An important application of the diophantine equation is given by the Pythagorean theorem, which connects the dovzhin x and y legs of the rectum tricutaneous with the dovzhin of the hypotenusus:
You, of course, have identified one of the miraculous solutions of this equation in natural numbers, and the Pythagorean trio of numbers itself x = 3, y = 4, z = 5. What are these threes?
There are quite a lot of Pythagorean trios and all of them have been known for a long time. The smells can be separated from familiar formulas, as you will learn from this paragraph.
Since the defiant zeal of the first and other levels is already supreme, then the nutritious zeal of the higher stages is still unacknowledged, regardless of the strength of the greatest mathematicians. At this time, for example, Fermat’s famous hypothesis about those who have no whole value n2 Rivnyannya
Whole numbers have no solution.
For the development of certain types of diophantine equals, such names can play a major role complex numbers. What is it? May the letter indicate any object that satisfies the mind i 2 = -1(It is clear that the same amount of activity does not satisfy the mind). Let's take a look at the sight α + iβ, where α and β are active numbers. Such expressions will be called complex numbers, since the operations of addition and multiplication are performed on them, as with binomials, and therefore there is no difference, which is expressed i 2 Replaced everywhere by the number -1:
7.1. There's a lot in just three
Let me know what it is x 0 , y 0 , z 0- Pythagorean trio, then triplets y 0 x 0 z 0і x 0 k, y 0 k, z 0 k at any value of the natural parameter it is also Pythagorean.
7.2. Private formulas
Turn it over, what kind of natural meanings are there? m>n three in mind
є Pythagorean. Say every Pythagorean triple x, y, z Can you see from this view that it is possible to rearrange the numbers x and y in the third?
7.3. Short threes
The Pythagorean trio of numbers, which represent a long term greater than 1, is called slow. It should be noted that the Pythagorean triple is not a quick coincidence, since even two of the numbers of the triple are mutually forgiven.
7.4. The power of slow trios
Prove that in any quick Pythagorean trio x, y, z, the number z and one of the numbers x or y are unpaired.
7.5. All non-fast threes
Show that the triad of numbers x, y, z is a quick Pythagorean triad and more so, if, up to the order of the first two numbers, the triad can be matched 2mn, m 2 - n 2, m 2 + n 2 de m>n- Mutually simple natural numbers of different pairs.
7.6. Zagalni formulas
Let us know that everything has been decided
natural numbers are specified with precision up to an order of magnitude unknown x and y formulas
where m>n and k are natural parameters (to turn off the duplication of any triplets, to choose numbers that are mutually simple and of different pairings).
7.7. First 10 threes
Find all Pythagorean triplets x, y, z, what pleases the mind x
7.8. The power of the Pythagorean trio
Let us know that for any Pythagorean trio x, y, z fair affirmation:
a) I want one of the numbers x or y to be a multiple of 3;
b) I want one of the numbers x or y to be a multiple of 4;
c) I want one of the numbers x, y or z to be a multiple of 5.
7.9. Zastosuvannya of complex numbers
Modulus of a complex number α + iβ called an unknown number
Turn it over, for any complex numbers α + iβі γ + iδ power comes to an end
Corrupting the power of complex numbers and their modules, it is argued that any two integers m and n satisfy equalities
to set a decision
whole numbers (refer to assignment 7.5).
7.10. Non-Pythagorean triplets
Studying the powers of complex numbers and their modules (div. problem 7.9), find formulas for any number of solutions:
a) x 2 + y 2 = z 3; b) x 2 + y 2 = z4.
7.1. Yakshcho x 0 2 + y 0 2 = z 0 2 That y 0 2 + x 0 2 = z 0 2 and for whatever natural value k may
what needed to be accomplished.
7.2. Of jealousy
fits what is indicated in the task three satisfies equals x 2 + y 2 = z 2 in natural numbers. However, not every Pythagorean trio x, y, z can be seen in such a look; For example, the triple 9, 12, 15 is Pythagorean, but the number 15 does not look like the sum of the squares of any two natural numbers m and n.
7.3. What are two numbers from the Pythagorean triad? x, y, z If there is a debtor d, then he will be a debtor on the third day (so, if x = x 1 d, y = y 1 d maєmo z 2 = x 2 + y 2 = (x 1 2 + y 1 2)d 2 stars z 2 is divided into d 2 and z is divided into d). Therefore, for the speed of the Pythagorean triple it is necessary that any two of the numbers of the triple be mutually forgivable,
7.4. Dear, one of the numbers x and y, say x, is a slow Pythagorean triple x, y, zє unpaired, the fragments of the same numbers x and y would be mutually forgivable (division 7.3). If the number y is also unpaired, then the number is offended
give a surplus of 1 when divided into 4 and the number z 2 = x 2 + y 2 gives, when divided into 4, a surplus of 2, then. It must be divisible by 2, but not divisible by 4, which we cannot do. Thus, the number y may be paired, and the number z, therefore, unpaired.
7.5. Let go of the Pythagorean trio x, y, z is not fast and for value the number x is paired, and the numbers y, z are not paired (div. problem 7.4). Todi
de numbers 
That is for the whole purpose. Let us prove that the numbers a and b are mutually simple. In fact, if there was a smell of a large debtor, more than 1, then such a debtor is small in number z = a + b, y = a - b, then the three would not be slow (div. problem 7.3). Now, when decomposing the numbers a and b to create simple multipliers, we respect that any simple multiplier must enter before 4ab = x 2 only in a paired world, and if you are allowed to enter the number a, then do not enter the number b and the same. Therefore, if any simple multiplier is included in the decomposition of the numbers a and b, it will be added to the paired world, and the numbers themselves are the squares of integers. We agree 
then jealousy is eliminated
Moreover, the natural parameters m>n are mutually simple (due to the mutual simplicity of numbers a and b) and have different pairings (due to the unparity of the number z = m 2 + n 2).
Now let natural numbers m>n of different pairs be mutually forgiven. Todi three x = 2mn, y = m2 - n2, z = m2 + n2, according to the assertions of problem 7.2, and Pythagorean. Let's see that it won't be quick. For this purpose, it is enough to verify that the numbers y and z do not have any meaning (div. problem 7.3). In fact, these numbers are not paired, because the type of numbers can be paired differently. Just as the numbers y and z are like a simple combination (and also necessarily unpaired), then the same sequence is likewise from the numbers and from them and from the numbers m and n, which supersedes their mutual simplicity.
7.6. By virtue of the firmament formulated in problems 7.1, 7.2, the assigned formulas are given only by Pythagorean triplets. On the other hand, be it the Pythagorean trio x, y, z After this, the shortening of k pairs of numbers x and y to the largest distance k is slow (div. problem 7.3) and, therefore, can be presented exactly to the order of numbers x and y in the form described in problem 7.5. Therefore, the Pythagorean triple is determined by meaningful formulas for certain values of parameters.
7.7. With uneasiness z and the formulas of problem 7.6 we can determine the estimate m 2 tobto. m≤5. Respectfully m = 2, n = 1і k = 1, 2, 3, 4, 5, Threes can be removed 3, 4, 5; 6, 8, 10; 9, 12, 15; 12,16,20; 15, 20, 25. Respectfully m = 3, n = 2і k = 1, 2, Threes can be removed 5, 12, 13; 10, 24, 26. Respectfully m = 4, n = 1, 3і k = 1, Threes can be removed 8, 15, 17; 7, 24, 25. Come on, respectfully m = 5, n = 2і k = 1, let's remove the three 20, 21, 29.
Worm Vitaly
Competition of science projects for schoolchildren
Within the framework of the regional scientific and practical conference “Eurika”
Small Academy of Sciences of Kuban
Research on Pythagorean numbers
Section of mathematics.
Cherv'yak Vitaly Gennadiyovich, 9th grade
MOBU ZOSH No. 14
Korenivskyi district
Art. Zhuravska
Scientific quarry:
Manko Galina Vasylivna
Math teacher
MOBU ZOSH No. 14
Korenivsk 2011 r
Cherv'yak Vitaly Gennadiyovych
Pythagorean numbers
Abstract.
Follow-up topic:Pythagorean numbers
Purposes of investigation:
Investigation department:
Investigation methods:
Visnovok:
Cherv'yak Vitaly Gennadiyovych
Krasnodar region, Zhuravska village, MOBU ZOSH No. 14, 9th grade
Pythagorean numbers
Science book: Manko Galina Vasylivna, mathematics teacher of MOBU ZOSH No. 14
2.1 Historical background……………………………………………………4
2.2 Proof of paired and unpaired legs………...................................5-6
2.3 Summary of the pattern of discovery
Pythagorean numbers………………………………………………………………………………7
2.4 Power of Pythagorean numbers ……………………………………………… 8
3. Visnovok………………………………………………………………………………9
4.List of wikis and literature…………………… 10
Program......................................................... ........................................................ ......eleven
Addendum I………………………………………………………………………………11
Addendum II…………………………………………………………………………………..13
Cherv'yak Vitaly Gennadiyovych
Krasnodar region, Zhuravska village, MOBU ZOSH No. 14, 9th grade
Pythagorean numbers
Science book: Manko Galina Vasylivna, mathematics teacher of MOBU ZOSH No. 14
Enter
I felt about Pythagoras and his life in the fifth grade in mathematics class, and I was hooked by the phrase “Pythagoras’s pants on all sides are equal.” With the introduction of the Pythagorean theorem, Pythagorean numbers were not used.meta research: Learn more about Pythagorean theorem and “Pythagorean numbers”.
Relevance by those. The value of the Pythagorean theorem and Pythagorean triplets has been demonstrated over many centuries. The problem about which I work in my work seems to be due to the simple fact that it is based on a mathematical statement, as everyone knows, - the Pythagorean theorem: if any rectilinear tricuput has a square, the impulses on the hypotenuse, the ancient sum squares formed on the sides. Now there are three natural numbers x, y, z, for which x 2 + y 2 = z 2 , commonly calledPythagorean triplets. It turns out that the Pythagorean triplets were already known in Babylonia. The Greek mathematicians quickly discovered them.
Meta tsієї roboti
It is important to note that the robot is placed low on the feet zavdan:
1. Read more about the history of the Pythagorean theorem;
2. Analysis of the universal powers of the Pythagorean triplets.
3. Analysis of the practical construction of Pythagorean triplets.
Object of investigation: Pythagorean triplets.
Subject of investigation: mathematics .
Investigation methods: - Wikipedia resources on the Internet; - development to pre-modern literature; -conducting the experiment;
Theoretical significance:the role played by the Pythagorean triplets in science; practical zastosuvannya Pythagoras's insight into human life.
Applied valueThe investigation lies in the analysis of literary sources and systematization of facts.
Cherv'yak Vitaly Gennadiyovych
Krasnodar region, Zhuravska village, MOBU ZOSH No. 14, 9th grade
Pythagorean numbers
Science book: Manko Galina Vasylivna, mathematics teacher of MOBU ZOSH No. 14
On the history of Pythagorean numbers.
Mathematical book Chu-pei:[ 2]
“If a straight line is laid out in a warehouse, then the line that connects the ends of its sides will be 5, if the base is 3 and the height is 4.”
Cantor (the greatest German historian of mathematics) appreciates that jealousy 3² + 4² = 5² It was already known to the Egyptians that it was still close to 2300 BC. e., for the king's hours Amenemheta (from papyrus 6619 to the Berlin Museum). On Kantor's thought harpedonapti and the “tensioning motuzoks” were straight-cut knits behind the help of straight-cut knitted knits with 3 sides; 4 and 5.
“The merit of the first Greek mathematicians, such as Thales, Pythagoras and the Pythagoreans, is not the discovery of mathematics, but rather the systematization and delineation. In our hands, calculative recipes, based on incredible phenomena, have been transformed into an exact science."
Although this theorem is connected with the names of Pythagoras, it was known long before.
In Babylonian texts, it is traced 1200 years before Pythagoras.
Obviously, we are the first to know this proof. In connection with this, the following note was made: “... if vin is crooked, so that the hypotenuse of the straight-cut tricutaneous fish has the appearance of cathetes, vin brought a sacrifice of bik, crushed from wheat dough.”
Cherv'yak Vitaly Gennadiyovych
Krasnodar region, Zhuravska village, MOBU ZOSH No. 14, 9th grade
Pythagorean numbers
Science book: Manko Galina Vasylivna, mathematics teacher of MOBU ZOSH No. 14
Investigation of Pythagorean numbers.
3 2 + 4 2 = 5 2.
Pythagorean triplets have been around for a long time. In the architecture of ancient Lysopotamian tombstones, there is an isosfemoral tricubitus, folded with two rectangular ones with sides of 9, 12 and 15 liters. The pyramids of Pharaoh Sneferu (XXVII century BC) were made from three-piece triangles with sides of 20, 21 and 29, as well as 18, 24 and 30 tens of Egyptian liters.[ 1 ]
The rectum tricucutineum with legs 3, 4 and hypotenuse 5 is called the Egyptian tricucutineum. The area of the tricutnik is close to the thorough number 6. The perimeter is close to 12 - the number, as it was respected as a symbol of happiness and prosperity.
For help, a skein divided by knots into 12 equal parts of the ancient Egyptians was a straight-cut tricot and a straight kut. It is a convenient and more accurate way to win by land surveyors while using perpendicular lines. It is necessary to take a cord and three strands; The cord is pulled out with a tricut, which has a straight cut.
This ancient method, which, obviously, has been stoked by thousands of years ago by the Egyptian pyramids, foundations on what a leather tricot, the sides of which are seen like 3: 4: 5, according to the Pythagorean theorem, rectilinear .
Euclid, Pythagoras, Diophantus and many others were engaged in the knowledge of the Pythagorean trios.[ 1]
I realized what it is (x, y, z ) is a Pythagorean triple, then for any natural k three (kx, ky, kz) will also be a Pythagorean three. Zokrema (6, 8, 10), (9, 12, 15) etc. - Pythagorean triplets.
In the world of that, as the numbers grow, the Pythagorean triplets grow ever closer and know them more and more importantly. The Pythagoreans found a way to
such triplets and, corystayuchis him, brought, scho pіthagorean triplets іsnuє more richly.
Triplets, which do not have double triplets larger than 1, are called the simplest.
Let's take a look at the power dynamics of the Pythagorean trios.[ 1]
Apparently, up to the Pythagorean theorem, ci numbers can be dozhinas of a single straight-cut tricutnik; Therefore, y are called “legs”, and s are called “hypotenuse”.
It was understood that if a, c, h are a triple of Pythagorean numbers, then i pa, p, pc, de p-an integer multiplier, are Pythagorean numbers.
Correct and turnaround!
To that end, there are only three mutually simple Pythagorean numbers (there is no need to multiply them by the integer multiplier p).
It is shown that in the skin of such triples a, b, c, one of the “catetivs” can be paired, and the other unpaired. Let us fade away “seemingly unacceptable.” If the offense is “side” a and guys, then the guy will be the number a 2 + y 2 , And that means “hypotenuse”. Ale tse superechit that scho numbers a, b And there are no compatible multipliers, so three paired numbers can be used as a 2-fold multiplier. In this way, one of the “cathetes” is wanted, and the other one is unpaired.
Another possibility is lost: the “leg” is unpaired, and the “hypotenuse” is paired. It doesn’t matter what we can’t say, since the “lines” look like 2 x + 1 and 2y + 1, then the sum of their squares is the same
4x 2 + 4x + 1 + 4y 2 + 4y +1 = 4 (x 2 + x + y 2 + y) +2, then. This is a number that, when divided by 4, gives a surplus of 2. Now the square of any paired number must be divided by 4 without a surplus.
Also, the sum of the squares of two unpaired numbers can be the square of a paired number; Otherwise, it seems, our three numbers are not Pythagorean.
VISNOVOK:
Well, from the “cathetes”, one is paired, and the other is unpaired. Therefore the number a 2 + y 2 unpaired, which means unpaired and “hypotenuse” p.
Pythagoras knows formulas that can be written in modern symbolism like this: a=2n+1, b=2n(n+1), c=2 n 2 +2n+1 de n – whole number.
These numbers are Pythagorean triplets.
Cherv'yak Vitaly Gennadiyovych
Krasnodar region, Zhuravska village, MOBU ZOSH No. 14, 9th grade
Pythagorean numbers
Science book: Manko Galina Vasylivna, mathematics teacher of MOBU ZOSH No. 14
A summary of the pattern of finding Pythagorean numbers.
The axis is the Pythagorean trio:
It is important to note that when the skin is multiplied by the numbers of the Pythagorean triad by 2, 3, 4, 5, etc. We take away the stepping threes.
They also smell like Pythagorean numbers/
Cherv'yak Vitaly Gennadiyovych
Krasnodar region, Zhuravska village, MOBU ZOSH No. 14, 9th grade
Pythagorean numbers
Science book: Manko Galina Vasylivna, mathematics teacher of MOBU ZOSH No. 14
The power of Pythagorean numbers.
Cherv'yak Vitaly Gennadiyovych
Krasnodar region, Zhuravska village, MOBU ZOSH No. 14, 9th grade
Pythagorean numbers
Science book: Manko Galina Vasylivna, mathematics teacher of MOBU ZOSH No. 14
Visnovok.
Geometry, like other sciences, has grown out of the needs of practice. The word “geometry” itself is Greek, but in translation it means “earthcraft”.
People became aware of the need to destroy land plots very early on. Already for 3-4 thousand. rocks BC The skin of the native land in the valleys of the Nile, Euphrates and Tigris, the river of China is of great importance for the life of people. It valued a great store of geometric and arithmetic knowledge.
Step by step, people began to admire and conquer the power of folding geometric figures.
And in Egypt and Babylonia colossal temples were built, the construction of which could only be carried out on the basis of advanced developments. There were also water pipes. Everything was about the chair and rozrakhunkiv. Until now, we were well aware of the implications of the Pythagorean theorem, who already knew that if we take three sides with sides x, y, z, then x, y, z are the same numbers that x 2 + y 2 = z 2 , then the tricuts will be straight-cut.
All this knowledge has completely stagnated in many spheres of human life.
So dosі great vіdkrittya vchennogo that philosopher of old Pіthagoras know directly zastosuvannya in our life.
The life of Budinki, roads, spaceships, automobiles, verstativs, oil pipelines, aircraft, tunnels, metro and more. Pythagorean triplets know directly zastosuvannya in the design of impersonal speeches, which will alienate us from everyday life.
And their minds will continue to search for new versions of proofs of the Pythagorean theorem.
Cherv'yak Vitaly Gennadiyovych
Krasnodar region, Zhuravska village, MOBU ZOSH No. 14, 9th grade
Pythagorean numbers
Science book: Manko Galina Vasylivna, mathematics teacher of MOBU ZOSH No. 14
Literature
4. Anosov D.V. A look at mathematics and what comes from it. - M.: MTsNMO, 2003.
5. Children's Encyclopedia. - M.: Branch of the Academy of Pedagogical Sciences of the RRFSR, 1959.
6. Stepanova L.L. Selections from the elementary theory of numbers. - M.: Prometheus, 2001.
7. V. Sierpinsky Pythagorean tricutaneous tissues. - M: Uchpedgiz, 1959. P.111
Follow-up Historical background; Pythagorean theorem; Let us know that one of the “cathetes” may be a guy, and the other is unpaired; Demonstration of regularities for finding Pythagorean numbers; Reveal the power of Pythagorean numbers;
I smelled about Pythagoras and his life in fifth grade math class, and I was intrigued by the phrase “Pythagoras’ pants on all sides.” With the introduction of the Pythagorean theorem, Pythagorean numbers became less important. I have set the goal of investigation: to find out more about the Pythagorean theorem and “Pythagorean numbers”.
The truth will be eternal if weak people know it! And Pythagoras’s theorem is true, as it was in this distant century
On the history of Pythagorean numbers. Ancient China Mathematical book Chu-pei: “If a straight cut is laid out in a warehouse, then the line that connects the ends of its sides will be 5, if the base is 3, and the height is 4.”
Pythagorean numbers among the ancient Egyptians Cantor (the greatest German historian of mathematics) notes that the equation 3 + 4 = 5 was already known to the Egyptians around 2300 BC. e.., for the hours of King Amenemhet (from papyrus 6619 to the Berlin Museum). According to Cantor, the harpedonapts, or the tensioners of the motorcycles, were directly behind the help of straight-cut tricutlets with 3 sides; 4 and 5.
Theorem of Pythagoras in Babylonia “The merit of the first Greek mathematicians, such as Thales, Pythagoras and the Pythagoreans, is not the discovery of mathematics, but rather the systematization and delineation. In our hands, calculative recipes, based on incredible phenomena, have been transformed into an exact science."
Kozhen trikutnik, the sides are connected as 3: 4: 5, consistent with the basic Pythagorean theorem, - straight-cut, fragments 3 2 + 4 2 = 5 2. The number of numbers 3,4 and 5 is, apparently, an infinite number including all positive numbers, in i s that satisfy the relationship A 2 + b 2 = z 2. These numbers are called Pythagorean numbers
Apparently, up to the Pythagorean theorem, ci numbers can be dozhinas of a single straight-cut tricutnik; therefore, a and c are called “legs”, and z - “Hypotenuse”. It is clear that a, b, c are a trio of Pythagorean numbers, those ra, pv, rs, where p is an integer multiplier, - Pythagorean numbers. Correct and turnaround! To that end, there are only three mutually simple Pythagorean numbers (there is no need to multiply them by the integer multiplier p).
Visnovok! Of the numbers, one is paired, the other is unpaired, and the third is unpaired.
The axis is the Pythagorean triad: 3, 4, 5; 9+16=25. 5, 12, 13; 25 +144 = 169. 7, 24, 25; 49 +576 = 625. 8, 15, 17; 64 +225 = 289. 9, 40, 41; 81 +1600 = 1681. 12, 35, 37; 144 +1225 = 1369. 20, 21, 29; 400 +441 = 841
It is important to note that when the skin is multiplied by the numbers of the Pythagorean triad by 2, 3, 4, 5, etc. We take away the stepping threes. 6, 8, 10; 9,12,15. 12, 16, 20; 15, 20, 25; 10, 24, 26; 18, 24, 30; 16, 30, 34; 21, 28, 35; 15, 36, 39; 24, 32, 40; 14, 48, 50; 30, 40, 50 etc. Also includes Pythagorean numbers
Powers of Pythagorean numbers When looking at Pythagorean numbers, I considered a number of powers: 1) One of the Pythagorean numbers is a multiple of three; 2) one of them may be a multiple of chotiri; 3) And other Pythagorean numbers are multiples of five;
More practical use of Pythagorean numbers
Abstract: As a result of my work, I was given 1. Learn more about Pythagoras, his life, the brotherhood of the Pythagoreans. 2. Get to know the history of the Pythagorean theorem. 3. Learn about Pythagorean numbers, their power, learn to find them. It has been confirmed experimentally that we can put a direct line using Pythagorean numbers.
Let's take a closer look different ways generation of effective Pythagorean triplets Pythagorean scientists were the first to find a simple method for generating Pythagorean triplets, a Vikorist formula, parts of which constitute a Pythagorean triple:
m 2 + ((m 2 − 1)/2) 2 = ((m 2 + 1)/2) 2 ,
De m- Neparne, m>2. True,
4m 2 + m 4 − 2m 2 + 1
m 2 + ((m 2 − 1)/2) 2 = ————————— = ((m 2 + 1)/2) 2 .
4
The ancient Greek philosopher Plato proposed a similar formula:
(2m) 2 + (m 2 − 1) 2 = (m 2 + 1) 2 ,
De m- Whatever the number. For m= 2,3,4,5 the following triplets are generated:
(16,9,25), (36,64,100), (64,225,289), (100,576,676).
As a matter of fact, these formulas cannot represent all possible primitive triads.
Let's look at the next polynomial, which decomposes into the sum of polynomials:
(2m 2 + 2m + 1) 2 = 4m 4 + 8m 3 + 8m 2 + 4m + 1 =
=4m 4 + 8m 3 + 4m 2 + 4m 2 + 4m + 1 = (2m(m+1)) 2 + (2m +1) 2 .
Here are the formulas for deriving primitive triplets:
a = 2m +1 , b = 2m(m+1) = 2m 2 + 2m , c = 2m 2 + 2m + 1.
These formulas generate triplets whose average number increases by one from the largest, so not all possible triplets are generated. Here the first threes become more advanced: (5,12,13), (7,24,25), (9,40,41), (11,60,61).
To identify the methods of generation of all primitive trios, trace their power. First of all, yakscho ( a, b, c) is a primitive three, then aі b, bі c, Aі c- The guilt will be forgiven mutually. Let's go aі b share with d. Todi a 2 + b 2 - also divisible by d. Apparently, c 2 ta c responsibility to share d. This is not a primitive three.
In another way, the middle of numbers a, b One may be a guy, and another may be an unpaired one. To be fair aі b- guys, then h being a guy, the numbers can be divided into 2. Since the stinks are not paired, they can be represented as 2 k+1 and 2 l+1, de k,l- Deyaki numbers. Todi a 2 + b 2 = 4k 2 +4k+1+4l 2 +4l+1, tobto, h 2, yak i a 2 + b 2 when divided by 4 there may be a surplus of 2.
Let's go h- whatever the number, then h = 4k+i (i= 0, ..., 3). Todi h 2 = (4k+i) 2 may be left over 0 or 1 and cannot be left over 2. In this manner, aі b can't be unpaired, then a 2 + b 2 = 4k 2 +4k+4l 2 +4l+1 and surplus in division h 2 by 4 may be 1, which means what h May be unpaired.
Such benefits to the elements of the Pythagorean triad satisfy the following numbers:
a = 2mn, b = m 2 − n 2 , c = m 2 + n 2 , m > n, (2)
De mі n- Mutually simple with different partners. First, these deposits became known from the father of Euclid, who was alive 2300 people. back.
Let us prove the fairness of deposits (2). Let's go A- guy, todi bі c- Neparni. Todi c + b i c − b- guys. Ix can be seen as c + b = 2uі c − b = 2v, de u,v- Days are whole numbers. Tom
a 2 = h 2 − b 2 = (c + b)(c − b) = 2u·2 v = 4uv
I that ( a/2) 2 = uv.
You can make it seem unacceptable that uі v- Forgive each other. Let's go uі v- share with d. Todi ( c + b) that ( c − b) are divided into d. And that cі b responsibility to share d, and it’s super important to think about the Pythagorean trio.
So yak uv = (a/2) 2 ta uі v- Simply put, it’s difficult to convey that uі v toss the buti with squares of yakihos numbers.
In this way, there are positive whole numbers mі n, so what u = m 2 ta v = n 2. Todi
A 2 = 4uv = 4m 2 n 2, so what?
A = 2mn; b = u − v = m 2 − n 2 ; c = u + v = m 2 + n 2 .
So yak b> 0, then m > n.
It's too late to show what mі n there's a hint of steaminess. Yakshcho mі n- guys, then uі v It’s up to you to be guys, but it’s impossible, because the stink is mutually simple. Yakshcho mі n- Neparni, then b = m 2 − n 2 ta c = m 2 + n 2 would be boys, it’s impossible, splinters cі b- Forgive each other.
With such an order, no matter how primitive the Pythagorean trio is, it is responsible for satisfying the mind (2). At this number mі n are called generating numbers primitive threes. For example, let’s not forget the primitive Pythagorean triple (120,119,169). In this case
A= 120 = 2 12 5, b= 119 = 144 − 25, that c = 144+25=169,
De m = 12, n= 5 - generating numbers, 12> 5; 12 and 5 - mutually simple and distinct pairs.
You can bring the soreness up to date m, n after formulas (2) give the primitive Pythagorean triple (a, b, c). True,
A 2 + b 2 = (2mn) 2 + (m 2 − n 2) 2 = 4m 2 n 2 + (m 4 − 2m 2 n 2 + n 4) =
= (m 4 + 2m 2 n 2 + n 4) = (m 2 + n 2) 2 = c 2 ,
Tobto ( a,b,c) - Pythagorean trio. Let's see what's going on a,b,c- The numbers are mutually simple. Let these numbers be divided into p> 1. Oskilki mі n there's a hint of steaminess, then bі c- unparni, then p≠ 2. Shards R divide bі c, That R May divide 2 m 2 and 2 n 2, but it’s impossible, so p≠ 2. Tom m, n- It’s mutually simple a,b,c- just forgive each other.
Table 1 shows all primitive Pythagorean triplets generated by formulas (2) m≤10.
Table 1. Primitive Pythagorean triplets for m≤10
| m | n | a | b | c | m | n | a | b | c |
| 2 | 1 | 4 | 3 | 5 | 8 | 1 | 16 | 63 | 65 |
| 3 | 2 | 12 | 5 | 13 | 8 | 3 | 48 | 55 | 73 |
| 4 | 1 | 8 | 15 | 17 | 8 | 5 | 80 | 39 | 89 |
| 4 | 3 | 24 | 7 | 25 | 8 | 7 | 112 | 15 | 113 |
| 5 | 2 | 20 | 21 | 29 | 9 | 2 | 36 | 77 | 85 |
| 5 | 4 | 40 | 9 | 41 | 9 | 4 | 72 | 65 | 97 |
| 6 | 1 | 12 | 35 | 37 | 9 | 8 | 144 | 17 | 145 |
| 6 | 5 | 60 | 11 | 61 | 10 | 1 | 20 | 99 | 101 |
| 7 | 2 | 28 | 45 | 53 | 10 | 3 | 60 | 91 | 109 |
| 7 | 4 | 56 | 33 | 65 | 10 | 7 | 140 | 51 | 149 |
| 7 | 6 | 84 | 13 | 85 | 10 | 9 | 180 | 19 | 181 |
Analysis of this table shows the presence of a number of patterns:
Born in 1971 American mathematicians Teigan and Hedwin, for the generation of triplets, proposed such low-profile parameters of the recticutaneous tricutaneous tissue as its height (height) h = c− b that excess (success) e = a + b − c. In Fig. 1. indications of the size of the ordinary tricutaneous plant.
Malyunok 1. Straight-cut trikutnik ta yogo zrostannya ta nadlishok
The name “oversupply” is similar to the fact that it is an additional step, because it is necessary to go along the sides of the tricubitus from one vertex to the prolongation, as it does not go along its diagonal.
Through the excess growth of the sides of the Pythagorean tricutule, one can express it as follows:
e 2 e 2
a = h + e, b = e + ——, c = h + e + ——, (3)
2h 2h
Not all combinations hі e may be similar to Pythagorean tricuputans. For a given h possible values e- Do this on the same date d. This number d May be called growth and reach h Let's advance with rank: d- This is the smallest positive integer number whose square is divisible by 2 h. So yak e multiple d, then it will be written as e = kd, de k- Positively.
Couples for help ( k,h) you can generate all the Pythagorean tricubitules, including the primitive ones and the regular ones, in this way:
(dk) 2 (dk) 2
a = h + dk, b = dk + ——, c = h + dk + ——, (4)
2h 2h
Moreover, the three is primitive, because kі h- Mutually simple and simple h =± q 2 at q- Unpaired.
In addition, this will be the Pythagorean trio, as k> √2· h/dі h > 0.
You need to know kі h z ( a,b,c), create the following:
For example, for three (8,15,17) we can h= 17−15 = 2 1, then p= 2 i q = 1, d= 2, i k= (8 − 2)/2 = 3. So this triple is given as ( k,h) = (3,2).
For three (459,1260,1341) we can h= 1341 − 1260 = 81, also p = 1, q= 9 i d= 18, stars k= (459 − 81)/18 = 21, so the code for this three is older ( k,h) = (21, 81).
Calling three people for help hі k May be low among the authorities. Parameter k dorivnyuє
k = 4S/(dP), (5)
De S = ab/2 is the area of the tricutaneous, and P = a + b + c- Yogo perimeter. It flows with zeal eP = 4S, which comes from the Pythagorean theorem.
For straight-cut tricutaneous e equal to the diameter of the stake inscribed in the tricube. This comes from the fact that the hypotenuse h = (A − r)+(b − r) = a + b − 2r, de r- Radius of the stake. Zvidsi h = c − b = A − 2rі e = a − h = 2r.
For h> 0 ta k > 0, kє serial number of triplets a-b-c in the sequence of Pythagorean tricuputans in growth h. Table 2 shows a number of options for triplets generated in pairs h, k, it is clear that this has increased k The size of the sides of the tricutaneous tree increases. Thus, instead of classical numbering, numbering in pairs h, k There is greater order in sequences of triplets.
Table 2. Pythagorean triplets generated by pairs h, k.
| h | k | a | b | c | h | k | a | b | c |
| 2 | 1 | 4 | 3 | 5 | 3 | 1 | 9 | 12 | 15 |
| 2 | 2 | 6 | 8 | 10 | 3 | 2 | 15 | 36 | 39 |
| 2 | 3 | 8 | 15 | 17 | 3 | 3 | 21 | 72 | 75 |
| 2 | 4 | 10 | 24 | 26 | 3 | 4 | 27 | 120 | 123 |
| 2 | 5 | 12 | 35 | 37 | 3 | 5 | 33 | 180 | 183 |
For h > 0, d satisfies anxiety 2√ h ≤ d ≤ 2h, in which the lower boundary can be reached when p= 1, and the upper one - at q= 1. That’s why d shodo 2√ h- as much as the number h distances from the square of a definite number.
Remnants of Rivnyanya x 2 + y 2 = z 2 uniformly, with multiplication x , yі z the same number yields another Pythagorean triple. The Pythagorean triple is called primitive Since they cannot be separated in this way, then they are mutually prime numbers.
These Pythagorean triplets (sorted by increasing maximum number, seen as primitive):
(3, 4, 5), (6, 8, 10), (5, 12, 13), (9, 12, 15), (8, 15, 17), (12, 16, 20), (15, 20, 25), (7, 24, 25), (10, 24, 26), (20, 21, 29), (18, 24, 30), (16, 30, 34), (21, 28, 35), (12, 35, 37), (15, 36, 39), (24, 32, 40), (9, 40, 41), (14, 48, 50), (30, 40, 50)…
Pythagorean triplets have been around for a long time. In the architecture of ancient Mesopotamian tombstones, there is an isosfemoral tricubitus, folded from two rectangular ones with sides of 9, 12 and 15 liters. The pyramids of Pharaoh Sneferu (XXVII century BC) were made from three-piece triangles with sides of 20, 21 and 29, as well as 18, 24 and 30 tens of Egyptian liters.
X All-Russian Symposium on Applied and Industrial Mathematics. St. Petersburg, May 19, 2009.
Algorithm for solving the Diophantine Rivne.
The robot examines the method of investigating the Diophantine levels and presenting the results using this method: - Fermat’s great theorem; - jokes about Pythagorean trios, etc. http://referats.protoplex.ru/referats_show/6954.html
Wikimedia Foundation. 2010.
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In mathematics, a Pythagorean triple is called a tuple of three natural numbers, which satisfies Pythagoras’s opinion: At which numbers, which create a Pythagorean triple, are called Pythagorean numbers. Place 1 Primitive threes… Wikipedia
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» Emeritus Professor of Mathematics at the University of Warwick, a prominent popularizer of science, Ian Stewart, dedicated to the role of numbers in human history and their relevance in our time.
The Pythagorean tricubitules are straight and have integral sides. In the simplest case, their found side has a dowzhin of 5, the other - 3 and 4. In all cases, there are 5 correct rich sides. The roots of the fifth step cannot be traced to the roots of the fifth step - or some other root. Gratis on the plain and in the trivial expanse do not bear the pentapelous symmetry of the wrapping, which is why such symmetries are daily and in crystals. However, stench can be present in almost every world wide space and in small structures known as quasicrystals.
The Pythagorean theorem says that the one side of a rectilinear triangle (the famous hypotenuse) is related to the other two sides of the triangle is very simple and beautiful: the square of the hypotenuse is equal to the sum of the squares of the other two sides.
This theorem is traditionally called after Pythagoras, but in fact the story is quite vague. The clay tablets suggest that the ancient Babylonians knew the Pythagorean theorem long before Pythagoras; The fame of the penman was brought to him by the mathematical cult of the Pythagoreans, whose adherents believed that the whole world of foundations was based on numerical laws. Ancient authors attributed to the Pythagoreans - and also to Pythagoras - various mathematical theorems, but in reality there are no statements about those that Pythagoras himself took up in mathematics. We don’t know whether the Pythagoreans could prove the Pythagorean theorem or simply believed that it was true. Or, what is most certain, they had extensive data about their truth, which, protely, would not be consistent with those that we value as proof today.
The first known proof of the Pythagorean theorem is found in Euclid’s “Cobs.” Here is a simple proof from the Victorian armchair, in which Victorian schoolchildren immediately recognized Pythagorean pants; The chair and the truth is that the underpants are drying on the motorbike. There are literally hundreds of other pieces of evidence, most of which are more obvious.
// Small 33. Pythagorean pants
One of the simplest proofs is a kind of mathematical puzzle. Take any straight-cut tricorn, make several copies of it and collect them from the middle of the square. With one placed bachimo, the square is on the hypotenuse; with the other - squares on the other two sides of the tricubitus. In this case, it is clear that the area in this and the other phase will become more mature.
// Small 34. Evil: a square on the hypotenuse (plus chotiri trikutniks). Right-handed: the sum of squares on the other two sides (plus those chotiri tricutniks). And now turn off the tricks
Rozsіchennya Perigal - another proof-puzzle.
// Small 35. Rose of Perigal
It is also a proof of the theorem with alternative ways of laying squares on a plane. It is possible that the Pythagoreans and their invisible predecessors themselves discovered this theorem. If you look at how an oblique square overlaps two other squares, you can do something like cut a large square into pieces, and then fold two smaller squares from them. You can also use straight-cut triangles, the sides of which give the dimensions of three squares.
// Small 36. Proof to the pavements
Є tsіkavі prove from vikoristannyam similar trikutnikіv in trigonometry. There seem to be less than fifty different proofs.
The number theorem, the Pythagorean theorem, became the kernel of a fruitful idea: to find integer solutions to algebraic equations. Pythagorean trinity - tse set of integers a, b and c, such that
Geometrically, such a three-piece signifies a straight-cut tri-cut from the whole sides.
The smallest hypotenuse of the Pythagorean trinity is 5.
The other two sides of this tricot fabric are 3 and 4. Here
32 + 42 = 9 + 16 = 25 = 52.
Comes after the value of the hypotenuse is more than 10, to that
62 + 82 = 36 + 64 = 100 = 102.
However, in fact, the same tricoutnik from the undersides. Coming after the value and in the right way, the hypotenuse is more expensive 13 for her
52 + 122 = 25 + 144 = 169 = 132.
Euclid knew that there were a number of different variants of the Pythagorean triplets, and gave what could be called a formula for finding them all. Later, Diophantus of Oleksandria proposed a simple recipe that avoids Euclidean.
Take two natural numbers and calculate:
Ex subdivisions;
the difference in their squares;
the sum of their squares.
The three numbers that come out will be the sides of the Pythagorean tricutnik.
Let’s take, for example, the numbers 2 and 1. Countable:
subdivision tvir: 2×2×1 = 4;
difference in squares: 22 - 12 = 3;
sum of squares: 22 + 12 = 5,
and we removed the tricuticle 3–4–5. If we take the place of this number 3 and 2, we cancel:
subdivision tvir: 2×3×2 = 12;
difference in squares: 32 - 22 = 5;
sum of squares: 32 + 22 = 13,
and the popular trikutnik 5 - 12 - 13 is eliminated. Let's try to take the numbers 42 and 23 and eliminated:
subdivision tvir: 2×42×23 = 1932;
difference in squares: 422 - 232 = 1235;
sum of squares: 422 + 232 = 2293,
no one ever heard about the trikutnik 1235–1932–2293.
All numbers are also pronounced:
12352 + 19322 = 1525225 + 3732624 = 5257849 = 22932.
The diophantine rule has one more feature, as we have already mentioned: having subtracted three numbers, we can take another sufficient number and multiply all of them by a new one. In this way, a 3-4-5 tricut can be converted into a 6-8-10 tricut by multiplying all sides by 2, or into a 15-20-25 tricut by multiplying everything by 5.
When switching to English algebra, the rule becomes clear: let u, v and k be natural numbers. Todi straight-cut tricut with sides
2kuv and k (u2 - v2) is the hypotenuse
Find other ways to express the basic ideas, rather than reduce them to the described thing. This method allows you to remove all Pythagorean triplets.
There are exactly five regular rich facets. A regular polyhedron (or polyhedron) is a volumetric figure with an end number of flat faces. The faces converge on one line, called edges; the edges sharpen at points called vertices.
The culmination of the Euclidean “Cobs” is the proof that there can be more than five regular richahedrons, or richahedrons, which have a skin edge correct rich man(Equal sides, equal sides), all faces are identical and all vertices are marked with an equal number of but expanded faces. Axis of five regular rich sides:
a tetrahedron with three triangulated faces, several vertices and six edges;
cube, or hexahedron, with 6 square faces, 8 vertices and 12 edges;
octahedron with 8 tricut faces, 6 vertices and 12 edges;
dodecahedron with 12 pentagonal faces, 20 vertices and 30 edges;
An icosahedron with 20 tributary faces, 12 vertices and 30 edges.
// Small 37. Five regular rich sides
The correct polyhedrons can be found in nature. In 1904 Ernst Haeckel published tiny tiny organisms known as radiolarians; From their shape you can guess the five regular rich facets. It is possible, however, by tweaking nature a little, and the little ones do not completely reflect the form of specific living things. The first three structures are also observed in crystals. You won’t find dodecahedrons and icosahedrons in crystals, although irregular dodecahedrons and icosahedrons are sometimes trapped there. The same dodecahedrons may have the appearance of quasicrystals, which are similar to crystals, due to the fact that their atoms do not form periodic compounds.
// Small 38. Haeckel’s babies: radiolarians in the form of regular rich-sided faces
// Small 39. Rostering of regular rich facets
It is easy to work with a model of regular rich facets from a paper, having first seen a set of faces connected to each other - this is called a rich facet part; Fold the pile along the ribs and glue the top ribs together. It is important to add an additional pad for glue to one of the edges of the skin, as shown in Fig. 39. If there is no such maidan, you can remove the sticky stitch.
There is no algebraic formula for the highest level of the 5th level.
In the traditional view, the level of the fifth stage looks like this:
ax5 + bx4 + cx3 + dx2 + ex + f = 0.
The problem is to know the formula for unleashing such jealousy (someone may have up to five decisions). Evidence of the relationship with square and cubic levels, as well as with the levels of the fourth stage, allows us to assume that such a formula is also valid for the levels of the fifth stage, and in it, according to the idea, the figures are to blame Take the root of the fifth, third and other stage. Again, we can safely admit that such a formula, as it appears, will appear even more complex.
This absolution was revealed to those who had mercy. True, there is no such formula; There is no known formula that consists of the coefficients a, b, c, d, e and f, folded with vicary folds, folded, multiplied and subdivided, and then twisted roots. In this manner, the middle of the 5th is completely special. The reasons for this unexpected behavior of the five are very deep, and it took them quite an hour for them to understand.
The first sign of the problem was that, no matter how mathematicians tried to come up with such a formula, no matter how reasonable they were, they invariably recognized failures. For many hours everyone has been paying attention to the fact that the reasons lie in the incredible complexity of the formula. It was important that no one simply could do well in this algebra. However, over time, many mathematicians began to doubt that such a formula was true, and in 1823 Niels Hendrick Abel decided to take it further. There is no such formula. Nezabar after this, Evarist Galois knows the best way to determine how you can determine the level of this or that level - the 5th, 6th, 7th, whatever - with the help of this kind of formula.
The symbol is simple: the number 5 is special. You can use the level of algebra (with the help of roots nth stage for different values n) for stages 1, 2, 3 and 4, but not for the 5th stage. This is where the obvious pattern ends.
No one is surprised that the level of steps above 5 is even tighter; However, the same complexity is associated with them: there are no cryptic formulas for their versatility. This does not mean that jealousy is not a solution; This also does not mean that it is impossible to know even more precisely the numerical values of these solutions. Everything on the right is surrounded by traditional algebraic tools. This reminds us of the impossibility of a trisection using a ruler and a compass. The answer is clear, but overdone methods are insufficient and do not allow us to determine what it is.
The crystals of two and three vimirs do not have a 5-promenian symmetry wrapping.
The atoms in the crystal create a structure that is periodically repeated in many independent directions. For example, babies on trellises are repeated after each roll; In addition, it is necessary to repeat this in a horizontal direction, at the same time as pulling one strip of trellis to the next step. In fact, the trellis is a two-worldly crystal.
There are 17 varieties of trellis crabs on the plain (section 17). They will compete for the types of symmetry, and for ways to brutally destroy the little ones in such a manner that the wine just lies on itself in the cob position. Up to the types of symmetry one can see, from a close view, different variations of the symmetry of wrapping, where the little ones turn the song cut towards the song point - the center of symmetry.
The order of symmetry of the wrap is the number of times you can rotate the body until all the parts of the baby are rotated in the first position. For example, a rotation of 90° is the symmetry of the 4th order wrapper. The list of possible types of symmetry of wrapping in a crystalline lattice again confirms the insignificance of the number 5: there is none. There is an option based on the symmetry of the 2nd, 3rd, 4th and 6th order wrappings, otherwise the trellis baby does not have the symmetry of the 5th order wrapping. Wrap symmetries of order greater than 6 still do not occur in crystals, but if the sequence is disrupted, they still appear at number 5.
Those same experiences with crystallographic systems in trivial space. Here the gratis repeat to themselves for three times with independent directness. There are 219 different types of symmetry, or 230, since we consider the mirror image of the baby as an option - despite the fact that in this type there is no mirror symmetry. Again, beware of the symmetry of the wrapping of orders 2, 3, 4 and 6, rather than 5. This fact is taken away from the name of crystallographic boundary.
The vast expanse of the lattice has 5th order symmetry; vzagalі, for ґrat dosit vysokї rozmіrnostі mozhlivy be-what ahead of tasks the order of symmetry of wrapping.
// Small 40. Crystalline pellets of kitchen salt. Dark bags represent sodium atoms, light ones represent chlorine atoms.
Although symmetry wrapping of the 5th order in two-world and three-world structures is impossible, it can be used in smaller regular structures, like quasi-crystals. Skipping Kepler's sketches, Roger Penrose deconstructed flat systems with a greater type of fivefold symmetry. They took away the name of quasicrystals.
Quasicrystals appear in nature. Born in 1984 Daniel Shekhtman discovered that an alloy of aluminum and manganese can create quasicrystals; Initially, crystallography was aware of this skepticism, but later the revelation was confirmed, and in 2011. Shekhtman was awarded Nobel Prize from chemistry. In 2009 A team of scientists under Luca Bindi's research discovered quasi-crystals in minerals from the Russian Koryak rock - connected to aluminum, mid and lye. Today this mineral is called icosahedrite. Using a mass spectrometer, instead of using acid in minerals of various isotopes, they have now shown that this mineral does not belong to the Earth. It was formed around 4.5 billion years ago, when sleepy system Only arose, and spent most of the hour in the asteroid belt, whirling around the Sun, until a storm changed its orbit and brought it to Earth.
// Small 41. Zliva: one of two quasi-crystalline gates with exact five-fold symmetry. Right-handed: atomic model of an icosahedral aluminum-paladium-manganese quasicrystal
