Prime Numbers Formula Discovery

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Prime numbers formula was discovered in 5th August 2003 by Seyyed Mohammad Reza Hashemi Moosavi English language                                                                                      Persian Language

   

 

 

 

 

 

 

 

 

 

 

 

 

Table of contents of the book

Table of Contents : Book of the discovery of prime numbers formula and its results

BySeyyed Mohammad Reza Hashemi Moosavi

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Now you can scan the QR picture to use Instagram for reading the book of The discovery of prime numbers formula and its results

@hashemi.moosavi.2023

The book of the discovery of prime numbers formula and its results

Contents

 

    Preface of author

    1. A brief of view of number theory

1.1   Number theory in ancient time........................................................

1.2   What is number theory?.................................................................

1.3   Prime numbers...............................................................................

1.4   The fundamental theorem and some of its applications.....................

1.5   Sieve of Eratosthense.....................................................................

1.6   Periodic sieve for small numbers.....................................................

1.7   The infinity of prime numbers..........................................................

1.8   Functions,  and ....................................................................

1.9    Perfect numbers............................................................................

1.10 Bertrandís principle and theorems of Chebyshev, Dirichlet and Poisson

1.11 Lagrangeís theorem.......................................................................

 

    2. On the history of forming prime numbers tables and determining the smallest divisor of composite numbers

2.1 Famous tables of prime numbers and divisors of composite numbers.

2.2 Calculation of tables.........................................................................

2.3 Stochasticís theorem........................................................................

2.4 Another research on stochastic theorem...........................................

2.5 Tables of divisors.............................................................................

2.6 Burkhardís tables............................................................................

 

  3. Decisive solution to the problem of forming of tables concerning divisors of composite numbers by regular loops in arithmetic progressions and successive cycles

3.1 "H.M" Matrix table (zero and one) for recognizing prime numbers and devisors of composite numbers           

3.2 "H.M" Loop table for recognizing prime numbers and divisors of composite numbers 

3.3 "H.M" Loop-cycle table for recognizing prime numbers and divisors of composite numbers    

 

  4. On the history of the problem of recognizing prime numbers by two sided theorem in particular Wilsonís theorem and its consequences

4.1 Wilsonís theorem ............................................................................

4.2 Remarks concerning Wilsonís theorem and its converse and corollaries         

4.3 Corollaries of Wilsonís theorem.......................................................

4.4 Some of if theorems for recognizing prime numbers ..........................

4.5 Factorization of composite numbers.................................................

 

  5. Decisive solution to the problem of recognizing prime numbers by a formula concerning recognizing of numbers ""

5.1 Determination of the formula for the characteristic function of numbers           

5.2 Formula for surjective characteristic function.....................................

 

  6. On the history of the problem of searching for finding generating function of prime numbers ""

6.1 A summery of the history of 2000 years old attempts for finding a formula for prime numbers 

6.2  Millsís theorem...............................................................................

6.3  Kuiperís theorem............................................................................

6.4  Nivenís theorem.............................................................................

6.5  Formulas generating prime numbers.................................................

6.6  Generalized Mills theorem...............................................................

6.7  Investigation into polynomials..........................................................

6.8 A formula presenting for generating of prime numbers by Wilsonís theorem    

 

  7. Decisive solution to the problem of finding the generator of prime numbers via discovering the surjective generating function of prime numbers

7.1 Determination of the formula for the surjective generating function of prime numbers   

7.2 Domain and range of the surjective generating function of prime numbers

 

  8. On the history of the problem of determining the number of prime and its related functions "" and ""

8.1 An introduction to the function ""and "li"........................................

8.2 Prime numbers theorem...................................................................

8.3 The function "li" or "the logarithmic integral"......................................

8.4 Meisselís formula for "".............................................................

 

  9. Decisive solution to the problem of determining of the precise number of  "" primes by characteristic function ""

9.1 Determining "" by ""........................................................

9.2 Comparing the precise formula for ""with Meisselís formula......

  10. On the history of determining "k-th" prime number by bounds for "" (determining lower and upper bounds for "")

10.1 Determining the bounds for "" (the k-th term of the sequence of prime numbers)   

10.2 Bounds for "" from below and above .........................................

10.3 Bonseís theorem............................................................................

10.4 Theorems concerning consecutive prime numbers ..........................

10.5 Theorems of Chebyshev................................................................

10.6 Theorems of Ishikavea...................................................................

 

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

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

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

 

  12. On the history of attempts for solving Riemann zeta equation and the low of rarity of prime numbers

12.1 Riemann zeta function and its celebrated equation ""............

12.2 An introductory method for finding a fundamental formula for ""

12.3 Statistical investigation into the fundamental formula for ""........

12.4 Separating intervals of prime numbers.............................................

 

  13. Decisive solution to Riemann zeta equation () by the determining function concerning the precise number of primes ()

13.1 Riemann zeta function ()..............................................................

13.2 Decisive solution to Riemann zeta equation ()......................

 

  14. On the history of searching for famous prime numbers and the factorizations of these numbers ()

14.1 Some of famous numbers...............................................................

14.2 Fermatís numbers..........................................................................

14.3 Special problems and Fermatís numbers........................................

14.4 Another proof for Euclidís theorem................................................

14.5 Speed of the growth of Fermatís numbers......................................

14.6 Fermatís numbers and the problem of inscribing regular polygons inside a circle       

14.7   Refutation of Fermatís assertion and factorization of Fermatís numbers

14.8   Mersenneís numbers....................................................................

14.9   Problems concerning Mersenneís numbers...................................

14.10 Perfect, imperfect and redundant numbers....................................

14.11 Historical remarks concerning (even) perfect numbers and Mersenneís numbers    

14.12 Role of computers in searching large prime numbers.....................

14.13 Odd perfect numbers...................................................................

14.14 Special problems concerning perfect numbers...............................

14.15 Problems on distinguishing Mersenneís prime numbers and Fermatís   numbers      

14.16 Problems concerning Fermat (), Mersenne (), perfect and  redundant numbers         

 

  15. Definition of the sets of Fermat, Mersenne, perfect prime numbers by the prime numbers formula

15.1 Some general facts concerning Fermatís numbers ()...................

15.2 Definition of the set of Fermatís prime numbers by the prime numberís formula        

15.3 Some general facts about Mersenneís numbers and even perfect numbers and the relation between them      

15.4 Definition of the sets of Mersenne, even perfect prime numbers by the prime numbers formula          

 

  16. On the history of attempts for proving Goldbach and Hardy conjectures

16.1 Goldbach and Hardy conjectures...................................................

16.2 Goldbach conjecture and other open problems related to it.............

16.3 Some unsolved problems and other conjectures concerning prime numbers  

16.4 Applied investigations into Goldbach and Hardy conjectures...........

16.5 Theoretical investigation into Goldbach conjecture..........................

 

  17. On the history of attempts for proving the conjecture of existence of infinity many twin prime numbers

17.1 Twin prime numbers .....................................................................

17.2 Clementís theorem ........................................................................

17.3 Approaching to the solution of infinity many twin prime numbers .....

17.4 The distances of prime numbers.....................................................

17.5 Problems concerning twin prime numbers.......................................

 

  18. Decisive solution to the problem of infinity many twin prime numbers and method of generating them and definition of twin prime numbers set by twin prime numbers formula  

18.1 Generation of twin prime numbers .................................................

18.2 There is infinity many twin prime numbers.......................................

 

  19. On the history of attempts for proving Fermatís last theorem and the fundamental role of prime numbers (regular) and its properties leading to solving Diophantine equation  

19.1 Diophantine equations....................................................................

19.2. An introduction to the Chronology of Fermatís theorem.................

19.3. Chronology of Fermatís theorem...................................................

19.4. Fermatís theorem, for exponent 4.................................................

19.5. Fermatís theorem, for exponent 3.................................................

 

  20. Fundamental role of prime numbers and its properties in a complete investigation into Diophantine equations in the sense of existence or
non-existent solution and presenting a general solution for the Diophantine equation

20.1 Investigation into extension Fermatís theorem ...........

20.2. Primitive, Algebraic and geometric methods...................................

20.3 An indirect proof of Fermatís theorem (elliptic curves)....................

20.4 Taniyama- Shimura Ė Weil conjecture and Fermat last theorem.......

20.5 Theorems of Wiles and Taylor-Wiles ............................................

20.6 Latest achievements and fundamental results concerning Fermatís last theorem and its extension (H.M)        

20.7 Reducibility law (H.M)...................................................................

20.8 Studying Diophantine equation of n-th order (similar exponents) (H.M)

              

20.9 Solving Diophantine equations having non-similar exponents (multi-equalities) (H.M)

20.10 Finding an answer for extension of Fermatís last theorem using the theorems related to prime numbers        

20.11 Determining an general answer for equation

20.12 Determining an general answer for equation

20.13 Determining an general answer for equation         

20.14 Determining an general answer for equation (H.M).......................

20.15.Determining an general answer for equation (H.M) ......................

20.16 Determining an general answer ....................................................

 

  21. The newest of methods of solving and calculation

Appendixes (I)

21.1 Solving congruence and Diophantine equations by "H.M" table ()

21.2 Solving Diophantine equation of order in by "H.M" table

21.3 A new and fast method for calculating determinant ("H.M" method)    

21.4 Definition of regular and ir-regular prime numbers by "H.M" determinant.     

21.5 New method of calculation of sum of  "k-th" power of the first "n" natural numbers by "H.M" determinant (Expressing "" by a determinant)

21.6 Determining the number of roots of perfect cubic degree equation directly by "H.M" method

21.7 Proof of a new and applied "H.M" theorem (Concerning the factorization of composite numbers)     

 

  22. The abstract of formulas and their software programs source code

Appendixes (II)

22.1 The abstract of the formula of the distinction function of the prime numbers.     

22.2 The program for distinction of the prime numbers..................

22.3 The abstract of the formula of the prime numbers generator ...

22.4. The final formula of the prime numbers generator..................

22.5 The program of the prime numbers generator.........................

22.6 The abstract of the formula of the determining of the "k-th" prime number          

22.7 The program for determining of prime number "k-th"................

22.8 The abstract of  solution Riemannís Zeta equation ............

22.9 The program for determining of the number of the prime numbers smaller than or equal any arbitrary number "p" exactly...........................................................................

22.10 The abstract of the definition of the prime numbers set by using the surjective generating function of the prime numbers (IP)................................................................................

22.11 The program for the definition of the prime numbers set.

22.12 The abstract of the definition of the Mersenneís prime numbers set by using the prime numbers generator.

22.13 The program for the determining of the Mersenneís prime numbers of M-digits (M: Arbitrary number)       

22.14 The New Mersenneís prime number as "42nd" known Mersenne prime found (February 2005)     

22.15 The determining of generating function of the prime numbers greater than the greatest prime number (by prime numbers formula)..........................................................................

 

l References on some historical parts of the book

 

 

 

 

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Product details

  • Publisher ‏ : ‎ Meraje Ghalam Publications (July 16, 2016)
  • Publication date ‏ : ‎ July 16, 2016
  • Language ‏ : ‎ English
  • Print Length ‏ : ‎ 344 pages
  • ISBN-10 ‏ : ‎ 6009446791
  • ISBN-13 ‏ : ‎ 9786009446797
  • Item Weight ‏ : ‎ 488 grams
  • Best Sellers Rank: #390631 www.primenumbersformula.com & www.komhm.com
  • Price: 121.00 $
  • Password of PDF file:  www.primenumbersformula.com

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                      Alireza Magic Square

          

Magic squares of Seyyed Alireza Hashemi Moosavi publish date: 2021/13/07

                   

 

 

Magic square of 3 by 3 by Seyyed Alireza Hashemi Moosavi

                   

Magic square of 4 by 4 by Seyyed Alireza Hashemi Moosavi

                   

Magic square of 5 by 5 by Seyyed Alireza Hashemi Moosavi

                  

Magic square of 6 by 6 by Seyyed Alireza Hashemi Moosavi

                    

Magic square of 7 by 7 by Seyyed Alireza Hashemi Moosavi

                   

Magic square of 8 by 8 by Seyyed Alireza Hashemi Moosavi

                   

Magic square of 9 by 9 by Seyyed Alireza Hashemi Moosavi

                    

Magic square of 10 by 10 by Seyyed Alireza Hashemi Moosavi

                     

Magic square of 11 by 11 by Seyyed Alireza Hashemi Moosavi

                      

Magic square of 12 by 12 by Seyyed Alireza Hashemi Moosavi

                       

Magic square of 13 by 13 by Seyyed Alireza Hashemi Moosavi

                  

Magic square of 14 by 14 by Seyyed Alireza Hashemi Moosavi

                     

Magic square of 15 by 15 by Seyyed Alireza Hashemi Moosavi

                     

Now let us check the prime numbers formula's in Alireza magic squares

: (you can obviously find the twin prime numbers and prime twin couples)

 

 

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                                                             You can check the address to find the list below

http://www.topmillion.net/domain-list-333 \

 

 

 Alexa ranking of top million list of 1,000,000 Websites of the World that www.primenumbersformula.com in the year of 2022 could

achieve the rank of #390631 in the top one million websites of the world.

                  

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You can  Freely download  prime numbers formula program  by Seyyed Alireza Hashemi Moosavi.

Using prime numbers formula software by Alireza is the best and quickest way to check numbers and find its prime or composite.

 

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                          The abstract of formulas and their software programs

 

†††† (Domain function) (Natural numbers set)

(Range function) †(Prime numbers set)

 

 

Contents:

Ÿ The Abstract of the formula of the prime numbers generator††

Ÿ The Program of the prime numbers generator

Ÿ The program for distinction of the prime numbers

Ÿ The abstract of the formula of the determining of the "k-th" prime number

Ÿ The Program for determining of prime number "k-th"

Ÿ The Abstract of Riemannís Zeta equation solution

 

Ÿ The Program for determining of the number of the prime numbers smaller than or equal any arbitrary number "p" exactly

 

Ÿ The abstract of the definition of the prime numbers set by using the surjective generating function of the prime numbers (IP).

٠ The program for the definition of the prime numbers set.

 

Ÿ The abstract of the definition of the Mersenneís prime numbers set by using the prime numbers generator.

Ÿ The program for the determining of the Mersenneís prime numbers of M-digits by using prime numbers formula†††††††††††††††††††††† (M: Arbitrary number).

Ÿ The New Mersenneís prime number as "42nd" known Mersenne prime found (February 2005).

Ÿ Determining of generating function of the prime numbers greater than the greatest prime number (by prime numbers formula)

Ÿ Determining of generating function of the prime numbers greater than the greatest prime number (by prime numbers formula)

  • . The abstract of the distinction formula

of the prime numbers

††† For Example 1:

†††††

 

n

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

...

...

3

5

7

9

11

13

15

17

19

21

23

25

27

29

31

...

...

1

1

1

0

1

1

0

1

1

0

1

0

0

1

1

...

1

...

  • . Testing the program o f"" for distinction of ""†

  • . The distinction program of the prime numbers

This program"" is for testing the formula.The purpose of this program is distinction of prime numbers. With running this program when the program require to enter a number, then enter a number, and press the enter key, so the program will tell you that this number is prime or not.

 

 

  • . The abstract of the formula of the prime numbers generator

  • . Example

 

n

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

...

3

5

7

p

11

13

p

17

19

p

23

p

p

29

31

...

  • . The final formula of the prime numbers generator

 

(Domain function)† †††††† (Natural numbers set)

(Range function) ††††††††† †(Prime numbers set)

  • . The definition of the prime numbers set by onto generating function

  • . The running of Eulerís special prime numbers formula[2]:††

†††

 

 

 

 

  • . The program of the prime numbers generator ()

  • . Example

  • . Program"":

†This program is for testing the relation †the purpose of this program is generating of prime numbers. After running this program at first the below message will appear:

"Enter Down Limit"

and then message

"Enter Up Limit"

In answer to this message we should enter the down limit and up limit of natural-subset that we want to enter the numbers of "1" up to "1000,000" after pressing the enter key. All of the prime numbers between "1" and "1000, 000" will be saved as a file to the name of [4]

  • . The abstract of the formula of the determining of the "k-th" prime number ():

 

  • . Example:

 

3

5

7

9

11

13

15

17

19

21

23

...

k

2

3

4

5

5

6

7

7

8

9

9

...

3

5

7

0

11

13

0

17

19

0

23

...

 

  • . Example:

 

††††††††††††††††††††††† ††††††††

 

k

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

...


 

  • . A program for determining the prime number "k-th" ()

This program ""is for testing the formula "" from part (22.6.1). The purpose of this program is calculating the prime number "k-th".

 

 


  • . The abstract of Riemann's zeta equation solution

  • . Riemann's Zeta Function

 

  • . Riemann's Zeta equation""

 

 

†The determining of the number of the prime numbers less than or equal any arbitrary number "p" exactly:

  • . Example

 

 

P

3

5

7

11

13

17

19

23

29

31

...

2

3

4

5

6

7

8

9

10

11

...

  • . A program for determining of the number of the prime numbers smaller than or equal any arbitrary number "p" exactly

This program () is for testing part (22.8.3).

The purpose of this program is determining the number of the prime numbers before. For example, if you enter a prime number like, the program will give you number "6" in answer that shows number †is †prime number.

  • . The abstract of the definition of the prime numbers set by using the surjective generating function of the prime numbers ()

 

  • . Example

 

 

P=2:

n

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

...

3

5

7

2

11

13

2

17

19

2

23

2

2

29

31

2

2

...

 


  • . A program for defining the prime numbers set ()

This program () is for testing (22.10). The purpose of this program is determining the prime number set. In below form:

 

 

 

 

 

 

  • . The abstract of the definition of the Mersenne's prime numbers set by using the prime numbers generator

 

 

  • . The new Mersenneís prime number as "42nd" known Mersenne prime found (February 2005)

On February 18, 2005, Dr. Martin Nowak from Germany, found the new largest known prime number,. The prime number has 7,816,230 digits! It took more than 50 days of calculations on Dr. Nowak's 2.4 GHz Pentium 4 computer. The new prime was independently verified in 5 days by Tony Reix of Grenoble, France using a 16 Itanium CPU Bull NovaScale 5000 HPC running the Glucas program by Guillermo Ballester Valor of Granada, Spain. A second verification was completed by Jeff Gilchrist of Elytra Enterprises Inc. In Ottawa, Canada using 15 days of time on 12 CPUs of a Compaq Alpha GS160 1.2 GHz CPU server at SHARCNET.

Dr. Martin Nowak, an eye surgeon in Michelfeld, Germany learned of GIMPS in April 1999 when he red an article his newspaper, the "Frankfurter Allgemeine Zeitung". Dr. Nowak, a math hobbyist, started with one PC and as his practice grew so did his participation in GIMPS. Six years later, he has 24 computers doing calculations for GIMPSóand one Mersenne Prime to his credit!

Perfectly Scientific, Dr. Crandall's company which developed the FFT algorithm used by GIMPS, makes a poster you can order containing the entire number. It is kind of pricey because accurately printing an over-sized poster in 1-point font is not easy! Makes a cool present for the serious math nut in your family.

Dr. Nowak could not have made this discovery alone. In recognition of contributions made by tens of thousands GIMPS volunteers, credit for this new discovery will go to "Nowak, Woltman, Kurowski, et al". The discovery is the eighth record prime for the GIMPS project. Join now and you could find the next record-breaking prime! You could even win some cash.

For more information on this latest prime discovery read the full press release.

  • . The determining of generating function of the prime numbers greater than the greatest prime number (by prime numbers formula)

For determining the generating function of an unknown large enough prime number on other word the generating function of the numbers greater than the greatest prime number which is the "42nd" known Mersenne prime namely "" is defined as follows:

†††††† (1)

 

Therefore, the following functions produce either, which is the greatest known prime number or the prime numbers after this number:

 

The function in (1) is introduced as the generating function of the unknown large enough prime numbers.

In order to find a prime number large enough by using a computer, there are a lot of algorithms. Each of these algorithms because of its digital inherence is not able to distinguish the prime numbers after a specific large number. So, we should define a function which is able to produce the numbers greater than the greatest known prime number. The following function is a kind of these functions:

††† (2)


Prime Numbers formula 8 part of source code of the programs of  above mentioned article of

The discovery of prime numbers formula and its results that for the first time in the world

 we are presenting the source code of the programs by Seyyed Alireza Hashemi Moosavi.

So we are beginning to represent the source code of programs as it is told in the Article of

The Discovery of on-to generating function of the prime numbers and its results.

The 2300 years old unsolvable problem is now solved forever.

Discoverer: Prof.Seyyed Mohammadreza Hashemi Moosavi

 

1.The distinction program of the prime numbers and the source code of the

program by Seyyed Alireza Hashemi Moosavi based on the Article by

Prof.Seyyed Mohammadreza Hashemi Moosavi.

 

This program"" is for testing the formula.The purpose of this program is distinction of

prime numbers. With running this program when the program require to enter a number, then

 enter a number, and press the enter key, so the program will tell you that this number is prime or not.

 

2. The program of the prime numbers generator ()and the source code of the program

by Seyyed Alireza Hashemi Moosavi based on the Article

by Prof.Seyyed Mohammadreza Hashemi Moosavi.

 

 

3.The definition of the prime numbers set by using the on-to generating function of the prime umbers

source code of the program by Seyyed Alireza Hashemi Moosavi based on the Article

by Prof.Seyyed Mohammadreza Hashemi Moosavi.

 

 

4.The definition of the Mersenne prime numbers set by using the prime numbers generator

source code of the program by Seyyed Alireza Hashemi Moosavi based on the Article

by Prof.Seyyed Mohammadreza Hashemi Moosavi.

 

 

 

5. The determining of the k-th prime number source code of the program

by Seyyed Alireza Hashemi Moosavi based on the Article

by Prof.Seyyed Mohammadreza Hashemi Moosavi.

 

 

6. Solution to the Riemann Zeta equation and determining of the numbers of the prime numbers

less than or equal to any arbitrary number p exactly for p>2 source code of the program

by Seyyed Alireza Hashemi Moosavi based on the Article

by Prof.Seyyed Mohammadreza Hashemi Moosavi.

 

7. The examination of the guesses of Goldbuch and hardy source cod of the program

by Seyyed Alireza Hashemi Moosavi based on the Article

by Prof.Seyyed Mohammadreza Hashemi Moosavi.

8.The proof of being infinite number of the prime twin couples source code of the program

by Seyyed Alireza Hashemi Moosavi based on the Article

by Prof.Seyyed Mohammadreza Hashemi Moosavi.

 

                              

This Program is for determining the Mersenneís prime numbers of M-digits (M: Arbitrary number)

Note. This program designed for determining of the power of new Mersenneís prime number (Power "p" in by using of previous Mersenneís prime numbers power). Here, look at the part of program and running its results. It dose contain "25" pages and about "500" lines.

 source code by Seyyed Alireza Hashemi Moosavi.

 

 

 Click here to freely download prime numbers formula program. Open source code by Seyyed Alireza Hashemi Moosavi based on H.M formula. Prime numbers formula was discovered by Professor Seyyed Mohammadreza Hashemi Moosavi on 5th August 2003 .

www.primenumbersformula.com

               

       

     Note : You are just permitted to use the subject with mentioning the reference address of the 

Site. All rights reserved to web designer Seyyed Alireza Hashemi Moosavi & www.primenumbersformula.com

 

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Note : Prime numbers formula is one of the on-to generating functions for the prime numbers that for every natural number of "m" it generates all the prime numbers in order (3,5,7,2,11,13,2,17,19, ...).

The presented "H.M"  functions by discoverer (Prof. Seyyed Mohammad Reza Hashemi Moosavi) are six number  that four functions are by Wilson's theorem and one of them by Euler's function( ) and another one by sigma( ) functions and bracket ( ) functions are discovered by discoverer and the software of this function is lately produced.

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Note :  In Year of 2007 (AAAS) " National Association Of Academies Of Science " (USA) Awarded A++ grade (Equals to excellent) to the prime numbers formula and its results by Prof S.M.R Hashemi Moosavi.  

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Note :  The discovery of prime numbers formula and its results has been published under an article in journal of "Roshd of Borhan "  associated with the Ministry of Education in Iran.

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 More explain about discovery of prime numbers formula

 Infinity proof of prime numbers being was propound 300 years B.C. by Euclid and since that time great mathematicians like Euler tried to discover a formula to produce prime numbers. Many of other mathematicians after many other researches finally found that discovery of prime numbers formula is impossible and this problem will be unsolvable. This discovery shows that one of the complicated and unsolvable problems of mathematics was solved and this discovery proves that there could be no unsolvable problem. I spent twenty years of my life researching and I found this fact that I can't comeback from this path which I came through and I promised myself to keep on searching for the rest of my life to find the solution even if I couldn't achieve the proof.

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How to use this formula in mathematics and other sciences?

 One of the results of discovering the prime numbers formula is to achieve the solution of Riemann Zeta equation which is on of the popular universal unsolvable problems in mathematics and it's solution needs to achieve the number of the prime numbers for any desirable number of  N carefully (with prime numbers formula). Another result is to determine k-th desirable prime number and other uses are definition of prime numbers set, proof of infinity of twin prime pairs considering  to the conjectures of Gold Bach and Hardy, find the generator formula for Mersenne prime numbers and also very unknown and big prime numbers and other problems related to prime numbers. In fact the main use of this formula is in coding and decoding that usually use from very big prime numbers for this reason and in the past it was necessary to gain them with complicated mathematics methods and had a hard way to pass through But with presenting of prime numbers formula, definition of coding and decoding systems became easy and convenient.

                               

After Euclidís theory about infinite prime numbers in 300 B.C. Most of the mathematicians and other researchers have been curious to find a formula which could generates prime numbers. After many years later some mathematicians like Euler and Fermat presented some formulas to generate prime numbers limitedly. Great mathematicians like Hardy and Courant and many other researchers finally officially announced that such a formula canít be found and to prove their wrong idea they started to publish some Algebraic theorems in their books. Furthermore, determining the number of prime numbers was very important problem. So Gauss and other mathematicians started to set some tables for them. We knew that so far there was no exact formula to determine the number of prime numbers exactly. This problem is known as Zeta Riemann equation which was one of the seven known unsolvable problems of the world that after my discovery on 5th August 2003, one of them is no more unsolvable with the prime numbers formula accurately you can absolutely generate all prime numbers to the nth one. Its consequent generate of prime numbers formula resulted in defining the set of prime numbers and so many other unbelievable results until now like breaking the code of RSA and AES by the use of prime numbers formula and other sets like Mersenne prime, perfect numbers and so many important sets and results just related to the field of number theory and basic sciences.

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    Discovery of prime numbers formula by Prof. Seyyed Mohammadreza Hashemi Moosavi caused so many results in basic sciences that we will mention part of it in follow:

  • 1.         Distinction of prime numbers.

  • 2.       Defining a formula for generating prime numbers.

  • 3.       Definition of prime numbers set by using the generating function of prime numbers.

  • 4.       Defining a formula to generate the Mersenne prime numbers.

  • 5.       Determination of Nth prime number.

  • 6.       Solving Riemann Zeta equation by using the determination of the number of prime number less than or equal to arbitrary number N exactly.

  • 7.       The proof of guesses of Gold Buch and Hardy.

  • 8.       The proof of infinity of the prime twin couples.

  • 9.       Determining a general series of answer for Diophantine equations.

  • 10.   This formula has so many unknown applications in Cryptography, generating Titan Mersenne prime numbers and other sciences like solving NP.

 

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