Prime number theorem

Statement of the prime number theory:

Let π(x) be the prime - counting function that gives the cardinal of primes which is less than or equal to x, for any absolute cardinal x. For example, π(10) = 4 because 2, 3, 5 and 7 are prime numbers which are less than or equal to 10.

According to prime number theorem,the limit of the quotient of the two functions π(x) and x / ln(x) as x approaches infinity is 1, which is expressed by the formula:

Using asymptotic notation the above formula can be written as

 

The theorem states that x/ln(x) approximates π(x) which means that the relative error of this approximation approaches 0 as x approaches infinity.

See the link for Prime and composite numbers

An example of the problem on prime number theory is given below: 

 

Solution: