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 Bertrands principle and theorems of Chebyshev, Dirichlet and Poisson.................................................................................... . .......35
1.11 Lagranges 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 Stochastics 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 Wilsons theorem and its consequences ................................................ . 66
4.1 Wilsons theorem ......................................................................................................................................................................................... . ....... 67
4.2 Remarks concerning Wilsons theorem and its converse and corollaries................................................................................... . .........69
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 . . .....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 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 German's 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" ........................................................................................... 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