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 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)
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 |
… |