A prime number is a natural number that is divisible only by itself and one.
The remaining numbers are called composite numbers.
Prime natural numbers
But not all natural numbers are prime numbers.
Prime natural numbers are only those that are divisible only by themselves and one.
Examples of prime numbers:
2; 3; 5; 7; 11; 13;...
Prime Integers
It follows that only natural numbers are prime numbers.
This means that prime numbers are necessarily natural numbers.
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 - the number 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 and by two, that is, such a number will always have three divisors, and possibly more.
Ilya's answer is correct, but not very detailed. In the 18th century, by the way, one was still considered a prime number. For example, such great mathematicians as Euler and Goldbach. Goldbach is the author of one of the seven problems of the millennium - the Goldbach hypothesis. The original formulation states that every even number can be represented as the sum of two prime numbers. Moreover, initially 1 was taken into account as a prime number, and we see this: 2 = 1+1. This is the smallest example that satisfies the original formulation of the hypothesis. Later it was corrected, and the formulation acquired a modern form: “every even number, starting with 4, can be represented as the sum of two prime numbers.”
Let's remember the definition. A prime number is a natural number p that has only 2 different natural divisors: p itself and 1. Corollary from the definition: a prime number p has only one prime divisor - p itself.
Now let's assume that 1 is a prime number. By definition, a prime number has only one prime divisor - itself. Then it turns out that any prime number greater than 1 is divisible by a prime number different from it (by 1). But two different prime numbers cannot be divided by each other, because otherwise they are not prime numbers, but composite numbers, and this contradicts the definition. With this approach, it turns out that there is only 1 prime number - the unit itself. But this is absurd. Therefore, 1 is not a prime number.
1, as well as 0, form another class of numbers - the class of neutral elements with respect to n-ary operations in some subset of the algebraic field. Moreover, with respect to the operation of addition, 1 is also a generating element for the ring of integers.
With this consideration, it is not difficult to discover analogues of prime numbers in other algebraic structures. Suppose we have a multiplicative group formed from powers of 2, starting from 1: 2, 4, 8, 16, ... etc. 2 acts as a formative element here. A prime number in this group is a number greater than the smallest element and divisible only by itself and the smallest element. In our group, only 4 have such properties. That’s it. There are no more prime numbers in our group.
If 2 were also a prime number in our group, then see the first paragraph - again it would turn out that only 2 is a prime number.
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 using both elementary methods and 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 the 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 this 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 constructed a graph indicating 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 “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 the right 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 available “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 broadly supported 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 calculus of infinitesimals by 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 series transformation 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 in 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 were 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 were 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 Göttingen, 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 studied 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.
- Translation
The properties of prime numbers were first studied by mathematicians of Ancient Greece. Mathematicians of the Pythagorean school (500 - 300 BC) were primarily interested in the mystical and numerological properties of prime numbers. They were the first to come up with ideas about perfect and friendly numbers.
A perfect number has a sum of its own divisors equal to itself. For example, the proper divisors of the number 6 are 1, 2 and 3. 1 + 2 + 3 = 6. The divisors of the number 28 are 1, 2, 4, 7 and 14. Moreover, 1 + 2 + 4 + 7 + 14 = 28.
Numbers are called friendly if the sum of the proper divisors of one number is equal to another, and vice versa - for example, 220 and 284. We can say that a perfect number is friendly to itself.
By the time of Euclid's Elements in 300 B.C. Several important facts about prime numbers have already been proven. In Book IX of the Elements, Euclid proved that there are an infinite number of prime numbers. This, by the way, is one of the first examples of using proof by contradiction. He also proves the Fundamental Theorem of Arithmetic - every integer can be represented uniquely as a product of prime numbers.
He also showed that if the number 2n-1 is prime, then the number 2n-1 * (2n-1) will be perfect. Another mathematician, Euler, was able to show in 1747 that all even perfect numbers can be written in this form. To this day it is unknown whether odd perfect numbers exist.
In the year 200 BC. The Greek Eratosthenes came up with an algorithm for finding prime numbers called the Sieve of Eratosthenes.
And then there was a big break in the history of the study of prime numbers, associated with the Middle Ages.
The following discoveries were made already at the beginning of the 17th century by the mathematician Fermat. He proved Albert Girard's conjecture that any prime number of the form 4n+1 can be written uniquely as the sum of two squares, and also formulated the theorem that any number can be written as the sum of four squares.
He developed a new method for factoring large numbers, and demonstrated it on the number 2027651281 = 44021 × 46061. He also proved Fermat's Little Theorem: if p is a prime number, then for any integer a it will be true that a p = a modulo p.
This statement proves half of what was known as the "Chinese conjecture" and dates back 2000 years: an integer n is prime if and only if 2 n -2 is divisible by n. The second part of the hypothesis turned out to be false - for example, 2,341 - 2 is divisible by 341, although the number 341 is composite: 341 = 31 × 11.
Fermat's Little Theorem served as the basis for many other results in number theory and methods for testing whether numbers are primes - many of which are still used today.
Fermat corresponded a lot with his contemporaries, especially with a monk named Maren Mersenne. In one of his letters, he hypothesized that numbers of the form 2 n +1 will always be prime if n is a power of two. He tested this for n = 1, 2, 4, 8 and 16, and was confident that in the case where n was not a power of two, the number was not necessarily prime. These numbers are called Fermat's numbers, and only 100 years later Euler showed that the next number, 2 32 + 1 = 4294967297, is divisible by 641, and therefore is not prime.
Numbers of the form 2 n - 1 have also been the subject of research, since it is easy to show that if n is composite, then the number itself is also composite. These numbers are called Mersenne numbers because he studied them extensively.
But not all numbers of the form 2 n - 1, where n is prime, are prime. For example, 2 11 - 1 = 2047 = 23 * 89. This was first discovered in 1536.
For many years, numbers of this kind provided mathematicians with the largest known prime numbers. That M 19 was proved by Cataldi in 1588, and for 200 years was the largest known prime number, until Euler proved that M 31 was also prime. This record stood for another hundred years, and then Lucas showed that M 127 is prime (and this is already a number of 39 digits), and after that research continued with the advent of computers.
In 1952 the primeness of the numbers M 521, M 607, M 1279, M 2203 and M 2281 was proven.
By 2005, 42 Mersenne primes had been found. The largest of them, M 25964951, consists of 7816230 digits.
Euler's work had a huge impact on the theory of numbers, including prime numbers. He extended Fermat's Little Theorem and introduced the φ-function. Factorized the 5th Fermat number 2 32 +1, found 60 pairs of friendly numbers, and formulated (but could not prove) the quadratic reciprocity law.
He was the first to introduce methods of mathematical analysis and develop analytical number theory. He proved that not only the harmonic series ∑ (1/n), but also a series of the form
1/2 + 1/3 + 1/5 + 1/7 + 1/11 +…
The result obtained by the sum of the reciprocals of prime numbers also diverges. The sum of n terms of the harmonic series grows approximately as log(n), and the second series diverges more slowly as log[ log(n) ]. This means that, for example, the sum of the reciprocals of all prime numbers found to date will give only 4, although the series still diverges.
At first glance, it seems that prime numbers are distributed quite randomly among integers. For example, among the 100 numbers immediately before 10000000 there are 9 primes, and among the 100 numbers immediately after this value there are only 2. But over large segments the prime numbers are distributed quite evenly. Legendre and Gauss dealt with issues of their distribution. Gauss once told a friend that in any free 15 minutes he always counts the number of primes in the next 1000 numbers. By the end of his life, he had counted all the prime numbers up to 3 million. Legendre and Gauss equally calculated that for large n the prime density is 1/log(n). Legendre estimated the number of prime numbers in the range from 1 to n as
π(n) = n/(log(n) - 1.08366)
And Gauss is like a logarithmic integral
π(n) = ∫ 1/log(t) dt
With an integration interval from 2 to n.
The statement about the density of primes 1/log(n) is known as the Prime Distribution Theorem. They tried to prove it throughout the 19th century, and progress was achieved by Chebyshev and Riemann. They connected it with the Riemann hypothesis, a still unproven hypothesis about the distribution of zeros of the Riemann zeta function. The density of prime numbers was simultaneously proved by Hadamard and Vallée-Poussin in 1896.
There are still many unsolved questions in prime number theory, some of which are hundreds of years old:
- The twin prime hypothesis is about an infinite number of pairs of prime numbers that differ from each other by 2
- Goldbach's conjecture: any even number, starting with 4, can be represented as the sum of two prime numbers
- Is there an infinite number of prime numbers of the form n 2 + 1?
- Is it always possible to find a prime number between n 2 and (n + 1) 2? (the fact that there is always a prime number between n and 2n was proven by Chebyshev)
- Is the number of Fermat primes infinite? Are there any Fermat primes after 4?
- is there an arithmetic progression of consecutive primes for any given length? for example, for length 4: 251, 257, 263, 269. The maximum length found is 26.
- Is there an infinite number of sets of three consecutive prime numbers in an arithmetic progression?
- n 2 - n + 41 is a prime number for 0 ≤ n ≤ 40. Is there an infinite number of such prime numbers? The same question for the formula n 2 - 79 n + 1601. These numbers are prime for 0 ≤ n ≤ 79.
- Is there an infinite number of prime numbers of the form n# + 1? (n# is the result of multiplying all prime numbers less than n)
- Is there an infinite number of prime numbers of the form n# -1 ?
- Is there an infinite number of prime numbers of the form n? + 1?
- Is there an infinite number of prime numbers of the form n? - 1?
- if p is prime, does 2 p -1 always not contain prime squares among its factors?
- does the Fibonacci sequence contain an infinite number of prime numbers?
The largest twin prime numbers are 2003663613 × 2 195000 ± 1. They consist of 58711 digits and were discovered in 2007.
The largest factorial prime number (of the type n! ± 1) is 147855! - 1. It consists of 142891 digits and was found in 2002.
The largest primorial prime number (a number of the form n# ± 1) is 1098133# + 1.