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 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.)

Beskrovny I.M. 1

1 OAO Angstrem-M

The aim of the work is to develop methods and algorithms for calculating Pythagorean triples of the form a2+b2=c2. The analysis process was carried out in accordance with the principles of a systematic approach. Along with mathematical models, graphical models are used that display each member of the Pythagorean triple in the form of composite squares, each of which consists of a set of unit squares. It has been established that an infinite set of Pythagorean triples contains an infinite number of subsets that distinguish by the difference between the values ​​b–c. An algorithm for the formation of Pythagorean triples with any predetermined value of this difference is proposed. It is shown that Pythagorean triples exist for any value 3≤a

Pythagorean triplets

system analysis

mathematical model

graphic model

1. Anosov D.N. A look at mathematics and something from it. - M.: MTSNMO, 2003. - 24 p.: ill.

2. Ayerland K., Rosen M. Classical introduction to modern number theory. – M.: Mir, 1987.

3. Beskrovny I.M. System Analysis and Information Technology in Organizations: Textbook. - M.: RUDN, 2012. - 392 p.

4. Simon Singh. Fermat's Last Theorem.

5. Ferma P. Studies in Number Theory and Diophantine Analysis. – M.: Nauka, 1992.

6. Yaptro. Ucoz, Available at: http://yaptro.ucoz.org/news/pifagorovy_trojki_chisel/2012-05-07-5.

Pythagorean triples are a cohort of three integers that satisfy the Pythagorean relation x2 + y2 = z2. Generally speaking, this is a special case of Diophantine equations, namely, systems of equations in which the number of unknowns is greater than the number of equations. They have been known for a long time, since the time of Babylon, that is, long before Pythagoras. And they acquired the name after Pythagoras proved his famous theorem on their basis. However, as follows from the analysis of numerous sources in which the question of Pythagorean triples is touched upon in one way or another, the question of the existing classes of these triples and the possible ways of their formation has not yet been fully disclosed.

So in the book of Simon Singh it says: - "The disciples and followers of Pythagoras ... told the world the secret of finding the so-called Pythagorean three k." However, following this we read: - “The Pythagoreans dreamed of finding other Pythagorean triples, other squares, from which it would be possible to add a third large square. …As the numbers increase, Pythagorean triples are becoming rarer and harder and harder to find. The Pythagoreans invented a method for finding such triplets and, using it, proved that there are infinitely many Pythagorean triplets.

Words that cause confusion are highlighted in the quote. Why "the Pythagoreans dreamed of finding ..." if they "invented a method for finding such triples ...", and why for large numbers "it becomes more and more difficult to find them ...".

In the work of the famous mathematician D.V. Anosov, the desired answer seems to be given. - “There are such triples of natural (i.e. positive integer) numbers x, y, z that

x2 + y2 = z2. (1)

…is it possible to find all solutions of the equation x2+y2=z2 in natural numbers? …Yes. The answer is that each such solution can be represented as

x=l(m2-n2), y=2lmn, z=l(m2+n2), (2),

where l, m, n are natural numbers, and m>n, or in a similar form in which x and y are interchanged. We can say a little more briefly that x, y, z from (2) with all possible naturals l and m > n are all possible solutions of (1) up to a permutation of x and y. For example, the triple (3, 4, 5) is obtained with l=1, m=2, n=1. ... Apparently, the Babylonians knew this answer, but how they arrived at it is not known.”

Usually mathematicians are known for their exactingness in the rigor of their formulations. But, in this quote, such rigor is not observed. So what exactly: find or imagine? Obviously, these are completely different things. Here is a line of "freshly baked" triples (obtained by the method described below):

12, 35, 37; 20, 21, 29; 44, 117, 125; 103, 5304, 5305.

There is no doubt that each of these triples can be represented in the form of relation (2) and then the values ​​of l, m, n can be calculated. But, this is after all the values ​​of the triples have been found. But what about before that?

It cannot be ruled out that the answers to these questions have long been known. But for some reason, they have not yet been found. Thus, the purpose of this work is a systematic analysis of the totality of known examples of Pythagorean triples, the search for system-forming relations in various groups of triples and the identification of systemic features characteristic of these groups, and then the development of simple efficient algorithms for calculating triples with a predetermined configuration. By configuration, we mean the relationship between the quantities that make up the triple.

As a toolkit, a mathematical apparatus will be used at a level that does not go beyond the framework of mathematics taught in high school, and system analysis based on the methods outlined in.

Model building

From the standpoint of system analysis, any Pythagorean triple is a system formed by objects, which are three numbers and their properties. Their totality, in which objects are placed in certain relationships and form a system with new properties that are not inherent in either individual objects or any other of their totality, where objects are placed in other relationships.

In equation (1), the objects of the system are natural numbers related by simple algebraic relations: to the left of the equal sign is the sum of two numbers raised to the power of 2, to the right is the third number, also raised to the power of 2. Individual numbers, to the left of the equality, being raised to the power of 2, do not impose any restrictions on the operation of their summation - the resulting sum can be anything. But, the equal sign placed after the summation operation imposes a system restriction on the value of this sum: the sum must be such a number that the result of the operation of extracting the square root is a natural number. And this condition is not satisfied for any numbers substituted into the left side of the equality. Thus, the equal sign put between two terms of the equation and the third one turns the triple of terms into a system. A new feature of this system is the introduction of restrictions on the values ​​of the original numbers.

Based on the form of writing, the Pythagorean triple can be considered as a mathematical model of a geometric system consisting of three squares interconnected by summation and equality relations, as shown in Fig. 1. Fig. 1 is a graphical model of the system under consideration, and its verbal model is the statement:

The area of ​​a square with side length c can be divided without remainder into two squares with side lengths a and b, such that the sum of their areas is equal to the area of ​​the original square, that is, all three quantities a, b, and c are related by the relation

Graphical model of the decomposition of a square

Within the framework of the canons of system analysis, it is known that if a mathematical model adequately reflects the properties of a certain geometric system, then the analysis of the properties of this system itself allows us to clarify the properties of its mathematical model, to know them deeper, clarify, and, if necessary, improve. This is the path we will follow.

Let us clarify that, according to the principles of system analysis, addition and subtraction operations can only be performed on composite objects, that is, objects composed of a set of elementary objects. Therefore, we will perceive any square as a figure made up of a set of elementary or unit squares. Then the condition for obtaining a solution in natural numbers is equivalent to accepting the condition that the unit square is indivisible.

A unit square is a square whose length of each side is equal to one. That is, when the area of ​​a unit square determines the following expression.

The quantitative parameter of a square is its area, which is determined by the number of unit squares that can be placed on a given area. For a square with an arbitrary value of x, the expression x2 determines the area of ​​the square formed by segments of length x unit segments. x2 unit squares can be placed on the area of ​​this square.

The above definitions may be perceived as trivial and obvious, but they are not. D.N. Anosov defines the concept of area in a different way: - “... the area of ​​​​a figure is equal to the sum of the areas of its parts. Why are we sure that this is so? ... We imagine a figure made of some kind of homogeneous material, then its area is proportional to the amount of matter contained in it - its mass. It is further understood that when we divide a body into several parts, the sum of their masses is equal to the mass of the original body. This is understandable, because everything consists of atoms and molecules, and since their number has not changed, their total mass has not changed either ... After all, in fact, the mass of a piece of homogeneous material is proportional to its volume; therefore, you need to know that the volume of the "sheet" that has the shape of a given figure is proportional to its area. In a word, ... that the area of ​​a figure is equal to the sum of the areas of its parts, in geometry it is necessary to prove this. ... In Kiselev's textbook, the existence of an area that has the very property that we are discussing now was honestly postulated as some kind of assumption, and it was said that this was actually true, but we will not prove this. So the Pythagorean theorem, if it is proved with areas, in a purely logical sense, will remain not completely proved.

It seems to us that the definitions of the unit square introduced above remove the indicated D.N. Anosov uncertainty. After all, if the area of ​​a square and a rectangle is determined by the sum of the unit squares that fill them, then when the rectangle is divided into arbitrary adjacent parts, the area of ​​the rectangle is naturally equal to the sum of all its parts.

Moreover, the introduced definitions remove the uncertainty of using the concepts "divide" and "add" in relation to abstract geometric figures. Indeed, what does it mean to divide a rectangle or any other flat figure into parts? If it is a sheet of paper, then it can be cut with scissors. If the land - put a fence. Room - put a partition. What if it's a drawn square? Draw a dividing line and declare that the square is divided? But, after all, D.I. Mendeleev: "... You can declare everything, but you - go ahead, demonstrate!"

And using the proposed definitions, “Divide a figure” means to divide the number of unit squares filling this figure into two (or more) parts. The number of unit squares in each of these parts determines its area. The configuration of these parts can be given arbitrary, but the sum of their areas will always be equal to the area of ​​the original figure. Perhaps, mathematicians will consider these arguments incorrect, then we will take them as an assumption. If such assumptions are acceptable in Kiselyov's textbook, then it would be a sin for us not to use such a technique.

The first step in system analysis is to identify the problem situation. At the beginning of this stage, several hundred Pythagorean triples found in various sources were looked through. At the same time, attention was drawn to the fact that the entire set of Pythagorean triples mentioned in publications can be divided into several groups that differ in configuration. We will consider the difference in the lengths of the sides of the original and subtracted squares, that is, the value c-b, as a sign of a specific configuration. For example, in publications, triples that satisfy the condition c-b=1 are often shown as an example. We assume that the entire set of such Pythagorean triples forms a set, which we will call "Class c-1", and we will analyze the properties of this class.

Consider the three squares shown in the figure, where c is the length of the side of the square to be reduced, b is the length of the side of the square to be subtracted, and a is the length of the side of the square formed from their difference. On fig. 1 it can be seen that when subtracting the area of ​​the subtracted square from the area of ​​the reduced square, two bands of unit squares remain in the remainder:

In order to form a square from this remainder, the condition must be satisfied

These relations allow us to determine the values ​​of all members of the triple by a single given number c. The smallest number c that satisfies relation (6) is c = 5. Thus, the lengths of all three sides of squares satisfying relation (1) were determined. Recall that the value b of the side of the mean square

was chosen when we decided to form a middle square by reducing the side of the original square by one. Then from relations (5), (6). (7) we obtain the following relation:

from which it follows that the chosen value c = 5 uniquely determines the values ​​b = 4, a = 3.

As a result, relations are obtained that allow representing any Pythagorean triple of the class "c - 1" in such a form, where the values ​​of all three members are determined by one specified parameter - the value c:

We add that the number 5 in the above example appeared as the minimum of all possible values ​​of c for which equation (6) has a solution in natural numbers. The next number that has the same property is 13, then 25, then 41, 61, 85, etc. As you can see, in this series of numbers, the intervals between adjacent numbers increase rapidly. So, for example, after a valid value , the next valid value is , and after , the next valid value is , that is, the valid value is more than fifty million from the previous one!

Now it’s clear where this phrase came from in the book: - “As the numbers increase, Pythagorean triples are less and less common, and it becomes more and more difficult to find them ...”. However, this statement is not correct. One has only to look at the Pythagorean triples corresponding to the above pairs of neighboring values ​​of c, as one feature immediately catches the eye - in both pairs, in which the values ​​of c are separated by such large intervals, the values ​​of a turn out to be neighboring odd numbers. Indeed, for the first pair we have

and for the second pair

So it’s not the triples themselves that are “less and less common”, but the intervals between neighboring values ​​of c are increasing. The Pythagorean triples themselves, as will be shown below, exist for any natural number.

Now consider the triples of the next class - "Class c-2". As can be seen from fig. 1, when subtracting from a square with side c a square with side (c - 2), the remainder is the sum of two unit bands. The value of this sum is determined by the equation:

From equation (10) we obtain a relationship that defines any of the infinite set of triples class "c-2":

The condition for the existence of a solution to equation (11) in natural numbers is any such value c for which a is a natural number. The minimum value of c for which a solution exists is c = 5. Then the “starting” triple for this class of triples is determined by the set a = 4, b = 3, c = 5. That is, again, the classical triple 3, 4, 5 is formed , only now the area of ​​the square to be subtracted is less than the area of ​​the remainder.

And finally, let's analyze the triples of the "s-8" class. For this class of triples, subtracting the area of ​​the square from the area c2 of the original square, we get:

Then, from equation (12) it follows:

The minimum value of c for which the solution exists is c = 13. The Pythagorean triple at this value will take the form 12, 5, 13. In this case, again, the area of ​​the square to be subtracted is less than the area of ​​the remainder. And rearranging the designations in places, we get the triple 5, 12, 13, which by its configuration belongs to the class "c - 1". It seems that further analysis of other possible configurations will not reveal anything fundamentally new.

Derivation of calculated ratios

In the previous section, the logic of analysis was developed in accordance with the requirements of system analysis in four of its five main stages: analysis of the problem situation, formation of goals, formation of functions and formation of structure. Now it's time to move on to the final, fifth stage - the test of feasibility, that is, the test of the extent to which the goals are achieved. .

Table 1 is shown below. 1, which shows the values ​​of Pythagorean triples belonging to the class "c - 1". Most triples are found in various publications, but triples for a values ​​equal to 999, 1001 have not been found in known publications.

Table 1

Pythagorean triples of class "c-1"

One can check that all triples satisfy relation (3). Thus, one of the goals set has been achieved. Relations (9), (11), (13) obtained in the previous section make it possible to form an infinite set of triples by setting the only parameter c, the side of the reduced square. This, of course, is a more constructive option than relation (2), for the use of which one should set arbitrarily three numbers l, m, n, having any value, then look for a solution, knowing only that in the end, a Pythagorean triple will certainly be obtained, and which one is unknown. In our case, the configuration of the triple being formed is known in advance and only one parameter needs to be set. But, alas, not every value of this parameter has a solution. And you need to know in advance its allowable values. So the result is good, but far from ideal. It is desirable to obtain such a solution that Pythagorean triples can be calculated for any arbitrarily given natural number. To this end, let us return to the fourth stage - the formation of the structure of the obtained mathematical relations.

Since the choice of the value c as the basic parameter for determining the remaining members of the triple turned out to be inconvenient, another option should be tried. As can be seen from Table. 1, the choice of the parameter a as the base one seems to be preferable, since the values ​​of this parameter are in a row in a series of odd natural numbers. After simple transformations, we bring relations (9) to a more constructive form:

Relations (14) allow us to find a Pythagorean triple for any preassigned odd value a. At the same time, the simplicity of the expression for b allows you to perform calculations even without a calculator. Indeed, choosing, for example, the number 13, we get:

And for the number 99, respectively, we get:

Relations (15) allow obtaining the values ​​of all three terms of the Pythagorean string for any given n, starting from n=1.

Now consider the Pythagorean triples of the class "c - 2". In table. 2 shows ten such triples as an example. Moreover, only three pairs of triples were found in known publications - 8, 15, 23; 12, 35, 36; and 16, 63, 65. This turned out to be enough to determine the patterns by which they are formed. The remaining seven were found from previously derived relations (11). For convenience of calculation, these ratios were transformed so that all parameters are expressed in terms of a. From (11) it obviously follows that all triples for the class "c - 2" satisfy the following relations:

table 2

Pythagorean triples of class "c-2"

As can be seen from Table. 2, the entire infinite set of triples of class "c - 2" can be divided into two subclasses. For triples where the value of a is divisible by 4 without a remainder, the values ​​of b and c are odd. Such triples, for which GCD = 1, are called primitive. For triples whose values ​​a are not divisible by 4 in integers, all three members of the triple a, b, c are even.

Now let's move on to reviewing the results of the analysis of the third of the selected classes - the class "c - 8". The calculated relations for this class, obtained from (13), have the form:

Relations (20), (21) are essentially identical. The difference is only in the choice of the sequence of actions. Or, in accordance with (20), the desired value of a is selected (in this case, this value is required to be divided by 4), then the values ​​of b and c are determined. Or, an arbitrary number is chosen, and then, from relations (21), all three members of the Pythagorean triple are determined. In table. 3 shows a number of Pythagorean triples calculated in this way. However, calculating the values ​​of Pythagorean triples is even easier. If at least one value is known, then all subsequent values ​​\u200b\u200bare determined very simply by the following relationships:

Table 3

The validity of relation (22) for all can be verified both by triples from Table. 2, as well as from other sources. As an example, in Table. 4 italicized triples from an extensive table of Pythagorean triples (10000 triples) calculated on the basis of a computer program by relation (2) and in bold type - triples calculated by relation (20). These values ​​were not in the specified table.

Table 4

Pythagorean triples of class "s-8"

Accordingly, for triples of the form, the following relations can be used:

And for triplets of the form<>, we have the ratio:

It should be emphasized that the above classes of triples "c - 1", "c - 2", "c - 8" make up more than 90% of the first thousand triples from the table given in. This gives reason to consider these classes as base. Let us add that when deriving relations (22), (23), (24), no special properties of numbers studied in number theory (prime, coprime, etc.) were used. The revealed regularities in the formation of Pythagorean triples are due only to the system properties of the geometric figures described by these triples - squares, consisting of a set of unit squares.

Conclusion

Now, as Andrew Wiles said in 1993, "I think I should stop there." The goal set has been fully achieved. It is shown that the analysis of the properties of mathematical models, the structure of which is associated with geometric figures, is greatly simplified if, in the process of analysis, along with purely mathematical calculations, the geometric properties of the models under study are also taken into account. Simplification is achieved, in particular, due to the fact that the researcher "sees" the desired results without performing mathematical transformations.

For example, equality

becomes obvious without transformations on its left side, one has only to look at fig. 1 for a graphical model of this equality.

As a result, on the basis of the analysis performed, it is shown that for any square with a side, squares with sides b and c can be found such that equality holds for them and relations are obtained that provide results with a minimum amount of calculations:

for odd values ​​a,

and - for even values.

Bibliographic link

Beskrovny I.M. SYSTEM ANALYSIS OF THE PROPERTIES OF PYTHAGOREAN TRIPLES // Modern science-intensive technologies. - 2013. - No. 11. - P. 135-142;
URL: http://site/ru/article/view?id=33537 (date of access: 03/20/2020). We bring to your attention the journals published by the publishing house "Academy of Natural History"

» 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 appear 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 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

An important example of a Diophantine equation is given by the Pythagorean theorem, which relates the lengths x and y of the legs of a right triangle to the length z of its hypotenuse:


Of course, you have come across one of the wonderful solutions of this equation in natural numbers, namely the Pythagorean triple of numbers x=3, y=4, z=5. Are there any other triplets?

It turns out that there are infinitely many Pythagorean triples, and all of them were found a long time ago. They can be obtained by well-known formulas, which you will learn about from this paragraph.

If Diophantine equations of the first and second degree have already been solved, then the question of solving equations of higher degrees still remains open, despite the efforts of leading mathematicians. At present, for example, Fermat's famous conjecture that for any integer value n2 the equation


has no solutions in integers.

For solving certain types of Diophantine equations, the so-called complex numbers. What it is? Let the letter i denote some object that satisfies the condition i 2 \u003d -1(it is clear that no real number satisfies this condition). Consider expressions of the form α+iβ, where α and β are real numbers. Such expressions will be called complex numbers, having defined the operations of addition and multiplication over them, as well as over binomials, but with the only difference that the expression i 2 everywhere we will replace the number -1:

7.1. Many of the three

Prove that if x0, y0, z0- Pythagorean triple, then triples y 0 , x 0 , z 0 And x 0 k, y 0 k, z 0 k for any value of the natural parameter k are also Pythagorean.

7.2. Private formulas

Check that for any natural values m>n trinity of the form

is Pythagorean. Is it any Pythagorean triple x, y, z can be represented in this form, if you allow to rearrange the numbers x and y in the triple?

7.3. Irreducible triplets

A Pythagorean triple of numbers that do not have a common divisor greater than 1 will be called irreducible. Prove that a Pythagorean triple is irreducible only if any two of the numbers in the triple are coprime.

7.4. Property of irreducible triples

Prove that in any irreducible Pythagorean triple x, y, z the number z and exactly one of the numbers x or y are odd.

7.5. All irreducible triples

Prove that a triple of numbers x, y, z is an irreducible Pythagorean triple if and only if it coincides with the triple up to the order of the first two numbers 2mn, m 2 - n 2, m 2 + n 2, Where m>n- coprime natural numbers of different parity.

7.6. General formulas

Prove that all solutions of the equation


in natural numbers are given up to the order of the unknown x and y by the formulas

where m>n and k are natural parameters (in order to avoid duplication of any triples, it is enough to choose numbers of type coprime and, moreover, of different parity).

7.7. First 10 triplets

Find all Pythagorean triples x, y, z satisfying the condition x

7.8. Properties of Pythagorean triplets

Prove that for any Pythagorean triple x, y, z statements are true:

a) at least one of the numbers x or y is a multiple of 3;

b) at least one of the numbers x or y is a multiple of 4;

c) at least one of the numbers x, y or z is a multiple of 5.

7.9. Application of complex numbers

The modulus of a complex number α + iβ called a non-negative number

Check that for any complex numbers α + iβ And γ + iδ property is executed

Using the properties of complex numbers and their moduli, prove that any two integers m and n satisfy the equality

i.e., they give a solution to the equation


integers (compare with Problem 7.5).

7.10. Non-Pythagorean triples

Using the properties of complex numbers and their moduli (see Problem 7.9), find formulas for any integer solutions of the equation:

a) x 2 + y 2 \u003d z 3; b) x 2 + y 2 \u003d z 4.

Solutions


7.1. If x 0 2 + y 0 2 = z 0 2 , That y 0 2 + x 0 2 = z 0 2 , and for any natural value of k we have

Q.E.D.

7.2. From equalities

we conclude that the triple indicated in the problem satisfies the equation x 2 + y 2 = z 2 in natural numbers. However, not every Pythagorean triple x, y, z can be represented in this form; for example, the triple 9, 12, 15 is Pythagorean, but the number 15 cannot be represented as the sum of the squares of any two natural numbers m and n.

7.3. If any two numbers from the Pythagorean triple x, y, z have a common divisor d, then it will also be a divisor of the third number (so, in the case x = x 1 d, y = y 1 d we have z 2 \u003d x 2 + y 2 \u003d (x 1 2 + y 1 2) d 2, whence z 2 is divisible by d 2 and z is divisible by d). Therefore, for a Pythagorean triple to be irreducible, it is necessary that any two of the numbers in the triple be coprime,

7.4. Note that one of the numbers x or y, say x, of an irreducible Pythagorean triple x, y, z is odd because otherwise the numbers x and y would not be coprime (see problem 7.3). If the other number y is also odd, then both numbers

give a remainder of 1 when divided by 4, and the number z 2 \u003d x 2 + y 2 gives a remainder of 2 when divided by 4, that is, it is divisible by 2, but not divisible by 4, which cannot be. Thus, the number y must be even, and the number z must therefore be odd.

7.5. Let the Pythagorean triple x, y, z is irreducible and, for definiteness, the number x is even, while the numbers y, z are odd (see Problem 7.4). Then

where are the numbers are whole. Let us prove that the numbers a and b are coprime. Indeed, if they had a common divisor greater than 1, then the numbers would have the same divisor z = a + b, y = a - b, i.e., the triple would not be irreducible (see Problem 7.3). Now, expanding the numbers a and b into products of prime factors, we notice that any prime factor must be included in the product 4ab = x2 only to an even degree, and if it is included in the expansion of the number a, then it is not included in the expansion of the number b and vice versa. Therefore, any prime factor is included in the expansion of the number a or b separately only to an even degree, which means that these numbers themselves are squares of integers. Let's put then we get the equalities

moreover, the natural parameters m>n are coprime (due to the coprimeness of the numbers a and b) and have different parity (due to the odd number z \u003d m 2 + n 2).

Let now natural numbers m>n of different parity be coprime. Then the troika x \u003d 2mn, y \u003d m 2 - n 2, z \u003d m 2 + n 2, according to Problem 7.2, is Pythagorean. Let us prove that it is irreducible. To do this, it suffices to check that the numbers y and z do not have common divisors (see Problem 7.3). In fact, both of these numbers are odd, since the type numbers have different parities. If the numbers y and z have some simple common divisor (then it must be odd), then each of the numbers and and with them each of the numbers m and n has the same divisor, which contradicts their mutual simplicity.

7.6. By virtue of the assertions formulated in Problems 7.1 and 7.2, these formulas define only Pythagorean triples. On the other hand, any Pythagorean triple x, y, z after its reduction by the greatest common divisor k, the pair of numbers x and y becomes irreducible (see Problem 7.3) and, therefore, can be represented up to the order of the numbers x and y in the form described in Problem 7.5. Therefore, any Pythagorean triple is given by the indicated formulas for some values ​​of the parameters.

7.7. From inequality z and the formulas of Problem 7.6, we obtain the estimate m 2 i.e. m≤5. Assuming m = 2, n = 1 And k = 1, 2, 3, 4, 5, we get triplets 3, 4, 5; 6, 8, 10; 9, 12, 15; 12,16,20; 15, 20, 25. Assuming m=3, n=2 And k = 1, 2, we get triplets 5, 12, 13; 10, 24, 26. Assuming m = 4, n = 1, 3 And k = 1, we get triplets 8, 15, 17; 7, 24, 25. Finally, assuming m=5, n=2 And k = 1, we get three 20, 21, 29.

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.

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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.…