Is their story that simple? Scientific work. Prime numbers are just finding new prime numbers
Prime and composite numbers. Signs of divisibility.
2014-02-01
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number divisor
multiple
even number
odd number
Prime number
composite number
Test for divisibility by 2
Test for divisibility by 4
Divisibility test by 5
Test for divisibility by 3 and 9
If $a$ and $b$ are natural numbers, and
$a=bq$,
where $q$ is also a natural number, then we say that $q$ is
the quotient of dividing the number $a$ by the number $b$, and write: $q = a/b$.
It is also said that $a$ is divisible by $b$ completely or without a trace.
Any number $b$ that divides $a$ without a remainder is called a divisor of $a$
Self
the number $a$ in relation to its divisor is called a multiple
Thus, the multiples of $b$ are the numbers $b, 2b, 3b, \cdots$.
Numbers that are multiples of 2 (i.e. divisible by 2 without a remainder) are called even
.Numbers that are not evenly divisible by 2 are called odd.
Every natural number is either even or odd.
If each of two numbers $a_(1), a_(2)$ is a multiple of $b$, then the sum $a_(1)+a_(2)$ is a multiple of $b$. This can be seen from the entry $a_(1)=bq_(1), a_(2)=bq_(2); a_(1)+a_(2)=bq_(1)+bq_(2)= b (q_(1)+q_(2))$.
Conversely, if $a_(1)$ and $a_(1)+a_(2)$ are multiples of $b$, then $a_(2)$ is also a multiple of $b$.
Every natural number other than one has at least two divisors: one and itself.
If a number has no divisors other than itself and one, it is called prime
.A number that has a divisor other than itself and one is called composite.
In number. It is customary to classify the unit as neither a prime nor a composite number. Here are the first few prime numbers, written in ascending order:
$2,3,5,7,11,13,17,\cdots$
The number 2 is the only even prime number; all other prime numbers are odd.
The fact that there is an infinite number of prime numbers was established in ancient times (Euclid, 3rd century BC).
The idea of Euclid's proof of the infinity of the set of prime numbers is quite simple. Let us assume that there are a finite number of prime numbers; Let's list them all, for example, arranging them in ascending order:
$2,3,5, \cdots , p$. (1)
Let's make a number equal to their product plus one:
$a = 2 \cdot 3 \cdot 5 \cdots p+1$.
Obviously, this number is not divisible by any of the numbers (1). Therefore, either it is itself prime, or, if it is composite, it has a prime factor different from the numbers in (1), which contradicts the assumption that notation (1) lists all the prime numbers.
This proof is of great interest because it provides an example of a proof of an existence theorem (of an infinite set of prime numbers) that does not involve actually finding the objects whose existence is being proven.
It can be proven that every composite number can be represented as a product of prime numbers. For example,
$1176 = 2 \cdot 2 \cdot 2 \cdot 3 \cdot 7 \cdot 7$ or $1176 = 2^(3) \cdot 3 \cdot 7^(2)$.
As can be seen from this example, when decomposing a given number into prime factors, some of them can be repeated several times.
In general, in the notation of the decomposition of the number $a$ into prime factors
$a = p^(k_(1))_(1) p^(k_(2))_(2) \cdots p^(k_(n))_(n)$ (2)
it is implied that all prime numbers $p_(1),p_(2), \cdots , p_(n)$ are different from each other (and $p_(1)$ is repeated by a factor of $k_(1)$ times, $p_(2 )$ is repeated by a factor of $k_(2)$ times, etc.). Under this condition, it can be proven that the expansion is unique up to the order in which the factors are written.
When decomposing a number into prime factors, it is useful to use divisibility tests, which allow you to find out whether a given number is divisible by some other number without a remainder, without performing the division itself. We will derive divisibility criteria for the numbers 2, 3, 4, 5, 9.
Divisibility test by 2. Those and only those numbers are divisible by 2 in which the last digit expresses an even number (0, 2, 4, 6 or 8).
Proof. Let's represent the number $\overline(c_(1)c_(2) \cdots c_(m))$ as $\overline(c_(1)c_(2) \cdots c_(m)) = \overline(c_(1 )c_(2) \cdots 0) + c_(m)$.
The first term on the right side is divisible by 10 and therefore even; the sum will be even if and only if $c_(m)$ is an even number.
Divisibility by 4 The number $\overline(c_(1)c_(2) \cdots c_(m))$ is divisible by 4 if and only if the two-digit number expressed by its last two digits is divisible by 4.
Proof. Let us represent the number $\overline(c_(1)c_(2) \cdots c_(m))$ in the form
$\overline(c_(1)c_(2) \cdots c_(m)) = \overline(c_(1)c_(2) \cdots 00) + \overline(c_(m-1)c_(m)) $
The first term is divisible by 100 and even more so by 4. The sum will be divisible by 4 if and only if $\overline(c_(m-1)c_(m))$ is divisible by 4.
Test for divisibility by 5. Those and only those numbers whose notation ends with the number 0 or the number 5 are divisible by 5.
Signs of divisibility by 3 and 9. A number is divisible by 3 (respectively by 9) if and only if the sum of its digits is divisible by 3 (respectively by 9).
Proof. Let us write down the obvious equalities
$10 = 9+1$,
$100 = 99 + 1$,
$1000 = 999+1$,
$\cdots$,
due to which the number $\overline(c_(1)c_(2) \cdots c_(m))$ can be represented as
$a_(m)=c_(1)(99 \cdots 9 + 1) + \cdots + c_(m-1) (9+1) + c_(m)$
or
$a_(m)=c_(1) \cdot 99 \cdots 9 + \cdots + c_(m-1) \cdot 9 + (c_(1) + c_(2) + \cdots + c_(m-1) + c_(m))$.
It can be seen that all terms, except perhaps the last bracket, are divisible by 9 (and even more so by 3). Therefore, a given number is divisible by 3 or 9 if and only if the sum of its digits $c_(1)+c_(2)+ \cdots + c_(m)$ is divisible by 3 or 9.
Numbers follow people everywhere. Even our body is in tune with their world - we have a certain number of organs, teeth, hair and skin cells. Counting has become a habitual, automatic action, so it is difficult to imagine that people once did not know numbers. In fact, the history of the emergence of numbers can be traced back to ancient times.
Numbers and primitive people
At some point, the person felt a great need to count. For this it is
Life itself pushed me. It was necessary to somehow organize the tribe, sending only a certain number of people to hunt or gather. Therefore, they used their fingers to count. There are still tribes that show one hand instead of the number “5”, and two instead of ten. With such a simple counting algorithm, the history of the emergence of numbers began to develop.
Prime numbers
The history of the emergence of numbers allows us to notice that people quite a long time ago discovered the difference between an odd and an even number, as well as various relationships within the numerical expressions themselves. Considerable contribution to such
research was introduced by the ancient Greeks. For example, the Greek scientist Eratosthenes created a fairly easy way to find prime numbers. To do this, he wrote down the required number of numbers in order, and then began to cross out - first all the numbers that could be divided by two, then by three. The result was a list of numbers that were not divisible by anything except one and itself. This method was called the “sieve of Eratosthenes” due to the fact that the Greeks did not cross out, but instead poked out unnecessary numbers on tablets covered with wax.
Thus, the history of the emergence of numbers is an ancient and profound phenomenon. According to scientists, it began about 30 thousand years ago. During this time, a lot has changed in a person’s life. But to this day it guides our existence.
Molokov Maxim
This year we studied the topic “Prime and Composite Numbers”, and I was wondering which of the scientists was studying them, how to get prime numbers other than those contained on the flyleaf of our textbook (from 1 to 1000), this became the goal of completing this work.
Tasks:
1. Study the history of the discovery of prime numbers.
2. Get acquainted with modern methods of finding prime numbers.
3. Find out in which scientific fields prime numbers are used.
4. Are there names among Russian scientists of those who studied prime numbers?
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History of prime numbers MBOU Sukhovskaya Secondary School Author: 6th grade student Molokov Maxim Supervisor: mathematics teacher Babkina L. A. p. Novosukhovy December 2013
This year we studied the topic “Prime and Composite Numbers”, and I was wondering which of the scientists was studying them, how to get prime numbers other than those contained on the flyleaf of our textbook (from 1 to 1000), this became the goal of completing this work. Objectives: 1. Study the history of the discovery of prime numbers. 2. Get acquainted with modern methods of finding prime numbers. 3. Find out in which scientific fields prime numbers are used. 4. Are there names among Russian scientists of those who studied prime numbers?
Anyone who studies prime numbers is fascinated and at the same time feels powerless. The definition of prime numbers is so simple and obvious; finding the next prime number is so easy; factoring into prime factors is such a natural action. Why do prime numbers so stubbornly resist our attempts to comprehend the order and patterns of their arrangement? Maybe there is no order in them at all, or are we so blind that we don’t see it? C. Userell.
Pythagoras and his students studied the question of the divisibility of numbers. They called a number equal to the sum of all its divisors (without the number itself) a perfect number. For example, the numbers 6 (6 = 1 + 2 +3), 28 (28 = 1+2+4+7+14) are perfect. The following perfect numbers are 496, 8128, 33550336.. Pythagoras (VI century BC)
The Pythagoreans only knew the first three perfect numbers. The fourth - 8128 - became known in the first century AD. The fifth - 33550336 - was found in the 15th century. By 1983, 27 perfect numbers were already known. But scientists still don’t know whether there are odd perfect numbers or whether there is a largest perfect number.
The interest of ancient mathematicians in prime numbers is due to the fact that any number is either prime or can be represented as a product of prime numbers, i.e. Prime numbers are like the bricks from which the rest of the natural numbers are built.
You probably noticed that prime numbers in the series of natural numbers occur unevenly - in some parts of the series there are more of them, in others there are fewer. But the further we move along the number series, the less common prime numbers are.
The question arises: is there a last (largest) prime number? The ancient Greek mathematician Euclid (3rd century BC) in his book (“Elements”), which was the main textbook in mathematics for 2000 years, proved that there are infinitely many prime numbers, i.e. behind every prime number there is a larger prime number Euclid (3rd century BC)
Another Greek mathematician Eratosthenes came up with this method to find prime numbers. He wrote down all the numbers from one to some number, and then crossed out one, which is not a prime or composite number, then crossed out through one all the numbers coming after the 2nd number, multiples of two, i.e. 4,6,8, etc.
The first remaining number after two was 3. Then, after two, all numbers coming after three (numbers multiples of 3, i.e. 6,9,12, etc.) were crossed out. In the end, only prime numbers remained uncrossed.
Since the Greeks made notes on wax-coated tablets or on drawn papyrus, and the numbers were not crossed out, but poked out with a needle, the table at the end of the calculations resembled a sieve. Therefore, Eratosthenes’ method is called the sieve of Eratosthenes: in this sieve, prime numbers are “sifted out” from composite numbers.
So, the prime numbers from 2 to 60 are 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, tables of prime numbers are compiled, but with the help of computers.
Euclid (3rd century BC) proved that between the natural number n and n! There must be at least one prime number. Thus, he proved that the natural series of numbers is infinite. In the middle of the 11th century. Russian mathematician and mechanic Pafnutiy Lvovich Chebyshev proved a stronger theorem than Euclid. Between the natural number n and a number 2 times larger than it, i.e. 2 n contains at least one prime number. That is, in Euclid’s theorem the number n! replaced by the number 2n. Pafnuty Lvovich Chebyshev (1821-1894) Russian mathematician and mechanic
The next question arises: “If it is so difficult to find the next prime number, then where and for what can these numbers be used in practice?” The most common use of prime numbers is in cryptography (data encryption). The most secure and difficult to decipher cryptography methods are based on the use of prime numbers with more than three hundred digits.
Conclusion The problem of the absence of patterns in the distribution of prime numbers has occupied the minds of mankind since the times of ancient Greek mathematicians. Thanks to Euclid, we know that there are infinitely many prime numbers. Erastophenes proposed the first algorithm for testing numbers for primality. Chebyshev and many other famous mathematicians tried and are still trying to solve the riddle of prime numbers. To date, many elegant algorithms and patterns have been found and proposed, but all of them are applicable only for a finite series of prime numbers or prime numbers of a special type. The cutting edge of science in the study of prime numbers at infinity is considered to be the proof of the Riemann hypothesis. It is one of the seven unsolved problems of the millennium, for the proof or refutation of which the Clay Mathematical Institute has offered a $1,000,000 prize.
Internet - sources and literature http://www.primenumb.ru/ http://www.bestpeopleofrussia.ru/persona/Pafnutiy-Chebyshev/bio/ http://uchitmatematika.ucoz.ru/index/vayvayvayjajavvvjavvvvva/0-7 Textbook “Mathematics” for the sixth grade of educational institutions /N.Ya. Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S.I. Schwarzburg - M. Mnemosyne 2010/
Department of Education and Youth Policy of the Administration
Yalchik district of the Chuvash Republic
Project
Prime numbers...
Is their story that simple?
Completed by a 7th grade student of the municipal educational institution "Novoshimkusskaya Secondary School of the Yalchik District of the Chuvash Republic" Efimova Marina
Head: Category I mathematics teacher, Municipal Educational Institution "Novoshimkus Secondary School, Yalchik District, Chuvash Republic" Kirillova S.M.
village New Shimkusy - 2007
Defining Prime Numbers 3
Merits of Euler 3
Fundamental Theorem of Arithmetic 4
Mersen primes 4
Fermat 5 primes
Sieve of Eratosthenes 5
Discovery of P.L. Chebyshev 6
Goldbach problem 7
I.M.Vinogradov 8
Conclusion 8
Literature 10
Interest in the study of prime numbers arose among people in ancient times. And it was caused not only by practical necessity. They were attracted by their extraordinary magical power. Numbers that can be used to express the quantity of any object. The unexpected and at the same time natural properties of natural numbers discovered by ancient mathematicians surprised them with their remarkable beauty and inspired new research.
It must have been one of the first properties of numbers discovered by man that some of them could be factored into two or more factors, e.g.
6=2*3, 9=3*3, 30=2*15=3*10, while others, such as 3, 7, 13, 37, cannot be expanded in this way.
When number c = Ab is the product of two numbers A and b , then the numbers a andb are called multipliers or dividers numbers s. Each number can be represented as a product of two factors. For example, with = 1 *c = c*1.
Simple is a number that is divisible only by itself and one.
A unit that has only one divisor is not a prime number. It does not apply to composite numbers either. The unit occupies a special position in the number series. The Pythagoreans taught that unit is the mother of all numbers, the spirit from which the entire visible world comes, it is reason, goodness, harmony.
At Kazan University, Professor Nikolsky, with the help of a unit, managed to prove the existence of God. He said: “Just as there cannot be a number without one, so the Universe cannot exist without a single Lord.”
One is indeed a number with unique properties: it is divisible only by itself, but any other number is divisible by it without a remainder, any of its powers is equal to the same number - one!
After dividing by it, not a single number changes, and if you divide any number by itself, you get one again! Isn't this surprising? After thinking about this, Euler said: “One must exclude the unit from the sequence of prime numbers; it is neither prime nor composite.”
This was already an essential ordering in the dark and complex question of prime numbers.
Euler's merits
Leonard Euler
(1707-1783)
Everyone studied with Euler - both in Western Europe and in Russia. The range of his creativity is wide: differential and integral calculus, algebra, mechanics, dioptrics, artillery, marine science, the theory of planetary and lunar motion, music theory - you can’t list everything. In this whole scientific mosaic is the theory of numbers. Euler devoted a lot of effort to it and achieved a lot. He, like many of his predecessors, was looking for a magic formula that would make it possible to isolate prime numbers from the infinite set of numbers in the natural series, that is, from all the numbers that can be imagined. Euler wrote more than a hundred works on number theory.
...It has been proven, for example, that the number of prime numbers is unlimited, that is: 1) there is no largest prime number; 2) there is no last prime number after which all numbers would be composite. The first proof of this position belongs to scientists of ancient Greece (V-III centuries BC), the second proof - Euler (1708-1783).
Fundamental Theorem of Arithmetic
Every natural number other than 1 is either prime or can be represented as a product of prime numbers, and uniquely, if you do not pay attention to the order of the factors.
Proof. Let's take a natural number n≠ 1. If n is prime, then this is the case mentioned in the conclusion of the theorem. Now suppose that n is composite. Then it is represented as a product n = ab,
where the natural numbers a and b are less than n. Again, either a and b are simple, then everything is proven, or at least one of them is composite, that is, made up of smaller factors, and so on; eventually we will get a prime factorization.
If the number n is not divisible by any prime not exceeding√n, then it is simple.
Proof. Assume the opposite, let n be composite and P = ab, where 1 ≤b and p is a prime divisor of the number A, hence the numbers n. By condition P is not divisible by any prime not exceeding √ n. Hence, р >√n. But then a >√n And √ n ≤ A≤ b ,
where n = ab = √ n√ n = P; came to a contradiction, the assumption was incorrect, the theorem was proven.
Example 1. If c = 91 then √ с = 9, ... check the prime numbers 2, 3, 5, 7. We find that 91 = 7 13.
Example 2. If c = 1973, then we find √ c = √ 1973 =44, ...
since no prime number before 43 does not divide with, then this number is prime.
Example 3. Find the prime number following the prime number 1973. Answer: 1979.
Mersen primes
For several centuries there was a pursuit of prime numbers. Many mathematicians have competed for the honor of being the discoverer of the largest known prime number.
Mersen prime numbers are prime numbers of a special form M p = 2 p - 1
Where R - another prime number.
These numbers have been part of mathematics for a long time; they appear in Euclidean reflections on modern numbers. They received their name in honor of the French monk Merenne Mersen (1589-1648), who spent a long time working on the problem of modern numbers.
If we calculate the numbers using this formula, we get:
M 2 = 2 2 – 1 = 3 – prime;
M 3 = 2 3 – 1 = 7 – simple;
M 5 = 2 5 – 1 = 31 – simple;
M 7 = 2 7 – 1 = 127 – simple;
M 11 = 2 11 – 1 = 2047 = 23 * 89
A general way to find large Mersen primes is to test all numbers M p for different primes R.
These numbers increase very quickly and the labor costs to find them increase just as quickly.
In the study of Mersen numbers, an early stage can be distinguished, culminating in 1750, when Euler established that the number M 31 is prime. By that time, eight Mersen prime numbers had been found: "r
R= 2, р= 3, р = 5 , р = 7, р= 13, p = 17, p = 19, R =31.
Euler's number M 31 remained the largest known prime number for over a hundred years.
In 1876, the French mathematician Lucas established that the huge number M 127 has 39 digits. The 12 Mersen primes were calculated using only pencil and paper, and the next ones were calculated using mechanical desktop adding machines.
The advent of electrically driven computers made it possible to continue the search, bringing it to R = 257.
However, the results were disappointing, and there were no new Mersen primes among them.
Then the task was transferred to the computer.
The largest currently known prime number has 3376 digits. This number was found on a computer at the University of Illinois (USA). The mathematics department of this university was so proud of their achievement that they depicted this number on their postmark, thus reproducing it on every letter they sent for public viewing.
Fermat primes
There is another type of prime number with a long and interesting history. They were first introduced by the French jurist Pierre Fermat (1601-1665), who became famous for his outstanding mathematical works.
Pierre Fermat (1601-1665)
Fermat's first prime numbers were numbers that satisfied the formula F n = 
+ 1.
F 0 = 
+ 1 = 3;
F 1 = 
+ 1 = 5;
F 2 = 
+ 1 = 17;
F 3 = 
+ 1 = 257;
F 4 = 
+ 1 = 65537.
However, this assumption was consigned to the archive of unjustified mathematical hypotheses, but after Leonhard Euler took one step further and showed that the next Fermat number F 5 = 641 6 700 417 is composite.
It is possible that the history of Fermat numbers would have been completed if Fermat numbers had not appeared in a completely different problem - constructing regular polygons using a compass and ruler.
However, not a single Fermat prime number has been found, and many mathematicians now tend to believe that they no longer exist.
Sieve of Eratosthenes
There are tables of prime numbers extending to very large numbers. How to approach compiling such a table? This problem was, in a sense, solved (about 200 BC) by Eratosthenes, a mathematician from Alexandria. -
His scheme is as follows. Let's write a sequence of all integers from 1 to the number with which we want to end the table.
Let's start with the prime number 2. We'll throw out every second number. Let's start with 2 (except for the number 2 itself), i.e. even numbers: 4, 6, 8, 10, etc., underline each of them.
After this operation, the first un-underlined number will be 3. It is prime, since it is not divisible by 2. Leaving the number 3 un-underlined, we will underline every third number after it, i.e. numbers 6, 9, 12, 15... Some of them have already been underlined because they are even. In the next step, the first ununderlined number will be the number 5; it is simple, since it is not divisible by either 2 or 3. Let's leave the number 5 ununderlined, but underline 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 unstressed number will be the number 7. It is prime because it is not divisible by any of its smaller primes 2, 3, 5. By repeating this process, we will eventually get a sequence of unstressed numbers; all of them (except number 1) are prime. This method of sifting numbers is known as the "sieve of Eratosthenes". Any table of prime numbers is created according to this principle.
Eratosthenes created a table of prime numbers from 1 to 120 more than 2000 years ago. He wrote on papyrus stretched over a frame, or on a wax tablet, and did not cross out, as we do, but pierced the composite numbers. The result was something like a sieve through which the composite numbers were “sifted.” Therefore, the table of prime numbers is called the “Sieve of Eratosthenes.”
How many prime numbers are there? Is there a last prime number, that is, one after which all numbers will be composite? If such a number exists, how to find it? All these questions have interested scientists since ancient times, but the answer to them was not so easy to find.
Eratosthenes was a very witty man. This contemporary and friend of Archimedes, with whom he constantly corresponded, was a mathematician, an astronomer, and a mechanic, which was considered natural for the great men of that time. He was the first to measure the diameter of the globe, without leaving the Alexandrian library where he worked. The accuracy of his measurements was amazingly high, even higher than that with which Archimedes measured the Earth.
Eratosthenes invented an ingenious device - mesolabite, with with the help of which he mechanically solved the well-known problem of doubling the cube, of which he was very proud, and therefore gave the order to depict this device on a column in Alexandria. Moreover, he corrected the Egyptian calendar by adding one day to four years - in a leap year.
The sieve of Eratosthenes is a primitive and at the same time ingenious invention, which Euclid did not even think of, suggesting the well-known idea that everything ingenious is simple.
The Eratosthenes sieve worked well for researchers of far from prime numbers. Time passed. There was a search for ways to catch prime numbers. A kind of competition began to find the largest prime number from ancient times to Chebyshev and even to the present day.
Discovery of P.L. Chebysheva
AND 
Thus, the number of prime numbers is infinite. We have already seen that prime numbers are arranged without any order. Let's look at it in more detail.
2 and 3 are prime numbers. This is the only pair of prime numbers that are adjacent.
Then come 3 and 5, 5 and 7, 11 and 13, 17 and 19, etc. These are the so-called adjacent primes or twins. There are many twins: 29 and 31, 41 and 43, 59 and 61, 71 and 73, 101 and 103, 827 and 829, etc. The largest pair of twins known now is: 10016957 and 10,016,959.
Panfutiy Lvovich Chebyshev
How are prime numbers distributed in a natural series in which there is not a single prime number? Is there any law in their distribution or not?
If so, which one? How to find it? But the answer to these questions has not been found for more than 2000 years.
The first and very big step in resolving these issues was made by the great Russian scientist Panfuty Lvovich Chebyshev. In 1850, he proved that between any natural number (not equal to 1) and a number twice its size (i.e., between n and 2n), there is at least one prime number.
Let's check this with simple examples. Let us take several arbitrary values of n for n .
and find the value 2n accordingly.
n = 12, 2n = 24;
n = 61, 2n = 122;
n = 37, 2n = 74.
We see that for the examples considered, Chebyshev’s theorem is true.
Chebyshev proved it for any case, for any n. For this theorem he was called the winner of prime numbers. The law of distribution of prime numbers discovered by Chebyshev was a truly fundamental law in number theory after the law discovered by Euclid about the infinity of the number of prime numbers.
Perhaps the kindest, most enthusiastic response to Chebyshev’s discovery came from England from the famous mathematician Sylvester: “...Further successes in the theory of prime numbers can be expected when someone is born who is as superior to Chebyshev in his insight and thoughtfulness as Chebyshev is superior to these qualities ordinary people."
More than half a century later, the German mathematician E. Landau, a prominent specialist in number theory, added the following to this statement: “The first after Euclid, Chebyshev took the right path in solving the problem of prime numbers and achieved important results.”
Goldbach's problem
Let's write down all the prime numbers from 1 to 50:
2, 3, 5, 7, 9, 11, 17, 19, 23, 29, 31, 37, 41, 43, 47.
Now let's try any number from 4 to 50 represent it as the sum of two or three prime numbers. Let's take a few numbers at random:
As you can see, we completed the task without difficulty. Is this always possible? Can any number be represented as the sum of several prime numbers? And if so, how many: two? three? ten?
In 1742, Goldbach, a member of the St. Petersburg Academy of Sciences, in a letter to Euler, suggested that any positive integer greater than five is the sum of at most three prime numbers.
Goldbach tested a lot of numbers and never met a number that could not be decomposed into the sum of two or three simple terms. But whether it will always be like this, he has not proven. Scientists have been studying this problem for a long time, which is called the “Goldbach problem” and is formulated as follows.
You need to prove or disprove the proposition:
Any number greater than one is the sum of at most three prime numbers.
For almost 200 years, outstanding scientists tried to solve the Goldbach-Euler problem, but without success. Many have come to the conclusion that it is impossible to solve it.
But its solution, almost completely, was found in 1937 by the Soviet mathematician I.M. Vinogradov.
THEM. Vinogradov
Ivan Matveevich Vinogradov is one of the greatest modern mathematicians. He was born on September 14, 1891 in the village of Milolub, Pskov province. In 1914 he graduated from St. Petersburg University and was left to prepare for a professorship.
His first scientific work I.M. Vinogradov wrote in 1915. Since then he has written more than 120 different scientific works. In them, he solved many problems that scientists around the world had been working on for tens and hundreds of years.
Ivan Matveevich Vinogradov
For services in the field of mathematics I.M. Vinogradov is recognized by all scientists of the world as one of the first mathematicians of our time, and was elected to the number of members of many academies around the world.
We are proud of our wonderful compatriot.
Conclusion.
From the classroom to the outer space
Let's start our conversation about prime numbers with a fascinating story about an imaginary journey from the classroom into outer space. This imaginary journey was invented by the famous Soviet mathematics teacher Professor Ivan Kozmich Andronov (born in 1894). “...a) mentally take a straight wire coming out of the classroom into the world space, piercing the earth’s atmosphere, going to where the Moon rotates, and then beyond the fireball of the Sun, and further into the world’s infinity;
b) mentally hang light bulbs on a wire every meter, numbering them starting from the nearest: 1, 2, 3, 4, ..., 100, ..., 1000, ..., 1,000,000...;
c) mentally turn on the current in such a way that all the light bulbs with prime numbers and only those with prime numbers light up; : .
d) mentally fly close to the wire.
The following picture will unfold before us.
1. Light bulb number 1 does not light up. Why? Because one is not a prime number.
2. The next two light bulbs numbered 2 and 3 are lit, since 2 and 3 are both prime numbers. Can two adjacent burning light bulbs meet in the future? No, they can't. Why? Every prime number except two is an odd number, and those adjacent to a prime on either side will be even numbers, and every even number other than two is a composite number, since it is divisible by two.
3. Next we observe a pair 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 occur less frequently; all pairs of twins, like pairs of prime numbers, have the form 6n ± 1; For example
6*3 ± 1 equals 19 and 17
or 6*5 ± 1 equals 31 and 29, ...;
but 6*20 ± 1 is equal to 121 and 119 - this pair is not a twin, since there is a pair of composite numbers.
We reach the pair of twins 10,016,957 and 10,016,959. Will there be further pairs of twins? Modern science does not yet provide an answer: it is not known whether there is a finite or an infinite number of pairs of twins.
4. But then the law of a large gap, filled only with composite numbers, begins to operate: we fly in the dark, look back - darkness, and no light is visible ahead. We remember the property discovered by Euclid and boldly move forward, since there should be luminous light bulbs ahead, and there should be an infinite number of them ahead.
5. Having flown into a place in the natural series, where several years of our movement have already passed in the dark, we remember the property proven by Chebyshev, and calm down, confident that in any case, we need to fly no more than we flew in order to see at least one luminous light bulb."
Literature
1.
The great master of induction, Leonhard Euler.
2. Behind the pages of a mathematics textbook.
3. Prudnikov N.I. P.L. Chebyshev.
4. Serbsky I. A. What we know and don't know about prime numbers.
5. Publishing house “First of September”. Mathematics No. 13, 2002
6. Publishing house “First of September”. Mathematics No. 4, 2006
Municipal educational institution "Chastoozersk secondary school"
Research work on the topic:
“Numbers rule the world!”
Work completed:
6th grade student.
Supervisor: ,
mathematic teacher.
With. Chastoozerye.
I. Introduction. -3 pages
II. Main part. -4 pages
· Mathematics among the ancient Greeks. - 4 pages
· Pythagoras of Samos. -6 pages
· Pythagoras and numbers. -8pp.
2. Numbers are simple and composite. -10pp.
3. Goldbach's problem. -12pp.
4. Signs of divisibility. -13pp.
5. Curious properties of natural numbers.-15pp.
6. Number tricks. -18pp.
III. Conclusion. -22pp.
IV. Bibliography. -23pp.
I. Introduction.
Relevance:
While studying the topic “Divisibility of Numbers” in mathematics lessons, the teacher suggested preparing a report on the history of the discovery of prime and composite numbers. When preparing the message, I was interested in the words of Pythagoras “Numbers rule the world!”
Questions have arisen:
· When did the science of numbers arise?
· Who contributed to the development of the science of numbers?
· The meaning of numbers in mathematics?
I decided to study in detail and summarize the material about numbers and their properties.
Purpose of the study: study prime and composite numbers and show their role in mathematics.
Object of study: prime and composite numbers.
Hypothesis: If, in the words of Pythagoras, “Numbers rule the world,
then what is their role in mathematics.
Research objectives:
I. Collect and summarize all kinds of information about prime and composite numbers.
II. Show the meaning of numbers in mathematics.
III. Show interesting properties of natural numbers.
Research methods:
· Theoretical analysis of literature.
· Method of systematization and processing of data.
II. Main part.
1. The history of the emergence of the science of numbers.
· Mathematics among the ancient Greeks.
In both Egypt and Babylon, numbers were used mainly to solve practical problems.
The situation changed when the Greeks took up mathematics. In their hands, mathematics changed from a craft to a science.
Greek tribes began to settle on the northern and eastern shores of the Mediterranean Sea about four thousand years ago.
Most of the Greeks settled on the Balkan Peninsula - where the state of Greece is now. The rest settled on the islands of the Mediterranean Sea and along the coast of Asia Minor.
The Greeks were excellent sailors. Their light, sharp-nosed ships plied the Mediterranean Sea in all directions. They brought dishes and jewelry from Babylon, bronze weapons from Egypt, animal skins and bread from the shores of the Black Sea. And of course, like other peoples, ships brought knowledge to Greece along with goods. But the Greeks are not just
learned from other peoples. Very soon they overtook their teachers.
Greek masters built palaces and temples of amazing beauty, which later served as a model for architects of all countries for thousands of years.
Greek sculptors created wonderful statues from marble. And not only “real” mathematics began with the Greek scientists, but also many other sciences that we study at school.
Do you know why the Greeks were ahead of all other nations in mathematics? Because they were good at arguing.
How can debate help science?
In ancient times, Greece consisted of many small states. Almost every city with surrounding villages was a separate state. Every time an important state issue had to be resolved, the townspeople gathered in the square and discussed it. They argued about how to do it better, and then voted. It is clear that they were good debaters: at such meetings they had to refute their opponents, reason, and prove that they were right. The ancient Greeks believed that argument helps to find the best. The most correct decision. They even came up with the following saying: “Truth is born in a dispute.”
And in science the Greeks began to do the same. Like at a people's meeting. They didn’t just memorize the rules, but looked for reasons: why it was right to do it this way and not otherwise. Greek mathematicians tried to explain every rule and prove that it was not true. They were arguing with each other. They reasoned and tried to find errors in the reasoning.
They will prove one rule - reasoning leads to another, more complex one, then to a third, to a fourth. Laws were made from rules. And the science of laws is mathematics.
As soon as it was born, Greek mathematics immediately moved forward by leaps and bounds. She was helped by wonderful walking boots, which other nations did not have before. They were called "reasoning" and "proof".
· Pythagoras of Samos.
The first to talk about numbers was the Greek Pythagoras, who was born on the island of Samos in the 6th century AD.
Therefore, he is often called Pythagoras of Samos. The Greeks told many legends about this thinker.
Pythagoras early showed an aptitude for science, and Father Mnesarchus took him to Syria, to Tire, so that the Chaldean sages could teach him there. She learns about the mysteries of the Egyptian priests. Burning with the desire to enter their circle and become an initiate, Pythagoras begins to prepare for a trip to Egypt. He spends a year in Phenicia, at the school of priests. Then he will visit Egypt, Heliopolis. But the local priests were unfriendly.
Having shown persistence and passed extremely difficult entrance tests, Pythagoras achieves his goal - he is accepted into the caste. He spent 21 years in Egypt, perfectly studied all types of Egyptian writing, and read many papyri. The facts known to the Egyptians in mathematics lead him to his own mathematical discoveries.
The sage said: “There are things in the world that you need to strive for. It is, firstly, beautiful and glorious, secondly, useful for life, thirdly, giving pleasure. However, pleasure is of two kinds: one, which satisfies our gluttony with luxury, is disastrous; the other is righteous and necessary for life.”
Numbers occupied a central place in the philosophy of students and adherents of Pythagoras:
« Where there is no number and measure, there is chaos and chimeras,”
"The wisest thing is a number"
"Numbers rule the world."
Therefore, many consider Pythagoras the father of numbering - a complex science shrouded in mystery, describing events in it, revealing the past and future, predicting the fate of people.
· Pythagoras and numbers.
The ancient Greeks, and with them Pythagoras and the Pythagoreans, thought of numbers visibly in the form of pebbles laid out on the sand or on a counting board - an abacus.
Pebble numbers were laid out in the form of regular geometric figures, these figures were classified, and this is how the numbers today called figured numbers arose: linear numbers (i.e. prime numbers) - numbers that are divisible by one and by themselves and, therefore, representable as a sequence dots lined up
https://pandia.ru/text/79/542/images/image006_30.jpg" width="312" height="85 src=">
solid numbers expressed by the product of three factors
https://pandia.ru/text/79/542/images/image008_20.jpg" width="446" height="164 src=">
square numbers:
https://pandia.ru/text/79/542/images/image010_15.jpg" width="323" height="150 src=">
And. etc. It is from figurative numbers that the expression “ Square or cube a number».
Pythagoras did not limit himself to flat figures. From points, he began to add pyramids, cubes and other bodies and study pyramidal, cubic and other numbers (see Fig. 1). By the way, the name cube of numbers We still use it today.
But Pythagoras was not satisfied with the numbers obtained from various figures. After all, he proclaimed that numbers rule the world. Therefore, he had to figure out how to use numbers to depict concepts such as justice, perfection, and friendship.
To depict perfection, Pythagoras began working on the divisors of numbers (he took the divisor 1, but did not take the number itself). He added all the divisors of the number, and if the sum was less than the number, it was declared insufficient, and if more, it was declared excessive. And only when the sum exactly equaled the number was it declared perfect. Friendship numbers were depicted in a similar way - two numbers were called friendly if each of them was equal to the sum of the divisors of the other number. For example, the number 6 (6=1+2+3) is perfect, the number 28 (1+2+4+7+17) is perfect. The next perfect numbers are 496, 8128, .
2. Numbers are simple and composite.
Modern mathematics remembers friendly or perfect numbers with a smile as a childhood hobby.
And the concepts of prime and composite numbers introduced by Pythagoras are still the subject of serious research, for which mathematicians receive high scientific awards.
From the experience of calculations, people knew that every number is either a prime number or the product of several prime numbers. But they didn't know how to prove it. Pythagoras or one of his followers found proof of this statement.
Now it’s easy to explain the role of prime numbers in mathematics: they are the building blocks from which other numbers are built using multiplication.
The discovery of patterns in a series of numbers is a very pleasant event for mathematicians: after all, these patterns can be used to build hypotheses, to test evidence and formulas. One of the properties of prime numbers that interests mathematicians is that they refuse to obey any pattern.
The only way to determine whether a number 100,895,598,169 is prime is to use the rather laborious "Sieve of Eratosthenes."
The table shows one of the options for this sieve.
In this table, all prime numbers less than 48 are circled. They are found like this: 1 has a single divisor - itself, therefore 1 is not considered a prime number. 2 is the smallest (and only even) prime number. All other even numbers are divisible by 2, which means they have at least three divisors; therefore they are not simple and can be crossed out. The next uncrossed number is 3; it has exactly two divisors, so it is prime. All other numbers that are multiples of three (i.e. those that can be divided by 3 without a remainder) are crossed out. Now the first number not crossed out is 5; it is simple, and all its multiples can be crossed out.
By continuing to cross out multiples, you can eliminate all prime numbers less than 48.
3. Goldbach's problem.
Any number can be obtained from prime numbers by multiplication. What happens if you add prime numbers?
The mathematician Goldbach, who lived in Russia in the 18th century, decided to add odd prime numbers only in pairs. He discovered an amazing thing: every time he was able to represent an even number as the sum of two prime numbers. (as was the case in Goldbach's time, we consider 1 to be a prime number).
4 = 1 +3, 6 = 3 + 3, 8 = 3 + 5. etc.
https://pandia.ru/text/79/542/images/image016_5.jpg" width="156" height="191 src=">
Goldbach wrote about his observation to the great mathematician
XVIII century to Leonhard Euler, who was a member of the St. Petersburg Academy of Sciences. After testing many more even numbers, Euler was convinced that they were all the sum of two prime numbers. But there are infinitely many even numbers. Therefore, Euler's calculations only gave hope that all numbers had the property that Goldbach noticed. However, attempts to prove that this will always be the case have led nowhere.
Mathematicians pondered Goldbach's problem for two hundred years. And only the Russian scientist Ivan Matveevich Vinogradov managed to take the decisive step. He established that any sufficiently large natural number is
the sum of three prime numbers. But the number from which Vinogradov’s statement is true is unimaginably large.
4. Signs of divisibility.
489566: 11 = ?
To find out whether a given number is prime or composite, you don’t always need to look at the table of prime numbers. Often for this it is enough to use the signs of divisibility.
· Test for divisibility by 2.
If a natural number ends in an even digit, then the number is even and is divisible by 2 without a remainder.
· Test for divisibility by 3.
If the sum of the digits of a number is divisible by 3, then the number is divisible by 3.
· Test for divisibility by 4.
A natural number containing at least three digits is divisible by 4 if the number formed by the last two digits of that number is divisible by 4.
· Test for divisibility by 5.
If a natural number ends in 0 or 5, then that number is divisible by 5 without a remainder.
· Test for divisibility by 7 (by 13).
A natural number is divisible by 7 (by 13) if the algebraic sum of the numbers forming faces of three digits (starting with the units digit), taken with the “+” sign for odd faces and with the “minus” sign for even faces, is divided by, we compose the algebraic sum of the faces, starting from the last face and alternating the + and - signs: + 254 = 679. The number 679 is divisible by 7, which means this number is also divisible by 7.
· Test for divisibility by 8.
A natural number containing at least four digits is divisible by 8 if the number formed by the last three digits is divisible by 8.
· Test for divisibility by 9.
If the sum of the digits of a number is divisible by 9, then the number itself is divisible by 9.
· Test for divisibility by 10.
If a natural number ends in 0, then it is divisible by 10.
· Divisibility test 11.
A natural number is divisible by 11 if the algebraic sum of its digits, taken with a plus sign if the digits are in odd places (starting with the ones digit), and taken with a minus sign if the digits are in even places, is divisible by, 7 – 1 + 5 = 11, divisible by 11).
· Test for divisibility by 25.
A natural number containing at least three digits is divisible by 25 if the number formed by the last two digits of that number is divisible by 25.
· Test for divisibility by 125.
A natural number containing at least four numbers is divisible by 125 if the number formed by the last three digits of that number is divisible by 125.
5. Curious properties of natural numbers.
Natural numbers have many interesting properties that are revealed when arithmetic operations are performed on them. But it is still easier to notice these properties than to prove them. Let us present several such properties.
1) Let's take some natural number at random, for example 6, and write down all its divisors: 1, 2, 3.6. For each of these numbers, write down how many divisors it has. Since 1 has only one divisor (the number itself), 2 and 3 have two divisors each, and 6 has 4 divisors, we get the numbers 1, 2, 2, 4. They have a remarkable feature: if you raise these numbers to cube and add up the answers, you get exactly the same amount that we would get by first adding these numbers and then squaring the sum, in other words,
https://pandia.ru/text/79/542/images/image019_3.jpg" width="554" height="140 src=">
Calculations show that both on the left and on the right the answer is the same, namely 324.
Whatever number we take, the property we noticed will be fulfilled. But it’s quite difficult to prove this.
2) . Let's take any four-digit number, for example 2519, and arrange its digits first in descending order, and then in ascending order: and From the larger number, subtract the smaller: =8262. Let's do the same with the resulting number: 86=6354. And one more similar step: 65 = 3087. Next, = 8352, = 6174. Aren't you tired of subtracting? Let's take one more step: =6174. Again it turned out to be 6174.
Now we are, as programmers say, “in a loop”: no matter how many times we subtract now, we will not get anything other than 6174. Maybe the fact is that this is how the original number 2519 was chosen? It turns out that it has nothing to do with it: no matter what four-digit number we take, after no more than seven steps we will definitely get the same number 6174.
3) . Let's draw several circles with a common center and write any four natural numbers on the inner circle. For each pair of adjacent numbers, subtract the smaller from the larger and write the result on the next circle. It turns out that if you repeat this enough times, on one of the circles all the numbers will be equal to zero, and therefore you will continue to get nothing but zeros. The figure shows this for the case when the numbers 25, 17, 55, 47 are written on the inner circle.
4) . Let's take any number (even a thousand-digit number) written in the decimal number system. Let's square all its numbers and add them up. Let's do the same with the amount. It turns out that after several steps we get either the number 1, after which there will be no other numbers, or 4, after which we have the numbers 4, 16, 37, 58, 89, 145, 42, 20 and again we get 4. This means there is no cycle avoid here too.
5. Let's create such an infinite table. In the first column we will write the numbers 4, 7, 10, 13, 16, ... (each next one is 3 more than the previous one). From the number 4 we draw a line to the right, increasing the numbers by 3 at each step. From the number 7 we draw a line, increasing the numbers by 5, from the number 10 - by 7, etc. The following table is obtained:
If you take any number from this table, multiply it by 2 and add 1 to the product, you will always get a composite number. If we do the same with a number not included in this table, we get a prime number. For example, let's take the number 45 from the table. The number 2*45+1=91 is composite, it is equal to 7*13. But the number 14 is not in the table, and the number 2*14+1=29 is prime.
This wonderful way to distinguish prime numbers from composite numbers was invented in 1934 by the Indian student Sundaram. Observations of numbers reveal other remarkable statements. The properties of the world of numbers are truly inexhaustible.
Number tricks.
https://pandia.ru/text/79/542/images/image022_2.jpg" width="226" height="71">
After all, if next to a three-digit number you write the same number again, then the original number will be multiplied by 1001 (for example, 289 289= 289https://pandia.ru/text/79/542/images/image024_3.jpg" width="304" height="74">
And four-digit numbers are repeated once and divided by 73,137. The solution is in equality
https://pandia.ru/text/79/542/images/image026_6.jpg" width="615" height="40 src=">
Note that the cubes of the numbers 0, 1, 4, 5, 6 and 9 end with the same number (for example, https://pandia.ru/text/79/542/images/image028_4.jpg" width="24" height= "24 src=">.jpg" width="389" height="33">
In addition, you need to remember the following table showing where the fifth powers of the following numbers begin:
https://pandia.ru/text/79/542/images/image032_2.jpg" width="200 height=28" height="28">This means that you need to add the number 3 to the five-digit number originally written on the board in front, and Subtract 3 from the resulting number.
To prevent the audience from guessing the trick, you can reduce the first digit of any of the numbers by several units and reduce the corresponding digit in total by the same number of units. For example, in the figure the first digit in the third term is reduced by 2 and the corresponding digit in the sum is reduced by the same amount.
Conclusion.
Having collected and summarized material about prime and composite numbers, I came to the following conclusion:
1. The study of numbers goes back to ancient times and has a rich history.
2. The role of prime numbers in mathematics is great: they are the building blocks from which all other numbers are built using multiplication.
3. Natural numbers have many interesting properties. The properties of the world of numbers are truly inexhaustible.
4. The material I prepared can be safely used in mathematics lessons and math circle classes. This material will help you prepare more deeply for various types of Olympiads.