What numbers are called prime? Prime numbers: the mundanity of an unsolved riddle

Prime number

a natural number greater than one and having no divisors other than itself and one: 2, 3, 5, 7, 11, 13... The number of prime numbers is infinite.

Prime number

whole positive number, greater than one, having no other divisors except itself and one: 2, 3, 5, 7, 11, 13,... The concept of a number is fundamental in the study of the divisibility of natural (positive integers) numbers; Namely, the main theorem of the theory of divisibility establishes that every positive integer, except 1, is uniquely decomposed in the product of a number of numbers (the order of the factors is not taken into account). There are infinitely many prime numbers (this proposition was known to ancient Greek mathematicians; its proof is available in the 9th book of Euclid’s Elements). Questions of the divisibility of natural numbers, and therefore questions related to prime numbers, are important in the study of groups; in particular, the structure of a group with a finite number of elements is closely related to the way in which this number of elements (the order of the group) is decomposed into prime factors. The theory of algebraic numbers deals with the issues of divisibility of algebraic integers; The concept of a partial number turned out to be insufficient for constructing a theory of divisibility; this led to the creation of the concept of an ideal. P. G. L. Dirichlet established in 1837 that the arithmetic progression a + bx for x = 1, 2,... with coprime integers a and b contains infinitely many prime numbers. Determining the distribution of prime numbers in the natural series of numbers is a very difficult problem in number theory. It is formulated as a study of the asymptotic behavior of the function p(x), which denotes the number of partial numbers not exceeding a positive number x. The first results in this direction belong to P.L. Chebyshev, who in 1850 proved that there are two constants a and A such that ═< p(x) < ═при любых x ³ 2 [т. е., что p(х) растет, как функция ]. Хронологически следующим значительным результатом, уточняющим теорему Чебышева, является т. н. асимптотический закон распределения П. ч. (Ж. Адамар, 1896, Ш. Ла Валле Пуссен, 1896), заключающийся в том, что предел отношения p(х) к ═равен

Subsequently, significant efforts of mathematicians were directed toward clarifying the asymptotic law of distribution of the P. number. Questions of the distribution of P. number are studied and elementary methods, and methods mathematical analysis. Particularly fruitful is the method based on the use of the identity

(the product extends to all P. h. p = 2, 3,...), first indicated by L. Euler; this identity is valid for all complex s with a real part greater than unity. On the basis of this identity, questions of the distribution of P. numbers are led to the study of a special function ≈ zeta function x(s), determined for Res > 1 by the series

This function was used in questions of the distribution of prime numbers for real s by Chebyshev; B. Riemann pointed out the importance of studying x(s) for complex values ​​of s. Riemann hypothesized that all roots of the equation x(s) = 0 lying in the right half-plane have a real part equal to 1/

This hypothesis has not been proven to date (1975); its proof would do a great deal in solving the problem of the distribution of prime numbers. Questions of the distribution of prime numbers are closely related to Goldbach’s problem, the still unsolved problem of “twins,” and other problems of analytic number theory. The problem of the “twins” is to find out whether the number of P. numbers differing by 2 (such as, for example, 11 and 13) is finite or infinite. Tables of P. numbers lying within the first 11 million natural numbers show the presence of very large “twins” (for example, 10006427 and 10006429), but this is not proof of the infinity of their number. Outside the compiled tables, individual partial numbers are known that admit of a simple arithmetic expression [for example, it was established (1965) that 211213 ≈1 is a regular number; it has 3376 digits].

Lit.: Vinogradov I.M., Fundamentals of Number Theory, 8th ed., M., 1972; Hasse G., Lectures on number theory, trans. from German, M., 1953; Ingham A.E., Distribution of prime numbers, trans. from English, M. ≈ L., 1936; Prahar K., Distribution of prime numbers, trans. from German, M., 1967; Trost E., Prime numbers, transl., from German, M., 1959.

Wikipedia

Prime number

Prime number- a natural number that has exactly two distinct natural divisors - and itself. In other words, the number x is prime if it is greater than 1 and is divisible without remainder only by 1 and x. For example, 5 is a prime number, and 6 is a composite number, since, in addition to 1 and 6, it is also divisible by 2 and 3.

Natural numbers that are greater than one and are not prime numbers are called composite numbers. So everything integers are divided into three classes: unit. Number theory studies the properties of prime numbers. In ring theory, prime numbers correspond to irreducible elements.

The sequence of prime numbers starts like this:

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 , 127 , 131 , 137 , 139 , 149 , 151 , 157 , 163 , 167 , 173 , 179 , 181 , 191 , 193 , 197 , 199 …

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

What numbers are called “simple” in English?

Very often, schoolchildren do not know how to answer one of the simplest questions in mathematics at first glance, about what a prime number is. They often confuse prime numbers with natural numbers (that is, the numbers that people use when counting objects, while in some sources they begin with zero, and in others with one). But these are completely two different concepts. Prime numbers are natural numbers, that is, integers and positive numbers that are greater than one and that have only 2 natural divisors. Moreover, one of these divisors is the given number, and the second is one. For example, three is a prime number because it cannot be divided without a remainder by any number other than itself and one.

Composite numbers

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

Subsequence

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

Theories about the properties of prime numbers

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

Prime numbers are the “building blocks” of natural numbers

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

Search for prime numbers. Simplicity tests

Many scientists from different times tried to find some principles (systems) for finding a list of prime numbers. Science knows systems called the Atkin sieve, the Sundartham sieve, and the Eratosthenes sieve. However, they do not produce any significant results, and a simple test is used to find the prime numbers. Mathematicians also created algorithms. They are usually called primality tests. For example, there is a test developed by Rabin and Miller. It is used by cryptographers. There is also the Kayal-Agrawal-Sasquena test. However, despite sufficient accuracy, it is very difficult to calculate, which reduces its practical significance.

Does the set of prime numbers have a limit?

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

What is the largest prime number?

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

Names of special prime numbers

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

1. Merssen.

4. Cullen.

6. Mills et al.

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

1. Luc-Lemaire.

2. Pepina.

3. Riesel.

4. Billhart - Lemaire - Selfridge and others.

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

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

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

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

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

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

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

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

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

• Translation

The properties of prime numbers were first studied by mathematicians 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 have already been proven important facts regarding prime numbers. 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 it happened 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 new method factorization of 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 earlier: the 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 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 amount reciprocals to 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 prime density 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 hypothesis: 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?
• does it exist arithmetic progression of consecutive prime numbers 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.

Prime number is a natural (positive integer) number that is divisible without a remainder by only two natural numbers: by and by itself. In other words, a prime number has exactly two natural divisors: and the number itself.

By definition, the set of all divisors of a prime number is two-element, i.e. represents a set.

The set of all prime numbers is denoted by the symbol. Thus, due to the definition of the set of prime numbers, we can write: .

The sequence of prime numbers looks like this:

Fundamental Theorem of Arithmetic

Fundamental Theorem of Arithmetic states that every natural number greater than one can be represented as a product of prime numbers, and in a unique way, up to the order of the factors. Thus, prime numbers are the elementary "building blocks" of the set of natural numbers.

Natural number expansion title="Rendered by QuickLaTeX.com" height="13" width="42" style="vertical-align: -1px;"> в произведение простых чисел называют !} canonical:

where is a prime number, and . For example, the canonical expansion of a natural number looks like this: .

Representing a natural number as a product of primes is also called factorization of a number.

Properties of Prime Numbers

Sieve of Eratosthenes

One of the most famous algorithms for searching and recognizing prime numbers is sieve of Eratosthenes. So this algorithm was named after the Greek mathematician Eratosthenes of Cyrene, who is considered the author of the algorithm.

To find all prime numbers less than a given number, following Eratosthenes' method, follow these steps:

Step 1. Write down all the natural numbers from two to , i.e. .
Step 2. Assign the value to the variable, that is, the value equal to the smallest prime number.
Step 3. Cross out in the list all numbers from to that are multiples of , that is, the numbers: .
Step 4. Find the first uncrossed number in the list greater than , and assign the value of this number to a variable.
Step 5. Repeat steps 3 and 4 until number is reached.

The process of applying the algorithm will look like this:

All remaining uncrossed numbers in the list at the end of the process of applying the algorithm will be the set of prime numbers from to .

Goldbach conjecture

Cover of the book “Uncle Petros and the Goldbach Hypothesis”

Despite the fact that prime numbers have been studied by mathematicians for quite a long time, many related problems remain unsolved today. One of the most famous unsolved problems is Goldbach's hypothesis, which is formulated as follows:

• Is it true that every even number greater than two can be represented as the sum of two prime numbers (Goldbach's binary hypothesis)?
• Is it true that every odd number, greater than 5, can be represented as a sum three simple numbers (ternary Goldbach hypothesis)?

It should be said that the ternary Goldbach hypothesis is a special case of the binary Goldbach hypothesis, or as mathematicians say, the ternary Goldbach hypothesis is weaker than the binary Goldbach hypothesis.

Goldbach's conjecture became widely known outside the mathematical community in 2000 thanks to a promotional marketing stunt by the publishing companies Bloomsbury USA (USA) and Faber and Faber (UK). These publishers, having released the book “Uncle Petros and Goldbach’s Conjecture,” promised to pay a prize of 1 million US dollars to anyone who proves Goldbach’s hypothesis within 2 years from the date of publication of the book. Sometimes the mentioned prize from publishers is confused with prizes for solving the Millennium Prize Problems. Make no mistake, Goldbach’s hypothesis is not classified by the Clay Institute as a “millennium challenge,” although it is closely related to Riemann hypothesis- one of the “millennium challenges”.

The book “Prime numbers. Long road to infinity"

Cover of the book “The World of Mathematics. Prime numbers. Long road to infinity"

Additionally, I recommend reading a fascinating popular science book, the annotation to which says: “The search for prime numbers is one of the most paradoxical problems in mathematics. Scientists have been trying to solve it for several millennia, but, growing with new versions and hypotheses, this mystery still remains unsolved. The appearance of prime numbers is not subject to any system: they appear spontaneously in the series of natural numbers, ignoring all attempts by mathematicians to identify patterns in their sequence. This book will allow the reader to trace the evolution scientific ideas from ancient times to the present day and will introduce you to the most interesting theories of searching for prime numbers.”

Additionally, I will quote the beginning of the second chapter of this book: “Prime numbers are one of the important topics that return us to the very origins of mathematics, and then, along a path of increasing complexity, lead to Front edge modern science. Thus, it would be very useful to trace the fascinating and complex history of prime number theory: exactly how it developed, exactly how the facts and truths that are now generally accepted were collected. In this chapter we will see how generations of mathematicians carefully studied the natural numbers in search of a rule that predicted the appearance of prime numbers - a rule that became increasingly elusive as the search progressed. We will also look in detail at the historical context: under what conditions mathematicians worked and to what extent their work involved mystical and semi-religious practices, which are not at all similar to scientific methods, used nowadays. Nevertheless, slowly and with difficulty, the ground was prepared for new views that inspired Fermat and Euler in the 17th and 18th centuries.”