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 …