**
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 Bertrand’s principle and theorems of Chebyshev,
Dirichlet and Poisson

1.11 Lagrange’s
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 Stochastic’s
theorem........................................................................

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

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

2.6 Burkhard’s
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 Wilson’s theorem
and its consequences **

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

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

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

6.3 Kuiper’s
theorem............................................................................

6.4 Niven’s
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
Wilson’s 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 Meissel’s 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
Meissel’s 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 Bonse’s
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 Fermat’s
numbers..........................................................................

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

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

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

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

14.7 Refutation of Fermat’s assertion and factorization of
Fermat’s numbers

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

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

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

14.11 Historical remarks concerning (even) perfect numbers
and Mersenne’s 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 Mersenne’s prime numbers
and Fermat’s 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 Fermat’s numbers (_{})...................

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

15.3 Some general facts about Mersenne’s 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 Clement’s 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 Fermat’s 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 Fermat’s
theorem.................

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

19.4. Fermat’s theorem, for exponent
4.................................................

19.5. Fermat’s 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 Fermat’s theorem _{
}...........

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

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

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

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

20.6 Latest achievements and fundamental results concerning
Fermat’s 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 Fermat’s 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 Riemann’s 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 Mersenne’s prime
numbers set by using the prime numbers generator.

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

22.14 The New Mersenne’s 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