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You can find every thing you want to know about me

                                                                                                                                                                           

Discoverer of prime numbers formula                                                 

  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 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

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Cover and table of contents of the book

"Cover of the book"

The discovery of prime numbers formula and its results

  "Table of contents"

   Preface of author

    1. A brief of view of number theory

1.1   Number theory in ancient time

1.2   What is number theory?

1.3   Prime numbers

1.4   The fundamental theorem and some of its applications

1.5   Sieve of Eratosthense

1.6   Periodic sieve for small numbers

1.7   The infinity of prime numbers

1.8   Functions,  and

1.9    Perfect numbers

1.10 Bertrand’s principle and theorems of Chebyshev, Dirichlet and Poisson

1.11 Lagrange’s theorem

2. On the history of forming prime numbers tables and determining the smallest divisor of composite numbers

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. Decisive solution to the problem of forming of tables concerning divisors of composite numbers by regular loops in                                                              arithmetic progressions and successive cycles

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. On the history of the problem of recognizing prime numbers by two sided theorem in particular Wilson’s theorem and its consequences

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. Decisive solution to the problem of recognizing prime numbers by a formula concerning recognizing of numbers ""

5.1 Determination of the formula for the characteristic function of numbers           

5.2 Formula for surjective characteristic function

        6. On the history of the problem of searching for finding generating function of prime numbers ""

6.1 A summery of the history of 2000 years old attempts for finding a formula for prime numbers 

6.2  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. Decisive solution to the problem of finding the generator of prime numbers via discovering the surjective generating function of prime numbers

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. On the history of the problem of determining the number of prime and its related functions "" and ""

8.1 An introduction to the function ""and "li

8.2 Prime numbers theorem

8.3 The function "li" or "the logarithmic integral"

8.4 Meissel’s formula for ""

 9. Decisive solution to the problem of determining of the precise number of  "" primes by characteristic function ""

9.1 Determining "" by ""

9.2 Comparing the precise formula for ""with Meissel’s formula

       10. On the history of determining "k-th" prime number by bounds for "" (determining lower and upper bounds for "")

10.1 Determining the bounds for "" (the k-th term of the sequence of prime numbers)   

10.2 Bounds for "" from below and above

10.3 Bonse’s theorem

10.4 Theorems concerning consecutive prime numbers

10.5 Theorems of Chebyshev

10.6 Theorems of Ishikawa

 11. Decisive solution to determining "k-th" prime number by determining function concerning the number of primes in a precise manner

11.1 Determination of  "" in a precise manner

11.2 Other formulas for determining "" in a precise manner

 12. On the history of attempts for solving Riemann zeta equation and the low of rarity of prime numbers

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. Decisive solution to Riemann zeta equation

() by the determining function concerning the precise number of primes ()

13.1 Riemann zeta function ()

13.2 Decisive solution to Riemann zeta equation ()

 14. On the history of searching for famous prime numbers and the factorizations of these numbers ()

14.1 Some of famous numbers

14.2 Fermat’s numbers

14.3 Special problems and Fermat’s numbers

14.4 Another proof for Euclid’s theorem

14.5 Speed of the growth of Fermat’s numbers

14.6 Fermat’s numbers and the problem of inscribing regular polygons inside a circle       

14.7   Refutation of Fermat’s assertion and factorization of Fermat’s numbers

14.8   Mersenne’s numbers

14.9   Problems concerning Mersenne’s numbers

14.10 Perfect, imperfect and redundant numbers

14.11 Historical remarks concerning (even) perfect numbers and Mersenne’s numbers    

14.12 Role of computers in searching large prime numbers

14.13 Odd perfect numbers

14.14 Special problems concerning perfect numbers

14.15 Problems on distinguishing Mersenne’s prime numbers and Fermat’s numbers      

14.16 Problems concerning Fermat (), Mersenne (), perfect and  redundant numbers         

 15. Definition of the sets of Fermat, Mersenne, perfect prime numbers by the prime numbers formula

15.1 Some general facts concerning Fermat’s numbers ()

15.2 Definition of the set of Fermat’s prime numbers by the prime number’s formula        

15.3 Some general facts about Mersenne’s numbers and even perfect numbers and the relation between them      

15.4 Definition of the sets of Mersenne, even perfect prime numbers by the prime numbers formula          

        16. On the history of attempts for proving Goldbach and Hardy  conjectures

16.1 Goldbach and Hardy conjectures

16.2 Goldbach conjecture and other open problems related to it

16.3 Some unsolved problems and other conjectures concerning prime numbers  

16.4 Applied investigations into Goldbach and Hardy conjectures

16.5 Theoretical investigation into Goldbach conjecture

 17. On the history of attempts for proving the conjecture of existence of infinity many twin prime numbers

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. 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 

18.1 Generation of twin prime numbers

18.2 There is infinity many twin prime numbers

 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  

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. 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 equation

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. The newest of methods of solving and calculation

                                   Appendixes (I)

21.1 Solving congruence and Diophantine equations

by "H.M" table  ()

21.2 Solving Diophantine equation of order in by "H.M" table

21.3 A new and fast method for calculating determinant ("H.M" method)    

21.4 Definition of regular and ir-regular prime numbers by "H.M" determinant.     

21.5 New method of calculation of sum of  "k-th" power of the first "n" natural numbers by "H.M" determinant (Expressing "" by a determinant)

21.6 Determining the number of roots of perfect cubic degree equation directly by "H.M" method

21.7 Proof of a new and applied "H.M" theorem (Concerning the factorization of composite numbers)     

 22. The abstract of formulas and their software programs

                                    Appendixes (II)

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)

 l References on some historical parts of the book

List of my books

           here is the ISBN of some of my books below :

  NOTICE : These are samples, and I have written many other books.

CHAPTER 1

A brief view of number theory

1.1. Number theory in ancient time

We should have a quick review to the past (before Fermat, in 17^th 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  which difficulty point, they are (3,4,5) to (12709,13500,18541).

    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 45^o to 31^o. 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 20^th 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  in especial cases many years before that, samples of this equation was solved by Archimedes and also by Diaphanous or one of his contemporaries.

    With decreasing the influence of Greeks and then advent of Roman imperial
(that had not present a new thing in mathematics), the center of civilization was transferred to Baghdad. Probably Harmonious Knowledge of Babylonians, Egyptians, Greeks and Hindus, Have been useful.

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 () or some of its subsets or sets that includes it in addition to their useful roles in measurement integer numbers or in relation to each other are interesting alone. Apparently the domain of this definition includes primary arithmetic, which in fact, it is in this manner except cases about exact and improvement aspect.

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 is true and probably one of the most interesting facts, but we can not name this truth as a theorem. Because it can be proved by testing a finite set 1, 2, …, 1729.

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  does not answer in positive integer numbers set. Fermat claimed that he proved this but as his habit, he did not express its proof. It seems, this is only a recorded example that he claimed a result which he never proved (Although he propounded a false guess about the prime number  that we will express below). Since there is no demonstration for this claim, modern mathematicians prefer to call it Fermat’s problem instead of Fermat’s theorem. This is the oldest and maybe the famous unsolved problem in mathematics. Although only a counter example is enough to discredit it, finding four numbers x, y, z and "n", if they exist, is out of the capacity of future and computers. Because, now, it is obvious that this equation does not have any answers for () and if the answer exists, x, y or z must be greater than.

(The known worlds can contain only  objects in proton size).

One of the basic concepts of number theory is prime numbers. Integer number "p" is prime if  and equation  does not have an answer with respect to integer number "a" and "b" except or. Therefore, we can say briefly that a prime number is an integer number that is the opposite of  and does not have any non–trivial divisor. Before, Euclid knew that the sequence of prime numbers 2, 3, 5, 7 …does not finish and the manner of their appearance is very irregular. Fermat and Mersenne also were seeking a kind of order but both of them guessed a wrong thing. Fermat guessed that all of numbers are prime numbers that in fact, this conjecture is true for n=0, 1, 2, 3, 4. After a while, it was clarified that, it stops soon, because Leonard Euler in 1739 showed that "F₅" is a divisor of "641". In fact, for  no prime value was found for F_n. Since false guesses about prime numbers are very frequent this story would not be very valuable if Fermat’s prime numbers appeared "200" years later in different situation again. Carl Fredrich Guass[2] searched one of ancient Greece’s problems. He proved that regular m-angle can be made by ruler and compass if and only if (IFF) "m" can be factorization in form, that k,n₁,…n_r are non-negative integer numbers and are Fermat’s distinct prime numbers. So it is interesting to know whether prime numbers more than this kind exist or not?

    Although portion of Mersenne in Mathematics was propagating the new results more than creating them, but he studied prime numbers among numbers in form.

    If , then "M_n" is divisible by "M_r" and also by "M_s". Therefore "M_n" can be prime Just for prime values of "n". In 1644, Mersenne expressed that from "55" numbers of "M_p" that, the ones which are prime number, are corresponding with . So, Mersenne committed "5" errors. Because he took into account 67 and 257 and he didn’t teak into account 61, 89 and 107. It is clear that he had wrong but it is important that he could gain information about numbers which have 78 digits without pocket calculator. Again we see numeral truth, not theorem, and frankly some cases must be hidden beyond these subjects, that "N is prime number if …" or "N isn’t prime number if …" and such cases are useful. Always there is strong interaction effect between intuitive truth and theorems of numbers theory. Calculations give information that we can infer from them some counter examples or theorem’s guesses and also subjects for useful theorems which lead to useful algorithms. (Algorithm means methods for calculation).

    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,  was indicator of the number of positive prime numbers not more than "x", then what Guass did, was searching the increments of  according to the increments of "x". He started by enumeration of prime numbers in intervals with constant lengths and obtained a table like the follow:

    The average of the number of prime numbers reduces in successive interval and Guass chose inverse of  and compared it with a different primary function. The following table is obtained about natural logarithm of "x":

    The wonderful match between these numbers strengthen, this guess  is almost equal to. Since  is equal to slope angle of a segment on curve  we must integrate the approximate equation of  so that  can be calculated, therefore Guass guessed:

    This Integral is not a primary function and usually is shown by. Its values are calculated easily and the modern calculations present the following comparison that  is given with its nearest integer number.

    What Gauss expressed in present ion, a guess that  is a good approximation of  for great "x", was not that, or even  was limit, but this was that its relative error is little:

Or:

                 (1)

    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".
It is not so difficult to show (1) result:

             (2)

    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  for them, is that he persisted on this important point that no calculation can substitute the demonstration. As the table clearly show’s always presents with abundant approximate. It means that at least the value of [] is positive and increasing to. But this isn’t permanent, because Littlewood in 1914 showed that the sign of [] changes infinite times. No body knows that when first sign changing occurs. But Skewes in 1955 proved that this subject happens for "x" that. Probably no determinate value will be found for "x" in future so that for it. Many questions exist about prime numbers that all of them remain unsolved (despite the attempts that have been done for two or more centuries) for example, is there infinite number of twin prime numbers like 17 and 19, 4967 and 4969 so that their difference is "2", and does every even number greater than 4 equal to the sum of two odd prime numbers?

    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 and it is justified up to.There are some branches of number theory that integer numbers exist in them less “a” the prime number theory.

    For example, in problems about nature of numbers like "" and "e", such a situation exists. This question that, whether every one of these numbers are rational or not, in fact is equivalent to, whether every one of them is an answer for a linear equation like  with integer coefficients or not? Lambert[3] Switzer, a mathematician, mentioned to this fact and later in 1761 proved that  is not a rational number. Generally, this question can be propounded that, are "e" and  algebraic or not? In other words, It there any of them that is adapted in a polynomial like () with integer coefficients like () or not?

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 "" is a prime when its divisors are only "1" and "p", each natural number except "1" is compound when it is not prime. This definition, results that "1" is neither prime nor composite. Also natural number like "n" is a composite number if and only if  that. Some of the prime numbers less than "100", written in Pythagorean’s table are as follow:

    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 ()=1 or , it means, "a" is
coprime with "p" or "p" aliquots it. Therefore, sometimes it is better to write  instead of.

    Now we suppose "p" is prime and "a" and "b" are two integer numbers, so that. If, then  and it results. Therefore, the products of two integer numbers are divisible by prime number "p", if and only if at least one of those two numbers is divisible by "p". This result can be extended to multiplication of a finite number of integer numbers. (By using this product symbol, we present the following General results.

1.3.2. Lemma

    Imagine "p" is prime number and "k" is natural number. If  are integer numbers, so that, then for one "i" and  we will have, now we express and prove the following lemma:

1.3.3. Lemma

    Every natural number has a prime factor.

    Proof. Suppose that  is a supposed natural number and "S" is a set of all divisors of "n" greater than "1". Since, then the set "S" is non-null and consequently "S" has a smaller member, namely "p". We prove that "p" is a prime number. Since each divisor of "p", is a divisor of "n", therefore if "p" is composite, then it is  necessary that "S" has a member which is less than "p", and this is impossible, therefore "p" is prime. The following theorem is from Euclid, and therefore this theorem is over 2000 years old.

1.3.4. Theorem

    The 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  are "n" different prime numbers. So we can write. According to lemma (1.3.3), "N" has a prime factor like "p". On the other hand, no value of  is the factor of "N", because in this condition, it is necessary that. Therefore "n+1", prime numbers of  are distinct. So we can prove the theorem’s demonstration, by induction.

    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 "" or "" that "n" is a non-zero integer number.

    But "2n" cannot be prime number except for "". Then each prime number is odd, except 2. Therefore we can obtain this fact with use of above theorem the number of odd prime numbers is infinite.

    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 and "n", is a natural  number. Again "" can not be a prime number, unless. Then each prime number except "3" is in form of or. In other words, every prim number except "3" is in form of. Therefore, the number of prime numbers is infinite to this form.

    Now in the next step we remind that each natural number is in these forms  and "n" is natural number.

    It is clear that "4n" is newer prime and is prime only if. Then each odd prime number is in one of these forms or. On the other hand, each odd prime number is in form. Therefore the numbers of prime numbers which are in this form are infinite.

    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, then the number of prime numbers in form of "" is infinite ("n" is the natural number).

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  or prime or we can write it in a form like multiplication of prime numbers. According to the lemma (1.3.3), number "a" has a prime factor like  and therefore, there is a natural number like  in that.  So, there is a natural number like  in that  or. If  then "a" is written as the multiplication of the primes  and. Now, if  then  must have had a factor like  and.Since  so this sequence can’t repeat to extreme infinite times and we must have  for a natural number like "r", and then. But it is not necessary for all of prime numbers  to be different results. In this way, we proved a part of the following theorem and this theorem has an important role in studying integer numbers which usually it is famous to the Arithmetic basic theorem. Before speaking about the theorem, we accept that there is a multiplication even with one factor. Therefore, it is not necessary that we express different theorems for the situation that "a" is prime itself.

1.4.1. Basic theorem

We can write each natural number  with only one way as multiplication of prime numbers (without paying attention to the order factors). A part of the theorem that is proved above, now we prove that showing each natural number is in form of multiplication of unique prime numbers. We suppose that it is possible to write "a" in two different ways as the form of multiplication of prime numbers, it means that  and  which "" and "" are prime and "" is not exactly to "" exactly, Then we have:

Now, if we delete equal prime numbers from both side of equality, we will have:

That  and. It results and from lemma (1.3.2) it is necessary that  aliquot one of the prime numbers. Therefore  must be equal to one of "" that is against of omitting the equal prime numbers from the both sides of above equation. So presupposition is paradox that we can write "a" in two different methods as multiplication of prime numbers. So, there fore, the theorem will be proved.

    In writing integer number , as multiplication of prime numbers, some times it is better to use symbol which shows all of the different prime factors of "a". Previous theorem indicated that we can write each natural number, only in one way:

          (1)

    that are the different prime factors of "a" and  . The right side of equation (1) named in form of standard factorization of integer number "a".

    We suppose  and (1) is in standard factorization of "a". Now, we suppose;  and. If we express each one of "c" or "d" as multiplication of prime numbers (it isn’t necessary for all of them to be different) and we put it instead of "c" and "d" in, then "a" will be written as multiplication of prime numbers. If we look at equal prime numbers together in this multiplication, then according to the basic theorem (1.4.6), number "a" is written exactly as form of (1). Therefore, we will have easy but important following theorem.

1.4.2. Theorem

    If 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:

       (2)

It is clear for number "a" that we can give  with selection of   in (2) also with Selection of  we can get "a" itself. Remember that there is no need for (2) to be the standard factorization of "d", because it is possible that some or all of the powers are zero. We can get the greatest common divisor and the smallest common multiplication of two numbers  and  from the standard factorization of "a" and "b". So, we change symbols a little. We suppose  are different prime factors of "a" or "b". Therefore:

  ,                  (3)

    that "" and "" could be zero. Nevertheless, for each, at least one of "" or "" is greater than zero. We show the greatest and smallest number of  respectively by and.

    It is obvious that when, then, with this expression, the following result is clear.

1.4.3. Theorem

    If "a" and "b" are in form of (3), then:

    for example, we suppose that  and . Standard factorization of these integer numbers are  and

    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  are in  which,  is one of these four values 0,1,2,3 and  is one of these three numbers 0,1,2  and  is one of these two numbers 0 and 1. So,  selection exist for  and. Therefore, the number of divisor of 360 is.

    Generally,  selections exist for  in (2) and so we can say:

    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  for clear expression. We can show that multiplication:

    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 , so that ,  and , namely equal to the sum of divisors of number 360, because:

    Therefore, the sum of divisors of number 360 is:

.

1.4.4. Attention

    Sometimes, it is better that we use symbol for the summation, that domain of "d" is positive divisors of "a".

    For example,  is sum of products all divisors of "a".

    It is interesting to remember that, we can factorize all of the numbers smaller than , it means 10000 to multiplication of prime factors with knowing the prime numbers smaller than 100. We suppose "N" is a non–prime number and smaller than 10000, we have in this situation. In which "a" and "b" are prime or non-prime. If "a" and "b" are greater than 100, in this state, multiplication of "ab" will be greater than 10000 and this is against our supposition, that "N" should be smaller than 10000. If for example "a" is a number smaller than 100, it means that there is a prime factor smaller than 100 (namely one of the 25 factors which are mentioned before). Therefore it is enough to know that the number of "N" is divisible by which of these 25 numbers and if "N" is not divisible by any of them, it is prime number. For example we consider number 7458:

    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:

           (1)

    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 (for 5 values of "n"), in this way we will have  numbers. But among these numbers which aren’t divisible by 2 or 3, there are only two numbers which are divisible by 5. These two numbers come from the multiple of 5 by two numbers less than 6 (which are coprime with 2 and 3). Subsequently the number of numbers which aren’t divisible by any numbers 2 and 3 and 5, are equal to:

    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, obtain from the relation (2) for seven values of "n", namely 0,1,2,3,4,5,6.

    By this method we get  numbers which all of them are smaller than "210" and none of them are divisible by 2, 3 or 5.

    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
 as if we omit the mentioned numbers in relation (3).

    If  is representative of every one of these 48 numbers, the numbers which are in this form:

        (4)

    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  numbers[4] of, which none of them are divisible by 2, 3, 5, 7, 11 and they are written in this form[5]  and we must only search for the prime number between them. Now if we consider the prime number like 13 too, then we will obtain:

    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  numbers which are divisible by one of these numbers 2, 3, 5, 7, 11 and 13; in the first step, it facilitates the job.

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  numbers which will not be obtained as the multiple of one of these prime numbers by the others.

    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 and

1.8.1. Definition

    If "n" is integer and positive, then we will show the number of positive divisors of "n" by and the sum of all of the positive divisors by.

    In the following theorem, we obtain a formula for calculation of  and by using the decomposition into the prime factors.

1.8.2. Theorem

    Suppose, therefore:

and also:

    Proof: suppose  is a positive divisor of "n". For each "i" we have, then, for all "", there are  different selections.
(Indeed. Therefore, we can select the powers of  in different ways. Then:

    For calculating, at first, note the following product:

    This multiplication is equal to the sum of all possible products of so that.

    But family of all these products is exactly equal to the sum of all of the positive divisors of "n" therefore. To complete the proof, at first we consider:

    (instead it is enough to multiply  by)

    Therefore, we have:

1.8.3. Remark

    The function  and  are examples of number theory functions. They have common and very important qualities. Both  and  are multiplicative, it means for both two coprime numbers "m" and "n", we have:

    Generally, the function "f ", which is defined on set of positive integer numbers, is called multiplication if and only if for all "n" and "m" that:

    To prove multiplicative forms of and, we can gain some results directly and with some calculations and by using some formulas.

    Function-Euler is another important multiplicative function that we acquaint of it here.

1.8.4. Conclusion

    If "P" is a prime number:

 (: Euler's Function)

    Proof. This assertion is clear for "k=1". Because if "P" be prime, then:

    If, since "P" is prime, then the numbers which are not coprimes with are as follow:

    Therefore, the number of numbers is not coprime when "P^k” is equal to "P^k-1" and the other numbers are coprime with to "P^k". The number of them is equal to ""or  that here, assertion is proved. Now we can consider it as a demonstration.

1.8.5. Result

    We know that if (a, b, c,…) =1, then:

    Therefore, we can write for (p, q, r, … are prime factors):

    So:

    For example, the number of numbers smaller than 100 that are coprime with it is calculated by the following method:

1.9. Perfect numbers

1.9.1. Definition

    We call integer number  as perfect, if it equals the sum of divisors smaller than itself. Therefore, "n" is perfect if and only if, that  is the sum of divisor of "n" (Containing itself). Mental principle of perfect numbers roots back to ancient Greece's times and we must search it in history. Greeks had a lot of secret properties for these numbers. Greek mathematicians had a lot of tendency to these numbers. Although they knew only 4 perfect numbers in Euclid themes that are: 6, 28, 496 and 8128.

    In spite of this little and deficit information, they guessed even perfect numbers finish with 6 or 8 that 5th and 6th number of perfect numbers are 33550336 and 8589869056 that both of them finish to "6". Of course this result is correct that perfect number finishes with 6 or 8. Euclid mentioned the following method for calculating perfect numbers in his book "preliminaries".

1.9.2. Theorem’s Euclid

    Suppose  is prime, then  is a perfect number.

    Proof. Suppose then. Since "p" is prime, then divisors of  are in or  form so that.

    Therefore:

    Therefore "N" is a perfect number.

    The natural question that propounded is that:

    Is the inverse of this Euclid theorem established? Are all of perfect numbers, in mentioned form of (1.9.2)?

    Almost, 2000 years after that, Euler answered this important question.

1.9.3. Remark

    Euclid algorithm is not an organized method to calculate the highest common divisor of two numbers. Euclid expressed and proved this algorithm in his book "Preliminaries". Of course, may be this algorithm was known before Euclid, below lemma is the key of understanding Euclid algorithm.

1.9.4. Lemma

    Suppose "m" and "n" are integer numbers that both of them are not "o" together. So, for every correct number:

1.9.5. Theorem’s Euler

    All of the even perfect numbers are in form which  is a prime number.

    Proof. Suppose "N" is an even perfect number. At that rate

    Put, which  and "m" is a prime number.

    Since and  is a multiplicative function, we will have:

    By solving the equivalence, with respect to  we will have:                                        

      (2)

    Therefore, is integer number. Then both "m" and   are divisors of "m". Because,, from this we know that "m" and  are only the positive divisors of "m". So, means   and the result of "m" is a prime number.

    Here, we remember two problems about perfect numbers.

    The first of them is this odd perfect number. So, we know if there is an odd perfect number, it must be greater than 10³⁰⁰ and therefore it has 8 distinctive prime factors at least.

    With these descriptions, we can result that there is no odd perfect number.

    The second problem is not responded until now that, Are the number of perfect numbers infinite? In the primary ages they have known four perfect numbers which has been mentioned before.

    But the fifth of these numbers has not been discovered until 15th century. Now, we know 42 even perfect numbers (2005). The first 21 of them have been discovered up to the year 1900.

    For example, the known perfect number 2⁸⁵⁹⁴³² (2⁸⁵⁹⁴³³-1) is an ogre of mathematics with 517430 Digit almost. Then existing infinite perfect numbers, has been remained an open problem till now. Now after determining the highest prime number of 21st century, in fact, the highest perfect number also calculates the:

    Highest known perfect = 2^25964950(2^25964951-1) number of the year 2005.

    We finish this section by mentioning some important theorems about prime numbers like Bertrand’s principle. We know that Bertrand’s principle propounds existing of prime numbers between numbers "n" and "2n".

    Joseph Bertrand propounded his guess in 1845 and searched numbers between "1" and "3,000,000" But Russian mathematician Chebyshev solved it logically. Although its proof is easier than prime numbers "1", remember only its stronger results.

1.10. Bertrand’s principle and theorems of

Chebyshev, Dirichlet and Poisson

1.10.1. Bertrand’s principle

    For every, a prime number exists between "n" and "2n".A stronger result of Bertrand’s principle is:

1.10.2. Result

    If "n>5", then at least two different prime numbers exist between "n" and "2n". Another obvious result is that inequality "" results

1.10.3. Generalization

    In 1892 Bertrand’s principle way extended by James Joseph Sylvester in the following form:

    If "m" and "n" are two positive integer numbers and, then at least, one of numbers  has a prime divisor greater than "n". (With thesis that, Bertrand’s principle is resulted)

    Another important theorem about prime numbers is Dirichlet’s theorem that if "a" and "b" (a0) are coprime, then infinite prime numbers exist in  form that kn. It is obvious that if "a" and "b" have common factors greater than "1", then for every integer number "k", is a compound number.

    For example, existing infinite prime numbers in forms of "4k +3" and "4k +1" is obvious.

1.10.4. Dirichlet’s theorem

    Suppose "", "" and, therefore infinite integer number "k" exists, so that is "a" prime number.

    Dirichlet’s theorem is the first important application of analytic methods in number theory. In fact, Dirichlet’s theorem and prime numbers theorem are two important theorems of primary theory of numbers that are solved by analytic methods. For both of these two theorems, primary proofs exist that it doesn’t use deep theorems of functions theory. But expressing their proof is so hard that it is not in the frame of this book.

    Now, we propound a combination of prime numbers theorem and Dirichlet’s theorem:

1.10.5. Poisson’s theorem

    Suppose "a" is a positive number and (a, b) =1 and it defines  that is the number of prime numbers in form and smaller than "x". Then we can approach quotient of  sufficiently to, if "x" is great sufficiently.

1.10.6. Remark

    Limit of does not depend on choosing "b" until "a" and "b" are coprime. Quotient tends to a limit that only depends on "a".

1.10.7. Result of theorem (1.10.5)

    If "" and "" are sets of digits so that e_n is odd and "", then there will be infinite prime numbers so that start with digits "" and finish with "". Theorem (1.10.5) has very beautiful interpolation that probably it doesn’t see obviously from above propositions.

    fore example, if "", then it is resulted from theorem (1.10.5) that (since "") half of prime numbers are in  form and another half is in form. (Exactly, we can approach the ratio of "" to ½ for "x" that is sufficiently great). Therefore, we can find out from theorem (1.10.5) that if "", then a quarter of all prime numbers are in  form and the last quarter in form. Generally, by supposing "", if "" (mod a), then "n" is in form if and only if "n" is in "" form.

    So, we can only calculate different values for "n" so that "a" and "b" are coprime. Therefore, it is resulted from theorem (1.10.5), that for every permissible value of "b", sequence of the numbers  includes the equal ratio of prime numbers; it means that the ratio of prime numbers is.

1.11. Lagrange’s theorem

    Suppose "p" is a prime number and "n" a natural number then the greatest power of prime number "p" in "n!" is equal to:

    (: Greatest integer part).

Proof. We want to count the number of prime factors of "p" in "n!" .

The number of integer numbers among the numbers "1, 2, … and  n" that "p"

aliquot them, is equal to .But some of these numbers are also divisible by "". Especially in the sequence 1, 2, ... and n, the number of numbers which is divisible by "" is exactly equal to and etc.

Therefore the sum of , is equal to the number of prime factors of "p" that exists in n! ; It is necessary to mention that this summation has always a finite number of non-zero terms. Because for supposed "n", there is "k", so that, therefore, .

On the other word:

We suppose numbers "n" and  as natural numbers and  as a prime number. It is clear that numbers of  sequence are divisible by must be in  form that  is a natural number and adapted in condition, in which. It is obvious that the number of   values is equal to. On the other hand, "t" ("p" power) that appears in factorization of "n!" to prime factors of n is obtained from sum of numbers, which are the number of  sequence’s terms divisible by  or or or ... .Therefore justification of relation (1) is verified.

1.11.1. Example

Calculate the greatest power of that aliquot "62!".

Solution. According to and considering this point that greatest power of "5" is smaller than the greatest power of "3" which aliquot "62!", it is enough to determine the greatest power of "5" which aliquot "62!":

It is obvious that the greatest power of "3" which aliquot 62! is the least "14". So the greatest power of "15" which aliquot "62!" is "14" (number).

1.11.2. Example

How many zeros the number 100! is ended to?

Solution. In fact, we must calculate he greatest power of in "100!". Since the greatest power of "5" is smaller than the greatest power of "2" which aliquots "100!", We determine only the greatest power of "5" that aliquots 100! :

Therefore, 100! is ended to "24" number of zeros.

1.11.3. Example

How many zeros the number  is ended to?

Solution. The greatest power of "10" that aliquots "340!" is:

and also, the greatest power of "10" that aliquots "170!" is:

Therefore, number is ended to "83-41=42" number of zeros.

1.11.4. Example

Find natural number "n" in which "n!" is ended to "20" number of zero.

Solution. It is clear that number "n!" is ended to "20" number of zeros if and only if the greatest power of "5" that aliquots "n!" is equal to "". On the other hand we want to find "n" that

If "", then and if, then "", therefore, it is clear that.

If, then, we can write:

If and, then:

   ; 

Therefore, values of "n" will be determined for  and "":

;

1.11.5. Example

If we suppose that "n!" finishes to  the number of zeros, shows that is close to  for great number of "n".

Solution. It is clear that is equal to the greatest power of "5" which aliquots "n!" (According to theorem 1.11):

          (1)

It is obvious that if, then  and it means that there is not more than "" nonzero terms in. Here, we consider the following geometric progression:

                        (2)

By comparing every term of (1) series with every term of (2) series:

In special case:

       (3)

The limit of infinite terms of geometric progression (2) is equal to, so, we have the following relation by substituting this value instead of:

(In fact, if "n" be great infinitely,  is great infinitely)

             (4)

Since the rate of changing of logarithmic function is slower than "n", then if we choose "n" great enough, we can approach the quotient of  to so that is equal to "n/4" approximately.

1.11.6. Example

Whether number "n!" can finish to 48 numbers of zeros?

Solution. It is clear that for solving this problem, it is enough to determine the greatest power of "5" which aliquots "n!”. For, we calculate:

On the other hand, this summation is equal to for. Therefore,  cannot finish to  number of zeros for no natural number of

1.11.7. Paractice. How many zero the number is ended to?