CHAPTER 11
Decisive solution to determining "k-th" prime
number by determining function concerning the number of prime 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 k^{th} prime number, then it is obvious:
_{} (2)
Therefore, If _{}is equal to zero, then "N" will be equal to the k^{th} prime number"_{}". If expression be non-zero, "N" will not be equal to k^{th }prime number. Now we consider the below expression:
_{}
Here, we define _{}with the below method:
_{} (3)
It has been proved that the k^{th }prime number is smaller than_{}:
_{} (4)
Therefore"_{}" (k^{th }prime number) is computable with the below relation_{}:
_{} (5)
Therefore, _{}(k^{th }prime number) is defined as_{}:
_{} (6)
We calculate"_{}" with "_{}" formula (_{}):
_{}
According to another formula for_{}, another formula for "_{}" can be presented as below (According to the primal conditions):
_{} (7)
_{} (8)
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 |
… |