Doom. interesting facts about numbers

We all know the numbers from 0 to 9. But how did they appear? Where did these familiar 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 come from, which we constantly use in Everyday life? What are they called and why do they have such a name? Let's dive into history and find out the answers to these and many other questions.

The history of the emergence of numbers

Even in ancient times, a person needed an account. Even when there were no letters and numbers yet, when ancient man did not know what two or five were, he had to perform simple actions on the division of prey, determining the number of people for hunting, and many others.

Initially, he used his hands, and sometimes even his legs, showed on his fingers. Do you remember the saying “I know like my own 5 fingers”? It is possible that it was invented in those distant times. It was the fingers that were the first tools for counting.

Life went on as usual, everything changed, people needed some other signs besides fingers. The numbers were getting bigger, it was difficult to keep them in my head, I had to somehow label and write them down. That's how the numbers came about. Moreover, different countries came up with their own. The first were the Egyptians, then the Greeks and the Romans. Now we sometimes use Roman numerals. However, the most popular and used by us to this day are the numbers invented in India before the beginning of the 5th century.

Why are they called that

Why are the usual numbers called Arabic, because they were invented in India? And all because they received distribution precisely thanks to the Arab countries, which began to actively use them. The Arabs took the Indian numbers, changed them a little and began to actively use them. Among those who helped the world discover the Arabic numerals we know so well were the Frenchman Alexander de Villiers, the British teacher John Halifax and the famous mathematician Fibonacci, who often traveled to the East and studied the works of Arab scientists.

The word "number" itself is of Arabic origin. The consonant Arabic word "sifr" denotes those icons that we are used to using 0.1, 2 ... 9.

Let's take a closer look at the numbers

Number 1

Guess the riddle:

Sister with a sly nose
The account will open ... ( unit)

That's right, this is the number 1. The very first number. It's easy to write. It is with her that acquaintance with numbers always begins. Any number can be made from units, for example, 1+1=2, etc. In China, one is the beginning of everything. However, so do we. The start of the school year is September 1, and New Year- January 1st.

The number 1 symbolizes the beginning, unity, integrity, like God, the sun, the universe, the cosmos. It is an indivisible and unique number.

Number 2

Next riddle:

Neck, tail and head
Like a swan figure ... ( two)

Number 2. Look at it carefully. She really looks like a swan. In some countries, the deuce is considered a symbol of opposites, and in some, on the contrary, a symbol of pairing. And also integrity. Millions of creations without a pair are not a whole... For example, two wings, two eyes, two ears and other parts of the body. Every family starts with two...

Often the number two is found in the literature. Remember Krylov's fables "Two Doves", "Two Dogs" or the fairy tale of the Brothers Grimm "Two Brothers", Nosov's fairy tale "Two Frosts". Two is the smallest prime number. Also the worst grade in school. In order not to get deuces, you need to study well.

Number 3

Let's solve another riddle:

What a miracle
What a number!
Every tomboy knows.
Even in our alphabet
She has a twin sister... three)

Number 3. You probably noticed that the number three is very common in many fairy tales: “Father had three sons”, “traveled for three days and three nights”, “spit three times”, “knock on wood three times”, “ clap your hands three times”, “turn around your axis three times”, “say something three times”, “three heroes”, “three wishes”, etc. It is believed that the number "three" is sacred. The number really looks like the letters of the Russian alphabet "З".

Number 4

I stand after the number 3,
And I'm a little behind the number five.
What kind of number am I?

Number 4. They say that the four is the most magical of the numbers. In most states, it is a symbol of integrity. But in Asian countries they treat it with apprehension. In life, we meet the number 4 very often: 4 seasons, 4 cardinal points, 4 natural elements, 4 times of the day, etc.

Number 5

How many fingers are on the hand
And a penny in a patch,
At the starfish rays,
The beaks of five rooks,
Blades near maple leaves
And the corners of the bastion
Tell about it all
The numbers will help us... (five)

Number 5. In most schools, this is the best score! Although, for example, in Germany, the top five is put on the contrary to those who do not try hard. Where can we meet the five? For example, there are 5 continents on Earth, and the symbol of the Olympic Games has 5 rings, and there are 5 fingers on the hands and feet.

Number 6

How many letters does the dragon have
And zeros for a million
Various chess pieces
Wings of three white hens,
May beetle legs
And the sides of the chest.
If we can't count ourselves
Will tell us number ... (six)

Number 6. The trickiest number. If you stand on your head, the number 6 will become a nine. A cube has 6 sides, all insects have 6 legs, many musical instruments have 6 holes each - these are examples of where the number 6 occurs in life.

Number 7

How many colors are in a bright rainbow?
How many wonders of the world are there on earth?
How many hills does Moscow have?
This figure is so suitable for us to answer!

Number 7. Easy to write, reminiscent of an ax or a question mark. Perhaps everyone knows that this figure is considered the most successful. Each week has 7 days, music has 7 notes, and the rainbow has 7 colors, world civilization has 7 wonders of the world. As you can see, the number 7 is also very common in life.

I also love the number 7. folk beliefs and loves to live in fairy tales. Well, who doesn’t know such favorite fairy tales as “The Wolf and the Seven Kids”, “The Seven-Flower Flower”, “Snow White and the Seven Dwarfs”, “The Tale of the Princess and the Seven Bogatyrs”.

The most desired word in the world also contains the number 7 - Family.

Number 8

It's necessary! We carry the number
On the nose, take a look, please.
This figure plus hooks -
Points are earned...

Number 8. The number 8 is an inverted infinity sign. For many peoples, this figure is special. For example, in China, it means prosperity and wealth. The famous mathematician Pythagoras also believed that the number 8 is harmony, balance and prosperity. Do you remember what holiday we celebrate on March 8? How many hooves do two cows have? How many legs does a spider have?

Number 9

A kitten walked across the bridge
He sat on the bridge and hung his tail.
"Meow! It makes me feel better…”
The kitten has become a number ...!

Number 9. Remember, we recently studied the number 6? Is it true that the number 9 looks like it? This is the last number in the row.

Number 0

The numbers stood up like a squad,
In a friendly numerical series.
First in order role
The number will play for us ...

The number 0. This is the only number that cannot be divided by. The number zero is neither positive nor negative. Al-Khwarizmi, a medieval Persian scientist, was the first to use the figure.

We have already found out that the history of numbers and numbers is as old as the world. For all the time of existence, figures and numbers have acquired a variety of myths and legends. There are many interesting facts associated with them. The most interesting of them are presented below.

Translated from Arabic, the word "figure" means "emptiness, zero." Agree, this is very symbolic.
Is it possible to write zero in Roman numerals? And here it is not. You cannot write “zero” in Roman numerals, it does not exist in nature. The Roman count starts from one.
Most big number on the this moment- centillon. It is a 1 followed by 600 zeros. It was first written down on paper back in 1852.
What do you associate the number 666 with? Did you know that this is the sum of all the numbers on the casino roulette?
Around the world, 13 is considered an unlucky number. In many countries, floor number "13" is skipped and the twelfth is followed by the fourteenth or, for example, 12A. But in Asian countries (China, Japan, Korea), the unlucky number is 4, so the floor is also skipped. In Italy, for some reason, another unloved number is 17.
On the contrary, 7 is considered to be the luckiest and most successful number.
The Arabs themselves write numbers from right to left, and not as we are used to doing from left to right.
An interesting theory of one mathematician is that the numerical value is directly related to the number of angles in writing the number. Indeed, earlier the figures were written in an angular way, they acquired their rounded habitual outlines over time.

Why are the floors with the number 4 missing in the houses in the east?

In China, Korea and Japan, the number 4 is considered unlucky, as it is consonant with the word "death". In these countries, floors with numbers ending in four are almost always missing.

Why do houses in some countries not have a 13th floor?

Because of the fear of the number 13, in many countries there is no 13th floor in houses (after the 12th floor, the 14th floor immediately follows), or it is designated differently, for example, 12A or M (the 13th letter of the alphabet).

How do Arabs write and read numbers?

The Arabs use their own signs to write numbers, although the Arabs of Europe and North Africa use the "Arabic" numbers familiar to us. However, whatever the signs of the numbers, the Arabs write them, like letters, from right to left, but starting with the lower digits. It turns out that if we meet familiar numbers in the Arabic text and read the number in the usual way from left to right, we will not be mistaken.

How many times has the main prize of Sportloto been won?

In the entire history of the Soviet lottery Sportloto, all 6 out of 49 numbers were guessed correctly 3 times.

How many flowers should be given to European girls?

In the USA, Europe and some Eastern countries, it is believed that an even number of flowers given brings happiness. In Russia, it is customary to bring an even number of flowers only for the funeral of the dead. In cases where there are many flowers in the bouquet, the evenness or oddness of their number no longer plays such a role.

How to check the authenticity of a euro banknote serial number?

The authenticity of a euro banknote can be verified by its serial number of letters and eleven digits. It is necessary to replace the letter with its serial number in the Latin alphabet, add this number to the rest, then add the digits of the result until we get one digit. If this number is 8, then the bill is genuine. Another way to check is to add numbers like this, but without a letter. The result of one letter and number must correspond to a certain country, since the euro is printed in different countries. For example, for Germany it is X2.

How many legs do centipedes have?

A centipede does not necessarily have 40 legs. Centipede is a common name different types arthropods, scientifically united in the superclass centipedes. Different species of centipedes have from 30 to 400 or more legs, and this number can be different even in individuals of the same species. In English, two names for these animals have been established - centipede ("centipede" in Latin) and millipede ("thousand-footed"). Moreover, the difference between them is significant - the centipedes are not dangerous to humans, and the centipedes bite very painfully.

Where did the Olympic Games take place, on the emblem of which the year of the event was indicated by five digits?

On the emblems of the Olympic Games, the year is usually indicated by two (for example, Barcelona-92) or four digits (for example, Beijing-2008). But once the year was indicated by five digits. It happened in 1960, when the Olympics were held in Rome - the number 1960 was written as MCMLX.

What a strange way the numbers 70, 80 and 90 are called in French?

In most European languages, the names of numbers from 20 to 90 are formed according to the standard scheme - consonant with the basic numbers from 2 to 9. However, in French, the names of some numbers have a strange logic. Thus, the number 70 is pronounced 'soixante-dix', which translates as "sixty and ten", 80 - 'quatre-vingts' ("four times twenty"), and 90 - 'quatre-vingt-dix' ("four times twenty and ten "). The situation is similar in Georgian and Danish. In the latter, the number 70 is literally translated as "halfway from three times twenty to four times twenty."

The name of which world-famous corporation was the result of a spelling mistake?

When Larry Page and Sergey Brin came up with the name of the new search engine, they wanted to express in it the huge amount of information that the system is capable of processing. Their colleague suggested the word "googol" - this is the name in mathematics for a number of one followed by a hundred zeros. He immediately checked the domain name for employment and, finding that it was free, registered it. Moreover, he made a mistake in writing the word: instead of the correct 'googol.com' he entered 'google.com', but Larry liked the newly invented word and established itself as a name.

On satellite images of which Ukrainian city can you see the number 666?

According to the plan, a block of residential buildings was to be built in the 522 microdistrict of Kharkov, so that from the air they would form the letters of the USSR. However, after the construction of three letters C and a vertical line of the letter P, changes were made to the plan. As a result, these houses can now be seen as the number 666.

What mathematical law of the distribution of numbers will allow you to check the reliability of financial data?

There is Benford's mathematical law, which states that the distribution of the first digits in the numbers of any data sets from the real world is uneven. The numbers from 1 to 4 in such sets (namely, birth or death statistics, house numbers, etc.) in the first position are much more common than the numbers from 5 to 9. The practical application of this law is that it can be check the accuracy of accounting and financial data, election results and much more. In some US states, the non-compliance of data with Benford's law is even formal evidence in court.

Why is the name of the number 40 knocked out of the same type of names "twenty", "thirty", "fifty", etc.?

In Russian, the names of numerals up to 100, divisible by 10, are formed by adding the name of the number and “ten”: twenty, thirty, fifty, etc. An exception to this series is the number “forty”. This is explained by the fact that in ancient times, a bundle of 40 fur skins was a conditional unit of trade in fur skins. The fabric in which these skins were wrapped was called “forty” (the word “shirt” comes from the same root). Thus, the name "forty" replaced the more ancient "four deste".

What number social connections is it optimal for a person?

English anthropologist Robert Dunbar has identified a relationship between the size of the neocortex of the cerebral hemispheres of primates and the size of their flock. Based on these data, he determined optimal size social ties for a person - 150. This number is confirmed in a variety of historical periods and locations: for example, this is the estimated number of inhabitants of a Neolithic settlement or the size of the basic unit of the Roman army. In 2010, Dunbar began researching the social network Facebook and came to the conclusion that his number also works there: despite the fact that some people have in social networks hundreds and thousands of friends, the average person is able to effectively interact with no more than 150 contacts.

Why does the numbering of trolleybuses in Budapest start from the 70th number?

Trolleybuses appeared in Budapest in 1949. The first trolleybus was immediately given the number 70, since this year was the 70th anniversary of Stalin. And now there are no trolleybuses up to number 70 in Budapest.

Why did there never exist a Pope John XX, although there were Johns XXI, XXII and XXIII?

The Portuguese Pedro Julian was elected pope in 1276 and took the name John. However, although the previous John bore the 19th serial number, this pope omitted one digit and declared himself John XXI. He believed that a mistake had crept into the list of his predecessors, and there was an extra John in the history of the papacy. Later it turned out that he was mistaken, and there was no mistake, but the numbering could no longer be reversed. Therefore, it turned out that John XX never existed, although today the list of Johns ends with number XXIII.

Why are the floors with the number 4 missing in the houses in the east?

In China, Korea and Japan, the number 4 is considered unlucky, as it is consonant with the word "death". In these countries, floors with numbers ending in four are almost always missing.

Why do houses in some countries not have a 13th floor?

How do Arabs write and read numbers?

The Arabs use their own signs for writing numbers, although the Arabs of Europe and North Africa use the “Arabic” numbers we are used to. However, whatever the signs of the numbers, the Arabs write them, like letters, from right to left, but starting with the lower digits. It turns out that if we meet familiar numbers in the Arabic text and read the number in the usual way from left to right, we will not be mistaken.

How many times has the main prize of Sportloto been won?

In the entire history of the Soviet lottery Sportloto, all 6 out of 49 numbers were guessed correctly 2 or 3 times.

How many flowers should be given to European girls?

How to check the authenticity of a euro banknote by serial number?

How many legs do centipedes have?

A centipede does not necessarily have 40 legs. The centipede is the household name for various types of arthropods, scientifically united in the superclass centipedes. Different species of centipedes have from 30 to 400 or more legs, and this number can be different even in individuals of the same species. In English, there are two names for these animals - centipede ("centipede" in Latin) and millipede ("thousand-footed"). Moreover, the difference between them is significant - centipedes are not dangerous to humans, and centipedes bite very painfully.

Where did the Olympic Games take place, on the emblem of which the year of the event was indicated by five digits?

What are some strange names for the numbers 70, 80 and 90 in French?

The name of which world-famous corporation was the result of a spelling mistake?

On satellite images of which Ukrainian city can you see the number 666?

What mathematical law of the distribution of numbers will allow you to check the reliability of financial data?

Why is the name of the number 40 knocked out of the same type of names "twenty", "thirty", "fifty", etc.?

What is the optimal number of social connections for a person?

English anthropologist Robert Dunbar has identified a relationship between the size of the neocortex of the cerebral hemispheres of primates and the size of their flock. Based on these data, he determined the optimal size of social ties for a person - 150. This number is confirmed in a variety of historical periods and locations: for example, this is the estimated number of inhabitants of a Neolithic settlement or the size of a basic unit of the Roman army. In 2010, Dunbar began researching the social network Facebook and came to the conclusion that his number also works there: despite the fact that some people have hundreds and thousands of friends on social networks, the average person can interact effectively with no more than 150 contacts.

Why do numbers increase from bottom to top on a calculator, and from top to bottom on a phone?

The numbers on the calculator increase from bottom to top, and on the phone keypad - from top to bottom. This is because calculators evolved from mechanical calculating machines, where the numbers are historically arranged from the bottom up. Phones long time were equipped with a disk, and when the release of push-button devices with tone dialing became possible, they decided to make the arrangement of numbers on the buttons by analogy with the disk - ascending from top to bottom with zero at the end.

Why does the numbering of trolleybuses in Budapest start from the 70th number?

Why did there never exist a Pope John XX, although there were Johns XXI, XXII and XXIII?

The Portuguese Pedro Julian was elected pope in 1276 and took the name John. However, although the previous John bore the 19th serial number, this pope omitted one digit and declared himself John XXI. He believed that a mistake had crept into the list of his predecessors, and there was an extra John in the history of the papacy. Later it turned out that he was mistaken and there was no mistake, but the numbering could no longer be reversed. Therefore, it turned out that John XX never existed, although today the list of Johns ends with number XXIII.

1. Eastern countries they are afraid of the number 4. Its pronunciation is very close to the word "death". The Japanese, Koreans and Chinese equated it with an "unlucky" number. If you pay attention to the number of floors in buildings, you will notice that the number "4" at the end of the floor is almost never registered.

2. A little trick (elementarily explained by mathematics and logic). Take your year of birth, more precisely the last 2 numbers. Remember how old you were in 2011? To these years, add the last digits from the year of birth. I bet you got 111?

3. If you square 111 111 111, the result will surprise you! You will get 12345678987654321. These are all numbers in order. First they increase, then they decrease.

4. Guess what happens when you add up all the numbers on the casino roulette? The number of the devil that many fear is 666.

5. Many people know about the various lotteries "6 out of 49" (as it used to be in Sportloto). Do you know how many times the jackpot has been hit in the history of the game? 3 times! The real lucky ones.

6. Everyone from school remembers the number Pi - 3.14. He even has 2 holidays. Unofficial, of course. In America, this is March 14 (03.14) and July 22 (22/7). Ask why July? Because when you divide the number by the digit of the month, you get exactly the number Pi. Funny idea.

7. The largest number has 600 zeros behind one. It has its own name. It is a centillon.

8. Interesting Facts about numbers and figures concern also scientists. An American math graduate student was late for class one day. Equations were written on the board. George Dantzig (that was the name of the graduate student) thought it was a homework assignment. After suffering for several days, puzzling over how such a difficult task was given, George solved it. What was his surprise when he found out that this is an “unsolvable” problem in statistics. Many scientists have strained their convolutions for many years to unravel the mystery of these problems.

9. Guess which is the most common woman's name? Anna. 100 million women are named after him.

10. Famous people, too, with their "cockroaches" in their heads and fears. For example, Sigmund Freud was terrified of the number 62. This went so far that Freud did not stay in hotels with more than 61 rooms. What if he, the lucky one, gets 62 out of all? And the composer Schoenberg Arnold was afraid of the devil's dozen. And he died on Friday the 13th at the age of 76 (do you know how much 7 + 6 is?). That's the magic of numbers. And he only says that thoughts are material. And you don’t need to create fears for yourself so that they don’t “finish off” you.

11. Another interesting fact about the devil's number. Imagine that in the USSR, architects wanted to create a microdistrict by building houses in it in such a way that the name of a great power could be read from space. However, the idea somehow fell out of favor or finances did not allow. But as a result, there is the 522nd microdistrict in Kharkov, where there are only 3 houses. And the satellite shows them on the map as "666".

12. In the Himalayas there is a sacred mountain with a height of 6666 m. Its name is Kailash. What is striking is that its height is the distance to the center of the North Pole and at the same time to Stonehenge. Some kind of mystic. But the mountain is actually very beautiful.

13. The centipede actually has far from 40 legs. People often call this a spider with long and thin "legs". It moves so fast it looks like 40 feet. However, some call centipedes centipedes, which in fact have up to 400 legs, and sometimes more. Those who count 100 legs should be wary of this insect. It bites painfully. But the so-called millenniums are generally harmless and harmless. Biology is an interesting science.

14. In Budapest, trolleybuses received numbers in 49. It was in that year that Stalin celebrated his anniversary - the seventh decade. And now the very first trolleybus was assigned No. 70 (although now there is no such route anymore). Since then, route numbers have been given after 70. There is neither the first, nor the twentieth, nor the fifty-third.

15. Is it possible to live a million days? Interesting. But if you count, it's 27 centuries. So many days have not yet passed since the beginning of our era. So the answer is unequivocal - no, you cannot live so many days for 1 person.

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.

Prime numbers are evenly divisible by 1 and themselves. They are the basis of arithmetic and of all natural numbers. That is, those that arise naturally when counting objects, for example, apples. Any natural number is the product of some prime numbers. And those and others - an infinite number.

Prime numbers other than 2 and 5 end in 1, 3, 7, or 9. They were thought to be randomly distributed. And a prime number ending, for example, in 1 can with equal probability - 25 percent - be followed by a prime number that ends in 1, 3, 7, 9.
Prime numbers are integers greater than one that cannot be represented as the product of two smaller numbers. So 6 is not a prime number because it can be represented as a product of 2?3, but 5 is a prime number because the only way to represent it as a product of two numbers is 1?5 or 5?1. If you have several coins, but you can't arrange them all in a rectangle, but can only line them up in a straight line, your number of coins is a prime number.

A perfect number has 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 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 the appearance of the work of Euclid's "Beginnings" in 300 BC. several have already been proven important facts about prime numbers. In Book IX of the Elements, Euclid proved that there are an infinite number of prime numbers. By the way, this is one of the first examples of the use of proof by contradiction. He also proves the Basic Theorem of Arithmetic - every integer can be represented in a unique way as a product of prime numbers.
He also showed that if the number 2 n -1 is prime, then the number 2 n-1 * (2 n -1) will be perfect. Another mathematician, Euler, in 1747 was able to show that all even perfect numbers can be written in this form. To this day, it is not known whether odd perfect numbers exist.

In the year 200 B.C. The Greek Eratosthenes came up with an algorithm for finding prime numbers called the Sieve of Eratosthenes.

No one knows for sure in which society the prime numbers were first considered. They have been studied for so long that scientists have no records of those times. There are speculations that some early civilizations had some understanding of prime numbers, but the first real evidence for this comes from Egyptian papyri records made over 3,500 years ago.

The ancient Greeks were most likely the first to study prime numbers as a subject of scientific interest, and they believed that prime numbers were important for purely abstract mathematics. Euclid's theorem is still taught in schools, despite being over 2,000 years old.

After the Greeks, serious attention was paid to prime numbers again in the 17th century. Since then, many famous mathematicians have made important contributions to our understanding of prime numbers. Pierre de Fermat made many discoveries and is best known for Fermat's Last Theorem, a 350-year-old prime number problem solved by Andrew Wiles in 1994. Leonhard Euler proved many theorems in the 18th century, and in the 19th century a big breakthrough was made by Carl Friedrich Gauss, Pafnuty Chebyshev and Bernhard Riemann, especially with regard to the distribution of prime numbers. All this culminated in the hitherto unsolved Riemann Hypothesis, which is often called the most important unsolved problem in all of mathematics. The Riemann Hypothesis makes it possible to very accurately predict the appearance of prime numbers, and also partly explains why they are so difficult for mathematicians.

Discoveries made in the early 17th century by the mathematician Fermat proved Albert Girard's conjecture that any prime number of the form 4n+1 can be written in a unique way as a sum of two squares, and also formulated a theorem that any number can be represented as a sum of four squares.
He developed new method factorization big 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, a p = a modulo p will be true.
This statement proves half of what was known as the "Chinese hypothesis" and dates back 2000 years earlier: an integer n is prime if and only if 2n-2 is divisible by n. The second part of the hypothesis turned out to be false - for example, 2341 - 2 is divisible by 341, although the number 341 is composite: 341 \u003d 31? eleven.

Fermat's Little Theorem was the basis for many other results in number theory and methods for testing whether numbers are prime, many of which are still in use today.
Fermat corresponded extensively with his contemporaries, especially with a monk named Marin Mersenne. In one of his letters, he conjectured 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 sure that when n is not a power of two, the number was not necessarily prime. These numbers are called Fermat numbers, and it wasn't until 100 years later that Euler showed that the next number, 232 + 1 = 4294967297, is divisible by 641 and therefore 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 actively studied them.

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 gave mathematicians the largest known primes. That the number M 19 was proved by Cataldi in 1588, and for 200 years was the largest known prime number, until Euler proved that M 31 is also prime. This record held 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 proved.
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 number theory, including prime numbers. He extended Fermat's Little Theorem and introduced the ?-function. Factorized the 5th Fermat number 2 32 +1, found 60 pairs of friendly numbers, and formulated (but failed to prove) the quadratic law of reciprocity.

He was the first to introduce the methods of mathematical analysis and developed the analytic theory of numbers. 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 +…
obtained by the sum of reciprocals of prime numbers also diverges. The sum of the n terms of the harmonic series grows approximately like log(n), while the second series diverges more slowly, like log[ log(n) ]. This means that, for example, the sum of the reciprocals of all the prime numbers found to date will give only 4, although the series still diverges.
At first glance, it seems that prime numbers are distributed among integers rather randomly. 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 on large segments, prime numbers are distributed fairly evenly. Legendre and Gauss dealt with 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 density of primes is 1/log(n). Legendre estimated the number of primes between 1 and n as
?(n) = n/(log(n) - 1.08366)
And Gauss - as a logarithmic integral
?(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 Numbers Theorem. They tried to prove it throughout the 19th century, and Chebyshev and Riemann made progress. They connected it with the Riemann Hypothesis, a hitherto unproven conjecture about the distribution of zeros of the Riemann zeta function. The density of primes was simultaneously proved by Hadamard and de la Vallée-Poussin in 1896.
In the theory of prime numbers, there are still many unresolved questions, some of which are many hundreds of years old:

twin prime hypothesis - about an infinite number of pairs of prime numbers that differ from each other by 2
Goldbach's hypothesis: any even number, starting from 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 proved by Chebyshev)
Is there an infinite number of Fermat primes? are there any Fermat primes after the 4th?
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 primes in an arithmetic progression?
n 2 - n + 41 is a prime number for 0 ? n? 40. Is the number of such prime numbers infinite? The same question for the formula n 2 - 79 n + 1601. Are these numbers 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! - one?
if p is prime, does 2 p -1 always not include among the factors of squared primes
Does the Fibonacci sequence contain an infinite number of primes?

Some people think that prime numbers are not worth deep study, but they are fundamental to mathematics. Each number can be represented in a unique way in the form of prime numbers multiplied by each other. This means that prime numbers are "atoms of multiplication", small particles from which something large can be built.

Since primes are the building blocks of integers that are obtained by multiplication, many integer problems can be reduced to prime number problems. Similarly, some problems in chemistry can be solved using atomic composition chemical elements involved in the system. Thus, if there were a finite number of primes, one could simply check one by one on a computer. However, it turns out that there are an infinite number of primes, which mathematicians do not understand well at the moment.

Prime numbers have a huge number of applications both in the field of mathematics and beyond. Prime numbers are used almost daily these days, although most often they are not aware of it. Prime numbers are of such importance to scientists because they are the atoms of multiplication. A lot of abstract problems about multiplication could be solved if we knew more about prime numbers. Mathematicians often break down one problem into several smaller ones, and prime numbers could help with this if they understood them better.

Outside of mathematics, the main applications of prime numbers are related to computers. Computers store all data as a sequence of zeros and ones, which can be expressed as an integer. Many computer programs multiply numbers associated with data. This means that just below the surface lie prime numbers. When a person makes any online purchases, he takes advantage of the fact that there are ways to multiply numbers that are difficult for a hacker to decipher, but easy for a buyer. This works due to the fact that prime numbers do not have special characteristics - otherwise, an attacker could get the bank card data.

One way to find prime numbers is by computer search. By repeatedly checking whether a number is a factor of 2, 3, 4, and so on, one can easily determine whether it is prime. If it is not a factor of any smaller number, it is prime. This is actually a very time consuming way of finding out if a number is prime. However, there are better ways to determine this. The performance of these algorithms for each number is the result of a theoretical breakthrough in 2002.

There are a lot of prime numbers, so if you take a large number and add one to it, you can stumble upon a prime number. In fact, many computer programs rely on the fact that prime numbers are not too hard to find. This means that if you randomly select a number from 100 digits, your computer will find a larger prime number in a few seconds. Since there are more 100-digit prime numbers than there are atoms in the universe, it is likely that no one will know for sure that this number is prime.

As a rule, mathematicians do not look for individual prime numbers on the computer, however, they are very interested in prime numbers with special properties. There are two well-known problems: is there an infinite number of primes that are one more than a square (for example, this matters in group theory), and is there an infinite number of pairs of primes that differ from each other by 2.

The largest prime number calculated by the GIMPS project can be found in the table on the project's official page.

The largest twin primes are 2003663613 ? 2195000 ± 1. They consist of 58711 digits and were found in 2007.

The largest factorial prime number (of the form 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.

It would take a book of more than 7,000 pages to write down the new prime number found by mathematicians. It - this is an unprecedentedly large number - consists of 23,249,425 digits. It was discovered thanks to the GIMPS (Great Internet Mersenne Prime Search) distributed computing project.

Prime numbers are those that are divisible by one and themselves. And nothing more. What has now been found also applies to the so-called Mersenne numbers, which have the form 2 to the power of n minus 1. The record number can be expressed as 2 to the power of 77232917 minus 1. It has become the 50th known Mersenne number.

Prime numbers are used in cryptography - for encryption. They cost a lot of money. For example, in 2009, a premium of $100,000 was paid for one of the prime numbers.

Despite the fact that prime numbers have been studied for more than three millennia and have a simple description, surprisingly little is known about prime numbers. For example, mathematicians know that the only pair of primes that differ by 1 are 2 and 3. However, it is not known whether there is an infinite number of pairs of primes that differ by 2. It is assumed that there is, but this has not yet been proven. This is a problem that can be explained to a child school age, but the greatest mathematical minds have been puzzling over it for more than 100 years.

Many of the most interesting questions about prime numbers, both from a practical and theoretical point of view, are how many primes have a particular property. The answer to a simple question - how many prime numbers of a certain size are there - can theoretically be obtained by solving the Riemann hypothesis. An additional incentive to prove the Riemann Hypothesis is a one million dollar prize offered by the Clay Mathematical Institute, as well as a place of honor among the outstanding mathematicians of all time.

There are now good ways to guess what the correct answer to many of these questions will be. At the moment, mathematicians' guesses pass all numerical experiments, and there are theoretical reasons to rely on them. However, it is extremely important for pure mathematics and the operation of computer algorithms that these guesses are actually correct. Mathematicians can only be fully satisfied if they have an undeniable proof.
The biggest challenge for practical application is the complexity of finding all the prime factors of a number. If you take the number 15, you can quickly determine that 15=5x3. But if you take a 1000-digit number, calculating all its prime factors will take more than a billion years even for the most powerful supercomputer in the world. Internet security largely depends on the complexity of such calculations, so for the security of communication it is important to know that someone cannot just come up with fast way find prime factors.

It is currently impossible to say how prime numbers will be used in the future. Pure mathematics (for example, the study of prime numbers) has repeatedly found applications that may have seemed completely unbelievable when the theory was first developed. Again and again, ideas that were perceived as a wonderful academic interest, unsuitable for real world, turned out to be surprisingly useful for science and technology. Godfrey Harold Hardy, a famous mathematician of the early 20th century, argued that prime numbers have no real application. Forty years later, the potential of prime numbers for computer communication was discovered, and they are now vital to everyday use of the Internet.

Because prime numbers are at the heart of the whole number problem, and since integers are constantly found in real life, prime numbers will have ubiquitous applications in the world of the future. This is especially true given how the Internet is permeating life, and technology and computers are playing a bigger role than ever before.

There is an opinion that certain aspects of the theory of numbers and prime numbers go far beyond the scope of science and computers. In music, prime numbers explain why some complex rhythmic patterns take a long time to repeat. This is sometimes used in modern classical music to achieve a specific sound effect. The Fibonacci sequence occurs all the time in nature, and it is hypothesized that cicadas have evolved to hibernate for a mere number of years to gain an evolutionary advantage. It is also assumed that the transmission of prime numbers over radio waves would be the best way to attempt to establish a connection with alien life forms, since prime numbers are completely independent of any notion of language, but at the same time complex enough that they cannot be confused with the result of some pure physical natural process.

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