4.3 Corollaries of Wilsons
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 Millss
theorem..............................................................................
87
6.3 Kuiperss
theorem.........................................................................
87
6.4 Nivens
theorem.............................................................................
88
6.5 Formulas of generating prime
numbers................................... . 88
6.6 Generalized Millss
theorem..................................................
.... 92
6.7 Investigation into
polynomials................................................... 94
6.8 A formula presenting for
generating prime numbers by Wilsons
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 Riemanns 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 Fermats
numbers...................................................................
149
14.3 Special problems and Fermats
numbers........................... 154
14.4 Another proof for Euclids
theorem......................................
157
14.5 Speed of the growth of Fermats
numbers............................ 157
14.6 Fermats numbers and the problem of
inscribing regular polygons inside a circle
158
14.7 Refutation of Fermats assertion
and factorization of Fermats
numbers
159
14.8 Mersennes
numbers...............................................................
159
14.9 Problems concerning Mersennes
numbers.......................... 160
14.10 Perfect, imperfect and redundant
numbers............................ 161
14.11 Historical remarks concerning (even)
perfect numbers and Mersennes
numbers
64
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
Mersennes prime numbers and Fermats 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
Fermats numbers ()......
...
178
15.2 Definition of the set of Fermats
prime numbers by the prime numbers formula
............................................................................................
178
15.3 Some general facts about Mersennes
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 Hardys
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 Clements 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 Fermats 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
Fermats last theorem
......................................................................................................................................203
19.3 Chronology of Fermats last
theorem........................................................................................................................................................................
. 204
19.4 Sophie Germains
theorem...........................................................................................................................................................................................218
19.5 Fermats last theorem for exponent
"4".....................................................................................................................................................................
221
19.6 Fermats 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 Fermats
last theorem ..........................................................................................................................
..
230
20.2. Primitive, Algebraic and geometric
methods.......................................................................................................................................................
.. 231
20.3 An indirect proof of Fermats 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 Fermats 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
Fermats 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
Wilsons, Fermats and Eulers 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 Riemanns 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 Mersennes prime numbers set by using the prime numbers
generator............333
22.13 A program for determining the
Mersennes prime numbers of M-digits (M: Arbitrary
number)............................................................................ 334
22.14 The New Mersennes 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