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12/06/11 |
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I am Seyyed Mohammad Reza Hashemi Moosavi and university professor that I chosen as a superior investigator in superiors and initiators festival in 1383. My academic course is telecommunication electric engineering and I received specialized doctorate of Education (PhD) from Boston university of America and doctorate of mathematics (PhD) from Spain.
I have started my researches in mathematics field when I was fourteen. My first research was a flash that one of my mathematics teachers in guidance school caused it. He pronounced a method of mental multiplication of numbers. Impetus of mental multiplication occupied my mind to research for several years. Till in first year of high school, I could obtain a mental multiplication method for M figures and N figures and it was my first success in research works. My second research which lasted around two years was obtain the method of algebraic and geometric solution in cubic equation that I obtain in fourth year in high school and published in 22nd copy of mathematics teaching development magazine from research and lessons programmer organization. After it I researched seriously. For example calculation of K-th strength for n prime number that I express Sk a determinant which it doesn't need to Bernoulli coefficients or analysis methods. This point published in 16th copy of mathematics teaching development magazine too and then in 1994 it published in spectrum (the university Sheffield) in England. I obtain the calculation of ellipse circumference which has many usages in calculation of integral function and ellipsoid integral in analysis in a perfectly analytic method and it will publish in spectrum magazine. I performed another research like congruence equation solution in table method that it gives answer of every congruence equation with optional coefficient in the shortest time which is possible. My other research was presented a new method with a highest race of calculation for N*N determinant and it published in spectrum magazine in 2003.It is necessary to mention this point that all of these research became pedestal for my next research like obtaining the prime number formula. My other researches are integral expansion to series and calculation of integrals which have N-th power and express in a returning method. These articles published in ''Acquaintance with mathematics '' too and also my other researches that published in different copy of this magazine. My another important and basic researches is solution of fluid equation in N-th degree that has a great usage in engineering sciences and researching center .of course I performed a lot of researches in algebra, analysis , number theory field and other mathematics branch that I can't express it in this short time.
My books My writings are more than 17 copies that all of them had published and of course another books (more than 19 copy) that I translated or edited scientifically. I have written other books in university and Olympia levels and one of the master pieces of mathematics works that was collected with collaboration of other mathematics cooperators group in "school publication" is mathematics dictionary. also more than 34 specialized articles printed in out side and in side creditable magazines. Lots of my researchable investigational books like "essentials of coding and decoding" and new researches in mathematics and other titles also are ready to publish like
" The discovery of prime numbers formula and its results " . NOTICE : If you want to request this book please contact at below addresses .
Address : POB 9000, 2300 PA Leiden, The Netherlands Tel : +31 71 535 3594 Fax : +31 71 531 7532 email : vsppub@brill.nl (please use this email address because of SPAM) Site : www.brill.nl Site : www.vsppub.com
"Cover of the book" The discovery of prime numbers formula and its results
Preface of author
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 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
6.1 A summery of the history of 2000 years old attempts for finding a formula for prime numbers 6.2 Mill'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 " 8.2 Prime numbers theorem 8.3 The function "li" or "the logarithmic integral"
8.4 Meissel’s formula for "
9.1 Determining "
9.2 Comparing the precise formula for "
10.1 Determining the bounds for "
10.2 Bounds for " 10.3 Bonse’s theorem 10.4 Theorems concerning consecutive prime numbers 10.5 Theorems of Chebyshev 10.6 Theorems of Ishikawa
11.1 Determination of "
11.2 Other formulas for determining "
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 (
14.1 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 (
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
16.1 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 a general answer for equation
20.12 Determining a general answer for equation 20.13 Determining a general answer for equation
20.14 Determining a general answer for equation (H.M)
20.15.Determining a general answer for equation (H.M)
20.16 Determining a general answer
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 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 " 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)
22.1 The abstract of the formula of the formula 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)
CHAPTER 1 A brief view of number theory 1.1. Number theory in ancient timeWe should
have a quick review to the past (before Fermat, in 17th century).
Mesopotamia civilization (2000-3000 B.C) is the first civilization,
which presented documents that indicates mathematical activities in that
time. There and the process of writing has been done on some tablets made of
clay with a kind of hard writing called cuneiform; there are calendars,
which determines that the beginning of this matter goes back to about (2000
B.C) and it shows that Summer ions had an understanding of topologic
measurements, simple and complex interest, the solution of the square
equations and their uses of negative numbers. The first convincing sign
which archeologist scientists found was in 1945 and it was the time that A.
Negiver and A. Sakhz analyzed a table which was known to Plimpton 322 (Plimpton
library of Colombia University). From the language that used in it, we can
comprehend the history of it a little closer to (1600-1900 B.C).
However there is a schedule in it , including 15 answers for equation
In addition, the sequence of numbers have been written in a special way, indeed, it is requested to reduce an angle of a right triangle with (x, y, z) sides from 45o to 31o. Evidently, Babylonians did not know only the Pythagoras theorem and eventually the sense of trigonometric functions, but they used a rule for finding the answers of Pythagoras equation. If we suppose that all of these are not extraordinary enough, we should say these people have done all of these acts without symbolic algebra and without sense of common demonstration. It does not seem that mathematics of Egypt that has remained on the parchment wholly shows the proceeding of Mesopotamia in mathematics. The obtained works from B.C. Indo china are very scattered, but the important thing is that the acts which were done in Indo china have not had any effect on the development of numbers theory. The subjects which are known as mathematics today like deduction, proof and theorem, started from Greeks. Probably conclusion has been found by Tales (548-624 B.C) and almost was used by the students of Pythagoras school. Pythagoras (500-580 B.C) traveled to Babylon, Egypt and probably India. He was a philosopher and a Gnostic that gave importance to counting and philosophy. Probably he and his followers were depended on the senses of pictorial number (triangles numbers 1, 3, 5, 10,…; square numbers, etc), perfect numbers (for example, 28 is a perfect number because it is equal to 1+2+4+7+14, the sum of its divisors are less than itself), amicable numbers (for example 220, 284 because each of them equals to sum of another real divisors). But, it is not obvious which one of them had proved the theorems in these cases. The first institute like University which was called "museum" established in Alexandria and its first scientific member was Euclid. However, Euclid was famous mathematicians, most of the subjects that he reviewed in his “principles” book, have been former’s works. The volumes, number IX, VIII and VII of principles book have considered number theory. As a unique decomposition theorem equals to theorem 14 of IX book. The existence of infinite numbers of prime number is 20th theorem of IX book. Among three famous mathematicians that created the golden era of Greek mathematics on (200-300 B.C). Among Euclid, Archimedend, Apollonius only Euclid is who seems to have done many researches in number theory. Most of the time, mechanics and geometricians paid more attention to it and it took time more than 3 centuries for Diophantus and Alexandrian to begin a new way with the "arithmetic". In his work about 13 volumes of treasure that there have been just 6 volumes of them remained started multi variables (unknown) equations, Equations with two or more unknown quantity which the answers belong to Q+ or (today) to Z. Also, these books include some theorems like, if two integer numbers which each of them equals to the sum of two squares, the product of them also equals to the sum of two squares. According to indirect evidences, it seems that Chinese have known much mathematical subjects before finding them out in else where, which includes Pascal’s triangle and simple magic squares. On the other side, probably because they had no relation to others, their portion in mathematics is considered just in remained Chinese theorem that belongs to some ancient countries. In India, Brahmagopta discovered general integer answer of linear Diophantus equation" ax+by=c". But Diophantus had verified only equations with higher degrees, because linear equations are obvious when rational answers were considered. He always binds himself to special and singular answers.
Some years later Bascara (1114-1185),
solved equation
With decreasing the
influence of Greeks and then advent of Roman imperial 1.2. What is number theory?
This
question is motivation of primary attempts to present the definition. Number
theory is the study of a set of integer numbers ( We return to 17-century, for taking an idea and knowing about the time that Pier Fermat’s[1] work started a new era in mathematics. One of the most beautiful Fermat’s theorems is that every positive integer number can be shown as the sum of squares of four integer numbers. For example:
(According to this point, the multiplication of two representable numbers is a representable number; this is enough to prove that every prime number "P" is representable). He propounded this
theorem in 1636, but the first printed proof was, presented in 1770 by
Joseph Luis Lagrange. This theorem has an ideal aspect in theorems of number
theory that is: beauty, fast understandability, revealing and exact and
unexpected relation between integer numbers. The best result in relation to
its kind (7 cannot be shown as sum of less number than squares) and it is a
proposition about infinite set of integer numbers. The last one is very
important, because it determines the difference between theorems and numeral
truth. This subject that 1729 was the smallest positive integer number, has
two representations as the sum of two cubes
On the other side, we consider that the proposition "only finite integer numbers that exist have two or more presentations of that kind" is seducer. It seems that this proposition expresses a subject about finite set, but in fact, we can not prove or reject every finite set with testing. Therefore, this proposition will be an important theorem if it is true (unfortunately, it is not always true).
Another more famous subject that is
attributed to Fermat and sometimes called his latest theorem, expresses that
if "n" is an integer number greater than "2", the equation
(The known
worlds can contain only One of the
basic concepts of number theory is prime numbers. Integer number "p"
is prime if Although portion of
Mersenne in Mathematics was propagating the new results more than creating
them, but he studied prime numbers among numbers in
If
Fermat's numbers and Mersenne's numbers are so scattered that if all of them be prime numbers, we can infer some information about distribution of prime numbers in them. Other useful and
prominent studying by Guass was started in 1792 by using a table of prime
numbers smaller than 102,000 that some years ago printed by John Lambert. If
as it is usual,
The average of the
number of prime numbers reduces in successive interval and Guass chose
inverse of
The wonderful match
between these numbers strengthen, this guess
This Integral is not a
primary function and usually is shown by
What Gauss expressed in
present ion, a guess that
Or:
He guessed this subject
in 1793 when he was "15" years old. But this subject wasn’t proved, till
more than a "100" years later Hadamard and Poisson proved it (independently
in 1896). Its demonstration is too hard that can be expressed in this book.
But it is possible to show that if the limit of (1) exists, its value will
be equal to "1".
This subject and its inverse are proved. Because of the basic situation that relation (1) or its formal form (2), have in number theory, it is known as "theorem of prime numbers". The Reason for studying
the information of latest table for great number of "x" which Gauss
did not calculate Now, to have an extra
example we express a question that is less famous and hasn’t been propounded
recently and it seems to be very hard. We form the following double infinite
array that the first row includes prime numbers and every number of the
following rows equals to absolute value of subtraction of its two upper
numbers. Is it true that every row except the first row starts by "1"? This
subject is true about a part of array that is written in below up to
For example, in
problems about nature of numbers like " Again in both cases, it is proved that the answer is negative; but their demonstrations are a little hard to understand. If we verify the problem on the other side, we can study the numbers that are algebraic, and therefore it seems that an strong theory can be built on its base, so that it will be an interesting subject and a useful instrument to study the integer numbers.
1.3. Prime numbers
In this book, we consider the positive divisor of numbers. Therefore, we speak about "divisor", it means positive divisors unless against this mater are emphasized. For example the number of 6 has only four divisors that are 1,2,3,6. 1.3.1. Definition Natural number " 2,3,5,7,11,13,17,19,23,29,…,97 Since the only positive
divisors of prime number "p" are "1" and "p", then for each
integer number like "a" we have ( Now we suppose "p"
is prime and "a" and "b" are two integer numbers, so that 1.3.2. Lemma Imagine "p" is
prime number and "k" is natural number. If
1.3.3. Lemma Every natural number Proof. Suppose
that 1.3.4. TheoremThe number of prime numbers is infinite. This theorem means
that; if we suppose every natural number like "n", the number of
prime numbers is more than "n". We suppose
There are so many easy
ways for classifying prime numbers. At the first stage it seems that each
natural number is either even or odd, it means that each number is in the
form of " But "2n" cannot
be prime number except for " In this way, because
the reminder of division of each integer number by 3 is 0,1,2, then each
natural number that is in one of these three forms of
Now in the next step we
remind that each natural number is in these forms
It is clear that "4n"
is newer prime and Also we can classify prime numbers according to residuals of dividing by each positive correct number and constant (as it used in 2, 3 and 4). The following theorem
is famous to Dirichlet, it is proved in some special conditions with the use
of elementary way, but we do not know easy demonstration for the general
state of this theorem. Therefore, we express the theorem without proof. Its
especial state is the urgent result of theorem (1.3.4) for 1.3.5. Theorem If "a" and "b"
are two natural numbers and
1.4. The fundamental theorem and some of its applications
In the same way,
we will show, it is completely clear that each integer number is
1.4.1. Basic theoremWe can
write each natural number
Now, if we delete equal prime numbers from both side of equality, we will have:
That
In writing integer
number
that
We suppose
1.4.2. TheoremIf standard factorization of number "a" is in form of (1); then the divisors of "a" will exactly be numbers like "d" in the following form:
It is
clear for number "a" that we can give
that " It is obvious that when 1.4.3. TheoremIf "a" and "b" are in form of (3), then:
for example, we suppose
that By using theorem (1.4.3) we will have:
Now, for finding the
number of one positive integer number’s divisor, we obtain an easy formula.
For example, according to theorem (1.4.2), divisors of
Generally,
Whenever (1) is a standard factorization of number "a", then the number of divisors of "a" is in the following form:
Also, we can gain a
formula to calculate the sum of all of divisors of supposition integer
number of "a". Here we suppose the number
as the sum of 24 terms is in the following form:
In fact, the above
multiplication is equal to the sum of all integer numbers in form of
Therefore, the sum of divisors of number 360 is:
1.4.4. Attention
Sometimes, it is better that we use
symbol For example,
It is interesting to
remember that, we can factorize all of the numbers smaller than
Number 1243 is not divisible by either 7 or 5, but we have:
So, we can not continue the following action, because 113 is smaller than the square of 11 and by paying attention to these calculations that we have done, it is not divisible by any number smaller than 11, therefore it is a prime number. Then, we can be assured by a general method that "N" is prime number, when it isn’t divisible by any prime numbers smaller than "p". This method that is about primality of a simple special number is very hard when it is used for the great number of numbers and it is obvious that it is impossible for millions of numbers. There is an easy method for this case. It has been very usual since many years ago. It is related to Eratosthene and in this method, we identify the non-prime numbers among the numbers smaller than 10000 or 100000 and etc, and then we determine the least divisor of them which is prime. This way is known as Eratosthene sieve. 1.5. Sieve of Eratosthenes
Suppose that we want to determine the prime numbers smaller than 100. At first we omit even numbers, then we write the odd numbers on the consecutive rows (for example 10 numbers in each row), in this way we will have:
Then, before doing any thing, we omit the multiples of 3 and this is very easy, because these multiples are three-to-three. When we omit the multiples of 3 there is no need to take into account the omitted even numbers, but after omitting the multiples of 3, if we want to omit the multiples of 5, we must consider the numbers five-to-five, without ignoring the multiples of 3 that have been omitted before. For omitting the considering multiplication, it is better to adjust the above table to the following form, that in the two ends of the rows there are decimal digits and in the top of the columns, there are mono-digits.
At first, we put the number 3 in the squares which adapt numbers divisible by 3 (except the number 3 which is prime number). We will do this for number 5 and then 7. Finally the squares which are empty show the prime numbers. It is clear that the number 3 have a regular order in this table, along some diagonals; there is the same order for numbers 5 and 7, especially if divisors of 5 and 7 exist in equerries? Quarry that also divisors of 3 exist, this situation is more obvious.
1.6. Periodic sieve for small numbers
If we note that sieve has a certain period, especially for the smaller prime divisors, then a lot of calculations can be done more easily. First of all we pay attention to divisors of 2 and 3. The multiple of these two numbers is equal to 6 and it results that if a number is not divisible by 2 and 3 it will be in one of the following forms:
It means that in every successive 6 numbers, there are two numbers that are not divisible by 2 and 3. Now we search for the numbers which are divisible by 2, 3 or 5. We can see, among these numbers, there are only three numbers 2, 3 and 5 which are prime numbers. Nevertheless we consider them as the numbers which are divisible by 2, 3 or 5. Among numbers of (1),
which are divisible neither by 2 nor by 3, we can obtain the numbers smaller
than 30 for
These 8 numbers are prime except "1":
Since the number 30 is divisible by 2, 3 and 5, the numbers which are in one of the following forms:
are not divisible by 2 , 3 and 5 and these numbers aren’t necessarily prime, but, we must look for the prime numbers among them. On the other hand, for a number to be prime, the condition (2) is necessary but not enough. Now, we consider the
prime number 7. The numbers smaller than or equal to multiple of By this method we get
Among these numbers, how many of these numbers are divisible by "7"? If these numbers are divisible by 7, quotient is a number between "1" to "30" and because none of these numbers are divisible by 2 , 3 or 5, Therefore their quotient by 7 is a number that is not divisible by 2,3 or 5 subsequently this quotient is one of eight numbers which we mentioned them before. Conversely the multiple of each of these eight numbers by 7: 7,49,77,91,119,133,161,203 (3) will be the numbers smaller than "210" and not divisible by 2, 3 and 5, but they are divisible by 7. Finally among the first 210 numbers, the number of numbers which are not divisible by 2, 3, 5 and 7 will be:
These 48 numbers are
numbers which are obtained from relation (2) for If
are not divisible by 2, 3, 5 and 7; and therefore they could be prime numbers. In fact, all of 48 numbers aren’t prime and among them, there are numbers which aren’t prime:
In this way, we can
study the first 2310 numbers using the prime number "11" which is
immediately after 7. Among these numbers,
these are
that its appearance form is also very simple and among these 30030 primary numbers there are:
Number which are not
divisible by any prime number smaller than 17. For obtaining the prime
numbers smaller than 30030, we must omit the multiples of 17, 19 … up to 173
(square root of 30030) among them. This is relatively a detailed work? but
since it omits
1.7. The infinity of prime numbers
From what is said up to now, we can
get to this conclusion easily that the prime numbers sequence is infinite,
In fact, by an easy calculation which has been done before , it is resulted
that if we consider only the prime numbers 2, 3, 5 and 7; among the first
210 numbers greater than "1", there are 48 numbers which we can not obtain
them by multiplication of one of these four prime factors by the others. If
we consider more prime numbers, but restricted, there we will find more
numbers which are not resulted from multiplication of these few prime
numbers. For example, if we consider the prime numbers 2, 3, 5, 7, 11 and 13
among the first integer 30030 numbers, there are
There is another reason to prove the infinity of prime numbers which is old and it is easier in some features, but it can not show clearly that how enormous the number of prime numbers. This reasoning is as follow: We prove, there is at least a number which is greater than an arbitrary integer number "n". If we show the multiplication of first "n" integer numbers by "n!" and identify "N" by the following relation: N=n! +1 (1) then if "N" is not a prime number, it must have at least one prime divisor like "p". This divisor of "p" can not be smaller or equal to "n", because according to the relation (1), if we divide "N" by a number like "a" which is between 2 and "n" then the residue of division will be equal to unit. It means that "N" is not divisible by "a", therefore, there is a number like "p" which is greater than "n". Since the number of prime numbers is infinite, the distance between two arbitrary consecutive prime numbers can be great. We will show how we can use the decomposition into the positive prime factors in calculating the number of divisors and the sum of them.
1.8. Functions
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