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 Missal'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 Missal'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 Bonze'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 Gold Bach and Hardy’s conjectures......................................................................................................................................................…….……...182

16.2 Gold Bach'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 German'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" ...........................................................................................   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