CHAPTER 7
Decisive solution to the problem of finding the generator of prime numbers via discovering the surjective generating function of prime numbers
_{}
_{}
_{}
Here by using the identification function_{}, we can write onto generator function of prime numbers.
For this purpose, at first we consider _{} and "p" is an arbitrary prime number. We know that "N" is prime or composite.
It is enough to use identification function_{}for definition "N" in view of primality (be prime number or be composite):
_{}
It is obvious that _{}always have two cases, "0" or "1". It means that if "N" be composite,_{}is "0" and if it is prime _{}is "1".
Therefore, below function produces all of odd prime numbers_{}:
_{}
In fact Domain and Range of function "_{}" is:
(Domain)_{}, (Range)_{}
Now by a transferring, it is possible to obtain the function "P" with below formula:
_{}
If "p" is considered as an arbitrary prime number, according to function "_{}" and translation of axes to _{}
_{}
Domain and range of new function "P" is:
_{}
_{}
Domain and Range of this function is:
_{}and _{}
Therefore, the function with below formula produces just prime numbers and in fact this range function is set of prime numbers:
_{}
Substituting_{}, the formula of prime numbers or _{}onto function is:
_{}
According to primary condition, we form below table:
_{} |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
… |
_{} |
3 |
5 |
7 |
2 |
11 |
13 |
2 |
17 |
19 |
2 |
23 |
2 |
2 |
29 |
… |
Another function can be made equal with (*) function, for example:
_{} ; _{}
Domains and ranges of these functions are:
_{} , _{}
Because:
_{}
Therefore, functions _{}and _{}also are equal to the function "P". Here by using onto function "P" (generator of all prime numbers) or functions _{}and_{}, set of prime numbers can be defined below in forms:
_{}
If we consider "p" as an arbitrary prime number, then, formula of prime numbers can be considered in one of two below general cases:
_{}
7.2.2. "H.M" theorem
If _{}_{}and_{}then _{} is prime and_{}:
_{}