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 along 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 the 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.

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 most simple 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 greater than one, but have not two, but a larger number of divisors. So, for example, the numbers 4, 6, 8, 9, etc. are natural, composite, but not prime numbers. As you can see, these are mostly even numbers, but not all. But “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 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. It should have been so, 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 departments. 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, 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 win the $250,000 prize by finding the largest prime number.

Since the time of the ancient Greeks, prime numbers have been very attractive to mathematicians. They are constantly looking for different ways to find them, but the most effective way to “catch” prime numbers is considered to be the method found by the Alexandrian astronomer and mathematician Eratosthenes. This method is already about 2000 years old.

Which numbers are prime

How to determine a prime number? Many numbers are divisible by other numbers without a remainder. The number by which an integer is divided is called a divisor.

In this case we are talking about division without a remainder. For example, the number 36 can be divided by 1, 2, 3, 4, 6, 9, 12, 18 and by itself, that is, by 36. This means that 36 has 9 divisors. The number 23 is divisible only by itself and 1, that is, this number has 2 divisors - this number is prime.

Numbers that have only two divisors are called prime numbers. That is, a number that is divisible without a remainder only by itself and one is called prime.

For mathematicians, discovering patterns in a series of numbers that can then be used to formulate hypotheses is a very rewarding experience. But prime numbers refuse to obey any pattern. But there is a way to determine prime numbers. This method was discovered by Eratosthenes, it is called the “sieve of Eratosthenes”. Let's look at a version of such a “sieve”, presented in the form of a table of numbers up to 48, and understand how it is compiled.

In this table, all prime numbers less than 48 are marked orange. They were found like this:

• 1 – has a single divisor and therefore is not a prime number;
• 2 is the smallest prime number and the only even one, since all other even numbers are divisible by 2, that is, they have at least 3 divisors, these numbers are reduced to purple column;
• 3 is a prime number, has two divisors, all other numbers that are divisible by 3 are excluded - these numbers are summarized in the yellow column. The column marked in both purple and yellow contains numbers divisible by both 2 and 3;
• 5 is a prime number, all numbers that are divisible by 5 are excluded - these numbers are circled in a green oval;
• 7 is a prime number, all numbers that are divisible by 7 are circled in a red oval - they are not prime;

All numbers that are not prime are marked in blue. Then you can compile this table yourself in the image and likeness.

• 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.

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 divisors.

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 the 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

(3)

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 . notice, that αi accepts α +1 values, β j accepts β +1 values, γ k accepts γ +1 values, ... .