by
Seyyed Mohammad Reza Hashemi Moosavi
Freely
download prime numbers formula
software by Seyyed Alireza Hashemi Moosavi
The book of the discovery
of prime numbers formula and its results
The discovery of prime numbers formula and its results
"Table of contents"
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 Bertrand’s principle and theorems of
Chebyshev, Dirichlet and Poisson…………………………………………………………….35
1.11 Lagrange’s
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 Stochastic’s
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 Wilson’s theorem and its consequences
66
4.1 Wilson’s theorem
......................................................................
67
4.2 Remarks concerning Wilson’s theorem
and its converse and corollaries 69
4.3 Corollaries of Wilson’s
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 Mills’s
theorem.........................................................................
87
6.3 Kuipers’s
theorem.....................................................................
87
6.4 Niven’s
theorem........................................................................
88
6.5 Formulas of generating prime
numbers..................................... 88
6.6 Generalized Mills’s
theorem..................................................... 92
6.7 Investigation into
polynomials.................................................. 94
6.8 A formula presenting for generating
prime numbers by Wilson’s 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 Meissel’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
Meissel’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 Bonse’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 Riemann’s 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 Fermat’s
numbers...................................................................
149
14.3 Special problems and Fermat’s
numbers................................ 154
14.4 Another proof for Euclid’s
theorem...................................... 157
14.5 Speed of the growth of Fermat’s
numbers............................ 157
14.6 Fermat’s numbers and the problem of
inscribing regular polygons inside a circle 158
14.7 Refutation of Fermat’s assertion
and factorization of Fermat’s numbers………………………………………………...159
14.8 Mersenne’s
numbers............................................................
159
14.9 Problems concerning Mersenne’s
numbers.......................... 160
14.10 Perfect, imperfect and redundant
numbers......................... 161
14.11 Historical remarks concerning (even)
perfect numbers and Mersenne’s
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
Mersenne’s prime numbers and Fermat’s 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
Fermat’s numbers ().........
178
15.2 Definition of the set of Fermat’s
prime numbers by the prime numbers formula 178
15.3 Some general facts about Mersenne’s
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 Goldbach and Hardy’s
conjectures....................................... 182
16.2 Goldbach'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 Clement’s 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 Fermat’s 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
Fermat’s last theorem 203
19.3 Chronology of Fermat’s last
theorem.................................... 204
19.4 Sophie Germain’s
theorem………………………………………...218
19.5 Fermat’s last theorem for exponent
"4"................................. 221
19.6 Fermat’s 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 Fermat’s
last theorem
230
20.2. Primitive, Algebraic and geometric
methods........................ 231
20.3 An indirect proof of Fermat’s 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 Fermat’s 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
Fermat’s 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
Wilson’s, Fermat’s and Euler’s 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 Riemann’s 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 Mersenne’s prime numbers set by using the prime numbers
generator...333
22.13 A program for determining the
Mersenne’s prime numbers of M-digits (M: Arbitrary
number) 334
22.14 The New Mersenne’s 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