Index
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.......................................................................
2.1 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............................................................................
3.1 "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
4.1 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.................................................
5.1 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
7.1 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
8.1 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 "
".............................................................
9.1
Determining "
" by "
"........................................................
9.2 Comparing
the precise formula for ""with Meissel’s
formula......
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...................................................................
11.1
Determination of "" in a precise
manner ....................................
11.2 Other
formulas for determining "" in a precise
manner.................
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.............................................
13.1 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..........................
17.1 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.......................................
18.1 Generation of twin prime numbers .................................................
18.2 There is infinity many twin prime numbers.......................................
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.................................................
20.1
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
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
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