Definition 1. Prime number is a natural number greater than 1 that is divisible only by itself and 1.

In other words, a number is prime if it has only two distinct natural divisors.

Definition 2. Any natural number that has other divisors besides itself and one is called composite number.

In other words, natural numbers that are not prime are called composite numbers. Definition 1 implies that a composite number has more than two natural divisors. The number 1 is neither prime nor composite. has only one divisor 1 and, besides this, many theorems about prime numbers do not hold for unity.

It follows from Definitions 1 and 2 that every positive integer greater than 1 is either a prime or a composite number.

Below is a program for displaying prime numbers up to 5000. Fill in the cells, click on the "Create" button and wait a few seconds.

Prime number table

Statement 1. If a p is a prime number and a any integer, then either a divided by p, or p and a relatively prime numbers.

Really. If a p prime number, then it is only divisible by itself and 1 if a not divisible by p, then the greatest common divisor a and p equals 1. Then p and a relatively prime numbers.

Statement 2. If the product of several numbers of numbers a 1 , a 2 , a 3 , ... is divisible by a prime number p, then at least one of the numbers a 1 , a 2 , a 3 , ... is divisible by p.

Really. If none of the numbers is divisible by p, then the numbers a 1 , a 2 , a 3 , ... would be relatively prime numbers with respect to p. But from Corollary 3 () it follows that their product a 1 , a 2 , a 3 , ... is also coprime with respect to p, which contradicts the condition of the assertion. Therefore, at least one of the numbers is divisible by p.

Theorem 1. Any composite number can always be represented, and moreover in a unique way, as a product of a finite number of prime numbers.

Proof. Let k composite number, and let a 1 is one of its divisors different from 1 and itself. If a a 1 is composite, then it has in addition to 1 and a 1 and another divider a 2. If a a 2 is a composite number, then it has, in addition to 1 and a 2 and another divider a 3 . Arguing in this way and taking into account that the numbers a 1 , a 2 , a 3 , ... decrease and this series contains a finite number of terms, we will reach some prime number p one . Then k can be represented as

Suppose there are two expansions of a number k:

Because k=p 1 p 2 p 3 ... is divisible by a prime number q 1 , then at least one of the factors, for example p 1 is divisible by q one . But p 1 is prime and is only divisible by 1 and itself. Consequently p 1 =q 1 (because q 1 ≠1)

Then from (2) we can exclude p 1 and q 1:

Thus, we make sure that any prime number that enters the first expansion as a factor one or more times enters the second expansion at least the same number of times and vice versa, any prime number that enters the second expansion as a factor one or several times also enters the first expansion at least as many times. Therefore, any prime number enters as a factor in both expansions the same number of times and, thus, these two expansions are the same.■

Decomposition of a composite number k can be written in the following form

(3)

where p 1 , p 2 , ... distinct prime numbers, α, β, γ ... integer positive numbers.

Decomposition (3) is called canonical decomposition numbers.

Prime numbers in the series of natural numbers occur unevenly. In some parts of the series there are more of them, in others - less. The further we move along the number series, the rarer the prime numbers. The question is, is there a largest prime number? The ancient Greek mathematician Euclid proved that there are infinitely many prime numbers. We present this proof below.

Theorem 2. The number of prime numbers is infinite.

Proof. Suppose there are a finite number of primes, and let the largest prime be p. Let's consider all the numbers p. By the assumption of the statement, these numbers must be composite and must be divisible by at least one of the prime numbers. Let's choose a number that is the product of all these primes plus 1:

Number z more p because 2p already more p. p is not divisible by any of these prime numbers, since when divided by each of them, it gives a remainder of 1. Thus we arrive at a contradiction. Therefore, there are an infinite number of prime numbers.

This theorem is a special case of a more general theorem:

Theorem 3. Let an arithmetic progression be given

Then any prime number in n, should also be included in m, so in n cannot include other prime factors that are not included in m and, moreover, these prime factors in n appear no more times than in m.

The reverse is also true. If every prime factor of a number n occurs at least the same number of times m, then m divided by n.

Statement 3. Let a 1 ,a 2 ,a 3 ,... various primes appearing in m so

where i=0,1,...α , j=0,1,...,β , k=0,1,..., γ . notice, that a i accepts α +1 values, β j accepts β +1 values, γ k takes γ +1 values, ... .

The division of natural numbers into prime and composite is attributed to the ancient Greek mathematician Pythagoras. And if you follow Pythagoras, then the set of natural numbers can be divided into three classes: (1) - a set consisting of one number - one; (2, 3, 5, 7, 11, 13, ) is the set of prime numbers; (4, 6, 8, 9, 10, 12, 14, 15, ) is the set of composite numbers.

Many different mysteries conceal the second set. But first, let's figure out what a prime number is. We open the “Mathematical Encyclopedic Dictionary” (Yu. V. Prokhorov, publishing house “Soviet Encyclopedia”, 1988) and read:

“A prime number is a positive integer greater than one that has no other divisors than itself and one: 2,3,5,7,11,13,

The concept of a prime number is fundamental in the study of the divisibility of natural numbers; namely, the fundamental theorem of arithmetic states that every positive integer, except for 1, can be uniquely decomposed into a product of prime numbers (the order of the factors is not taken into account). There are infinitely many prime numbers (this proposition, called Euclid's theorem, was already known to ancient Greek mathematicians, its proof can be found in book 9 of Euclid's Elements). P. Dirichlet (1837) established that in an arithmetic progression a+bx at x=1. ,2,с with coprime integers a and b also contains infinitely many primes.

To find prime numbers from 1 to x, the well-known from the 3rd century is used. BC e. sieve of Eratosthenes. Considering the sequence (*) of primes from 1 to x shows that as x increases, it becomes rarer on average. There are arbitrarily long segments of a series of natural numbers, among which there is not a single prime number (Theorem 4). At the same time, there are such prime numbers, the difference between which is equal to 2 (the so-called twins). Until now (1987) it is unknown whether the set of such twins is finite or infinite. Tables of primes within the first 11 million natural numbers show very large twins (for example, 10,006,427 and 10,006,429).

Elucidation of the distribution of prime numbers in the natural series of numbers is a very difficult problem in number theory. It is posed as the study of the asymptotic behavior of a function denoting the number of primes not exceeding a positive number x. It is clear from Euclid's theorem that at. L. Euler introduced the zeta function in 1737.

He also proved that

Where the summation is carried out over all natural numbers, and the product is taken over all primes. This identity and its generalizations play a fundamental role in the theory of the distribution of prime numbers. Proceeding from this, L. Euler proved that the series and the product in prime p diverge. Moreover, L. Euler established that there are “many” prime numbers, because

And at the same time, almost all natural numbers are composite, since at.

and, for any (i.e., what grows as a function). Chronologically, the next significant result that refines Chebyshev's theorem is the so-called. the asymptotic law of distribution of prime numbers (J. Hadamard, 1896, Ch. La Vallee Poussin, 1896), which consisted in the fact that the limit of the ratio to is equal to 1. Subsequently, significant efforts of mathematicians were directed to clarifying the asymptotic law of distribution of prime numbers. Questions of the distribution of prime numbers are studied both by elementary methods and by methods of mathematical analysis.

Here it makes sense to prove some of the theorems given in the article.

Lemma 1. If gcd(a, b)=1, then there are integers x, y such that.

Proof. Let a and b be relatively prime numbers. Consider the set J of all natural numbers z, representable in the form, and choose the smallest number d in it.

Let us prove that a is divisible by d. Divide a by d with remainder: and let. Since it has the form, therefore,

We see that.

Since we assumed that d is the smallest number in J, we have a contradiction. So a is divisible by d.

In the same way, we prove that b is divisible by d. So d=1. The lemma is proven.

Theorem 1. If the numbers a and b are coprime and the product bx is divisible by a, then x is divisible by a.

Proof 1. We have to prove that ax is divisible by b and gcd(a,b)=1, then x is divisible by b.

By Lemma 1, there are x, y such that. Then, obviously, is divisible by b.

Proof 2. Consider the set J of all natural numbers z such that zc is divisible by b. Let d be the smallest number in J. It is easy to see that. Similarly to the proof of Lemma 1, we prove that a is divisible by d and b is divisible by d

Lemma 2. If the numbers q,p1,p2,pn are prime and the product is divisible by q, then one of the numbers pi is equal to q.

Proof. First of all, note that if a prime number p is divisible by q, then p=q. This immediately implies the assertion of the lemma for n=1. For n=2 it follows directly from Theorem 1: if p1p2 is divisible by a prime number q u, then p2 is divisible by q (ie).

We prove the lemma for n=3 as follows. Let p1 p2 p3 be divisible by q. If p3 = q, then everything is proved. If, then according to Theorem 1, p1 p2 is divisible by q. Thus, we have reduced the case n=3 to the already considered case n=2.

Similarly, from n=3 we can go to n=4, then to n=5, and in general, assuming that n=k the assertion of the lemma is proved, we can easily prove it for n=k+1. This convinces us that the lemma is true for all n.

Fundamental theorem of arithmetic. Every natural number can be decomposed into prime factors in a unique way.

Proof. Suppose that there are two factorizations of the number a into prime factors:

Since the right side is divisible by q1, the left side of the equality must also be divisible by q1. According to Lemma 2, one of the numbers is equal to q1. Let us cancel both sides of the equality by q1.

Let us carry out the same reasoning for q2, then for q3, for qi. In the end, all factors on the right will be reduced and 1 will remain. Naturally, nothing will remain on the left except one. Hence we conclude that the two expansions and can differ only in the order of the factors. The theorem has been proven.

Euclid's theorem. The number of prime numbers is infinite.

Proof. Assume that the series of prime numbers is finite, and denote the last prime number by the letter N. Compose the product

Let's add 1 to it. We get:

This number, being an integer, must contain at least one prime factor, that is, it must be divisible by at least one prime number. But all prime numbers, by assumption, do not exceed N, and the number M + 1 is not divisible without a remainder by any of the prime numbers less than or equal to N - each time the remainder is 1. The theorem is proved.

Theorem 4. Sections of composite numbers between prime numbers can be of any length. We will now prove that the series consists of n consecutive composite numbers.

These numbers go directly one after another in the natural series, since each next one is 1 more than the previous one. It remains to prove that they are all composite.

First number

Even, since both of its terms contain a factor of 2. And any even number greater than 2 is composite.

The second number consists of two terms, each of which is a multiple of 3. Therefore, this number is composite.

Similarly, we establish that the next number is a multiple of 4, and so on. In other words, each number in our series contains a factor that is different from one and itself; it is therefore composite. The theorem has been proven.

Having studied the proofs of the theorems, we continue the consideration of the article. In her text, the sieve of Eratosthenes was mentioned as a way to find prime numbers. Let's read about this method from the same dictionary:

“The sieve of Eratosthenes is a method developed by Eratosthenes that allows you to sift out composite numbers from the natural series. The essence of the sieve of Eratosthenes is as follows. The unit is crossed out. The number two is simple. All natural numbers divisible by 2 are crossed out. Number 3 - the first uncrossed number will be prime. Further, all natural numbers that are divisible by 3 are crossed out. The number 5 - the next uncrossed number - will be simple. Continuing similar calculations, one can find an arbitrarily long segment of a sequence of prime numbers. The sieve of Eratosthenes as a theoretical method for studying number theory was developed by W. Brun (1919).

Here is the largest number currently known to be prime:

This number has about seven hundred decimal places. The calculations by which it was found that this number is prime were carried out on modern computers.

“The Riemann zeta function, -function, is an analytic function of a complex variable, for σ>1, determined by an absolutely and uniformly convergent Dirichlet series:

For σ>1, the representation in the form of the Euler product is valid:

(2) where p runs through all prime numbers.

The identity of series (1) and product (2) is one of the main properties of the zeta function. It allows one to obtain various relationships that connect the zeta function with the most important number-theoretic functions. Therefore, the zeta function plays a big role in number theory.

The zeta function was introduced as a function of a real variable by L. Euler (1737, publ. 1744), who indicated its location in the product (2). Then the zeta function was considered by P. Dirichlet and especially successfully by P. L. Chebyshev in connection with the study of the law of distribution of prime numbers. However, the most profound properties of the zeta function were discovered after the works of B. Riemann, who for the first time in 1859 considered the zeta function as a function of a complex variable, he also introduced the name "zeta function" and the designation """.

But the question arises: what practical application is there for all this work on prime numbers? Indeed, there is almost no use for them, but there is one area where prime numbers and their properties are applied to this day. This is cryptography. Here, prime numbers are used in encryption systems without the transfer of keys.

Unfortunately, this is all that is known about prime numbers. There are still many mysteries left. For example, it is not known whether the set of prime numbers representable as two squares is infinite.

"NON-SIMPLE PRIME NUMBERS".

I decided to do a little research in order to find answers to some questions about prime numbers. First of all, I have compiled a program that prints out all consecutive prime numbers less than 1,000,000,000. In addition, I have compiled a program that determines whether the entered number is prime. To study the problems of prime numbers, I built a graph that marks the dependence of the value of a prime number on the ordinal number. As a further research plan, I decided to use the article by I. S. Zeltser and B. A. Kordemsky "Amusing flocks of prime numbers." The authors identified the following research paths:

1. 168 places of the first thousand natural numbers are occupied by prime numbers. Of these, 16 numbers are palindromic - each is equal to the reverse: 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929.

There are only 1061 four-digit primes, and none of them are palindromic.

There are many five-digit simple palindromic numbers. They include such beauties: 13331, 15551, 16661, 19991. Undoubtedly, there are flocks of this type: ,. But how many copies are in each such flock?

3+x+x+x+3 = 6+3x = 3(2+x)

9+x+x+x+9 = 18+3x =3(6+x)

It can be seen that the sum of the digits of the numbers and is divisible by 3, therefore these numbers themselves are also divisible by 3.

As for the numbers of the form, among them the numbers 72227, 75557, 76667, 78887, 79997 are prime.

2. In the first thousand numbers there are five "quartets" consisting of consecutive prime numbers, the last digits of which form the sequence 1, 3, 7, 9: (11, 13, 17, 19), (101, 103, 107, 109 ), (191, 193, 197, 199), (211, 223, 227, 229), (821, 823, 827, 829).

How many such quartets are there among n-digit primes for n>3?

Using the program I wrote, I found the quartet missed by the authors: (479, 467, 463, 461) and quartets for n = 4, 5, 6. For n = 4, there are 11 quartets

3. A flock of nine primes: 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879 - is attractive not only because it is an arithmetic progression with a difference of 210, but also because it can fit into nine cells so that that a magic square is formed with a constant equal to the difference of two prime numbers: 3119 - 2:

The next, tenth member of the progression under consideration, 2089, is also a prime number. If you remove the number 199 from the flock, but include 2089, then in this composition the flock can form a magic square - a topic for search.

It should be noted that there are other magic squares consisting of prime numbers:

1847 6257 6197 3677 1307 1877 2687

2267 1427 5987 5927 1667 2027 4547

2897 947 2357 4517 3347 5867 3917

3557 4157 4397 3407 2417 2657 3257

4337 5717 3467 2297 4457 1097 2477

4817 4767 827 887 5147 5387 1997

4127 557 617 3137 5507 4937 4967

The proposed square is curious because

1. It is a 7x7 magic square;

2. It contains a 5x5 magic square;

3. A 5x5 magic square contains a 3x3 magic square;

4. All these squares have one common central number - 3407;

5. All 49 numbers included in the 7x7 square end in the number 7;

6. All 49 numbers included in the 7x7 square are prime numbers;

7. Each of the 49 numbers included in a 7x7 square can be represented as 30n + 17.

The programs used were written by me in the Dev-C++ programming language and I provide their texts in the appendix (see files with the .cpp extension). In addition to all of the above, I wrote a program that decomposes consecutive natural numbers into prime factors (see Divisors 1. cpp) and a program that decomposes only the entered number into prime factors (see Divisors 2. cpp). Since these programs in compiled form take up too much space, only their texts are given. However, anyone can compile them if they have the right program.

BIOGRAPHIES OF SCIENTISTS INVOLVED IN THE PROBLEM OF PRIME NUMBERS

EUCLIDES

(about 330 BC - about 272 BC)

Very little reliable information has been preserved about the life of the most famous mathematician of Antiquity. It is believed that he studied in Athens, which explains his brilliant command of the geometry developed by the school of Plato. However, apparently, he was not familiar with the writings of Aristotle. He taught in Alexandria, where he earned high praise for his teaching activities during the reign of Ptolemy I Soter. There is a legend that this king demanded to reveal to him a way to achieve rapid success in mathematics, to which Euclid replied that there were no royal ways in geometry (a similar story, however, is also told about Menchem, who was allegedly asked about the same by Alexander the Great). Tradition has preserved the memory of Euclid as a benevolent and modest person. Euclid is the author of treatises on various topics, but his name is associated mainly with one of the treatises called "Beginnings". It is about a collection of works by mathematicians who worked before him (the most famous of them was Hippocrates of Kos), the results of which he brought to perfection thanks to his ability to generalize and diligence.

EULER (EULER) LEONARD

(Basel, Switzerland 1707 - St. Petersburg, 1783)

Mathematician, mechanic and physicist. Born in the family of a poor pastor Paul Euler. He received his education first from his father, and in 1720–24 at the University of Basel, where he attended lectures on mathematics by I. Bernoulli.

At the end of 1726, Euler was invited to the St. Petersburg Academy of Sciences and in May 1727 arrived in St. Petersburg. In the newly organized academy, Euler found favorable conditions for scientific activity, which allowed him to immediately start studying mathematics and mechanics. During the 14 years of the first Petersburg period of his life, Euler prepared about 80 works for publication and published more than 50. In St. Petersburg, he studied Russian.

Euler participated in many activities of the St. Petersburg Academy of Sciences. He gave lectures to students of the academic university, participated in various technical examinations, worked on compiling maps of Russia, and wrote the publicly available "Guide to Arithmetic" (1738–40). On special instructions from the Academy, Euler prepared for publication Naval Science (1749), a fundamental work on the theory of shipbuilding and navigation.

In 1741, Euler accepted the offer of the Prussian king Frederick II to move to Berlin, where the reorganization of the Academy of Sciences was to take place. At the Berlin Academy of Sciences, Euler took the post of director of the mathematics class and a member of the board, and after the death of its first president, P. Maupertuis, for several years (since 1759) he actually directed the academy. For 25 years of his life in Berlin, he prepared about 300 works, among them a number of large monographs.

While living in Berlin, Euler did not stop working intensively for the St. Petersburg Academy of Sciences, retaining the title of its honorary member. He conducted an extensive scientific and scientific-organizational correspondence, in particular, he corresponded with M. Lomonosov, whom he highly valued. Euler edited the mathematical department of the Russian academic scientific body, where during this time he published almost as many articles as in the "Memoirs" of the Berlin Academy of Sciences. He actively participated in the training of Russian mathematicians; future academicians S. Kotelnikov, S. Rumovsky and M. Sofronov were sent to Berlin to study under his leadership. Euler provided great assistance to the St. Petersburg Academy of Sciences, acquiring scientific literature and equipment for it, negotiating with candidates for positions at the academy, etc.

On July 17 (28), 1766, Euler and his family returned to St. Petersburg. Despite his advanced age and the almost complete blindness that befell him, he worked productively until the end of his life. During the 17 years of his second stay in St. Petersburg, he prepared about 400 works, among them several large books. Euler continued to participate in the organizational work of the academy. In 1776, he was one of the experts on the project of a single-arch bridge across the Neva, proposed by I. Kulibin, and out of the entire commission, he alone gave broad support to the project.

Euler's merits as a prominent scientist and organizer of scientific research were highly appreciated during his lifetime. In addition to the St. Petersburg and Berlin academies, he was a member of the largest scientific institutions: the Paris Academy of Sciences, the Royal Society of London and others.

One of the hallmarks of Euler's work is his exceptional productivity. Only during his lifetime, about 550 of his books and articles were published (the list of Euler's works contains about 850 titles). In 1909, the Swiss Natural Science Society began publishing Euler's complete works, which was completed in 1975; it consists of 72 volumes. Of great interest is Euler's colossal scientific correspondence (about 3,000 letters), which has so far been published only partially.

Euler's circle of studies was unusually wide, covering all departments of contemporary mathematics and mechanics, the theory of elasticity, mathematical physics, optics, music theory, machine theory, ballistics, marine science, insurance business, etc. About 3/5 of Euler's works belong to mathematics, the remaining 2/5 mainly to its applications. The scientist systematized his results and the results obtained by others in a number of classical monographs, written with amazing clarity and provided with valuable examples. These are, for example, “Mechanics, or the Science of Motion, Stated Analytically” (1736), “Introduction to Analysis” (1748), “Differential Calculus” (1755), “Theory of Motion of a Rigid Body” (1765), “Universal Arithmetic” (1768-69), which went through about 30 editions in 6 languages, "Integral Calculus" (1768-94), etc. In the XVIII century. and partly in the 19th century. The publicly available Letters on Various Physical and Philosophical Matters, written to a certain German princess, gained immense popularity. (1768–74), which went through over 40 editions in 10 languages. Most of the content of Euler's monographs was then included in the textbooks for higher and partially secondary schools. It is impossible to list all the theorems, methods and formulas of Euler hitherto used, of which only a few appear in the literature under his name [for example, Euler's broken line method, Euler's substitutions, Euler's constant, Euler's equations, Euler's formulas, Euler's function, Euler's numbers, Euler's formula - Maclaurin, Euler-Fourier formulas, Euler characteristic, Euler integrals, Euler angles].

In "Mechanics" Euler first expounded the dynamics of a point with the help of mathematical analysis: the free movement of a point under the action of various forces both in a vacuum and in a medium with resistance; movement of a point along a given line or along a given surface; movement under the influence of central forces. In 1744 he first correctly formulated the mechanical principle of least action and showed its first applications. In The Theory of Motion of a Rigid Body, Euler developed the kinematics and dynamics of a rigid body and gave the equations for its rotation around a fixed point, laying the foundation for the theory of gyroscopes. In his theory of the ship, Euler made a valuable contribution to the theory of stability. Significant are Euler's discoveries in celestial mechanics (for example, in the theory of the motion of the moon), and continuum mechanics (the basic equations of motion of an ideal fluid in the form of Euler and in the so-called Lagrange variables, gas oscillations in pipes, etc.). In optics, Euler (1747) gave the formula for a biconvex lens and proposed a method for calculating the refractive index of a medium. Euler adhered to the wave theory of light. He believed that different colors correspond to different wavelengths of light. Euler proposed ways to eliminate chromatic aberrations of lenses and gave methods for calculating the optical components of a microscope. Euler devoted an extensive series of works, begun in 1748, to mathematical physics: problems of the vibration of a string, plate, membrane, etc. All these studies stimulated the development of the theory of differential equations, approximate methods of analysis, and special. functions, differential geometry, etc. Many of Euler's mathematical discoveries are contained precisely in these works.

Euler's main work as a mathematician was the development of mathematical analysis. He laid the foundations of several mathematical disciplines that were only in their infancy or were completely absent in the infinitesimal calculus of I. Newton, G. Leibniz, and the Bernoulli brothers. Thus, Euler was the first to introduce functions of a complex argument and to study the properties of the basic elementary functions of a complex variable (exponential, logarithmic, and trigonometric functions); in particular, he derived formulas relating trigonometric functions to exponential. Euler's work in this direction marked the beginning of the theory of functions of a complex variable.

Euler was the creator of the calculus of variations, described in the work “Method for finding curved lines with maximum or minimum properties. » (1744). The method by which Euler in 1744 derived the necessary condition for the extremum of a functional, the Euler equation, was a prototype of the direct methods of the calculus of variations of the 20th century. Euler created the theory of ordinary differential equations as an independent discipline and laid the foundations for the theory of partial differential equations. Here he owns a huge number of discoveries: the classical method of solving linear equations with constant coefficients, the method of variation of arbitrary constants, elucidation of the basic properties of the Riccati equation, the integration of linear equations with variable coefficients using infinite series, criteria for special solutions, the doctrine of the integrating factor, various approximate methods and a number of techniques for solving partial differential equations. Euler compiled a significant part of these results in his "Integral Calculus".

Euler also enriched differential and integral calculus in the narrow sense of the word (for example, the theory of change of variables, the theorem on homogeneous functions, the concept of the double integral, and the calculation of many special integrals). In the "Differential Calculus", Euler expressed and supported with examples his conviction in the expediency of using divergent series and proposed methods for the generalized summation of series, anticipating the ideas of the modern rigorous theory of divergent series, created at the turn of the 19th and 20th centuries. In addition, Euler obtained many concrete results in the theory of series. He opened the so-called. the Euler-Maclaurin summation formula, proposed the transformation of series that bears his name, determined the sums of a huge number of series, and introduced new important types of series into mathematics (for example, trigonometric series). Euler's studies on the theory of continued fractions and other infinite processes adjoin here.

Euler is the founder of the theory of special functions. He first began to consider the sine and cosine as functions, and not as segments in a circle. He obtained almost all the classical expansions of elementary functions into infinite series and products. In his works, the theory of the γ-function was created. He investigated the properties of elliptic integrals, hyperbolic and cylindrical functions, the ζ-function, some θ-functions, the integral logarithm, and important classes of special polynomials.

According to P. Chebyshev, Euler laid the foundation for all the research that constitutes the general part of number theory. Thus, Euler proved a number of statements made by P. Fermat (for example, Fermat's little theorem), developed the foundations of the theory of power residues and the theory of quadratic forms, discovered (but did not prove) the quadratic reciprocity law, and studied a number of problems in Diophantine analysis. In works on the division of numbers into terms and on the theory of prime numbers, Euler was the first to use the methods of analysis, thus being the creator of analytic number theory. In particular, he introduced the ζ-function and proved the so-called. Euler's identity relating prime numbers to all natural numbers.

Euler's merits are also great in other areas of mathematics. In algebra, he owns works on the solution of equations of higher degrees in radicals and on equations in two unknowns, as well as the so-called. Euler's four-square identity. Euler made significant progress in analytic geometry, especially in the theory of second-order surfaces. In differential geometry, he studied in detail the properties of geodesic lines, for the first time applied the natural equations of curves, and most importantly, he laid the foundations of the theory of surfaces. He introduced the concept of principal directions at a point on a surface, proved their orthogonality, derived a formula for the curvature of any normal section, began studying developable surfaces, etc.; in one posthumously published work (1862), he partially anticipated the research of K. Gauss on the intrinsic geometry of surfaces. Euler also dealt with individual questions of topology and proved, for example, an important theorem on convex polyhedra. Euler the mathematician is often described as a brilliant "calculator". Indeed, he was an unsurpassed master of formal calculations and transformations; in his works, many mathematical formulas and symbols received a modern look (for example, he owns the designations for e and π). However, Euler also introduced a number of deep ideas into science, which are now strictly substantiated and serve as a model for the depth of penetration into the subject of research.

According to P. Laplace, Euler was a teacher of mathematicians in the second half of the 18th century.

DIRICHLET PETER GUSTAV

(Düren, now Germany, 1805 - Göttingen, ibid., 1859)

He studied in Paris, maintained friendly relations with outstanding mathematicians, in particular with Fourier. Upon receiving his degree, he was a professor at the universities of Breslau (1826 - 1828), Berlin (1828 - 1855) and Göttingen, where he became head of the department of mathematics after the death of the scientist Carl Friedrich Gauss. His most outstanding contribution to science concerns number theory, primarily the study of series. This allowed him to develop the series theory proposed by Fourier. Created his own version of the proof of Fermat's theorem, used analytic functions to solve arithmetic problems, and introduced convergence criteria for series. In the field of mathematical analysis he improved the definition and concept of a function, in the field of theoretical mechanics he focused on the study of the stability of systems and on the Newtonian concept of potential.

CHEBYSHEV PAFNUTIY LVOVYCH

Russian mathematician, founder of the St. Petersburg scientific school, academician of the St. Petersburg Academy of Sciences (1856). Chebyshev's works laid the foundation for the development of many new branches of mathematics.

Chebyshev's most numerous works are in the field of mathematical analysis. He was, in particular, the subject of a dissertation for the right to lecture, in which Chebyshev investigated the integrability of certain irrational expressions in algebraic functions and logarithms. Chebyshev also devoted a number of other works to the integration of algebraic functions. In one of them (1853), a well-known theorem on integrability conditions in elementary functions of a differential binomial was obtained. An important area of ​​research in mathematical analysis is his work on the construction of a general theory of orthogonal polynomials. The reason for its creation was the parabolic interpolation by the least squares method. Chebyshev's investigations on the problem of moments and on quadrature formulas adjoin the same circle of ideas. With the reduction of calculations in mind, Chebyshev proposed (1873) to consider quadrature formulas with equal coefficients (approximate integration). Research on quadrature formulas and on the theory of interpolation were closely connected with the tasks that were set for Chebyshev in the artillery department of the military scientific committee.

In the theory of probability, Chebyshev is credited with the systematic introduction to the consideration of random variables and the creation of a new technique for proving the limit theorems of probability theory - the so-called. method of moments (1845, 1846, 1867, 1887). He proved the law of large numbers in a very general form; At the same time, his proof is striking in its simplicity and elementarity. Chebyshev did not complete the study of the conditions for the convergence of distribution functions of sums of independent random variables to the normal law. However, A. A. Markov managed to do this with some addition of Chebyshev's methods. Without rigorous derivations, Chebyshev also outlined the possibility of refinements of this limit theorem in the form of asymptotic expansions of the distribution function of the sum of independent terms in powers of n21/2, where n is the number of terms. Chebyshev's work on probability theory constitutes an important stage in its development; in addition, they were the basis on which the Russian school of probability theory grew up, which at first consisted of direct students of Chebyshev.

Riemann Georg Friedrich Bernhard

(Breselenz, Lower Saxony, 1826 - Selaska, near Intra, Italy 66)

German mathematician. In 1846 he entered the University of Göttingen: he listened to the lectures of K. Gauss, many of whose ideas were developed by him later. In 1847–49 he attended lectures at the University of Berlin; in 1849 he returned to Göttingen, where he became close friends with Gauss' collaborator, physicist W. Weber, who aroused in him a deep interest in questions of mathematical natural science.

In 1851 he defended his doctoral thesis "Fundamentals of the general theory of functions of one complex variable." From 1854 Privatdozent, from 1857 professor at the University of Göttingen.

Riemann's work had a great influence on the development of mathematics in the second half of the 19th century. and in the 20th century. In his doctoral dissertation, Riemann laid the foundation for the geometric direction of the theory of analytic functions; he introduced the so-called Riemann surfaces, which are important in the study of multivalued functions, developed the theory of conformal mappings and, in connection with this, gave the basic ideas of topology, studied the conditions for the existence of analytic functions inside regions of various types (the so-called Dirichlet principle), etc. Methods developed by Riemann were widely used in his further works on the theory of algebraic functions and integrals, on the analytic theory of differential equations (in particular, equations defining hypergeometric functions), on analytic number theory (for example, Riemann indicated the connection between the distribution of prime numbers and the properties of the ζ-function, in in particular with the distribution of its zeros in the complex domain - the so-called Riemann hypothesis, the validity of which has not yet been proven), etc.

In a number of papers, Riemann investigated the expansion of functions into trigonometric series and, in connection with this, determined the necessary and sufficient conditions for integrability in the sense of Riemann, which was important for the theory of sets and functions of a real variable. Riemann also proposed methods for integrating partial differential equations (for example, using the so-called Riemann invariants and the Riemann function).

In his famous 1854 lecture "On the Hypotheses Underlying Geometry" (1867), Riemann gave the general idea of ​​a mathematical space (in his words, "manifolds"), including functional and topological spaces. Here he considered geometry in a broad sense as the doctrine of continuous n-dimensional manifolds, i.e., collections of any homogeneous objects, and, generalizing the results of Gauss on the intrinsic geometry of a surface, he gave the general concept of a linear element (the differential of the distance between points of a manifold), thereby defining what are called Finsler spaces. In more detail, Riemann considered the so-called Riemannian spaces, generalizing the spaces of the geometries of Euclid, Lobachevsky and Riemann's elliptic geometry, characterized by a special type of linear element, and developed the theory of their curvature. Discussing the application of his ideas to physical space, Riemann raised the question of the "causes of the metric properties" of it, as if anticipating what had been done in the general theory of relativity.

The ideas and methods proposed by Riemann opened up new paths in the development of mathematics and found application in mechanics and the general theory of relativity. The scientist died in 1866 from tuberculosis.

Numbers are different: natural, natural, rational, integer and fractional, positive and negative, complex and prime, odd and even, real, etc. From this article you can learn what prime numbers are.

What numbers are called the English word "simple"?

Very often, schoolchildren do not know how to answer one of the most seemingly simple questions in mathematics, about what a prime number is. They often confuse prime numbers with natural numbers (that is, the numbers that people use when counting objects, while in some sources they start from zero, and in others - from one). But these are two completely different concepts. Prime numbers are natural numbers, that is, integer and positive numbers that are greater than one and that have only 2 natural divisors. In this case, one of these divisors is a given number, and the second is a unit. For example, three is a prime number because it is not evenly divisible by any number other than itself and one.

Composite numbers

The opposite of prime numbers are composite numbers. They are also natural, also greater than one, but have not two, but more divisors. So, for example, the numbers 4, 6, 8, 9, etc. are natural, composite, but not prime numbers. As you can see, these are mostly even numbers, but not all. But the “two” is an even number and the “first number” in a series of prime numbers.

Subsequence

To build a series of prime numbers, it is necessary to make a selection from all natural numbers, taking into account their definition, that is, you need to act by contradiction. It is necessary to consider each of the natural positive numbers on the subject of whether it has more than two divisors. Let's try to build a series (sequence) that consists of prime numbers. The list starts with two, then comes three, since it's only divisible by itself and one. Consider the number four. Does it have divisors other than four and one? Yes, that number is 2. So four is not a prime number. Five is also prime (besides 1 and 5, it is not divisible by any other number), but six is ​​divisible. And in general, if you follow all the even numbers, you will notice that apart from “two”, none of them is prime. From this we conclude that even numbers, except for two, are not prime. Another discovery: all numbers that are divisible by three, except for the triple itself, whether even or odd, are also not prime (6, 9, 12, 15, 18, 21, 24, 27, etc.). The same applies to numbers that are divisible by five and seven. All their set is also not simple. Let's summarize. So, all odd numbers, except for one and nine, belong to simple single-digit numbers, and only “two” from even ones. The tens themselves (10, 20,... 40, etc.) are not prime. Two-digit, three-digit, etc. prime numbers can be defined based on the above principles: if they have no other divisors than themselves and one.

Theories about the properties of prime numbers

There is a science that studies the properties of integers, including prime ones. This is a branch of mathematics, which is called higher. In addition to the properties of integers, she also deals with algebraic, transcendental numbers, as well as functions of various origins related to the arithmetic of these numbers. In these studies, in addition to elementary and algebraic methods, analytical and geometric ones are also used. Specifically, the study of prime numbers deals with "Number Theory".

Prime numbers are the “building blocks” of natural numbers

In arithmetic there is a theorem called the main theorem. According to it, any natural number, except for unity, can be represented as a product, the factors of which are prime numbers, and the order of the factors is unique, which means that the representation method is unique. It is called the decomposition of a natural number into prime factors. There is another name for this process - factorization of numbers. Proceeding from this, prime numbers can be called “building material”, “blocks” for constructing natural numbers.

Search for prime numbers. Simplicity Tests

Many scientists of different times tried to find some principles (systems) for finding a list of prime numbers. Science knows systems called Atkin's sieve, Sundartam's sieve, Eratosthenes' sieve. However, they do not give any significant results, and a simple test is used to find prime numbers. Algorithms were also created by mathematicians. They are called primality tests. For example, there is a test developed by Rabin and Miller. It is used by cryptographers. There is also a Kayala-Agrawala-Saskena test. However, despite its sufficient accuracy, it is very difficult to calculate, which diminishes its practical value.

Does the set of primes have a limit?

The fact that the set of primes is infinity was written in the book "Beginnings" by the ancient Greek scientist Euclid. He said this: “Let's imagine for a moment that prime numbers have a limit. Then let's multiply them with each other, and add one to the product. The number obtained as a result of these simple operations cannot be divisible by any of the series of prime numbers, because the remainder will always be one. And this means that there is some other number that is not yet included in the list of prime numbers. Therefore, our assumption is not true, and this set cannot have a limit. In addition to Euclid's proof, there is a more modern formula given by the eighteenth-century Swiss mathematician Leonhard Euler. According to him, the sum, the reciprocal of the sum of the first n numbers, grows indefinitely with the growth of the number n. And here is the formula of the theorem regarding the distribution of prime numbers: (n) grows like n / ln (n).

What is the largest prime number?

All the same Leonard Euler was able to find the largest prime number for his time. This is 2 31 - 1 = 2147483647. However, by 2013, another most accurate largest in the list of prime numbers was calculated - 2 57885161 - 1. It is called the Mersenne number. It contains about 17 million decimal digits. As you can see, the number found by a scientist from the eighteenth century is several times smaller than this. It should have been so, because Euler did this calculation manually, while our contemporary was probably helped by a computer. Moreover, this number was obtained at the Department of Mathematics in one of the American departments. Numbers named after this scientist pass through the Luc-Lehmer primality test. However, science does not want to stop there. The Electronic Frontier Foundation, which was founded in 1990 in the United States of America (EFF), has offered a monetary reward for finding large primes. And if until 2013 the prize was given to those scientists who will find them from among 1 and 10 million decimal numbers, today this figure has reached from 100 million to 1 billion. Prizes range from 150 to 250 thousand US dollars.

Names of special prime numbers

Those numbers that were found thanks to algorithms created by certain scientists and passed the simplicity test are called special. Here is some of them:

1. Mersin.

4. Cullen.

6. Mills et al.

The simplicity of these numbers, named after the above scientists, is established using the following tests:

1. Lucas-Lemer.

2. Pepina.

3. Riesel.

4. Billhart - Lehmer - Selfridge and others.

Modern science does not stop there, and probably in the near future the world will know the names of those who were able to win a prize of 250,000 dollars by finding the largest prime number.

List of divisors. By definition, the number n is prime only if it is not evenly divisible by 2 and any integers other than 1 and itself. The above formula removes unnecessary steps and saves time: for example, after checking if a number is divisible by 3, there is no need to check if it is divisible by 9.

• The floor(x) function rounds x to the nearest integer less than or equal to x.

Learn about modular arithmetic. The operation "x mod y" (mod is an abbreviation of the Latin word "modulo", that is, "module") means "divide x by y and find the remainder". In other words, in modular arithmetic, upon reaching a certain value, which is called module, the numbers "turn" back to zero. For example, a clock measures time in modulus 12: it shows 10, 11, and 12 o'clock and then returns to 1.

• Many calculators have a mod key. The end of this section shows how to manually calculate this function for large numbers.

Learn about the pitfalls of Fermat's Little Theorem. All numbers for which the test conditions are not met are composite, but the remaining numbers are only probably are considered simple. If you want to avoid incorrect results, look for n in the list of "Carmichael numbers" (composite numbers that satisfy this test) and "pseudo-prime Fermat numbers" (these numbers meet the conditions of the test only for some values a).

If convenient, use the Miller-Rabin test. Although this method is rather cumbersome for manual calculations, it is often used in computer programs. It provides acceptable speed and gives less errors than Fermat's method. A composite number will not be taken as a prime number if calculations are made for more than ¼ values a. If you randomly select different values a and for all of them the test will give a positive result, we can assume with a fairly high degree of confidence that n is a prime number.

For large numbers, use modular arithmetic. If you don't have a mod calculator handy, or if your calculator isn't designed to handle such large numbers, use the power properties and modular arithmetic to make your calculations easier. Below is an example for 3 50 (\displaystyle 3^(50)) mod 50:

• Rewrite the expression in a more convenient form: mod 50. When calculating manually, further simplifications may be necessary.
• (3 25 ∗ 3 25) (\displaystyle (3^(25)*3^(25))) mod 50 = mod 50 mod 50) mod 50. Here we have taken into account the property of modular multiplication.
• 3 25 (\displaystyle 3^(25)) mod 50 = 43.
• (3 25 (\displaystyle (3^(25)) mod 50 ∗ 3 25 (\displaystyle *3^(25)) mod 50) mod 50 = (43 ∗ 43) (\displaystyle (43*43)) mod 50.
• = 1849 (\displaystyle =1849) mod 50.
• = 49 (\displaystyle=49).

A prime number is a natural number that is only divisible by itself and one.

The rest of the numbers are called composite.

Simple natural numbers

But not all natural numbers are prime.

Simple natural numbers are only those that are divisible only by themselves and by one.

Examples of prime numbers:

2; 3; 5; 7; 11; 13;...

Simple integers

It follows that only natural numbers are prime numbers.

This means that prime numbers are necessarily natural.

But all natural numbers are also integers.

Thus, all prime numbers are integers.

Examples of prime numbers:

2; 3; 5; 7; 11; 13; 17; 19; 23;...

Even prime numbers

There is only one even prime number, and that is two.

All other prime numbers are odd.

Why can't an even number greater than two be a prime number?

But because any even number greater than two will be divisible by itself, not by one, but by two, that is, such a number will always have three divisors, and possibly more.