by Seyyed Mohammad Reza Hashemi Moosavi

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

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................................................ 12

1.2   What is number theory?........................................................... 14

1.3   Prime numbers......................................................................... 20

1.4   The fundamental theorem and some of its applications.......... 22

1.5   Sieve of Eratosthense.............................................................. 26

1.6   Periodic sieve for small numbers............................................. 27

1.7   The infinity of prime numbers................................................. 29

1.8   Functions,  and .............................................................. 30

1.9   Perfect numbers....................................................................... 33

1.10 Bertrand’s principle and theorems of Chebyshev, Dirichlet and Poisson…………………………………………………………….35

1.11 Lagrange’s theorem................................................................. 38

2. On the history of forming prime numbers tables and determining the smallest divisor of composite numbers                                                       42

2.1 Famous tables of prime numbers and divisors of composite numbers        43

2.2 Calculation of tables.................................................................. 44

2.3 Stochastic’s theorem.................................................................. 45

2.4 Another research on stochastic theorem.................................... 54

2.5 Table of divisors of Burkhard.................................................... 55

3. Decisive solution to the problem of forming of tables concerning divisors of composite numbers by regular loops in arithmetic progressions and successive cycles                                                                                           58

3.1 "H.M" matrix table (zero and one) for recognizing prime numbers and devisors of composite numbers ................................................................................................ 59

3.2 "H.M" Loop table for recognizing prime numbers and divisors of composite numbers      60

3.3 "H.M" Loop-cycle table for recognizing prime numbers and divisors of composite numbers         63

4. On the history of the problem of recognizing prime numbers by two sided theorem in particular Wilson’s theorem and its consequences      66

4.1 Wilson’s theorem ...................................................................... 67

4.2 Remarks concerning Wilson’s theorem and its converse and corollaries     69

4.3 Corollaries of Wilson’s theorem................................................ 70

4.4 Some of if theorems for recognizing prime numbers ................ 73

4.5 Factorization of composite numbers.......................................... 77

5. Decisive solution to the problem of recognizing prime numbers by a formula concerning recognizing of numbers ""                              79

5.1 Determination of the formula for the characteristic function of numbers    80

5.2 Formula for surjective characteristic function........................... 83

6. On the history of the problem of searching for finding generating function of prime numbers ""                                                             85

6.1 A summery of the history of "2000" years old attempts for finding a formula for prime numbers 86

6.2  Mills’s theorem......................................................................... 87

6.3  Kuipers’s theorem..................................................................... 87

6.4  Niven’s theorem........................................................................ 88

6.5  Formulas of generating prime numbers..................................... 88

6.6  Generalized Mills’s theorem..................................................... 92

6.7  Investigation into polynomials.................................................. 94

6.8  A formula presenting for generating prime numbers by Wilson’s theorem………………………………………………….97

7. Decisive solution to the problem of finding the generator of prime numbers via discovering the surjective generating function of prime numbers     99

7.1 Determination of the formula for the surjective generating function of prime numbers    100

7.2 Domain and range of the surjective generating function of prime numbers…………………………………………………....100

8. On the history of the problem of determining the number of prime  and its related functions "" and ""                                           103

8.1 An introduction to the function ""and "li"............................ 104

8.2 Prime numbers theorem........................................................... 107

8.3 The function "li" or "the logarithmic integral"......................... 109

8.4 Meissel’s formula for "".................................................... 110

9. Decisive solution to the problem of determining of the precise number of "" primes by characteristic function ""            113

9.1 Determining "" by ""............................................. 114

9.2 Comparing the precise formula for ""with Meissel’s formula 116

10. On the history of determining "k-th" prime number by bounds for "" (determining lower and upper bounds for "")                  118

10.1 Determining the bounds for "" (the "k-th" term of the sequence of prime numbers)  119

10.2 Bounds for "" from below and above ............................... 120

10.3 Bonse’s theorem.................................................................... 121

10.4 Theorems concerning consecutive prime numbers ................ 121

10.5 Theorems of Chebyshev........................................................ 122

10.6 Theorems of Ishikawa............................................................ 123

11. Decisive solution to determining "k-th" prime number by determining function concerning the number of prime in a precise manner           125

11.1 Determination of "" in a precise manner ........................... 126

11.2 Other formulas for determining "" in a precise manner..... 127

12. On the history of attempts for solving Riemann zeta equation  "" and the low of rarity of prime numbers                                130

12.1 Riemann's zeta function and its celebrated equation ""….           131

12.2 An introductory method for finding a fundamental formula  for ""         132

12.3 Statistical investigation into the fundamental formula for "" ...140

12.4 Separating intervals of prime numbers................................... 140

13. Decisive solution to Riemann zeta equation () by the determining function concerning the precise number of primes ()                   143

13.1 Riemann’s zeta function ().................................................. 144

13.2 Decisive solution to Riemann's zeta equation ()  Millennium prize problem……………………………………….145

14. On the history of searching for famous prime numbers and the factorizations of these numbers ()                                           147

14.1 Some of famous numbers...................................................... 148

14.2 Fermat’s numbers................................................................... 149

14.3 Special problems and Fermat’s numbers................................ 154

14.4 Another proof for Euclid’s theorem...................................... 157

14.5 Speed of the growth of Fermat’s numbers............................ 157

14.6 Fermat’s numbers and the problem of inscribing regular polygons inside a circle          158

14.7   Refutation of Fermat’s assertion and factorization of Fermat’s numbers………………………………………………...159

14.8   Mersenne’s numbers............................................................ 159

14.9   Problems concerning Mersenne’s numbers.......................... 160

14.10 Perfect, imperfect and redundant numbers......................... 161

14.11 Historical remarks concerning (even) perfect numbers and Mersenne’s numbers……………………………………………..164

14.12 Role of computers in searching large prime numbers.......... 166

14.13 Odd perfect numbers........................................................... 168

14.14 Special problems concerning perfect numbers..................... 168

14.15 Problems on distinguishing Mersenne’s prime numbers and Fermat’s numbers       170

14.16 Problems concerning Fermat (), Mersenne (), perfect and redundant numbers     174

15. Definition of the sets of Fermat, Mersenne, perfect prime numbers by the prime numbers formula                                                        177

15.1 Some general facts concerning Fermat’s numbers ()......... 178

15.2 Definition of the set of Fermat’s prime numbers by the prime numbers formula       178

15.3 Some general facts about Mersenne’s numbers and even perfect numbers and the relation between them.............................................................................................. 179

15.4 Definition of the sets of Mersenne, even perfect prime numbers by the prime numbers formula   179

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

conjectures                                                                                         181

16.1 Goldbach and Hardy’s conjectures....................................... 182

16.2 Goldbach's conjecture and other open problems related to it 182

16.3 Some unsolved problems and other conjectures concerning  prime numbers        184

16.4 Applied investigations into Goldbach and Hardy conjectures 186

16.5 Theoretical investigation into Goldbach conjecture.............. 186

17. On the history of attempts for proving the conjecture of existence of infinity many twin prime numbers.........................188

17.1 Twin prime numbers ............................................................. 189

17.2 Clement’s theorem ................................................................ 190

17.3 Approaching to the solution of problem of infinity many twin prime numbers         190

17.4 The distances of prime numbers............................................ 190

17.5 Definitions and notes……………………………………………....191

17.6 Problems concerning twin prime numbers............................. 193

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             194

18.1 Generation of twin prime numbers ....................................... 195

18.2 There is infinity many twin prime numbers........................... 199

18.3 Set of prime twin pairs…………………………………………….201

19. On the history of attempts for proving Fermat’s last theorem and the fundamental role of prime numbers (regular) and its properties

leading to solving Diophantine equation                             202

19.1 Diophantine equations........................................................... 203

19.2 An introduction to the chronology of Fermat’s last theorem 203

19.3 Chronology of Fermat’s last theorem.................................... 204

19.4 Sophie Germain’s theorem………………………………………...218

19.5 Fermat’s last theorem for exponent "4"................................. 221

19.6 Fermat’s last theorem for exponent "3"................................. 225

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 equations
229

20.1 Investigation into extension Fermat’s last theorem 230

20.2. Primitive, Algebraic and geometric methods........................ 231

20.3 An indirect proof of Fermat’s last theorem (elliptic curves).. 234

20.4 Taniyama- Shimura – Weil conjecture and Fermat's last theorem 250

20.5 Frey-Serre-Ribet theorem…………………………………………251

20.6 Wiles and Taylor-Wiles theorem …………………………………251

20.7 Latest achievements and fundamental results concerning Fermat’s last theorem and its extension (H.M).................................................................................... 252

20.8 Reducibility law (H.M).......................................................... 256

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

……………………………………...…258

20.10 Solving Diophantine equations having non-similar exponents (multi-equalities) (H.M)…….....………………………….……..267

20.11 Finding an answer for extension of Fermat’s last theorem using the theorems related to prime numbers (H.M) …...……….269

20.12 Determining a general answer for equation (H.M)………………270

,

20.13 Determining a general answer for equation (H.M)………………272

20.14 Determining a general answer for equation (H.M)………………273

(the Fermat-catalan and the Beal conjectures "")

20.15 Determining a general series of answer for Diophantine equations with arbitrary degree by using Wilson’s, Fermat’s and Euler’s theorems and

the role of prime numbers formula in arising of Algebraic identities (H.M).................................................................... 275

21. The newest of methods of solving and calculating

Appendixes (I)                                                                                     287

21.1 Solving congruence and Diophantine equations by  "H.M" table ()……………………………………………...288

21.2 Solving linear Diophantine equations by "H.M" table

……………………………………………..294

21.3 Solving "n-th" degree Diophantine equation by "H.M" table ……………………………………………………..298

21.4 A new and fast method for calculating of determinant ("H.M" method)................................................................... 302

21.5 Definition of regular and irregular prime numbers by "H.M" determinant.          309

21.6 New method of calculation of sum of "k-th" power of the first "n" natural numbers by "H.M" determinant (expressing "" by a determinant)……………………………………………………...314

21.7 Determining the number of roots of perfect cubic degree equation directly by "H.M" method    316

21.8 Proof of a new and applied "H.M" theorem (Concerning the factorization of composite numbers)        318

22. The abstract of formulas and their software programs

Appendixes (II)                                                                                  320

22.1 The abstract of the distinction formula of the prime numbers. 321

22.2 The distinction program of the prime numbers............ 322

22.3 The abstract of the formula of the prime numbers generator           323

22.4. The final formula of the prime numbers generator...... 323

22.5 The program of the prime numbers generator............... 325

22.6 The abstract of the formula of the determining of the "k-th" prime number      ……………………………………………...327

22.7 A program for determining the prime number "k-th"………...328

22.8 The abstract of Riemann’s zeta equation solution .. 329

22.9 A Program for determining of the number of the prime numbers smaller than or equal any arbitrary number "p" exactly............................................... 330

22.10 The abstract of the definition of the prime numbers set by using  the surjective generating function of the prime numbers (IP)..................................................... 331

22.11 A program for defining the prime numbers set (IP)............ 332

22.12 The abstract of the definition of the Mersenne’s prime numbers set by using the prime numbers generator...333

22.13 A program for determining the Mersenne’s prime numbers of M-digits (M: Arbitrary number)            334

22.14 The New Mersenne’s prime number as "42nd" known Mersenne prime found (February 2005)           337

22.15 The determining of generating function of the prime numbers greater than the greatest prime number (by prime numbers formula)........................................................ 338

You can freely download prime numbers formula software. Open source code of Seyyed Alireza Hashemi Moosavi's program is in follow that is based on H.M formula of prime numbers that was discovered in the year of 2003  Prof. Seyyed Mohammad Reza Hashemi Moosavi.

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

The presented "H.M"  functions by Discoverer (Prof. Seyyed Mohammad Reza Hashemi Moosavi) are 6 numbers that 4 functions are by Wilson's theorem and one of them by Euler's function( ) and one of them by functions and functions are discovered by discoverer (Prof.S.M.R.Hashemi Moosavi). The software of this function is produced.

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

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

 More explain about discovery of prime numbers formula

Infinity proof of prime numbers was propound 300 years before the Christian era by Euclid and since that time great mathematician like Euler try to discover a formula for production of prime numbers.

Euler could define a quadratic function which give prime numbers for forty prime number which are uninterrupted and also Fermat presented a formula to obtain prime numbers and later, it breached by Euler for n = 5.

Many of other mathematicians achieved to violating and especially formula and finally they found that discovery of prime numbers formula is impossible and this problem will be unsolvable.

In fact this discovery means that one of complicated and unsolvable mathematics problem was solved and this discovery give this fact to man that earthy human can solve other unsolvable problems with research and effort.

I have worked about this problem around 20 years and I found this fact that I can't comeback from this path which I came and I promised that search about this un solvable problem till end of my life even if I couldn't achieve the final solution.

Of course my research had result with patronage under god and trust in god and I discovered the formula of prime numbers.

 How is the usage of this formula in mathematics  and other sciences?

A result for discovery of prime numbers formula is the solution of Riemann Zeta equation which is on of seven universal unsolvable problems in mathematics millennium and it's solution need to gain the number determination equation of prime number for any desirable number n carefully (with prime numbers formula).

Another result is determination of Kth desirable prime number and other usages are definition of prime number set, proof of infinity of prime twin pairs, considering of the guesses of Goldbuch and Hardy, gaining the generator formula of Mersenne prime numbers and also very unknown and big prime numbers and other problems related to prime numbers.

But the basic and cardinal usage of this formula is in coding and decoding that usually use from very big prime numbers for this and before it is necessary to gain them with complicated mathematics methods. But with presenting of this formula, definition of coding and decoding system became easy and I invent a system for coding with this formula that I presented this system inventions registration organization.

After Euclid’s 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 officially announced that such a formula can’t be found and in follow to prove their wrong idea they started to publish some Algebraic theorems in their books.

Meanwhile, Niven and Mills in relation to prime numbers function proved the above theorem. But their parameters have never been determined.

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 there is 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.

• Discovery of prime numbers formula by Prof. Seyyed Mohammadreza Hashemi Moosavi caused so many results in basic sciences that we will mention a little part 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 Goldbuch 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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