{\pi(x)}{x/ln(x)}= 1$
Hence, prime number theorem can be restated as $\pi(n)\sim\frac{\ (n)}{ln(n)}$
This means that $\frac{\ (n)}{ln(n)}$is a good approximation for $\pi(n)$
It could be restated in another way also as
$\pi {(n)\sim \int_{0}^{x}\frac{dt}{ln(t)} }$


This was discovered by Gauss and so its also called Gauss prime number theoem.

Chebyshev Prime Number Theorem

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To prove the prime number theorem, Chebyshev  introduced a new function
$\theta (x)=\sum_{p\leqslant x}ln(p)$  and $\psi  (x)=\sum_{p^{m}\leqslant x}ln(p)$
and hence  estimated that
$\lim_{x\to \propto }\frac{\theta (x)}{x}=1$
$\lim_{x\to \propto }\frac{\psi(x)}{x}=1

Prime Numbers- Fundamental Arithmetic Theorem

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Based on the prime numbers, there is a fundamental theorem of arithmetic that is satisfied by all prime numbers.
It states that any integer which is greater than 1 , can be expressed as a product of one or more primes numbers in a unique manner(except for prime).

By formula, if n is any natural number greater than 1, then n can be written
as
$n=p1*p2*p3…..pk$ where p1, p2…pk are the prime factors of n.
This is called the prime factorization of n
For example, 72 = 2*2*3*3
here 2 and 3 are prime numbers.
This can be written also as 2*3*3*2 or 3*2*3*2
So it can be seen that for 72, the prime factorization  is the same, even thought its order of arrangements are different
Hence,.
As all the numbers in algebra satisfies the commutative property, the fundamental theorem of arithmetic can be stated in another way:
Any natural number greater than 1 can be written in a unique form of a factorization of prime numbers, except for the order of factors.
For example lets consider the number 2552
It can be factored as $2534= 2*2*2*11*29$
Here 2, 11 and 29 are prime numbers.
Here also $2534$ can be written as 2*11*29*2*2 or 11*2*2*2*29.
In all these forms the prime factors used will be the same, even though the arrangement changes.