The history of the occurrence of simple and constituent numbers. The story of simple numbers
Department of Education and Youth Policy Administration
Yalchik district Chuvash Republic
Project
Simple numbers ...
Is there a simple story?
Completed the student of the 7th grade MOU "Novoshikuskaya Soshchik district of the Chuvash Republic" Efimova Marina
Leader: Mathematics Teacher I Category MOU "Novoshevsk School of Yalchik district Chuvash Republic" Cyrilova S.M.
s.Novy Shimkusov - 2007
Determination of prime numbers 3
The merits of Euler 3.
The main theorem of arithmetic 4
Simple numbers of Mersene 4
Simple numbers farm 5
Devolo Eratosthene 5.
Opening P.L. Chibyshev 6.
Goldbach problem 7.
I.M.Vinogradov 8.
Conclusion 8.
Literature 10
Interest in the study of prime numbers originated in people in ancient times. And he was called not only a practical necessity. Attracted their extraordinary magical power. The numbers that can be expressed by the number of any items. Unexpected and at the same time natural properties of natural numbers found by ancient mathematicians, surprised them with their wonderful beauty and inspired new studies.
It must be one of the first properties of the numbers open by man, was that some of them could be decomposed into two or more factors, for example,
6 \u003d 2 * 3, 9 \u003d 3 * 3, 30 \u003d 2 * 15 \u003d 3 * 10, while others, for example, 3, 7, 13, 37, cannot be decomposed in a similar way.
When the number C \u003d butb. is the work of two numbers but and B. , The numbers aIb. called multipliers or dividers numbers with. Each number can be represented as a work of two favors. For example, S. = 1 * C \u003d C * 1.
Simple It is called the number that is divided only by itself and per unit.
The unit having only one divider does not apply to simple numbers. It does not apply to constituent numbers. The unit occupies a special position in the numerical row. Pythagoreans taught that the unit is the mother of all numbers, the Spirit, from which the entire visible world takes place, it is a mind, good, harmony.
In Kazan University, Professor Nikolsky, with the help of the unit, managed to prove the existence of God. He said: "Therefore, there can be no number without one, so the universe cannot exist without a single lord."
Unit and in fact - the number is unique by properties: it only is divided into herself, but any other number is divided into it without a rest, any degree is equal to the same number - one!
After dividing on it, no number changes, and if we share any number on the very whole, it will turn out again a unit! Is it not surprising? After thinking about it, Euler said: "It is necessary to exclude a unit from the sequence of prime numbers, it is neither simple or composite."
It has already been a substantially important ordering in a dark and challenging issue about the simple numbers.
Merit Eilera
Leonard Euler
(1707-1783)
Euler studied everything - Western Europe, and in Russia. The range of its creativity is wide: differential and integral calculus, algebra, mechanics, dioptric, artillery, sea science, the theory of motion planets and the moon, music theory is not listed. In the whole of this scientific mosaic there is also the theory of numbers. Euler gave her a lot of strength and considerable. He, like many of his predecessors, was looking for a magic formula that would make it possible to highlight the simple numbers from an infinite set of numbers of a natural row, that is, of all the numbers, which you can imagine. Euler wrote more than one hundred writings on the theory of numbers.
... It is proved, for example, that the number of prime numbers is unlimited, i.e.: 1) there is no largest simple number; 2) There is no last simple number, after which all numbers would be composite. The first proof of this provision belongs to scientists of ancient Greece (Vs. BC. BC), the second evidence - Euler (1708-1783).
The main theorem of arithmetic
Any natural number, different from 1, is either simple, or can be represented as a product of prime numbers, and unambiguously, if you do not pay attention to the procedure for finding multipliers.
Evidence. Take a natural number p ≠ 1. If N is simple, then this is the case that is stated in the conclusion of the theorem. Now suppose that N is composite. Then it is represented as a work p \u003d A.b.,
where natural numbers a and b are less than n. Again either a and b - simple, then everything is proven, or at least one composite one of them, i.e., composed of smaller factors and so on; In the end, we will get a decomposition on simple multipliers.
If the number n is not divided into one simple, not exceeding √ n, it is simple.
Evidence. Suppose nasty, let n, composite and p = oh, where 1 ≤b. and p - simple divider number but, So, numbers n. By condition p not divided by any simple, not exceeding √ n. . Hence, p\u003e √n.. But then √n. and √ n. ≤ but ≤ B. ,
from p \u003d A.b. = √ n.√ n. = p; They came to the contradiction, the assumption was incorrect, the theorem was proved.
Example 1. If a c \u003d. 91, T. √ c \u003d 9, ... Check simple numbers 2, 3, 5, 7. Find that 91 = 7 13.
Example 2. If C \u003d 1973, then we find √ c. = √ 1973 =44, ...
since no simple number to 43 Does not divide C, then this number is simple.
Example 3. Find a simple number next to the thumbnail of 1973. Answer: 1979.
Simple numbers of Mermen
For several centuries, the pursuit was taken over simple numbers. Many mathematics fought for the honor of becoming openers of the largest of the famous prime numbers.
Mersene's simple numbers are simple numbers of a special species M p \u003d 2 P - 1
where r - Another simple number.
These numbers entered the mathematics for a long time, they appear even in Euclidean reflections on modern numbers. They received their name in honor of the French monk Merenna Mersen (1589-1648), which has long been engaged in the problem of modern numbers.
If you calculate the numbers for this formula, we get:
M 2 \u003d 2 2 - 1 \u003d 3 - simple;
M 3 \u003d 2 3 - 1 \u003d 7 - simple;
M 5 \u003d 2 5 - 1 \u003d 31- Eastern;
M 7 \u003d 2 7 - 1 \u003d 127 - simple;
M 11 \u003d 2 11 - 1 \u003d 2047 \u003d 23 * 89
The general method of finding large simple numbers of Mersene is to check all numbers M p for various simple numbers r.
These numbers increase very quickly and the labor costs for their finding are equally rapidly increasing.
In the study of Mersene's numbers, it is possible to allocate the early stage, which has reached its climax in 1750, when Euler has established that the number M 31 is simple. By that time, eight simple mercen numbers were found: "G
r \u003d 2, p \u003d 3, p \u003d 5 , p \u003d 7, p \u003d 13, p \u003d 17, p \u003d 19, r =31.
Euler's number M 31 remained the largest of the famous prime numbers over a hundred years.
In 1876, the French mathematician Lucas found that a huge number M 127 - with 39 digits. 12 simple mercen numbers were calculated using only pencil and paper, and mechanical desktop counting machines have already been used to calculate the following.
The appearance of computing machines with an electric drive allowed to continue the search by bringing them to r = 257.
However, the results were disappointing, among them were not new simple numbers of Mermen.
Then the task was shifted on a computer.
The most famous currently simple number has 3376 digits. This number was found on a computer in Illinois University (USA). The mathematical faculty of this university was so proud of its achievement, which depicted this number on his mail stamp, thus reproducing it on each sent letter for universal ferris.
Simple farm numbers
There is another type of simple numbers with a big and interesting story. They were first introduced by the French lawyer Pierre Farm (1601-1665), which was famous for his outstanding mathematics.
Pierre Farm (1601-1665)
The first simple farm numbers were numbers that satisfy the formula F n \u003d 
+ 1.
F 0 \u003d. 
+ 1 = 3;
F 1 \u003d. 
+ 1 = 5;
F 2 \u003d. 
+ 1 = 17;
F 3 \u003d. 
+ 1 = 257;
F 4 \u003d. 
+ 1 = 65537.
However, this assumption was put into the archive of unreasonable mathematical hypotheses, but after Leonard Euler took another step and showed that the next number of farm F. 5 \u003d 641 6 700 417 is composite.
It is possible that the history of the farm numbers would be completed if the numbers of the farm did not appear in a completely different task - to build the right polygons with the help of a circular and a ruler.
However, not a single simple number of the farm was found, and now many mathematics tend to believe that there are no more.
Swelto Eratosthen
There are tables of prime numbers extending to very large numbers. How to approach the compilation of such a table? This task was, in a certain sense, resolved (about 200 BC) Eratosthen, Mathematics from Alexandria. -
Its scheme is as follows. Write the sequence of all integers from 1 to the number we want to finish the table.
Let's start with a simple number 2. We will throw away each second number. Let's start with 2 (except for the number 2), i.e. even numbers: 4, 6, 8, 10, etc., we emphasize each of them.
After this operation, the first inextricant number will be 3. It is simple, since it is not divided by 2. Leaving the number 3 inappropriate, we will emphasize every third number after it, i.e. the numbers 6, 9, 12, 15 ... Some of them They were already emphasized because they are even. In the next step, the first inextricant number will be the number 5; It is simple, since it is not divided into 2, nor 3. Let us leave the number 5 inappropriate, but we emphasize every fifth number after it, i.e. the numbers 10, 15, 20 ... as before, some of them turned out to be underlined . Now the smallest inextricated number is the number 7. It is simple, since it is not divided into one of its smaller simple numbers 2, 3, 5. Repeating this process, we eventually get a sequence of inextful numbers; All of them (other than 1) are simple. This method of chipping numbers is known as "Deuto Eratosphen". Any table of prime numbers is created on this principle.
Eratosthen created a table of prime numbers from 1 to 120 more than 2,000 years ago. He wrote on a papyrus, stretched on a frame, or in a wax plank, and did not cross, as we do, but pierced the constituent numbers. It turned out something like a solid, through which "sieved" compound numbers. Therefore, the table of prime numbers is called "Eratosthena Race".
How many prime numbers? Is there a last simple number, i.e. is that, after which all the numbers are composite? If there is such a number, how to find it? All these questions were interested in scientists in ancient times, but the answer to them was not so easy to find.
Eratosthen was a witty man. This contemporary and friend of Archimedes, with whom he constantly corresponded, was both mathematician and astronomer, and a mechanic, which was considered natural for great husbands of that time. He first measured the diameter of the globe, and without leaving the Alexandria library, where he worked. The accuracy of its measurement was strikingly high, even above the one with the land of Archimedes.
Eratosthene invented a cunning device - mesolabit, S. The help of which mechanically solved the well-known task about the doubling of the cube, which was very proud of, and therefore gave the order to portray this device on the column in Alexandria. Moreover, he corrected the Egyptian calendar, adding one day to four years - in a leap year.
Swelto Eratosthene is a primitive and at the same time a brilliant invention, which did not think of the Euclidean, - brings to the well-known idea that everything is simply genius.
Eratosthenovo decided well for researchers far from no simple numbers. There was time. The search for methods of catching prime numbers was searched. A peculiar competition began to find the greatest simple number from ancient times to Chebyshev and even to the present day.
Opening P.L. Chebyshev
AND 
so, the number of prime numbers is infinite. We have already seen that simple numbers are posted without any order. Follow in more detail.
2 and 3 are simple numbers. This is the only couple of prime numbers standing nearby.
Then go 3 and 5, 5 and 7, 11 and 13, 17 and 19, etc. These are the so-called adjacent simple numbers or twins. Twins Many: 29 and 31, 41 and 43, 59 and 61, 71 and 73, 101 and 103, 827 and 829, etc. The largest twin pair is now: 10016957 and 10 016 959.
Panfuti Lvovich Chebyshev
How are the simple numbers in a natural row, in which there will be no simple number? Is there any law in their distribution or not?
If there is, what? How to find it? But the answer to these questions was not more than 2,000 years.
The first and very big step in the resolution of these issues was made by the great Russian scientist Polandi Lvovich Chebyshev. In 1850, he proved that between any natural number (not equal to 1) and the number, twice as many of it (i.e., between N and 2N), there is at least one simple number.
Check it on simple examples. We will take a few arbitrary values \u200b\u200bn .
and find a 2n value accordingly.
n \u003d 12, 2n \u003d 24;
n \u003d 61, 2n \u003d 122;
n \u003d 37, 2N \u003d 74.
We see that for the considered examples of the Chebyshev Theorem is true.
Chebyshev proved it for any occasion for any n. For this theorem, he was called the winner of prime numbers. Open Chebyshev Act of the distribution of prime numbers was truly a fundamental law in the theory of numbers after the law, open by Euclide, about infinity of the number of prime numbers.
It is hardly the most kind, the most enthusiastic response to the opening of Chebyshev came from England from the famous mathematics of Sylvester: "... The future success of the theory of prime numbers can be expected when someone is born, so superior Chebyshev with its insight and thoughtfulness, as far as Chebyshev exceeds these qualities ordinary people. "
More than half a century later, the German mathematician E. Landau, a major specialist in the theory of numbers, added to this statement the following: "First after Euclid, Chebyshev went properly by solving the problem of prime numbers and achieved important results."
Goldbach problem
Drink all the simple numbers from 1 to 50:
2, 3, 5, 7, 9, 11, 17, 19, 23, 29, 31, 37, 41, 43, 47.
And now we will try any number from 4 to 50 Present in the form of two or three simple numbers. Take several numbers at random:
As we can see the task, we did easily. Is it always possible? Is any number can be represented as a sum of several prime numbers? And if possible, how many: two? three? ten
In 1742, the member of the St. Petersburg Academy of Sciences Goldbach in a letter to Euler suggested that any whole positive number, more than five, is the amount of no more than three simple numbers.
Goldbach experienced a lot of numbers and never met such a number that could not be decomposed in the amount of two or three ordinary terms. But will it always be so, he has not proven. Long scientists were engaged in this task, which is called "the problem of Goldbach" and is formulated as follows.
It is required to prove or refute the offer:
any number, more units, is the sum of no more than three simple numbers.
Almost 200 years of outstanding scientists tried to resolve the problem of Goldbach Euler, but unsuccessfully. Many have come to the conclusion about the impossibility of solving it.
But her decision, and almost completely, was found in 1937 by the Soviet mathematician I.M. Vinogradov.
THEM. Vinogradov
Ivan Matveyevich Vinogradov is one of the largest modern mathematicians. He was born on September 14, 1891 in the village of Milolub Pskov province. In 1914 he graduated from the University of St. Petersburg and was left to prepare for the professorship.
His first scientific work I.M. Vinogradov wrote in 1915. Since then, more than 120 different scientific works have been written. In them, he allowed many tasks over which the scientists of the whole world worked dozens and hundreds of years.
Ivan Matveyevich Vinogradov
For merit in the field of mathematics IM Vinogradov by all scientists of the world recognized as one of the first mathematicians of our time, elected to the number of members of many academies of the world.
We are proud of our wonderful compatriot.
Conclusion.
From class - to world space
Conversation about the simple numbers Let's start a fascinating story about the imaginable journey from the class in the world space. This imaginary journey came up with the well-known Soviet teacher-mathematician Professor Ivan Kozmich Andronov (born in 1894). "... (a) We mentally take a straight line wire leaving the classroom in the world space, punching the earth's atmosphere, going to where the moon makes rotation, and further - for the fireball of the sun, and further - in world infinity;
b) mentally suspended on the wire through each meter, the electric light bulbs, numbering them, starting with the nearest: 1, 2, 3, 4, ..., 100, ..., 1000, ..., 1 000 000 ...;
c) mentally turning on the current with such a calculation so that all the light bulbs with simple numbers caught fire and only with simple numbers; :.
d) mentally rolling near the wire.
The next picture will unfold before us.
1. Light bulb with number 1 does not burn. Why? Because the unit is not a simple number.
2. The two next bulbs with numbers 2 and 3 are lit, as 2 and 3 are both simple numbers. Can there be two adjacent burning light bulbs in the future? No, can not. Why? Any simple number, besides two, there is an odd number, and adjacent to the simple one and the other side will be numbers even, and any even other, different from two, is a constant number, as it is divided into two.
3. Further observing a couple of light bulbs burning through one light bulb with numbers 3 and 5, 5 and 7, etc. It is clear why they are burning: these are twins. We notice that in the future they are less common; All pairs of twins, as well as pairs of prime numbers, have a form 6n ± 1; eg
6*3 ± 1 equal to 19 and 17
or 6 * 5 ± 1 equal to 31 and 29, ...;
but 6 * 20 ± 1 is 121 and119- This pair is not a twin, since there is a pair of components.
We arrive to the twin pair of 10 016 957 and 10 016 959. Will there be more couples for twins? Modern science does not give a response: it is unknown, there is a finite or infinite multiple twins pairs.
4. But now the law of a large range, filled with only components: we fly in the dark, look back - darkness, and there is no light in front. We remember the property open by Euclide, and be safely moving forward, as there should be glowing lights in front, and there should be an infinite set in front.
5. Flowing into such a natural range, where for several years of our movement passes in the dark, remember the property, proven Chebyshev, and calm down, confident that in any case, you need to fly no more of what flew to see at least one luminous Light bulb. "
Literature
1.
Great Master of Induction Leonard Euler.
2. Behind the pages of the textbook of mathematics.
3. Prudnikov N.I. P.L. Chebyshev.
4. Serbian I. A. What we know and what we do not know about the simple numbers.
5. Publishing House "First September". Mathematics №13, 2002
6. Publishing House "First September". Mathematics №4, 2006
Milok Maxim
This year we studied the topic "Simple and constituent numbers", and it became interesting to me who of scientists dealt with their study, how to get simple numbers, except those contained on the mound of our textbook (from 1 to 1000), this was the purpose of execution This work.
Tasks:
1. Examine the history of the opening of prime numbers.
2. Get acquainted with modern methods for finding prime numbers.
3. To learn how simple numbers are applied in which scientific fields.
4. Are there among Russian scientists the names of those who engaged in the study of prime numbers.
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History of prime numbers MBOU Sukhovskaya Sosh Author: Student 6 Class Milk Maxim Head: Math teacher Babkina L. A. Novosuhovy December 2013
This year we studied the topic "Simple and constituent numbers", and it became interesting to me who of scientists dealt with their study, how to get simple numbers, except those contained on the mound of our textbook (from 1 to 1000), this was the purpose of execution This work. Tasks: 1. Examine the history of the opening of prime numbers. 2. Get acquainted with modern methods for finding prime numbers. 3. To learn how simple numbers are applied in which scientific fields. 4. Are there among Russian scientists the names of those who engaged in the study of prime numbers.
Everyone who studies simple numbers is fascinated and at the same time feels their own impotence. The definition of simple numbers is so simple and obvious; Find another simple number so easy; Decomposition of simple factors is such a natural action. Why are the simple numbers so hard resist our attempts to comprehend the order and patterns of their location? Maybe there is no order in them at all, or are we so blind that we do not see it? C. Uterlell.
Pythagoras and his disciples studied the question of the divisibility of numbers. A number equal to the sum of all its divisors (without the number), they called the perfect number. For example, numbers 6 (6 \u003d 1 + 2 +3), 28 (28 \u003d 1 + 2 + 4 + 7 + 14) perfect. The following are the following numbers - 496, 8128, 33550336 .. Pythagoras (VI century BC)
Pythagoreans knew only the first three perfect numbers. Fourth - 8128 - became famous in the first century AD. Fifth - 33550336 - was found in the XV century. By 1983, 27 perfect numbers were already known. But so far, scientists do not know whether there are odd perfect numbers, whether there is a largest perfect number.
The interest of the ancient mathematicians to simple numbers is due to the fact that any number or simple, or can be represented as a product of prime numbers, i.e. Simple numbers are like bricks from which the remaining natural numbers are being built.
You probably noticed that simple numbers in a row of natural numbers are unevenly found in some parts of the series more, in others - less. But the farther we are moving around the numerical row, the less simple numbers are found.
The question arises: does the last one (the biggest) simple number? Ancient Greek mathematician Euclidean (III century BC) in his book ("beginning"), formerly for 2000 years the main textbook of mathematics, proved that simple numbers are infinitely a lot, i.e. Each simple number there is a greater simple number of Euclidean (III century. BC)
To find prime numbers, another Greek mathematician Eratosphen invented such a way. He recorded all the numbers from one to some number, and then he finished the unit, which is not a simple, not a constituent number, then highlighted through one all numbers going after 2 numbers, multiple two, i.e. 4,6,8, etc.
The first remaining number after two was 3. Further was shut down in two all numbers, reaching after three (numbers of multiple 3, i.e. 6,9,12, etc.). In the end, only simple numbers remained unsecured.
Since the Greeks made entries on the wax tables covered or on a pulling papyrus, and the numbers were not deducted, but they pumped the needle, the table at the end of the calculations resembled the sieve. Therefore, the Eratosphen method is called Eratosthena Reshet: in this sift the simple numbers from the composite.
So, with simple numbers from 2 to 60 are the 17 numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59. In this way and in Currently, there are tables of prime numbers, but already with the help of computing machines.
Euclid (III century BC) proved that between the natural number of N and N! Be sure to be at least one simple number. Thus, he proved that the natural number of numbers is infinite. In the middle of the first x in. The Russian mathematician and the mechanic of Paphunia Lvovich Chebyshev proved a stronger theorem than Euclide. Between the natural number N and the number 2 times more of it, i.e. 2 N contains at least one simple number. That is, in the Euclide's theorem Number N! Replaced the number 2n. Pafnutius Lvovich Chebyshev (1821-1894) Russian mathematician and mechanic
The following question arises: "If it's so hard to find the next simple number, then where and for which these numbers can be used in practice?" The most common example of using prime numbers is the use of them in cryptography (data encryption). The safest and hardly decoded methods of cryptography are based on the use of prime numbers having more than three hundred numbers.
Conclusion The problem of the lack of regularities of the distribution of prime numbers occupies the minds of humanity since the times of ancient Greek mathematicians. Thanks to Euclide, we know that simple numbers are infinitely a lot. Erasopen proposed the first algorithm for testing numbers to simplicity. Chebyshev and many other famous mathematicians tried and try to unravel the mystery of prime numbers to this day. At the moment, many elegant algorithms were found and proposed, but they all apply only for a finite number of prime numbers or simple numbers of a special type. The front edge of the science in studies of prime numbers on infinity is the proof of the Riemann hypothesis. It enters the seven of the unresolved millennium problems, for the proof or refutation of which the Mathematical Institute of Clai proposed a premium at $ 1.000.000.
Internet - sources and literature http://www.primenumb.ru/ http://www.bestpeopleofrussia.ru/persona/pafnutiy-chebyshev/bio/ http://uchitmatematika.ucoz.ru/index/vayvayvayjajavvvjavvvva/0-7 Tutorial "Mathematics" for the sixth class of educational institutions / N.I. Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S.I. Schwarzburg - M. Mnemozina 2010 /
Introduction
A simple number is a natural number that has exactly two different natural divisors: a unit and yourself. All other numbers, except for the unit, are called composite. Thus, all natural numbers, bombing units are divided into simple and composite. The study of the properties of simple numbers is engaged in the theory of numbers.
The main theorem of arithmetic claims that every natural number, more units, represents in the form of a work of prime numbers, and the only way to the order of the fact of the factory. Thus, simple numbers are elementary "building blocks" of natural numbers.
The representation of a natural number in the form of a piece of simple is called decomposition into a simple or factorization of the number.
From the history of prime numbers
Greek mathematician Eratosthen, who lived in more than 2,000 years before AD, was the first table of prime numbers. Eratosthen was born in the city of Kiren, he received an education in Alexandria under the leadership of Callimac and Lisania, in Athens, he listened to the philosophers of Ariston Hios and Arkesila, closely close to the Plato's school. In 246 T.N.E., after the death of Callimakh, the king of Ptolemy Everghet called Eratosthena from Athens and instructed him to manage the Alexandria library. Eratosthen worked in many areas of science: philology, grammar, history, literature, mathematics, chronology, astronomy, geography and music.
To find the simple numbers, Eratosthen invented such a way. He recorded all the numbers from 1 to some number, and then crossed out the unit, which is neither a simple or constant number, then tracked through one all numbers going after 2 (numbers, multiple 2, i.e. 4.6 , 8, etc.). The first remaining number after 2 was 3. Next, all numbers of multiples 3 were drawn out, i.e. 6,9,12, etc. In the end, only simple numbers remained unsecured. (Fig.1)
Since the Greeks made the records on the wax tables covered or on a tensioned papyrus, and the numbers were not shifted, but they spoiled the needle, the table at the end of the calculations resembled the solido. Therefore, the Eratosphen method is called Eratosthena Reshet: in this sift the simple numbers from the composite. In this way, the tables of prime numbers are currently constituted, but already with the help of computing machines.
Simple numbers in nature and their use by man
1) Periodic Cicks
People changed the world around us, built incredible cities, and developed impressive technologies that led to the emergence of the modern world. Hidden under the outer shell of the planet, where we live, the invisible world consists of numbers, sequences and geometry. Mathematics is a code that gives the meaning of the whole universe.
In the forests of Tennessee, this summer part of the code, which is in question, in the literal sense, rose straight from the ground. Every 13 years about 6 weeks, insect choir enchants anyone who becomes a witness to this rare natural phenomenon. The survival of these cicades, which can be found only in the eastern regions of North America depends on the strange properties of some of the most fundamental numbers in mathematics - simple numbers, numbers that are divided only on themselves and others.
Cycades appear here periodically, but their appearance always occurs in those years, the numbers of which consist of simple numbers. In the case of brood, which appeared around Nashville this year, then from the moment of their past appearance, 13 years have passed. The choice of the 13th-year cycle does not seem random. In different parts of North America there are two more broods, the life cycle of which is also 13 years. They arise in different regions and in different years, but there are exactly 13 years old between the emerges of these living beings. In addition, there are still 12 insect broods, which appear every 17 years.
You can take these numbers for completely random. But it is very curious that there is no cycade with a cycle of life, equal to 12, 14, 15, 16 or 18 years. However, look at these cycades, the mathematics and the picture begins to clarify. Because the numbers 13 and 17 are both indivisible, it gives cicades the evolutionary benefits between other animals whose life cycles are periodic, and not simple numbers. Take, for example, a predator that appears in the forests every six years. Then eight or nine-year-old cycad cycles will coincide with the cycles of the predator's life, while the seven-year life cycles will coincide with the cycle of the predator's life much less often.
These insects intervened in the mathematical code to survive.
2) cryptography
Tsicada found the use of prime numbers for their survival, but people realized that these numbers are not only the key to survival, but also a huge number of building material in mathematics. Each number, in essence, is a combination of prime numbers, and the set of numbers is mathematics, and from mathematics you will get a whole scientific world.
Simple numbers find hidden in nature, but humanity learned to use them.
Understanding the fundamental nature of these numbers and the use of their properties by people, literally put them in the basis of all codes that world cyber secrets are guarded.
Cryptography, thanks to which our credit cards remain safe when we buy something online, uses the same numbers that protect Cycades in North America - simple numbers. Every time you enter your credit card number on the website, you rely on the fact that simple numbers will retain your secrets and information about you in the Security. To coding your credit card, your computer receives a public number from a website that will be used to perform operations with your credit card.
It mixes your data so that the encoded letter can be sent via the Internet. The website uses the simple numbers to be divided by the N N to decompress the message. Although H is an open number, the simple numbers from which it consists are secret keys that decipher the data. The reason why such coding is so secure is that it is very easy to multiply the simple numbers among themselves, but it is almost impossible to expand the number to simple.
3) mysteries of prime numbers
Simple numbers are the atoms of arithmetic, hydrogen and the oxygen of the world of numbers. But contrary to their fundamental nature, they are also one of the biggest mysteries of mathematics. Because, passing through the universe numbers, it is almost impossible to predict where you will find the next simple number.
We know that the number of simple numbers goes into infinity, but the search for the laws of the appearance of prime numbers is the biggest mystery of mathematics. The prize of a million dollars is promised to someone who can reveal the secret of these numbers. The riddle about when the cicada first began to use simple numbers to survive is the same complex as the mystery itself is simple.
Simple numbers are "capricious". Tables of prime numbers detect large "incorrectness" in the distribution of prime numbers
The distruting patterns of the distribution of primes increase even more if noted that there are pairs of prime numbers that are separated in a natural row of only one number ("Gemini"). For example. 3 and 5, 5 and 7, 11 and 13, 10016957 and 10016959. On the other hand, there are pairs of prime numbers, between which there are many composite. For example, all 153 numbers from 4652354 to 465,2506 are composite.
For finding simple numbers from more than 100,000,000 and 1,000,000,000 decimal digits, the EFF has appointed monetary prizes, respectively, 150,000 and $ 250,000.
MOU "Freactive Secondary General School"
Research work on the topic:
"Numbers rule the world!"
Work completed:
student 6A class.
Leader:,
mathematic teacher.
from. Freorozer.
I. Introduction. -3pl.
II. Main part. -4pro.
· Mathematics in the ancient Greeks. - 4 times.
· Pythagora Samossky. -6st.
· Pythagoras and numbers. -8st.
2. Numbers are simple and composite. -10Tr.
3. Goldbach problem. -12pro.
4. Signs of divisibility. -13.
5. Curious properties of natural numbers. -15pro.
6. Numeric tricks. -18.
III. Conclusion. -22pro.
IV. Bibliography. -23.
I. Introduction.
Relevance:
Studying in the lessons of mathematics the topic "The divisibility of numbers", the teacher proposed to prepare a message about the history of the opening of ordinary and constituent numbers. When preparing a message, I was interested in the words of Pythagora "Numbers rule the world!"
There were questions:
· When did the science of numbers arise?
· Who made contribution to the development of science of numbers?
· The value of numbers in mathematics?
I decided to study in detail and summarize the material about the numbers and their properties.
Purpose of the study:examine simple and constituent numbers and show their role in mathematics.
Object of study:simple and constituent numbers.
Hypothesis: If, according to Pythagora "Numbers rule the world,
what is their role in mathematics.
Research tasks:
I. Collect and summarize all kinds of information about simple and components.
II. Show the value of numbers in mathematics.
III. Show curious properties of natural numbers.
Research methods:
· Theoretical analysis of literature.
· Method of systematization and data processing.
II. Main part.
1. The history of the emergence of science of numbers.
· Mathematics in the ancient Greeks.
And in Egypt, and in Babylon, the numbers used mainly to solve practical problems.
The situation has changed when the Greeks were engaged in mathematics. In their hands, mathematics from the craft became science.
Greek tribes began to settle on the northern and eastern banks of the Mediterranean sea about four thousand years ago.
Most of the Greeks asshel on the Balkan Peninsula - where the state is now Greece. The rest settled through the Islands of the Mediterranean Sea and on the banks of Malaya Asia.
Greeks were excellent sailors. Their lungs of stormy ships in all directions of the Mediterranean Sea are furious. They were brought dishes and decorations from Babylon, bronze weapons from Egypt, the skins of animals and bread from the shores of the Black Sea. And of course, like other nations, along with goods, ships brought knowledge to Greece. But the Greeks are not just
studied from other nations. Very soon they overtook their teachers.
Greek masters built amazing beauty Palaces and temples, which then served as a model for architects of all countries.
Greek sculptors created wonderful statues from marble. And from Greek scientists, not only "real" mathematics began, but also very many other sciences that we study at school.
Do you know why the Greeks overtake all other nations in mathematics? Because they knew how to argue well.
How can disputes help science?
In ancient times, Greece consisted of many small states. Like every city with surrounding villages was a separate state. Every time I had to solve some important state question, the townspeople gathered on the square, discussed him. Argued about how to do better, and then voted. It is clear that they were good debaters: at such meetings it had to refute the opponents, arguing, to prove their right. The ancient Greeks believed that the dispute helps to find the best. The most correct solution. They even came up with such a saying: "Truth is born in the dispute."
And in the science of the Greeks began to do the same. As at the People's Assembly. They did not just memorize the rules, but the reasons were drawn: why to do it right, and not otherwise. Every rule, Greek mathematicians tried to explain, to prove that it is not true. They argued with each other. Reasoned, tried to find in the arguments of the error.
Prove one rule - reasoning lead to another, more complex, then to the third, to the fourth. From the rules there were laws. And from the laws - science mathematics.
Already born, Greek mathematics immediately seven-year steps went forward. It was helped by wonderful boots, which were not before other peoples. They were called "reasoning" and "proof".
· Pythagora Samossky.
The first of the numbers began to reason the Greek Pythagoras, who was born on the island of the pets in the VI century and our era.
Therefore, it is often referred to as Pythagorea Samos. Many legends told the Greeks about this thinker.
Pythagoras early showed the ability to science, and the father of Mr. District him in Syria, in a shooting gallery so that the Chaldean wise men were taught there. She will learn about the sacraments of the Egyptian priests. Falling by the desire to enter their circle and become dedicated to, Pythagoras begins to prepare for traveling to Egypt. He holds a year in Phenicia, in the school of priests. Then he will be in Egypt, in Helioolis. But local priests were irrevocable.
showing the perseverance and withstood extremely difficult entry tests, Pythagoras seeks his own - he was taken in Casta.21. He stayed in Egypt, he studied all kinds of Egyptian letters in Egypt, read many papyrus. Facts known to the Egyptians in mathematics are pushing it to their own mathematical discoveries.
The sage said: "There are in the world with things to which you need to strive. This, firstly, beautiful and glorious, secondly, useful for life, third, delight. However, pleasure is a double kind: one, quenching the luxurious of our gluttony, is disastrous; Another righteous and necessary for life. "
The central place in the philosophy of pupils and adherents of Pythagora was occupied by the number:
« Where there is no number and measures - there is chaos and chimeras, "
"The wise is the number",
"Numbers manage the world."
Therefore, many consider Pythagora by the father of the numbering - a complex, shrouded in the secret science describing in it the events revealing the past and the future predicting the fate of people.
· Pythagoras and numbers.
The numbers of the ancient Greeks, and together with them Pythagorea and Pythagoreans, showed visibous in the form of pebbles laid out on the sand or on a counting board - Abaca.
The numbers of pebbles were laid out in the form of the correct geometric figures, these figures were classified, so the numbers were found, today called figured: linear numbers (i.e., simple numbers) - the numbers that are divided into one and itself and, therefore, represent in the form of a sequence Points lined in line
https://pandia.ru/text/79/542/images/image006_30.jpg "width \u003d" 312 "height \u003d" 85 src \u003d "\u003e
body numbers expressed by the work of three facilities
https://pandia.ru/text/79/542/images/image008_20.jpg "width \u003d" 446 "height \u003d" 164 src \u003d "\u003e
square numbers:
https://pandia.ru/text/79/542/images/image010_15.jpg "width \u003d" 323 "height \u003d" 150 src \u003d "\u003e
and. etc. It was from curly numbers that the expression went " Build a number in a square or cube».
Pythagoras did not limit himself with flat figures. From the points, he began to fold the pyramids, cubes and other bodies and study pyramidal, cubic and other numbers (see Fig. 1). By the way, the name cube number We also use today.
But the numbers received from different figures, Pythagoras was not satisfied. After all, he proclaimed that the numbers rule the world. Therefore, he had to invent, as with the help of numbers, to depict such concepts as justice, perfection, friendship.
To portray perfection, Pythagoras began for dividers of numbers (while the divider 1 he took, and the number itself did not take). All dividers of the number he folded, and if the amount turned out to be less than the number, it was not sufficient, and if more - excessive. And only in the case when the amount exactly was equal to the number, it was declared perfect. Similarly, portrayed the numbers of friendship - two numbers were called friendly if each of them was equal to the sum of the dividers of another number. For example, the number 6 (6 \u003d 1 + 2 + 3) is an indicable, the number 28 (1 + 2 + 4 + 7 + 17) is completely. The following perfect numbers are 496, 8128 ,.
2. Sheets are simple and composite.
About friendly or perfect numbers, modern mathematics recalls with a smile as childhood hobby.
The concepts of simple and constituent numbers introduced by Pythagorean are the subject of serious research, for which mathematicians receive high scientific awards.
From the experience of calculations, people knew that each number was either simple or by the work of several prime numbers. But they did not know how to prove it. Pythagorad or someone from his followers found the proof of this allegation.
Now it is easy to explain the role of prime numbers in mathematics: they are those bricks from which the other numbers are built using multiplication.
The discovery of regularities in a number of numbers is a very pleasant event for mathematicians: because these patterns can be used to build hypotheses, to test evidence and formulas. One of the mathematicians of the properties of prisons of prisons is that they refuse to obey at least some regularity.
The only way to determine whether the number is 100,895,598,69,69, - to take advantage of the rather time-consuming "eratosphen breathe".
The table presents one of the options for this solitude.
In this table, all the simple numbers smaller than 48 are circled. They found so: 1 has a single divider - herself, therefore 1 is not considered a simple number. 2 - the smallest (and the only one one) is a simple number. All other readers are divided into 2, and therefore have at least three divisors; Therefore, they are not simple and can be crossed out. The next unsecured number is 3; It has exactly two divisors, so it is simple. All other numbers, multiple three (i.e., such that can be divided into 3 without a residue) are crossed out. Now the first unsecured number is 5; It is simple, and all of its multiple can be deleted.
Continuing to cross multiple, you can cut down all the simple numbers, less than 48.
3. Goldbach problem.
Of the prime numbers, you can get any number using multiplication. What will happen if you fold simple numbers?
Mathematics Mathematics, who lived in Russia in the XVIII century, decided to put odd simple numbers only in pairs. He discovered an amazing thing: every time he managed to present an even number as the sum of two prime numbers. (As it was in the time of Goldbach, we consider 1 simple number).
4 \u003d 1 +3, 6 \u003d 3 + 3, 8 \u003d 3 + 5. etc.
https://pandia.ru/text/79/542/images/image016_5.jpg "width \u003d" 156 "height \u003d" 191 src \u003d "\u003e
Gold Mathematics wrote about his observation of Goldbach
The XVIII century Leonardu Eilor, who was a member of the St. Petersburg Academy of Sciences. Checking many more even numbers, Euler was convinced that all of them are the sums of two simple numbers. But even numbers are infinitely a lot. Therefore, the calculations of Euler were given only the hope that the property that goldbach noticed was all numbers. However, attempts to prove that it will always be so, have not led to anything.
Two hundred years have reflected mathematics over the problem of Goldbach. And only the Russian scientist Ivan Matveevich Vinogradov managed to make a decisive step. He found that any large natural number is
the sum of three simple numbers. But the number starting with which the statement of Vinogradov is correct, unimaginably great.
4. Signs of divisibility.
489566: 11 = ?
To find out how this number is simple or composite, you do not always need to look into the table of prime numbers. Often, it is enough to use signs of divisibility.
· Sign of divisibility by 2.
If the recording of a natural number ends with an even digit, then this number is even divided into 2 without a residue.
· Sign of divisibility by 3.
If the amount of numbers is divided by 3, then the number is divided by 3.
· Sign of divisibility by 4.
A natural number containing at least three digits is divided into 4, if it is divided into 4 numbers formed by the two last digits of this number.
· Sign of divisibility by 5.
If the name of the natural number ends with a number 0 or 5, then this number is divided by 5 without a residue.
· Sign of divisibility at 7 (B13).
The natural number is divided into 7 (by 13), if the algebraic amount of the numbers forming the verge of three digits (starting with the numbers of units) taken with the "+" sign for odd faces and with the "minus" sign for even faces, was divided into The algebraic amount of the faces, starting from the last face and alternating signs + and -: + 254 \u003d 679. The number 679 is divided into 7, it means that the number is divided by 7.
· Sign of divisibility by 8.
A natural number containing at least four digits is divided into 8, if it is divided into 8 numbers formed by three last figures.
· Sign of divisibility by 9.
If the amount of numbers is divided into 9, then the number itself is divided by 9.
· Sign of divisibility by 10.
If the natural number ends 0, then it is divided into 10.
· Sign of divisibility 11.
The natural number is divided into 11 if the algebraic amount of his numbers taken with the "Plus" sign, if the numbers are on odd places (starting with the numbers of units), and taken with the "minus" sign, if the numbers are in even places, is divided into, 7 - 1 + 5 \u003d 11, divided by 11).
· Sign of divisibility by 25.
A natural number containing at least three digits is divided into 25, if it is divided into 25 numbers formed by two last figures of this number.
· Sign of divisibility by 125.
A natural number containing at least four numbers is divided into 125 if the number formed by three last digits of this number is divided.
5. Curious properties of natural numbers.
Natural numbers have many curious properties that are detected when performing arithmetic action on them. But it is still easier to notice these properties than to prove them. We give several such properties.
1) . At the time of at random some kind of natural number, for example 6, and write all its divisors: 1, 2, 3.6. For each of these numbers we will write down how much dividers has. Since 1 only one divider (this number itself), in 2 and 3 two divisors, and in 6 we have 4 divisors, then we get numbers 1, 2, 2, 4. They have a wonderful feature: if we build these numbers in Cube and fold the answers, it will turn out exactly the same amount that we would get, first laying these numbers, and then erecting the sum in the square, in other words,
https://pandia.ru/text/79/542/images/image019_3.jpg "width \u003d" 554 "height \u003d" 140 src \u003d "\u003e
Cultures show that both the same answer is the same, namely 324.
Whatever number we have taken, the property noticed will be performed. That's just prove it is quite difficult.
2) . Take any four-digit number, for example 2519, and put it on the numbers first in descending order, and then in ascending order: and from a larger number, the smaller: \u003d 8262. With the number obtained, we do the same: 86 \u003d 6354. And one more is the same step: 65 \u003d 3087. Next, \u003d 8352, \u003d 6174. Are you not tired of subtracting? We still do one more step: \u003d 6174. It turned out 6174 again.
Now we, as programmers say, "looked around": how many times we have not read now, nothing but 6174, we will not get. Maybe the fact is that the initial number 2519 was chosen? It turns out that it is not with what: whatever four-digit number we have taken, after nothing more than seven steps will definitely be the same number 6174.
3) . Draw several circles with a common center and on the inner circle we write any four natural numbers. For each pair of adjacent numbers, I will read out of more smaller and write the result on the next circle. It turns out that if it repeat it quite many times, on one of their circles all numbers will be equal to zero, and therefore nothing except zeros will not be possible. The figure shows this for the case when numbers 25, 17, 55, 47 are written on the inner circle.
4) . Take any number (even thousandth) recorded in a decimal number system. Erend all his numbers into the square and fold. With the amount we will do the same. It turns out that after several steps we obtain either the number 1, after which there will be no other numbers, or 4, after which we have numbers 4, 16, 37, 58, 89, 145, 42, 20 and again we get 4. So, the cycle is not Avoid here.
5. We will make such an endless table. In the first column, we write numbers 4, 7, 10, 13, 16, ... (each next to 3 more than the previous one). From among the number 4, we will do the right string, increasing the number at each step by 3. From the number 7 we will behave, increasing the numbers by 5, from the number 10- to 7, etc. It turns out such a table:
If you take any number of this table, multiply it to 2 and add 1 to the work, it will always be a composite number. If you do the same with the number that is not included in this table, we get a simple number. For example, we take the number 45 from the table. The number 2 * 45 + 1 \u003d 91 is composite, it is 7 * 13. There are no numbers in the table in the table, and the number 2 * 14 + 1 \u003d 29 is simple.
This wonderful way to distinguish simple numbers from the composite invented in 1934 the Indian student Sundar. Observations for numbers allow you to open other wonderful allegations. The properties of the world numbers are truly inexhaustible.
Numeric tricks.
https://pandia.ru/text/79/542/images/image022_2.jpg "width \u003d" 226 "height \u003d" 71 "\u003e
After all, if you write the same number next to the three-digit number, then the initial number will multiply by 1001 (for example, 289 289 \u003d 289https: //pandia.ru/text/79/542/images/image024_3.jpg "width \u003d" 304 " Height \u003d "74"\u003e
And the four-digit numbers repeat once and divided by 73 137. The randering of equality
https://pandia.ru/text/79/542/images/image026_6.jpg "width \u003d" 615 "height \u003d" 40 src \u003d "\u003e
Note that the cubes of numbers 0, 1, 4, 5, 6 and 9 ends with the same digit (for example, https://pandia.ru/text/79/542/images/image028_4.jpg "width \u003d" 24 "height \u003d "24 src \u003d"\u003e. Jpg "width \u003d" 389 "height \u003d" 33 "\u003e
In addition, you need to remember the following table, showing where the fifth degrees of the following numbers begin:
https://pandia.ru/text/79/542/images/image032_2.jpg "width \u003d" 200 height \u003d 28 "height \u003d" 28 "\u003e means it is necessary to attribute to the initially written on the board five-digit number in front of the figure 3, and From the resulting number 3.
So that the audiences do not solve the focus, you can reduce the first digit of some of the numbers to several units and to the same units to reduce the corresponding digit in the amount. For example, in the figure is reduced, on 2 first digit in the third term and the same corresponding digit in the amount.
Conclusion.
Collecting and summarizing the material about the simple and constituent numbers, concluded:
1. The doctrine of numbers goes in ancient times and has a rich history.
2. The role of prime numbers in mathematics is great: they are the bricks, from which all other numbers are built using multiplication.
3. Natural numbers have many curious properties. The properties of the world numbers are truly inexhaustible.
4. Material prepared by me can be safely used in mathematics lessons and classes of a mathematical circle. This material will help more deeply prepare for various types of Olympics.
Properties of prime numbers for the first time began to study mathematics of ancient Greece. Mathematics 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 to ideas about perfect and friendly numbers.
In the perfect number, the sum of his own divisors is equal to him. For example, its own divisors of the number 6: 1, 2 and 3. 1 + 2 + 3 \u003d 6. In the number 28 dividers are 1, 2, 4, 7 and 14. At the same time, 1 + 2 + 4 + 7 + 14 \u003d 28.
Numbers are called friendly if the sum of its own divisors of the same number is equal to the other, and on the contrary - for example, 220 and 284. It can be said that the perfect number is friendly for himself.
By the time of the work of Euclida "Beginning" in 300 BC. There were already proven several important facts regarding prime numbers. In the book IX "began", Euclide proved that the simple numbers are an infinite amount. This, by the way, is one of the first examples of using evidence from the opponent. It also proves the main theorem of arithmetic - every integer can be submitted the only way in the form of a product of prime numbers.
He also showed that if the number 2 N -1 is simple, then the number 2 n-1 * (2 n -1) will be perfect. Another mathematician, Euler, in 1747 managed to show that all the most accurate numbers can be recorded in this form. To this day it is not known whether there are odd numbers.
In year 200 BC Greek Eratosthene came up with an algorithm for finding prime numbers called "Deuto Eratosthena".
And then there was a big break in the history of the study of prime numbers associated with the average centuries.
The following discoveries were made already at the beginning of the 17th century Mathematics Farm. He proved the hypothesis of Albert Girar, that any simple number of the type 4N + 1 can be recorded a unique way in the form of the sum of two squares, and also formulated the theorem that any number can be represented as the sum of four squares.
He developed a new method of factoring of large numbers, and demonstrated it at 2027651281 \u003d 44021? 46061. He also proved a small farm theorem: if P is a simple number, then for any whole A, it will be true a p \u003d a modulo p.
This statement proves half of what was known as the "Chinese hypothesis", and dates back to 2000 earlier: an integer n is simple then and only if 2 N -2 is divided into n. The second part of the hypothesis turned out to be false - for example, 2 341 - 2 is divided by 341, although the number 341 is composite: 341 \u003d 31? eleven.
The small farm farm served as the basis for many other results in the theory of numbers and methods for checking numbers to belong to simple - many of which are used to this day.
The farm rewrite a lot with his contemporaries, especially with a monk named Marren Meresenne. In one of the letters, he expressed the hypothesis that the numbers of the form 2 n +1 will always be simple if N is a degree of twos. He checked it for n \u003d 1, 2, 4, 8 and 16, and was confident that in the case when N is not a degree of twos, the number was not necessarily simple. These numbers are called farm numbers, and only after 100 years, Euler showed that the following number, 2 32 + 1 \u003d 4294967297 is divided by 641, and therefore it is not easy.
The numbers of the form 2 n - 1 also served as a subject matter, since it is easy to show that if N is a composite, then the number itself is also composite. These numbers are called mercine numbers, since he studied them actively.
But not all numbers of the form 2 n - 1, where N is simple, are simple. For example, 2 11 - 1 \u003d 2047 \u003d 23 * 89. For the first time, it was discovered in 1536.
For many years, the number of this species gave mathematicians the greatest well-known simple numbers. That the number M 19, Cataldi was proved in 1588, and for 200 years was the largest known one by one, until Euler proved that M 31 is also simple. This record lasted for another hundred years, and then the Lucas showed that M 127 is simple (and this is the number of 39 digits), and after it the research continued with the advent of computers.
In 1952, the simplicity of numbers M 521, M 607, M 1279, M 2203 and M 2281 was proved.
By 2005, 42 ordinary numbers were found. The greatest of them, M 25964951, consists of 7816230 digits.
The work of Euler had a huge impact on the theory of numbers, including simple. Did he expanded the small farm theorem and introduced? -Function. Factorized the 5th number of the farm 2 32 +1, there were 60 pairs of friendly numbers, and formulated (but could not prove) the quadratic law of reciprocity.
He first introduced the methods of mathematical analysis and developed the analytical theory of numbers. Did he proven that not only a harmonious row? (1 / N), but also a number of types
1/2 + 1/3 + 1/5 + 1/7 + 1/11 +…
The amount obtained by the amounts back to simple numbers is also diverged. The sum of N members of the harmonic series increases approximately as log (N), and the second row is descended slower than Log [Log (N)]. This means that, for example, the amount of reverse values \u200b\u200bto all the simply found numbers will give only 4, although the row diverges anyway.
At first glance, it seems that simple numbers are distributed among as much as accidentally. For example, among the 100 numbers running right in front of 10,000,000, 9 simple, and among the 100 numbers coming immediately after this value - only 2. But on large segments, simple numbers are distributed quite evenly. Lena and Gauss have been issued by their distribution. Gauss somehow described a friend that in any free 15 minutes he always counts the number of simple in the next 1000 numbers. By the end of his life, he counted all the simple numbers in the interval to 3 million. Lena and Gauss equally calculated that for large N, the density of prime numbers is 1 / log (n). Lenaland estimated the number of prime numbers in the interval from 1 to N, as
? (n) \u003d n / (log (n) - 1.08366)
And Gauss - as a logarithmic integral
? (n) \u003d? 1 / LOG (T) DT
With interval of integration from 2 to n.
The assertion of the density of prime numbers 1 / log (n) is known as the theorem on the distribution of prime numbers. She was trying to prove during the entire 19th century, and progress reached Chebyshev and Roman. They tied it with the hypothesis of Riemann - in this course of the non-proven hypothesis about the distribution of zelie-functions of Riemann. The density of prime numbers was simultaneously proved by Adamar and Valle Pussen in 1896.
In the theory of prime numbers there are still many unsolved issues, some of which have many hundreds of years:
- hypothesis about prime-twin numbers - about the infinite number of pairs of prime numbers, differing from each other by 2
- goldbach Hypothesis: Anyone number, starting with 4, can be represented as the sum of two simple numbers.
- is the number of prime numbers of the form N 2 + 1 infinite?
- can there always be a simple number between N 2 and (n + 1) 2? (the fact that between n and 2n there is always a simple number, it was proved by Chebyshev)
- is the number of simple farm numbers infinitely? Are there any simple farm numbers after the 4th?
- is there an arithmetic progression of consecutive simple numbers for any given length? For example, for a length of 4: 251, 257, 263, 269. The maximum of the found length is 26.
- is the number of sets of three consecutive simple numbers in arithmetic progression?
- n 2 - N + 41 - A simple number for 0? n? 40. Is infinitely the number of such prime numbers? The same question for formula N 2 - 79 n + 1601. These numbers are simple for 0? n? 79.
- is the number of prime numbers infinite the n # + 1 species? (N # - the result of multiplying all prime numbers smaller than N)
- is the number of prime numbers infinite the n # -1 species?
- is the number of simple numbers of the form n! + 1?
- is the number of simple numbers of the form n! - one?
- if p is simple, whether there is always 2 p -1, it does not contain among the multipliers of simple numbers
- does Fibonacci sequence contain an infinite number of prime numbers?
The biggest twins among the prime numbers are 2003663613? 2 195000 ± 1. They consist of 58711 digits, and were found in 2007.
The largest factorial simple number (species N! ± 1) is 147855! - 1. It consists of 142891 digits and was found in 2002.
The largest primorial simple number (the number of n # ± 1) is 1098133 # + 1.
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