Pythagorean fives. Professor Stewart's Incredible Numbers. Application of complex numbers

A convenient and very accurate method used by land surveyors to draw perpendicular lines on the ground is as follows. Let it be required to draw a perpendicular to the line MN through point A (Fig. 13). Lay off from A in the direction of AM three times some distance a. Then three knots are tied on the cord, the distances between which are 4a and 5a. Attaching the extreme knots to points A and B, pull the cord over the middle knot. The cord will be located in a triangle, in which the angle A is a right one.

This ancient method, apparently used thousands of years ago by the builders of the Egyptian pyramids, is based on the fact that each triangle, the sides of which are related as 3:4:5, according to the well-known Pythagorean theorem, is right-angled, since

3 2 + 4 2 = 5 2 .

In addition to the numbers 3, 4, 5, there is, as is known, an uncountable set of positive integers a, b, c, satisfying the relation

A 2 + b 2 \u003d c 2.

They are called Pythagorean numbers. According to the Pythagorean theorem, such numbers can serve as the lengths of the sides of some right triangle; therefore, a and b are called "legs", and c is called the "hypotenuse".

It is clear that if a, b, c is a triple of Pythagorean numbers, then pa, pb, pc, where p is an integer factor, are Pythagorean numbers. Conversely, if the Pythagorean numbers have a common factor, then by this common factor you can reduce them all, and again you get a triple of Pythagorean numbers. Therefore, we will first study only triples of coprime Pythagorean numbers (the rest are obtained from them by multiplying by an integer factor p).

Let us show that in each of such triplets a, b, c one of the "legs" must be even and the other odd. Let's argue "on the contrary". If both "legs" a and b are even, then the number a 2 + b 2 will be even, and hence the "hypotenuse". This, however, contradicts the fact that the numbers a, b, c do not have common factors, since three even numbers have a common factor of 2. Thus, at least one of the "legs" a, b is odd.

There remains one more possibility: both "legs" are odd, and the "hypotenuse" is even. It is easy to prove that this cannot be. Indeed, if the "legs" have the form

2x + 1 and 2y + 1,

then the sum of their squares is

4x 2 + 4x + 1 + 4y 2 + 4y + 1 \u003d 4 (x 2 + x + y 2 + y) + 2,

i.e., it is a number that, when divided by 4, gives a remainder of 2. Meanwhile, the square of any even number must be divisible by 4 without a remainder. So the sum of the squares of two odd numbers cannot be the square of an even number; in other words, our three numbers are not Pythagorean.

So, from the "legs" a, b, one is even and the other is odd. Therefore, the number a 2 + b 2 is odd, which means that the "hypotenuse" c is also odd.

Suppose, for definiteness, that odd is "leg" a, and even b. From equality

a 2 + b 2 = c 2

we easily get:

A 2 \u003d c 2 - b 2 \u003d (c + b) (c - b).

The factors c + b and c - b on the right side are coprime. Indeed, if these numbers had a common prime factor other than one, then the sum would also be divisible by this factor.

(c + b) + (c - b) = 2c,

and difference

(c + b) - (c - b) = 2b,

and work

(c + b) (c - b) \u003d a 2,

i.e. the numbers 2c, 2b and a would have a common factor. Since a is odd, this factor is different from two, and therefore the numbers a, b, c have the same common factor, which, however, cannot be. The resulting contradiction shows that the numbers c + b and c - b are coprime.

But if the product of coprime numbers is an exact square, then each of them is a square, i.e.

Solving this system, we find:

C \u003d (m 2 + n 2) / 2, b \u003d (m 2 - n 2) / 2, and 2 \u003d (c + b) (c - b) \u003d m 2 n 2, a \u003d mn.

So, the considered Pythagorean numbers have the form

A \u003d mn, b \u003d (m 2 - n 2) / 2, c \u003d (m 2 + n 2) / 2.

where m and n are some coprime odd numbers. The reader can easily verify the opposite: for any odd type, the written formulas give three Pythagorean numbers a, b, c.

Here are some triplets of Pythagorean numbers obtained with various types:

For m = 3, n = 1 3 2 + 4 2 = 5 2 for m = 5, n = 1 5 2 + 12 2 = 13 2 for m = 7, n = 1 7 2 + 24 2 = 25 2 for m = 9, n = 1 9 2 + 40 2 = 41 2 at m = 11, n = 1 11 2 + 60 2 = 61 2 at m = 13, n = 1 13 2 + 84 2 = 85 2 at m = 5 , n = 3 15 2 + 8 2 = 17 2 for m = 7, n = 3 21 2 + 20 2 = 29 2 for m = 11, n = 3 33 2 + 56 2 = 65 2 for m = 13, n = 3 39 2 + 80 2 = 89 2 at m = 7, n = 5 35 2 + 12 2 = 37 2 at m = 9, n = 5 45 2 + 28 2 = 53 2 at m = 11, n = 5 55 2 + 48 2 = 73 2 at m = 13, n = 5 65 2 + 72 2 = 97 2 at m = 9, n = 7 63 2 + 16 2 = 65 2 at m = 11, n = 7 77 2 + 36 2 = 85 2

(All other triples of Pythagorean numbers either have common factors or contain numbers greater than one hundred.)

educational: to study a number of Pythagorean triples, develop an algorithm for their application in various situations, draw up a memo on their use.

Educational: the formation of a conscious attitude to learning, the development of cognitive activity, the culture of educational work.

Educational: development of geometric, algebraic and numerical intuition, ingenuity, observation, memory.

During the classes

I. Organizational moment

II. Explanation of new material

Teacher: The mystery of the attractive power of Pythagorean triples has long worried mankind. The unique properties of Pythagorean triples explain their special role in nature, music, and mathematics. The Pythagorean spell, the Pythagorean theorem, remains in the brains of millions, if not billions, of people. This is a fundamental theorem, which every schoolchild is forced to memorize. Although comprehensible to ten-year-olds, the Pythagorean theorem is the inspiring start to the problem that the greatest minds in the history of mathematics have failed to solve: Fermat's Theorem. Pythagoras from the island of Samos (cf. Appendix 1 , slide 4) was one of the most influential yet enigmatic figures in mathematics. Since there are no reliable records of his life and work, his life has become shrouded in myths and legends, and historians find it difficult to separate fact from fiction. There is no doubt, however, that Pythagoras developed the idea of ​​the logic of numbers and that it is to him that we owe the first golden age of mathematics. Thanks to his genius, numbers were no longer used only for counting and calculations and were first appreciated. Pythagoras studied the properties of certain classes of numbers, the relationships between them, and the figures that form numbers. Pythagoras realized that numbers exist independently of the material world, and therefore the inaccuracy of our senses does not affect the study of numbers. This meant that Pythagoras gained the ability to discover truths independent of anyone's opinion or prejudice. Truths are more absolute than any previous knowledge. Based on the studied literature concerning Pythagorean triples, we will be interested in the possibility of using Pythagorean triples in solving trigonometry problems. Therefore, we will set ourselves the goal: to study a number of Pythagorean triples, to develop an algorithm for their application, to compile a memo on their use, to conduct a study on their application in various situations.

Triangle ( slide 14), whose sides are equal to Pythagorean numbers, is rectangular. Moreover, any such triangle is Heronian, i.e. one in which all sides and area are integers. The simplest of them is the Egyptian triangle with sides (3, 4, 5).

Let's make a series of Pythagorean triples by multiplying the numbers (3, 4, 5) by 2, by 3, by 4. We will get a series of Pythagorean triples, sort them in ascending order of the maximum number, select primitive ones.

(3, 4, 5), (6, 8, 10), (5, 12, 13) , (9, 12, 13), (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).

III. During the classes

1. Let's spin around the tasks:

1) Using the relationships between trigonometric functions of the same argument, find if

it is known that .

2) Find the value of the trigonometric functions of the angle?, if it is known that:

3) The system of training tasks on the topic “Addition formulas”

knowing that sin = 8/17, cos = 4/5, and are the angles of the first quarter, find the value of the expression:

knowing that and are the angles of the second quarter, sin = 4/5, cos = - 15/17, find:.

4) The system of training tasks on the topic “Double angle formulas”

a) Let sin = 5/13, be the angle of the second quarter. Find sin2, cos2, tg2, ctg2.

b) It is known that tg? \u003d 3/4, - the angle of the third quarter. Find sin2, cos2, tg2, ctg2.

c) It is known that , 0< < . Найдите sin, cos, tg, ctg.

d) It is known that , < < 2. Найдите sin, cos, tg.

e) Find tg( + ) if it is known that cos = 3/5, cos = 7/25, where and are the angles of the first quarter.

f) Find , is the angle of the third quarter.

We solve the problem in the traditional way using basic trigonometric identities, and then we solve the same problems in a more rational way. To do this, we use an algorithm for solving problems using Pythagorean triples. We compose a memo for solving problems using Pythagorean triples. To do this, we recall the definition of the sine, cosine, tangent and cotangent, the acute angle of a right triangle, depict it, depending on the conditions of the problem on the sides of the right triangle, we correctly arrange the Pythagorean triples ( rice. one). We write down the ratio and arrange the signs. The algorithm has been developed.

Picture 1

Problem solving algorithm

Repeat (study) theoretical material.

Know by heart the primitive Pythagorean triples and, if necessary, be able to construct new ones.

Apply the Pythagorean theorem for points with rational coordinates.

Know the definition of sine, cosine, tangent and cotangent of an acute angle of a right triangle, be able to draw a right triangle and, depending on the condition of the problem, correctly arrange Pythagorean triples on the sides of the triangle.

Know the signs of sine, cosine, tangent and cotangent depending on their location in the coordinate plane.

Required requirements:

1. know what signs sine, cosine, tangent, cotangent have in each of the quarters of the coordinate plane;
2. know the definition of sine, cosine, tangent and cotangent of an acute angle of a right triangle;
3. know and be able to apply the Pythagorean theorem;
4. know the basic trigonometric identities, addition formulas, double angle formulas, half argument formulas;
5. know the formulas of reduction.

Based on the above, fill in the table ( Table 1). It must be filled in following the definition of sine, cosine, tangent and cotangent or using the Pythagorean theorem for points with rational coordinates. In this case, it is constantly necessary to remember the signs of the sine, cosine, tangent and cotangent, depending on their location in the coordinate plane.

Table 1

		Triplets of numbers		sin	cos	tg	ctg
		(3, 4, 5)	I hour
		(6, 8, 10)	II hour			-	-
		(5, 12, 13)	3rd hour	-	-
		(8, 15, 17)	IV hour	-		-	-
		(9, 40, 41)	I hour

For successful work, you can use the memo of using Pythagorean triples.

table 2

(3, 4, 5), (6, 8, 10), (5, 12, 13) , (9, 12, 13), (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), …

2. We decide together.

1) Task: find cos, tg and ctg, if sin = 5/13, if - the angle of the second quarter.

» Honored Professor of Mathematics at the University of Warwick, a well-known popularizer of science Ian Stewart, dedicated to the role of numbers in the history of mankind and the relevance of their study in our time.

Pythagorean hypotenuse

Pythagorean triangles have a right angle and integer sides. In the simplest of them, the longest side has a length of 5, the rest are 3 and 4. There are 5 regular polyhedra in total. A fifth-degree equation cannot be solved with fifth-degree roots - or any other roots. Lattices in the plane and in three-dimensional space do not have a five-lobe rotational symmetry; therefore, such symmetries are also absent in crystals. However, they can be in lattices in four-dimensional space and in interesting structures known as quasicrystals.

Hypotenuse of the smallest Pythagorean triple

The Pythagorean theorem states that the longest side of a right triangle (the notorious hypotenuse) correlates with the other two sides of this triangle in a very simple and beautiful way: the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Traditionally, we call this theorem after Pythagoras, but in fact its history is rather vague. Clay tablets suggest that the ancient Babylonians knew the Pythagorean theorem long before Pythagoras himself; the glory of the discoverer was brought to him by the mathematical cult of the Pythagoreans, whose supporters believed that the universe was based on numerical patterns. Ancient authors attributed to the Pythagoreans - and hence to Pythagoras - a variety of mathematical theorems, but in fact we have no idea what kind of mathematics Pythagoras himself was engaged in. We don't even know if the Pythagoreans could prove the Pythagorean Theorem, or if they simply believed it was true. Or, more likely, they had convincing data about its truth, which nevertheless would not have been enough for what we consider proof today.

Evidence of Pythagoras

The first known proof of the Pythagorean theorem is found in Euclid's Elements. This is a rather complicated proof using a drawing that Victorian schoolchildren would immediately recognize as "Pythagorean pants"; the drawing really resembles underpants drying on a rope. Literally hundreds of other proofs are known, most of which make the assertion more obvious.

// Rice. 33. Pythagorean pants

One of the simplest proofs is a kind of mathematical puzzle. Take any right triangle, make four copies of it and collect them inside the square. With one laying, we see a square on the hypotenuse; with the other - squares on the other two sides of the triangle. It is clear that the areas in both cases are equal.

// Rice. 34. Left: square on the hypotenuse (plus four triangles). Right: the sum of the squares on the other two sides (plus the same four triangles). Now eliminate the triangles

The dissection of Perigal is another puzzle piece of evidence.

// Rice. 35. Dissection of Perigal

There is also a proof of the theorem using stacking squares on the plane. Perhaps this is how the Pythagoreans or their unknown predecessors discovered this theorem. If you look at how the oblique square overlaps the other two squares, you can see how to cut the large square into pieces and then put them together into two smaller squares. You can also see right-angled triangles, the sides of which give the dimensions of the three squares involved.

// Rice. 36. Proof by paving

There are interesting proofs using similar triangles in trigonometry. At least fifty different proofs are known.

Pythagorean triplets

In number theory, the Pythagorean theorem became the source of a fruitful idea: to find integer solutions to algebraic equations. A Pythagorean triple is a set of integers a, b and c such that

Geometrically, such a triple defines a right triangle with integer sides.

The smallest hypotenuse of a Pythagorean triple is 5.

The other two sides of this triangle are 3 and 4. Here

32 + 42 = 9 + 16 = 25 = 52.

The next largest hypotenuse is 10 because

62 + 82 = 36 + 64 = 100 = 102.

However, this is essentially the same triangle with doubled sides. The next largest and truly different hypotenuse is 13, for which

52 + 122 = 25 + 144 = 169 = 132.

Euclid knew that there were an infinite number of different variations of Pythagorean triples, and he gave what might be called a formula for finding them all. Later, Diophantus of Alexandria offered a simple recipe, basically the same as Euclidean.

Take any two natural numbers and calculate:

their double product;

difference of their squares;

the sum of their squares.

The three resulting numbers will be the sides of the Pythagorean triangle.

Take, for example, the numbers 2 and 1. Calculate:

double product: 2 × 2 × 1 = 4;

difference of squares: 22 - 12 = 3;

sum of squares: 22 + 12 = 5,

and we got the famous 3-4-5 triangle. If we take the numbers 3 and 2 instead, we get:

double product: 2 × 3 × 2 = 12;

difference of squares: 32 - 22 = 5;

sum of squares: 32 + 22 = 13,

and we get the next famous triangle 5 - 12 - 13. Let's try to take the numbers 42 and 23 and get:

double product: 2 × 42 × 23 = 1932;

difference of squares: 422 - 232 = 1235;

sum of squares: 422 + 232 = 2293,

no one has ever heard of the triangle 1235–1932–2293.

But these numbers work too:

12352 + 19322 = 1525225 + 3732624 = 5257849 = 22932.

There is another feature in the Diophantine rule that has already been hinted at: having received three numbers, we can take another arbitrary number and multiply them all by it. Thus, a 3-4-5 triangle can be turned into a 6-8-10 triangle by multiplying all sides by 2, or into a 15-20-25 triangle by multiplying everything by 5.

If we switch to the language of algebra, the rule takes the following form: let u, v and k be natural numbers. Then a right triangle with sides

2kuv and k (u2 - v2) has a hypotenuse

There are other ways of presenting the main idea, but they all boil down to the one described above. This method allows you to get all Pythagorean triples.

Regular polyhedra

There are exactly five regular polyhedra. A regular polyhedron (or polyhedron) is a three-dimensional figure with a finite number of flat faces. Facets converge with each other on lines called edges; edges meet at points called vertices.

The culmination of the Euclidean "Principles" is the proof that there can be only five regular polyhedra, that is, polyhedra in which each face is a regular polygon (equal sides, equal angles), all faces are identical, and all vertices are surrounded by an equal number of equally spaced faces. Here are five regular polyhedra:

tetrahedron with four triangular faces, four vertices and six edges;

cube, or hexahedron, with 6 square faces, 8 vertices and 12 edges;

octahedron with 8 triangular faces, 6 vertices and 12 edges;

dodecahedron with 12 pentagonal faces, 20 vertices and 30 edges;

icosahedron with 20 triangular faces, 12 vertices and 30 edges.

// Rice. 37. Five regular polyhedra

Regular polyhedra can also be found in nature. In 1904, Ernst Haeckel published drawings of tiny organisms known as radiolarians; many of them are shaped like the same five regular polyhedra. Perhaps, however, he slightly corrected nature, and the drawings do not fully reflect the shape of specific living beings. The first three structures are also observed in crystals. You will not find a dodecahedron and an icosahedron in crystals, although irregular dodecahedrons and icosahedrons sometimes come across there. True dodecahedrons can occur as quasicrystals, which are like crystals in every way, except that their atoms do not form a periodic lattice.

// Rice. 38. Drawings by Haeckel: radiolarians in the form of regular polyhedra

// Rice. 39. Developments of Regular Polyhedra

It can be interesting to make models of regular polyhedra out of paper by first cutting out a set of interconnected faces - this is called a polyhedron sweep; the scan is folded along the edges and the corresponding edges are glued together. It is useful to add an additional area for glue to one of the edges of each such pair, as shown in Fig. 39. If there is no such platform, you can use adhesive tape.

Equation of the fifth degree

There is no algebraic formula for solving equations of the 5th degree.

In general, the equation of the fifth degree looks like this:

ax5 + bx4 + cx3 + dx2 + ex + f = 0.

The problem is to find a formula for solving such an equation (it can have up to five solutions). Experience in dealing with quadratic and cubic equations, as well as with equations of the fourth degree, suggests that such a formula should also exist for equations of the fifth degree, and, in theory, the roots of the fifth, third and second degree should appear in it. Again, one can safely assume that such a formula, if it exists, will turn out to be very, very complicated.

This assumption ultimately turned out to be wrong. Indeed, no such formula exists; at least there is no formula consisting of the coefficients a, b, c, d, e and f, composed using addition, subtraction, multiplication and division, as well as taking roots. Thus, there is something very special about the number 5. The reasons for this unusual behavior of the five are very deep, and it took a lot of time to figure them out.

The first sign of a problem was that no matter how hard mathematicians tried to find such a formula, no matter how smart they were, they always failed. For some time, everyone believed that the reasons lie in the incredible complexity of the formula. It was believed that no one simply could understand this algebra properly. However, over time, some mathematicians began to doubt that such a formula even existed, and in 1823 Niels Hendrik Abel was able to prove the opposite. There is no such formula. Shortly thereafter, Évariste Galois found a way to determine whether an equation of one degree or another - 5th, 6th, 7th, generally any - is solvable using this kind of formula.

The conclusion from all this is simple: the number 5 is special. You can solve algebraic equations (using nth roots for different values ​​of n) for powers of 1, 2, 3, and 4, but not for powers of 5. This is where the obvious pattern ends.

No one is surprised that equations of powers greater than 5 behave even worse; in particular, the same difficulty is connected with them: there are no general formulas for their solution. This does not mean that the equations have no solutions; it does not mean also that it is impossible to find very precise numerical values ​​of these solutions. It's all about the limitations of traditional algebra tools. This is reminiscent of the impossibility of trisecting an angle with a ruler and a compass. There is an answer, but the listed methods are not sufficient and do not allow you to determine what it is.

Crystallographic limitation

Crystals in two and three dimensions do not have 5-beam rotational symmetry.

The atoms in a crystal form a lattice, that is, a structure that repeats periodically in several independent directions. For example, the pattern on the wallpaper is repeated along the length of the roll; in addition, it is usually repeated in the horizontal direction, sometimes with a shift from one piece of wallpaper to the next. Essentially, the wallpaper is a two-dimensional crystal.

There are 17 varieties of wallpaper patterns on the plane (see chapter 17). They differ in the types of symmetry, that is, in the ways of rigidly shifting the pattern so that it lies exactly on itself in its original position. The types of symmetry include, in particular, various variants of rotational symmetry, where the pattern should be rotated through a certain angle around a certain point - the center of symmetry.

The order of symmetry of rotation is how many times you can rotate the body to a full circle so that all the details of the picture return to their original positions. For example, a 90° rotation is 4th order rotational symmetry*. The list of possible types of rotational symmetry in the crystal lattice again points to the unusualness of the number 5: it is not there. There are variants with rotational symmetry of 2nd, 3rd, 4th and 6th orders, but no wallpaper pattern has 5th order rotational symmetry. There is also no rotational symmetry of order greater than 6 in crystals, but the first violation of the sequence still occurs at the number 5.

The same happens with crystallographic systems in three-dimensional space. Here the lattice repeats itself in three independent directions. There are 219 different types of symmetry, or 230 if we consider the mirror reflection of the pattern as a separate version of it - moreover, in this case there is no mirror symmetry. Again, rotational symmetries of orders 2, 3, 4, and 6 are observed, but not 5. This fact is called the crystallographic constraint.

In four-dimensional space, lattices with 5th order symmetry exist; in general, for lattices of sufficiently high dimension, any predetermined order of rotational symmetry is possible.

// Rice. 40. Crystal lattice of table salt. Dark balls represent sodium atoms, light balls represent chlorine atoms.

Quasicrystals

While 5th order rotational symmetry is not possible in 2D and 3D lattices, it can exist in slightly less regular structures known as quasicrystals. Using Kepler's sketches, Roger Penrose discovered flat systems with a more general type of fivefold symmetry. They are called quasicrystals.

Quasicrystals exist in nature. In 1984, Daniel Shechtman discovered that an alloy of aluminum and manganese can form quasi-crystals; Initially, crystallographers greeted his message with some skepticism, but later the discovery was confirmed, and in 2011 Shechtman was awarded the Nobel Prize in Chemistry. In 2009, a team of scientists led by Luca Bindi discovered quasi-crystals in a mineral from the Russian Koryak Highlands - a compound of aluminum, copper and iron. Today this mineral is called icosahedrite. By measuring the content of various oxygen isotopes in the mineral with a mass spectrometer, scientists showed that this mineral did not originate on Earth. It formed about 4.5 billion years ago, at a time when the solar system was just emerging, and spent most of its time in the asteroid belt, orbiting the sun, until some kind of disturbance changed its orbit and brought it eventually to Earth.

// Rice. 41. Left: one of two quasi-crystalline lattices with exact fivefold symmetry. Right: Atomic model of an icosahedral aluminum-palladium-manganese quasicrystal

Pythagorean triples of numbers

creative work

student 8 ”A” class

MAOU "Gymnasium No. 1"

Oktyabrsky district of Saratov

Panfilova Vladimir

Supervisor - teacher of mathematics of the highest category

Grishina Irina Vladimirovna

Content

Introduction……………………………………………………………………………………3

Theoretical part of the work

Finding the basic Pythagorean triangle

(formulas of the ancient Hindus)…………………………………………………………………4

Practical part of the work

Composing Pythagorean triples in various ways……………………........ 6

An important property of Pythagorean triangles………………………………………...8

Conclusion………………………………………………………………………………....9

Literature………………………………………………………………………………...10

Introduction

This academic year, in mathematics lessons, we studied one of the most popular theorems in geometry - the Pythagorean theorem. The Pythagorean theorem is applied in geometry at every step, it has found wide application in practice and everyday life. But, besides the theorem itself, we also studied the theorem inverse to the Pythagorean theorem. In connection with the study of this theorem, we have become acquainted with Pythagorean triples of numbers, i.e. with sets of 3 natural numbersa , b andc , for which the relation is valid: = + . Such sets include, for example, the following triplets:

3,4,5; 5,12,13; 7,24,25; 8,15,17; 20,21,29; 9,40,41; 12,35,37

I immediately had questions: how many Pythagorean triples can you come up with? And how to compose them?

In our geometry textbook, after presenting the theorem converse to the Pythagorean theorem, an important remark was made: it can be proved that the legsa andb and hypotenusewith right-angled triangles, the lengths of whose sides are expressed in natural numbers, can be found by the formulas:

a = 2km b = k( - )c = k( + , (1)

wherek , m , n are any natural numbers, andm > n .

Naturally, the question arises - how to prove these formulas? And is it only by these formulas that Pythagorean triples can be formed?

In my work, I have attempted to answer the questions that have arisen in my mind.

Theoretical part of the work

Finding the main Pythagorean triangle (formulas of the ancient Hindus)

Let us first prove formulas (1):

Let us denote the lengths of the legs throughX andat , and the length of the hypotenuse throughz . By the Pythagorean theorem, we have the equality:+ = .(2)

This equation is called the Pythagorean equation. The study of Pythagorean triangles is reduced to solving equation (2) in natural numbers.

If each side of some Pythagorean triangle is increased by the same number of times, then we get a new right-angled triangle similar to the given one with sides expressed in natural numbers, i.e. again the Pythagorean triangle.

Among all similar triangles, there is the smallest one, it is easy to guess that this will be a triangle whose sidesX andat expressed in coprime numbers

(gcd (x,y )=1).

We call such a Pythagorean trianglemain .

Finding the main Pythagorean triangles.

Let the triangle (x , y , z ) is the main Pythagorean triangle. NumbersX andat are coprime, and therefore cannot both be even. Let us prove that they cannot both be odd. For this, we note thatThe square of an odd number when divided by 8 gives a remainder of 1. Indeed, any odd natural number can be represented as2 k -1 , wherek belongsN .

From here: = -4 k +1 = 4 k ( k -1)+1.

Numbers( k -1) andk are consecutive, one of them must be even. Then the expressionk ( k -1) divided by2 , 4 k ( k -1) is divisible by 8, which means when divided by 8, the remainder is 1.

The sum of the squares of two odd numbers gives a remainder of 2 when divided by 8, therefore, the sum of the squares of two odd numbers is an even number, but not a multiple of 4, and therefore this numbercannot be the square of a natural number.

So equality (2) cannot hold ifx andat both are odd.

Thus, if the Pythagorean triangle (x, y, z ) - the main one, then among the numbersX andat one must be even and the other must be odd. Let the number y be even. NumbersX andz odd (oddz follows from equality (2)).

From the equation+ = we get that= ( z + x )( z - x ) (3).

Numbersz + x andz - x as the sum and difference of two odd numbers are even numbers, and therefore (4):

z + x = 2 a , z - x = 2 b , wherea andb belongN .

z + x =2 a , z - x = 2 b ,

z = a+b , x = a - b. (5)

From these equalities it follows thata andb are relatively prime numbers.

We prove this by arguing from the contrary.

Let GCD (a , b )= d , whered >1 .

Thend z andx , and hence the numbersz + x andz - x . Then, based on equality (3) would be a divisor . In this cased would be a common divisor of numbersat andX , but the numbersat andX must be coprime.

Numberat is known to be even, soy = 2s , wherewith - natural number. Equality (3) based on equality (4) takes the following form: =2a*2 b , or =ab.

It is known from arithmetic thatif the product of two coprime numbers is the square of a natural number, then each of those numbers is also the square of a natural number.

Means,a = andb = , wherem andn are coprime numbers, because they are divisors of coprime numbersa andb .

Based on equality (5) we have:

z = + , x = - , = ab = * = ; c = mn

Theny = 2 mn .

Numbersm andn , because are coprime, cannot be even at the same time. But they cannot be odd at the same time, because in this casex = - would be even, which is impossible. So one of the numbersm orn is even and the other is odd. Obviously,y = 2 mn is divisible by 4. Therefore, in every main Pythagorean triangle, at least one of the legs is divisible by 4. It follows that there are no Pythagorean triangles, all sides of which would be prime numbers.

The results obtained can be expressed as the following theorem:

All major triangles in whichat is an even number, are obtained from the formula

x = - , y =2 mn , z = + ( m > n ), wherem andn - all pairs of coprime numbers, one of which is even and the other odd (it doesn't matter which one). Every basic Pythagorean triple (x, y, z ), whereat – even, is determined uniquely in this way.

Numbersm andn cannot be both even or both odd, because in these cases

x = would be even, which is impossible. So one of the numbersm orn even and the other oddy = 2 mn divisible by 4).

Practical part of the work

Composing Pythagorean triples in various ways

In Hindu formulasm andn - coprime, but can be numbers of arbitrary parity and it is quite difficult to make Pythagorean triples using them. Therefore, let's try to find a different approach to compiling Pythagorean triples.

= - = ( z - y )( z + y ), whereX - odd,y - even,z – odd

v = z - y , u = z + y

= UV , whereu - odd,v – odd (coprime)

Because the product of two odd coprime numbers is the square of a natural number, thenu = , v = , wherek andl are coprime, odd numbers.

z - y = z + y = k 2 , whence, adding the equalities and subtracting from one another, we get:

2 z = + 2 y = - i.e

z= y= x = cl

k

l

x

y

z

37

9

1

9

40

41 (szeros)*(100…0 (szeros) +1)+1 =200…0 (s-1zeros) 200…0 (s-1zeros) 1

An important property of Pythagorean triangles

Theorem

In the main Pythagorean triangle, one of the legs is necessarily divisible by 4, one of the legs is necessarily divisible by 3, and the area of ​​the Pythagorean triangle is necessarily a multiple of 6.

Proof

As we know, in any Pythagorean triangle at least one of the legs is divisible by 4.

Let us prove that one of the legs is also divisible by 3.

To prove this, suppose that in the Pythagorean triangle (x , y , z x ory multiple of 3.

Now we prove that the area of ​​the Pythagorean triangle is divisible by 6.

Any Pythagorean triangle has an area expressed as a natural multiple of 6. This follows from the fact that at least one of the legs is divisible by 3 and at least one of the legs is divisible by 4. The area of ​​the triangle, determined by the half-product of the legs, must be expressed by a multiple of 6 .

Conclusion

In work

- proven formulas of the ancient Hindus

- conducted a study on the number of Pythagorean triples (there are infinitely many of them)

- methods for finding Pythagorean triples are indicated

- Studied some properties of Pythagorean triangles

For me it was a very interesting topic and finding answers to my questions became a very interesting activity. In the future, I plan to consider the connection of Pythagorean triples with the Fibonacci sequence and Fermat's theorem and learn many more properties of Pythagorean triangles.

Literature

L.S. Atanasyan "Geometry. 7-9 grades" M .: Education, 2012.

V. Serpinsky “Pythagorean triangles” M.: Uchpedgiz, 1959.

Saratov

2014

Properties

Since the equation x 2 + y 2 = z 2 homogeneous, when multiplied x , y and z for the same number you get another Pythagorean triple. The Pythagorean triple is called primitive, if it cannot be obtained in this way, that is - relatively prime numbers.

Examples

Some Pythagorean triples (sorted in ascending order of maximum number, primitive ones are highlighted):

(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)…

Based on the properties of Fibonacci numbers, you can make them, for example, such Pythagorean triples:

.

Story

Pythagorean triples have been known for a very long time. In the architecture of ancient Mesopotamian tombstones, an isosceles triangle is found, made up of two rectangular ones with sides of 9, 12 and 15 cubits. The pyramids of Pharaoh Snefru (XXVII century BC) were built using triangles with sides of 20, 21 and 29, as well as 18, 24 and 30 tens of Egyptian cubits.

see also

Links

• E. A. Gorin Powers of prime numbers in Pythagorean triples // Mathematical education. - 2008. - V. 12. - S. 105-125.

Wikimedia Foundation. 2010 .

See what "Pythagorean numbers" are in other dictionaries:

Triples of natural numbers such that a triangle whose side lengths are proportional (or equal) to these numbers is right-angled, e.g. triple of numbers: 3, 4, 5… Big Encyclopedic Dictionary

Triples of natural numbers such that a triangle whose side lengths are proportional (or equal) to these numbers is rectangular, for example, a triple of numbers: 3, 4, 5. * * * PYTHAGORAN NUMBERS PYTHAGORAN NUMBERS, triples of natural numbers such that ... ... encyclopedic Dictionary

Triples of natural numbers such that a triangle whose side lengths are proportional (or equal) to these numbers is a right triangle. According to the theorem, the inverse of the Pythagorean theorem (see Pythagorean theorem), for this it is enough that they ... ...

Triplets of positive integers x, y, z satisfying the equation x2+y 2=z2. All solutions of this equation, and consequently, all P. p., are expressed by the formulas x=a 2 b2, y=2ab, z=a2+b2, where a, b are arbitrary positive integers (a>b). P. h ... Mathematical Encyclopedia

Triples of natural numbers such that a triangle, the lengths of the sides to which are proportional (or equal) to these numbers, is rectangular, for example. triple of numbers: 3, 4, 5… Natural science. encyclopedic Dictionary

In mathematics, Pythagorean numbers (Pythagorean triple) is a tuple of three integers that satisfy the Pythagorean relation: x2 + y2 = z2. Contents 1 Properties 2 Examples ... Wikipedia

Curly numbers are the general name of numbers associated with a particular geometric figure. This historical concept goes back to the Pythagoreans. Presumably, the expression “Square or cube” arose from curly numbers. Contents ... ... Wikipedia

Curly numbers are the general name of numbers associated with a particular geometric figure. This historical concept goes back to the Pythagoreans. There are the following types of curly numbers: Linear numbers are numbers that do not decompose into factors, that is, their ... ... Wikipedia

- The “pi paradox” is a joke on the topic of mathematics, which was in circulation among students until the 80s (in fact, before the mass distribution of microcalculators) and was associated with the limited accuracy of calculating trigonometric functions and ... ... Wikipedia

- (Greek arithmetika, from arithmys number) the science of numbers, primarily of natural (positive integer) numbers and (rational) fractions, and operations on them. Possession of a sufficiently developed concept of a natural number and the ability to ... ... Great Soviet Encyclopedia

Books

• Archimedean summer, or the history of the community of young mathematicians. Binary number system, Bobrov Sergey Pavlovich. Binary number system, "Tower of Hanoi", knight's move, magic squares, arithmetic triangle, curly numbers, combinations, concept of probabilities, Möbius strip and Klein bottle.…