- 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 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 proven 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.
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 we know 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 in 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, just 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 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 cipherography. 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.
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 the method of Eratosthenes, you need to follow these steps:
Step 1. Write down all the natural numbers from two to , i.e. .
Step 2. Assign the variable the value , 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 the sum of three prime numbers (Goldbach's ternary 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 publishing houses, 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 of scientific concepts from ancient times to the present day and introduce 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 take us back to the very origins of mathematics, and then, along a path of increasing complexity, lead us to the forefront of 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: the conditions under which mathematicians worked and the extent to which their work involved mystical and semi-religious practices, which are quite different from the scientific methods used in our time. Nevertheless, slowly and with difficulty, the ground was prepared for new views that inspired Fermat and Euler in the 17th and 18th centuries.”
All other natural numbers are called composite. The natural number 1 is neither prime nor composite.
Example
Exercise. Which of the natural numbers written below are prime:
Answer.
Factoring a number
Representation of a natural number as a product of natural numbers is called factorization. If in the factorization of a natural number all factors are prime numbers, then such a factorization is called prime factorization.
Theorem
(Fundamental Theorem of Arithmetic)
Every natural number other than 1 can be factorized into prime factors, and in a unique way (if we identify the factorizations and , where and are prime numbers).
By combining identical prime factors in the decomposition of a number, we obtain the so-called canonical decomposition of a number:
where , are various prime numbers, and are natural numbers.
Example
Exercise. Find the canonical expansion of numbers:
Solution. To find the canonical decomposition of numbers, you first need to factor them into prime factors, and then combine the same factors and write their product as a power with a natural exponent:
Answer.
Historical background
How to determine which number is prime and which is not? The most common method for finding all prime numbers in any number range was proposed in the 3rd century. BC e. Eratosthenes (the method is called the “sieve of Eratosthenes”). Suppose we need to determine which numbers are prime. Let's write them out in a row and cross out every second number from those following the number 2 - they are all composite, since they are multiples of the number 2. The first of the remaining uncrossed out numbers - 3 - is prime. Let's cross out every third number from those following the number 3; the next of the uncrossed numbers - 5 - will also be prime. Using the same principle, we will cross out every fifth number from those following the number 5 and, in general, every single one from those following the number . All remaining uncrossed numbers will be prime numbers.
As prime numbers increase, they gradually become less and less common. However, the ancients were already well aware of the fact that there are infinitely many of them. His proof is given in Euclid's Elements.
The article discusses the concepts of prime and composite numbers. Definitions of such numbers are given with examples. We provide a proof that the number of prime numbers is unlimited and we will record it in the table of prime numbers using the method of Eratosthenes. Evidence will be given to determine whether a number is prime or composite.
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Prime and Composite Numbers - Definitions and Examples
Prime and composite numbers are classified as positive integers. They must be greater than one. Divisors are also divided into simple and composite. To understand the concept of composite numbers, you must first study the concepts of divisors and multiples.
Definition 1
Prime numbers are integers that are greater than one and have two positive divisors, that is, themselves and 1.
Definition 2
Composite numbers are integers that are greater than one and have at least three positive divisors.
One is neither a prime nor a composite number. It has only one positive divisor, so it is different from all other positive numbers. All positive integers are called natural numbers, that is, used in counting.
Definition 3
Prime numbers are natural numbers that have only two positive divisors.
Definition 4
Composite number is a natural number that has more than two positive divisors.
Any number that is greater than 1 is either prime or composite. From the property of divisibility we have that 1 and the number a will always be divisors for any number a, that is, it will be divisible by itself and by 1. Let's give a definition of integers.
Definition 5
Natural numbers that are not prime are called composite numbers.
Prime numbers: 2, 3, 11, 17, 131, 523. They are only divisible by themselves and 1. Composite numbers: 6, 63, 121, 6697. That is, the number 6 can be decomposed into 2 and 3, and 63 into 1, 3, 7, 9, 21, 63, and 121 into 11, 11, that is, its divisors will be 1, 11, 121. The number 6697 is decomposed into 37 and 181. Note that the concepts of prime numbers and coprime numbers are different concepts.
To make it easier to use prime numbers, you need to use a table:
A table for all existing natural numbers is unrealistic, since there are an infinite number of them. When numbers reach sizes of 10000 or 1000000000, then you should consider using the Sieve of Eratosthenes.
Let's consider the theorem that explains the last statement.
Theorem 1
The smallest positive divisor other than 1 of a natural number greater than one is a prime number.
Evidence 1
Let us assume that a is a natural number that is greater than 1, b is the smallest non-one divisor of a. It is necessary to prove that b is a prime number using the method of contradiction.
Let's assume that b is a composite number. From this we have that there is a divisor for b, which is different from 1 as well as from b. Such a divisor is denoted as b 1. It is necessary that condition 1< b 1 < b was completed.
From the condition it is clear that a is divided by b, b is divided by b 1, which means that the concept of divisibility is expressed as follows: a = b q and b = b 1 · q 1 , from where a = b 1 · (q 1 · q) , where q and q 1 are integers. According to the rule of multiplication of integers, we have that the product of integers is an integer with an equality of the form a = b 1 · (q 1 · q) . It can be seen that b 1 is the divisor for the number a. Inequality 1< b 1 < b Not corresponds, because we find that b is the smallest positive and non-1 divisor of a.
Theorem 2
There are an infinite number of prime numbers.
Evidence 2
Presumably we take a finite number of natural numbers n and denote them as p 1, p 2, …, p n. Let's consider the option of finding a prime number different from those indicated.
Let us take into consideration the number p, which is equal to p 1, p 2, ..., p n + 1. It is not equal to each of the numbers corresponding to prime numbers of the form p 1, p 2, ..., p n. The number p is prime. Then the theorem is considered to be proven. If it is composite, then you need to take the notation p n + 1 and show that the divisor does not coincide with any of p 1, p 2, ..., p n.
If this were not so, then, based on the divisibility property of the product p 1, p 2, ..., p n , we find that it would be divisible by pn + 1. Note that the expression p n + 1 dividing the number p equals the sum p 1, p 2, ..., p n + 1. We obtain that the expression p n + 1 The second term of this sum, which equals 1, must be divided, but this is impossible.
It can be seen that any prime number can be found among any number of given prime numbers. It follows that there are infinitely many prime numbers.
Since there are a lot of prime numbers, the tables are limited to the numbers 100, 1000, 10000, and so on.
When compiling a table of prime numbers, you should take into account that such a task requires sequential checking of numbers, starting from 2 to 100. If there is no divisor, it is recorded in the table; if it is composite, then it is not entered into the table.
Let's look at it step by step.
If you start with the number 2, then it has only 2 divisors: 2 and 1, which means it can be entered into the table. Same with the number 3. The number 4 is composite; it must be decomposed into 2 and 2. The number 5 is prime, which means it can be recorded in the table. Do this until the number 100.
This method is inconvenient and time-consuming. It is possible to create a table, but you will have to spend a lot of time. It is necessary to use divisibility criteria, which will speed up the process of finding divisors.
The method using the sieve of Eratosthenes is considered the most convenient. Let's look at the example tables below. To begin with, the numbers 2, 3, 4, ..., 50 are written down.
Now you need to cross out all the numbers that are multiples of 2. Perform sequential strikethroughs. We get a table like:
We move on to crossing out numbers that are multiples of 5. We get:
Cross out numbers that are multiples of 7, 11. Ultimately the table looks like
Let's move on to the formulation of the theorem.
Theorem 3
The smallest positive and non-1 divisor of the base number a does not exceed a, where a is the arithmetic root of the given number.
Evidence 3
It is necessary to denote b the smallest divisor of a composite number a. There is an integer q, where a = b · q, and we have that b ≤ q. Inequalities of the form are unacceptable b > q, because the condition is violated. Both sides of the inequality b ≤ q should be multiplied by any positive number b not equal to 1. We get that b · b ≤ b · q, where b 2 ≤ a and b ≤ a.
From the proven theorem it is clear that crossing out numbers in the table leads to the fact that it is necessary to start with a number that is equal to b 2 and satisfies the inequality b 2 ≤ a. That is, if you cross out numbers that are multiples of 2, then the process begins with 4, and multiples of 3 with 9, and so on until 100.
Compiling such a table using Eratosthenes' theorem suggests that when all composite numbers are crossed out, prime numbers will remain that do not exceed n. In the example where n = 50, we have that n = 50. From this we get that the sieve of Eratosthenes sifts out all composite numbers whose value is not greater than the value of the root of 50. Searching for numbers is done by crossing out.
Before solving, you need to find out whether the number is prime or composite. Divisibility criteria are often used. Let's look at this in the example below.
Example 1
Prove that the number 898989898989898989 is composite.
Solution
The sum of the digits of a given number is 9 8 + 9 9 = 9 17. This means that the number 9 · 17 is divisible by 9, based on the divisibility test by 9. It follows that it is composite.
Such signs are not able to prove the primeness of a number. If verification is needed, other actions should be taken. The most suitable way is to enumerate numbers. During the process, prime and composite numbers can be found. That is, the numbers should not exceed a in value. That is, the number a must be factorized into prime factors. if this is satisfied, then the number a can be considered prime.
Example 2
Determine the composite or prime number 11723.
Solution
Now you need to find all the divisors for the number 11723. Need to evaluate 11723 .
From here we see that 11723< 200 , то 200 2 = 40 000 , and 11 723< 40 000 . Получаем, что делители для 11 723 меньше числа 200 .
For a more accurate estimate of the number 11723, you need to write the expression 108 2 = 11 664, and 109 2 = 11 881 , That 108 2 < 11 723 < 109 2 . It follows that 11723< 109 . Видно, что любое число, которое меньше 109 считается делителем для заданного числа.
When expanding, we find that 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 are all prime numbers. This entire process can be depicted as division by a column. That is, divide 11723 by 19. The number 19 is one of its factors, since we get division without a remainder. Let's represent the division as a column:
It follows that 11723 is a composite number, because in addition to itself and 1 it has a divisor of 19.
Answer: 11723 is a composite number.
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