CHAPTER 11

Decisive solution to determining "k-th" prime number by determining function concerning the number of prime in a precise manner

## 11.1. Determination of "" in a precise manner

It has been proved that the number of prime numbers not more than "N", is exactly defined by the below relation:

(1)

If we consider "" as kth prime number, then it is obvious:

(2)

Therefore, If is equal to zero, then "N" will be equal to the kth prime number"". If expression be non-zero, "N" will not be equal to kth prime number. Now we consider the below expression:

Here, we define with the below method:

(3)

It has been proved that the kth prime number is smaller than:

(4)

Therefore"" (kth prime number) is computable with the below relation:

(5)

Therefore, (kth prime number) is defined as:

(6)

### 11.1.1. Example

We calculate"" with "" formula ():

## 11.2. Other formulas for determining ""in a precise manner

According to another formula for, another formula for "" can be presented as below (According to the primal conditions):

(7)

(8)

### 11.2.1. Remark

For calculating "" it is possible use the equation"":

()

Therefore, we define the below simple expression:

Here, we can define  using:

(9)

Therefore,  is computable by the below relation:

(10)

So we can use every relations (5), (8) and (10) for calculating"", but because of"", calculating "" may take too length and time. To solve this problem, we can calculate instead of. It means that a program can be written for calculating "".

According to the condition  and every mentioned relation for  and, we form the below table:

 3 5 7 9 11 13 15 17 19 21 23 … 2 3 4 5 5 6 7 7 8 9 9 … 3 5 7 0 11 13 0 17 19 0 23 …

and also:

 2 3 4 5 6 7 8 9 10 11 12 13 14 … 3 5 7 11 13 17 19 23 29 31 37 41 43 …