6 is a prime number. Prime numbers: history and facts

Prime numbers are one of the most interesting mathematical phenomena, which have attracted the attention of scientists and ordinary citizens for more than two millennia. Despite the fact that we now live in the age of computers and the most modern information programs, many riddles of prime numbers have not yet been solved; there are even some that scientists do not know how to approach.

Prime numbers are, as is known from the course of elementary arithmetic, those that are divisible without a remainder only by one and itself. By the way, if a natural number is divisible, in addition to those listed above, by any other number, then it is called composite. One of the most famous theorems states that any composite number can be represented as a unique possible product of prime numbers.

Some interesting facts. Firstly, the unit is unique in the sense that, in fact, it does not belong to either prime or composite numbers. At the same time, in the scientific community it is still customary to classify it specifically as belonging to the first group, since formally it fully satisfies its requirements.

Secondly, the only even number squeezed into the “prime numbers” group is, naturally, two. Any other even number simply cannot get here, since by definition, in addition to itself and one, it is also divisible by two.

Prime numbers, the list of which, as stated above, can begin with one, represent an infinite series, as infinite as the series of natural numbers. Based on the fundamental theorem of arithmetic, we can come to the conclusion that prime numbers are never interrupted and never end, since otherwise the series of natural numbers would inevitably be interrupted.

Prime numbers do not appear randomly in the natural series, as they might seem at first glance. Having carefully analyzed them, you can immediately notice several features, the most interesting of which are associated with the so-called “twin” numbers. They are called that because in some incomprehensible way they ended up next to each other, separated only by an even delimiter (five and seven, seventeen and nineteen).

If you look closely at them, you will notice that the sum of these numbers is always a multiple of three. Moreover, when dividing the left one by three, the remainder always remains two, and the right one always remains one. In addition, the very distribution of these numbers over the natural series can be predicted if we imagine this entire series in the form of oscillatory sinusoids, the main points of which are formed when numbers are divided by three and two.

Prime numbers are not only the object of close consideration by mathematicians all over the world, but have long been successfully used in the compilation of various series of numbers, which is the basis, among other things, for cryptography. It should be recognized that a huge number of mysteries associated with these wonderful elements are still waiting to be solved; many questions have not only philosophical, but also practical significance.

All natural numbers, except one, are divided into prime and composite. A prime number is a natural number that has only two divisors: one and itself. All others are called composite. The study of the properties of prime numbers is carried out by a special branch of mathematics - number theory. In ring theory, prime numbers are related to irreducible elements.

Here is a sequence of prime numbers starting from 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, ... etc.

According to the fundamental theorem of arithmetic, every natural number that is greater than one can be represented as a product of prime numbers. At the same time, this is the only way to represent natural numbers up to the order of the factors. Based on this, we can say that prime numbers are the elementary parts of natural numbers.

This representation of a natural number is called decomposition of a natural number into prime numbers or factorization of a number.

One of the most ancient and effective ways to calculate prime numbers is the “sieve of Erasstophenes”.

Practice has shown that after calculating prime numbers using the sieve of Erastophenes, it is necessary to check whether the given number is prime. For this purpose, special tests have been developed, the so-called simplicity tests. The algorithm of these tests is probabilistic. They are most often used in cryptography.

By the way, for some classes of numbers there are specialized effective primality tests. For example, to check the primality of Mersenne numbers, the Luc-Lehmer test is used, and to check the primality of Fermat numbers, the Pepin test is used.

We all know that there are infinitely many numbers. The question rightly arises: how many prime numbers are there then? There are also an infinite number of prime numbers. The most ancient proof of this proposition is Euclid's proof, which is set out in the Elements. Euclid's proof looks like this:

Let's imagine that the number of prime numbers is finite. Let's multiply them and add one. The resulting number cannot be divided by any of the finite set of prime numbers, because the remainder of division by any of them gives one. Thus, the number must be divisible by some prime number not included in this set.

The prime number distribution theorem states that the number of prime numbers less than n, denoted π(n), grows as n / ln(n).

After thousands of years of studying prime numbers, the largest known prime number is 243112609 − 1. This number has 12,978,189 decimal digits and is the Mersenne prime number (M43112609). This discovery was made on August 23, 2008 at the Faculty of Mathematics of uCLA University as part of the distributed search for Mersenne prime numbers project GIMPS.

The main distinguishing feature of Mersenne numbers is the presence of a highly effective Luc-Lemaire primality test. With its help, the Mersenne primes are, over a long period of time, the largest known primes.

However, to this day, many questions regarding prime numbers have not received precise answers. At the 5th International Congress of Mathematics, Edmund Landau formulated the main problems in the field of prime numbers:

Goldbach's problem or Landau's first problem is that it is necessary to prove or disprove that every even number greater than 2 can be represented as the sum of two primes, and every odd number greater than 5 can be represented as the sum of three primes numbers.
Landau's second problem requires finding an answer to the question: is there an infinite set of “prime twins” - prime numbers whose difference is 2?
Legendre's conjecture or Landau's third problem is: is it true that between n2 and (n + 1)2 there is always a prime number?
Landau's fourth problem: is the set of prime numbers of the form n2 + 1 infinite?
In addition to the above problems, there is the problem of determining an infinite number of prime numbers in many integer sequences such as the Fibonacci number, Fermat number, etc.

The division of natural numbers into prime and composite numbers 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, ) – set of prime numbers; (4, 6, 8, 9, 10, 12, 14, 15, ) – a set of composite numbers.

The second set hides many different mysteries. 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, which has no divisors other 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 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 known to ancient Greek mathematicians; its proof can be found in book 9 of Euclid’s Elements). P. Dirichlet (1837) established that in the arithmetic progression a + bx for x = 1. ,2,c with coprime integers a and b also contains infinitely many prime numbers.

To find prime numbers from 1 to x, it is known from the 3rd century. BC e. Eratosthenes' sieve method. An examination of the sequence (*) of prime numbers from 1 to x shows that as x increases it becomes, on average, rarer. 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 (so-called twins). It is still unknown (1987) whether the set of such twins is finite or infinite. Tables of prime numbers within the first 11 million natural numbers show the presence of very large twins (for example, 10,006,427 and 10,006,429).

Finding out the distribution of prime numbers in the natural series of numbers is a very difficult problem in number theory. It is formulated as the study of the asymptotic behavior of a function denoting the number of prime numbers not exceeding a positive number x. From Euclid's theorem it is clear that when. L. Euler introduced the zeta function in 1737.

He also proved that when

Where the summation is carried out over all natural numbers, and the product is taken over all prime numbers. This identity and its generalizations play a fundamental role in the theory of distribution of prime numbers. Based on this, L. Euler proved that the series and the product with respect to 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, C. La Vallée Poussin, 1896), which stated that the limit of the ratio to is equal to 1. Subsequently, significant efforts of mathematicians were directed to clarify 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 provide a proof of some of the theorems given in the article.

Lemma 1. If GCD(a, b)=1, then there exist 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 get a contradiction. This means a is divisible by d.

Let us prove in the same way that b is divisible by d. So d=1. The lemma is proven.

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

Proof1. We must prove that ax is divisible by b and gcd(a,b)=1, then x is divisible by b.

By Lemma 1, there exist x, y such that. Then obviously it 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. Similar to the proof of Lemma 1, it is proved 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 follows the statement of the lemma for n=1. For n=2 it follows directly from Theorem 1: if p1p2 is divisible by a prime number q and, then p2 is divisible by q(i.e).

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

In the same way, from n=3 we can go to n=4, then to n=5, and in general, assuming that the n=k statement of the lemma is proven, 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 factorized in a unique way.

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

Since the right side is divisible by q1, then the left side of the equality must 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's carry out the same reasoning for q2, then for q3, for qi. In the end, all the factors on the right will cancel and 1 will remain. Naturally, on the left there will be nothing left except one. From here 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 series of prime numbers is infinite.

Proof. Suppose that the series of prime numbers is finite, and we denote the last prime number by the letter N. Let us compose the product

Let's add 1 to it. We get:

This number, being an integer, must contain at least one prime factor, i.e. 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 proven.

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

These numbers come directly after each other 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 every even number greater than 2 is composite.

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

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

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

“The Eratosthenes sieve 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. Number two is prime. All natural numbers divisible by 2 are crossed out. Number 3 – the first uncrossed out number will be prime. Next, all natural numbers that are divisible by 3 are crossed out. The number 5, the next uncrossed out number, will be prime. Continuing similar calculations, you 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 V. 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 established that this number is prime were carried out on modern computers.

“The Riemann zeta function, -function, is an analytical function of a complex variable, for σ>1 determined absolutely and uniformly by a 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 us to obtain various relationships connecting 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 work 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 used to this day. This is cryptography. Here prime numbers are used in encryption systems without transferring 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.

"DIFFICULT PRIMES".

I decided to do a little research to find answers to some questions about prime numbers. First of all, I compiled a program that produces all consecutive prime numbers less than 1,000,000,000. In addition, I compiled a program that determines whether the entered number is prime. To study the problems of prime numbers, I built a graph indicating the dependence of the size 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 “Interesting flocks of prime numbers.” The authors identified the following research paths:

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

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

There are many five-digit prime palindromic numbers. They include such beauties: 13331, 15551, 16661, 19991. Undoubtedly, there are flocks of this type: ,. But how many specimens are there 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 numbers is divisible by 3, therefore these numbers themselves are also divisible by 3.

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

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

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

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 interesting because

1. It is a 7x7 magic square;

2. It contains a 5x5 magic square;

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

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

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

6. All 49 numbers included in a 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 extension . srr). In addition to all of the above, I wrote a program that decomposes consecutive natural numbers into prime factors (see Divisors 1. срр) and a program that decomposes only the entered number into prime factors (see Divisors 2. срр). Since these programs take up too much space in compiled form, only their texts are given. However, anyone can compile them if they have a suitable program.

BIOGRAPHIES OF SCIENTISTS INVOLVED IN THE PROBLEM OF PRIMES

EUCLIDES

(c. 330 BC – c. 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 mastery of geometry, developed by the school of Plato. However, apparently, he was not familiar with the works 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 that he discover a way to achieve quick success in mathematics, to which Euclid replied that there are 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 the Elements. 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 hard work.

EULER LEONARD

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

Mathematician, mechanic and physicist. Born into 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 he arrived in St. Petersburg. In the newly organized academy, Euler found favorable conditions for scientific activity, which allowed him to immediately begin studying mathematics and mechanics. During the 14 years of the first St. Petersburg period of his life, Euler prepared about 80 works for publication and published over 50. In St. Petersburg, he studied the Russian language.

Euler participated in many areas of activity of the St. Petersburg Academy of Sciences. He lectured to students at the academic university, participated in various technical examinations, worked on compiling maps of Russia, and wrote a publicly accessible “Manual to Arithmetic” (1738–40). On special instructions from the Academy, Euler prepared for publication “Nautical 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 (from 1759) he actually led the academy. Over the 25 years of his life in Berlin, he prepared about 300 works, including a number of large monographs.

While living in Berlin, Euler did not stop working intensively for the St. Petersburg Academy of Sciences, maintaining the title of its honorary member. He conducted 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, purchasing scientific literature and equipment for it, negotiating with candidates for positions at the academy, etc.

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, including 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 of the entire commission, he was the only one who provided broad support for the project.

Euler's merits as a major 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 distinctive aspects of Euler's work is his exceptional productivity. During his lifetime alone, about 550 of his books and articles were published (the list of Euler's works contains approximately 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. Euler's colossal scientific correspondence (about 3,000 letters) is also of great interest; it has so far only been partially published.

Euler's range of activities 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, etc. About 3/5 of Euler's works relate 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 classic monographs, written with amazing clarity and supplied with valuable examples. These are, for example, “Mechanics, or the Science of Motion, Presented Analytically” (1736), “Introduction to Analysis” (1748), “Differential Calculus” (1755), “Theory of Rigid Body Motion” (1765), “Universal Arithmetic” (1768–69), which went through about 30 editions in 6 languages, “Integral Calculus” (1768–94), etc. In the 18th century. , and partly in the 19th century. The publicly available “Letters on various physical and philosophical matters, written to a certain German princess,” became extremely popular. "(1768–74), which went through over 40 editions in 10 languages. Most of the content of Euler's monographs was then included in textbooks for higher and partially secondary schools. It is impossible to list all of Euler's theorems, methods and formulas still in use, 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 outlined the dynamics of a point using mathematical analysis: the free movement of a point under the influence of various forces both in emptiness and in a medium with resistance; movement of a point along a given line or 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 Rigid Body Motion,” 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 valuable contributions to the theory of stability. Euler's discoveries were significant in celestial mechanics (for example, in the theory of the motion of the Moon), continuum mechanics (the basic equations of motion of an ideal fluid in Euler's form and in the so-called Lagrange variables, gas oscillations in pipes, etc.). In optics, Euler gave (1747) 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 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 techniques. functions, differential geometry, etc. Many of Euler’s mathematical discoveries are contained in these works.

Euler's main work as a mathematician was the development of mathematical analysis. He laid the foundations of several mathematical disciplines, which were only in their rudimentary form 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 investigate the properties of the basic elementary functions of a complex variable (exponential, logarithmic and trigonometric functions); in particular, he derived formulas connecting trigonometric functions with exponential functions. Euler's work in this direction laid the foundation for the theory of functions of a complex variable.

Euler was the creator of the calculus of variations, outlined in the work “Method of finding curved lines that have the properties of a maximum or minimum. "(1744). The method with which Euler in 1744 derived the necessary condition for the extremum of a functional - the Euler equation - was the 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 is responsible for a huge number of discoveries: the classical method of solving linear equations with constant coefficients, the method of varying arbitrary constants, elucidating the basic properties of the Riccati equation, integrating 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 collected 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 doctrine of changes of variables, the theorem on homogeneous functions, the concept of double integral and the calculation of many special integrals). In “Differential Calculus,” Euler expressed and supported with examples his belief in the advisability of using divergent series and proposed methods for generalized summation of series, anticipating the ideas of the modern strict theory of divergent series, created at the turn of the 19th and 20th centuries. In addition, Euler obtained many concrete results in series theory. He discovered 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 important new types of series into mathematics (for example, trigonometric series). This also includes Euler's research on the theory of continued fractions and other infinite processes.

Euler is the founder of the theory of special functions. He was the first to consider sine and cosine as functions, and not as segments in a circle. He obtained almost all classical expansions of elementary functions into infinite series and products. His works created the theory of the γ-function. He studied 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 makes up 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 his works on the division of numbers into terms and on the theory of prime numbers, Euler was the first to use methods of analysis, thereby becoming the creator of the analytic theory of numbers. In particular, he introduced the ζ-function and proved the so-called. Euler's identity connecting prime numbers with all natural numbers.

Euler also made great achievements in other areas of mathematics. In algebra, he wrote works on solving equations of higher degrees in radicals and on equations with two unknowns, as well as the so-called. Euler's four-square identity. Euler significantly advanced analytical geometry, especially the doctrine of second-order surfaces. In differential geometry, he studied in detail the properties of geodesic lines, was the first to apply natural equations of curves, and most importantly, 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 the study of developable surfaces, etc.; in one posthumously published work (1862), he partially anticipated the research of K. Gauss on the internal geometry of surfaces. Euler also dealt with certain questions of topology and proved, for example, an important theorem on convex polyhedra. Euler the mathematician is often characterized as a brilliant “calculator.” Indeed, he was an unsurpassed master of formal calculations and transformations; in his works, many mathematical formulas and symbolism received a modern look (for example, he owned the notation for e and π). However, Euler also introduced a number of profound ideas into science, which are now strictly substantiated and serve as an example of the depth of penetration into the subject of research.

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

DIRICHLET PETER GUSTAV

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

He studied in Paris and maintained friendly relations with outstanding mathematicians, in particular with Fourier. Upon receiving his academic 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 theory of series 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 Newton’s concept of potential.

CHEBYSHEV PAFNUTY LVOVICH

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. In particular, a dissertation for the right to give lectures was devoted to him, 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 consists of his work on the construction of a general theory of orthogonal polynomials. The reason for its creation was parabolic interpolation using the least squares method. Chebyshev’s research on the problem of moments and quadrature formulas is adjacent to this same circle of ideas. With a view to reducing calculations, Chebyshev proposed (1873) to consider quadrature formulas with equal coefficients (approximate integration). Research on quadrature formulas and the theory of interpolation was closely related to the tasks that were posed to Chebyshev in the artillery department of the military scientific committee.

In probability theory, Chebyshev is credited with systematically introducing random variables into the consideration and creating a new technique for proving limit theorems in probability theory - the so-called. method of moments (1845, 1846, 1867, 1887). He proved the law of large numbers in a very general form; Moreover, his proof is striking in its simplicity and elementaryness. Chebyshev did not bring the study of the conditions for the convergence of distribution functions of sums of independent random variables to the normal law to complete completion. However, through some addition to Chebyshev’s methods, A. A. Markov managed to do this. Without strict conclusions, Chebyshev also outlined the possibility of clarifying 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, initially consisting of Chebyshev’s direct students.

RIEMANN GEORG FRIEDRIGG BERNHARD

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

German mathematician. In 1846 he entered the University of Göttingen: he listened to lectures by 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 Gottingen, where he became close to Gauss’s collaborator, the physicist W. Weber, who aroused in him a deep interest in questions of mathematical science.

In 1851 he defended his doctoral dissertation “Fundamentals of the general theory of functions of one complex variable.” Since 1854, privatdozent, since 1857, professor at the University of Göttingen.

Riemann's works had a great influence on the development of mathematics in the 2nd 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 multi-valued 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 domains 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 analytical theory of differential equations (in particular, equations defining hypergeometric functions), on analytical 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 region - the so-called Riemann hypothesis, the validity of which has not yet been proven), etc.

In a number of works, Riemann investigated the decomposability of functions into trigonometric series and, in connection with this, determined necessary and sufficient conditions for integrability in the Riemannian sense, 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 Which Underlie Geometry” (1867), Riemann gave a general idea of ​​mathematical space (in his words, “manifolds”), including functional and topological spaces. Here he considered geometry in a broad sense as the study of continuous n-dimensional manifolds, i.e., collections of any homogeneous objects and, generalizing the results of Gauss on the internal geometry of a surface, he gave the general concept of a linear element (the differential of the distance between points of the manifold), thereby defining what are called Finsler spaces. Riemann examined in more detail the so-called Riemannian spaces, generalizing the spaces of Euclidean, Lobachevsky and Riemannian elliptic geometry, characterized by a special type of linear element, and developed the doctrine 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 was 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.

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

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

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

In other words, natural numbers that are not prime numbers are called composite numbers. From Definition 1 it follows that a composite number has more than two natural factors. The number 1 is neither prime nor composite because has only one divisor 1 and, in addition, many theorems regarding prime numbers do not hold for unity.

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

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

Prime numbers table

Statement 1. If p- prime number and a any integer, then either a divided by p, or p And a coprime numbers.

Really. If p A prime number is divisible only by itself and 1 if a not divisible by p, then the greatest common divisor a And p is equal to 1. Then p And a coprime 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, ...divisible by p.

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

Theorem 1. Any composite number can always be represented, and 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 1 is composite, then has in addition to 1 and a 1 and another divisor a 2. If a 2 is a composite number, then it has, in addition to 1 and a 2 and another divisor a 3. Reasoning 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 1. Then k can be represented in the form

Suppose there are two decompositions of a number k:

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

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

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

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

Where p 1 , p 2, ... various prime numbers, α, β, γ ... positive integers.

Expansion (3) is called canonical expansion numbers.

Prime numbers occur unevenly in the series of natural numbers. In some parts of the row there are more of them, in others - less. The further we move along the number series, the less common prime numbers are. The question arises, 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 prime numbers, and let the largest prime number be p. Let's consider all numbers greater p. By 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 prime numbers plus 1:

Number z more p because 2p already more p. p is not divisible by any of these prime numbers, because when divided by each of them gives a remainder of 1. Thus we come to 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 included in n, should be included in m, therefore in n other prime factors that are not included in m and, moreover, these prime factors in n are included no more times than in m.

The opposite is also true. If every prime factor of a number n included at least as many times in the number m, That m divided by n.

Statement 3. Let a 1 ,a 2 ,a 3,... various prime numbers included in m So

Where i=0,1,...α , j=0,1,...,β , k=0,1,..., γ . Note that α i accepts α +1 values, β j accepts β +1 values, γ k accepts γ +1 values, ... .

In this article we will explore prime and composite numbers. First, we will give definitions of prime and composite numbers, and also give examples. After this we will prove that there are infinitely many prime numbers. Next, we will write down a table of prime numbers, and consider methods for compiling a table of prime numbers, paying particular attention to the method called the sieve of Eratosthenes. In conclusion, we will highlight the main points that need to be taken into account when proving that a given number is prime or composite.

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Prime and Composite Numbers - Definitions and Examples

The concepts of prime numbers and composite numbers refer to numbers that are greater than one. Such integers, depending on the number of their positive divisors, are divided into prime and composite numbers. So to understand definitions of prime and composite numbers, you need to have a good understanding of what divisors and multiples are.

Definition.

Prime numbers are integers, large units, that have only two positive divisors, namely themselves and 1.

Definition.

Composite numbers are integers, large ones, that have at least three positive divisors.

Separately, we note that the number 1 does not apply to either prime or composite numbers. Unit has only one positive divisor, which is the number 1 itself. This distinguishes the number 1 from all other positive integers that have at least two positive divisors.

Considering that positive integers are , and that one has only one positive divisor, we can give other formulations of the stated definitions of prime and composite numbers.

Definition.

Prime numbers are natural numbers that have only two positive divisors.

Definition.

Composite numbers are natural numbers that have more than two positive divisors.

Note that every positive integer greater than one is either a prime or a composite number. In other words, there is not a single integer that is neither prime nor composite. This follows from the property of divisibility, which states that the numbers 1 and a are always divisors of any integer a.

Based on the information in the previous paragraph, we can give the following definition of composite numbers.

Definition.

Natural numbers that are not prime are called composite.

Let's give examples of prime and composite numbers.

Examples of composite numbers include 6, 63, 121, and 6,697. This statement also needs clarification. The number 6, in addition to positive divisors 1 and 6, also has divisors 2 and 3, since 6 = 2 3, therefore 6 is truly a composite number. Positive factors of 63 are the numbers 1, 3, 7, 9, 21 and 63. The number 121 is equal to the product 11·11, so its positive divisors are 1, 11 and 121. And the number 6,697 is composite, since its positive divisors, in addition to 1 and 6,697, are also the numbers 37 and 181.

In conclusion of this point, I would also like to draw attention to the fact that prime numbers and coprime numbers are far from the same thing.

Prime numbers table

Prime numbers, for the convenience of their further use, are recorded in a table called a table of prime numbers. Below is prime numbers table up to 1,000.

A logical question arises: “Why did we fill the table of prime numbers only up to 1,000, isn’t it possible to create a table of all existing prime numbers”?

Let's answer the first part of this question first. For most problems that require the use of prime numbers, prime numbers within a thousand will be sufficient. In other cases, most likely, you will have to resort to some special solution techniques. Although we can certainly create a table of prime numbers up to an arbitrarily large finite positive integer, be it 10,000 or 1,000,000,000, in the next paragraph we will talk about methods for creating tables of prime numbers, in particular, we will look at a method called.

Now let's look at the possibility (or rather, the impossibility) of compiling a table of all existing prime numbers. We cannot make a table of all the prime numbers because there are infinitely many prime numbers. The last statement is a theorem that we will prove after the following auxiliary theorem.

Theorem.

The smallest positive divisor other than 1 of a natural number greater than one is a prime number.

Proof.

Let a is a natural number greater than one, and b is the smallest positive divisor of a that is different from one. Let us prove that b is a prime number by contradiction.

Let's assume that b is a composite number. Then there is a divisor of the number b (let's denote it b 1), which is different from both 1 and b. If we also take into account that the absolute value of the divisor does not exceed the absolute value of the dividend (we know this from the properties of divisibility), then condition 1 must be satisfied

Since the number a is divisible by b according to the condition, and we said that b is divisible by b 1, the concept of divisibility allows us to talk about the existence of integers q and q 1 such that a=b q and b=b 1 q 1 , from where a= b 1 ·(q 1 ·q) . It follows that the product of two integers is an integer, then the equality a=b 1 ·(q 1 ·q) indicates that b 1 is a divisor of the number a. Taking into account the above inequalities 1

Now we can prove that there are infinitely many prime numbers.