Index

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 rograms

Appendixes (II)

22.1 The abstract of the formula of theformula of the function distinction 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