Pythagorean numbers. Pythagorean triples of numbers (Creative student work) All primitive Pythagorean triples up to 200

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.)

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)…

History

Pythagorean triplets 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.

X All-Russian Symposium on Applied and Industrial Mathematics. St. Petersburg, May 19, 2009

Report: Algorithm for solving Diophantine equations.

The paper considers the method of studying Diophantine equations and presents the solutions solved by this method: - grand theorem Farm; - search for Pythagorean triplets, etc. http://referats.protoplex.ru/referats_show/6954.html

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 triples" are in other dictionaries:

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

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 a right triangle. According to the theorem, the inverse of the Pythagorean theorem (see Pythagorean theorem), for this it is enough that they ... ... Great Soviet Encyclopedia

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

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

In mathematics, a Pythagorean triple is a tuple of three natural numbers satisfying the Pythagorean relation: In this case, the numbers that form a Pythagorean triple are called Pythagorean numbers. Contents 1 Primitive triples ... Wikipedia

The Pythagorean theorem is one of the fundamental theorems of Euclidean geometry, establishing the relationship between the sides of a right triangle. Contents 1 ... Wikipedia

The Pythagorean theorem is one of the fundamental theorems of Euclidean geometry, establishing the relationship between the sides of a right triangle. Contents 1 Statements 2 Proofs ... Wikipedia

This is an equation of the form where P is an integer function (for example, a polynomial with integer coefficients), and the variables take integer values. Named after the ancient Greek mathematician Diophantus. Contents 1 Examples ... Wikipedia

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:

History

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 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.…

Next, we consider the well-known methods for generating effective Pythagorean triples. The students of Pythagoras were the first to invent a simple way to generate Pythagorean triples, using a formula whose parts represent a Pythagorean triple:

m 2 + ((m 2 − 1)/2) 2 = ((m 2 + 1)/2) 2 ,

Where m- unpaired, m>2. Really,

4m 2 + m 4 − 2m 2 + 1
m 2 + ((m 2 − 1)/2) 2 = ————————— = ((m 2 + 1)/2) 2 .
4

A similar formula was proposed by the ancient Greek philosopher Plato:

(2m) 2 + (m 2 − 1) 2 = (m 2 + 1) 2 ,

Where m- any 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 you can see, these formulas cannot give all possible primitive triples.

Consider the following polynomial, which is decomposed into a 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 .

Hence the following formulas for obtaining primitive triples:

a = 2m +1 , b = 2m(m+1) = 2m 2 + 2m , c = 2m 2 + 2m + 1.

These formulas generate triples in which the average number differs from the largest by exactly one, that is, not all possible triples are also generated. Here the first triples are: (5,12,13), (7,24,25), (9,40,41), (11,60,61).

To determine how to generate all primitive triples, one must examine their properties. First, if ( a,b,c) is a primitive triple, then a And b, b And c, but And c— must be coprime. Let be a And b are divided into d. Then a 2 + b 2 is also divisible by d. Respectively, c 2 and c should be divided into d. That is, it is not a primitive triple.

Secondly, among the numbers a, b one must be paired and the other unpaired. Indeed, if a And b- paired, then from will be paired, and the numbers can be divided by at least 2. If they are both unpaired, then they can be represented as 2 k+1 i 2 l+1, where k,l- some numbers. Then a 2 + b 2 = 4k 2 +4k+1+4l 2 +4l+1, that is, from 2 , as well as a 2 + b 2 has a remainder of 2 when divided by 4.

Let be from- any number, that is from = 4k+i (i=0,…,3). Then from 2 = (4k+i) 2 has a remainder of 0 or 1 and cannot have a remainder of 2. Thus, a And b cannot be unpaired, that is a 2 + b 2 = 4k 2 +4k+4l 2 +4l+1 and remainder from 2 by 4 should be 1, which means that from should be unpaired.

Such requirements for the elements of the Pythagorean triple are satisfied by the following numbers:

a = 2mn, b = m 2 − n 2 , c = m 2 + n 2 , m > n, (2)

Where m And n are coprime with different pairings. For the first time, these dependencies became known from the works of Euclid, who lived 2300 r. back.

Let us prove the validity of dependencies (2). Let be but- double, then b And c- unpaired. Then c + b i c − b- couples. They can be represented as c + b = 2u And c − b = 2v, where u,v are some integers. That's why

a 2 = from 2 − b 2 = (c + b)(c − b) = 2u 2 v = 4UV

And therefore ( a/2) 2 = UV.

It can be proven by contradiction that u And v are coprime. Let be u And v- are divided into d. Then ( c + b) And ( c − b) are divided into d. And therefore c And b should be divided into d, and this contradicts the condition for the Pythagorean triple.

Because UV = (a/2) 2 and u And v coprime, it is easy to prove that u And v must be squares of some numbers.

So there are positive integers m And n, such that u = m 2 and v = n 2. Then

but 2 = 4UV = 4m 2 n 2 so
but = 2mn; b = u − v = m 2 − n 2 ; c = u + v = m 2 + n 2 .

Because b> 0, then m > n.

It remains to show that m And n have different pairings. If m And n- paired, then u And v must be paired, but this is impossible, since they are coprime. If m And n- unpaired, then b = m 2 − n 2 and c = m 2 + n 2 would be paired, which is impossible because c And b are coprime.

Thus, any primitive Pythagorean triple must satisfy conditions (2). At the same time, the numbers m And n called generating numbers primitive triplets. For example, let's have a primitive Pythagorean triple (120,119,169). In this case

but= 120 = 2 12 5, b= 119 = 144 − 25, and c = 144+25=169,

Where m = 12, n= 5 - generating numbers, 12 > 5; 12 and 5 are coprime and of different pairings.

It can be proved that the numbers m, n formulas (2) give a primitive Pythagorean triple (a,b,c). Really,

but 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 ,

I.e ( a,b,c) is a Pythagorean triple. Let us prove that while a,b,c are coprime numbers by contradiction. Let these numbers be divided by p> 1. Since m And n have different pairings, then b And c- unpaired, that is p≠ 2. Since R divides b And c, then R must divide 2 m 2 and 2 n 2 , which is impossible because p≠ 2. Therefore m, n are coprime and a,b,c are also coprime.

Table 1 shows all primitive Pythagorean triples generated by formulas (2) for m≤10.

Table 1. Primitive Pythagorean triples for m≤10

Analysis of this table shows the presence of the following series of patterns:

• or a, or b are divided by 3;
• one of the numbers a,b,c is divisible by 5;
• number but is divisible by 4;
• work a· b is divisible by 12.

In 1971, the American mathematicians Teigan and Hedwin proposed such little-known parameters of a right-angled triangle as its height (height) to generate triplets h = c− b and excess (success) e = a + b − c. In Fig.1. these quantities are shown on a certain right triangle.

Figure 1. Right triangle and its growth and excess

The name “excess” is derived from the fact that this is the additional distance that must be traveled along the legs of the triangle from one vertex to the opposite, if you do not go along its diagonal.

Through excess and growth, the sides of the Pythagorean triangle can be expressed as:

e 2 e 2
a = h + e, b = e + ——, c = h + e + ——, (3)
2h 2h

Not all combinations h And e may correspond to Pythagorean triangles. For a given h possible values e is the product of some number d. This number d is called growth and refers to h in the following way: d is the smallest positive integer whose square is divisible by 2 h. Because e multiple d, then it is written as e = kd, where k is a positive integer.

With the help of pairs ( k,h) you can generate all Pythagorean triangles, including non-primitive and generalized, as follows:

(dk) 2 (dk) 2
a = h + dk, b = dk + ——, c = h + dk + ——, (4)
2h 2h

Moreover, a triple is primitive if k And h are coprime and if h =± q 2 at q- unpaired.
Moreover, it will be exactly a Pythagorean triple if k> √2 h/d And h > 0.

To find k And h from ( a,b,c) do the following:

• h = c − b;
• write down h how h = pq 2 , where p> 0 and such that is not a square;
• d = 2pq if p- unpaired and d = pq, if p is paired;
• k = (a − h)/d.

For example, for the triple (8,15,17) we have h= 17−15 = 2 1, so p= 2 and q = 1, d= 2, and k= (8 − 2)/2 = 3. So this triple is given as ( k,h) = (3,2).

For the triple (459,1260,1341) we have h= 1341 − 1260 = 81, so p = 1, q= 9 and d= 18, hence k= (459 − 81)/18 = 21, so the code of this triple is ( k,h) = (21, 81).

Specifying triples with h And k has a number of interesting properties. Parameter k equals

k = 4S/(dP), (5)

Where S = ab/2 is the area of ​​the triangle, and P = a + b + c is its perimeter. This follows from the equality eP = 4S, which comes from the Pythagorean theorem.

For a right triangle e equals the diameter of the circle inscribed in the triangle. This comes from the fact that the hypotenuse from = (but − r)+(b − r) = a + b − 2r, where r is the radius of the circle. From here h = c − b = but − 2r And e = a − h = 2r.

For h> 0 and k > 0, k is the ordinal number of triplets a-b-c in a sequence of Pythagorean triangles with increasing h. From table 2, which shows several options for triplets generated by pairs h, k, it can be seen that with increasing k the sides of the triangle increase. Thus, unlike classical numbering, numbering in pairs h, k has a higher order in sequences of triplets.

Table 2. Pythagorean triples generated by pairs h, k.

For h > 0, d satisfies the inequality 2√ h ≤ d ≤ 2h, in which the lower bound is reached at p= 1, and the upper one, at q= 1. Therefore, the value d with respect to 2√ h is a measure of how much h far from the square of some number.

The study of the properties of natural numbers led the Pythagoreans to another “eternal” problem of theoretical arithmetic (number theory) - a problem whose germs made their way long before Pythagoras in Ancient Egypt and Ancient Babylon, and a general solution has not been found to this day. Let's start with the problem, which in modern terms can be formulated as follows: solve the indefinite equation in natural numbers

Today this task is called problem of Pythagoras, and its solutions - triples of natural numbers satisfying equation (1.2.1) - are called Pythagorean triplets. Due to the obvious connection of the Pythagorean theorem with the Pythagorean problem, the latter can be given a geometric formulation: find all right triangles with integer legs x, y and integer hypotenuse z.

Particular solutions of the Pythagorean problem were known in ancient times. In a papyrus from the time of Pharaoh Amenemhat I (c. 2000 BC), stored in the Egyptian Museum in Berlin, we find a right triangle with the ratio of sides (). According to the largest German historian of mathematics M. Kantor (1829 - 1920), in ancient Egypt there was a special profession harpedonapts- "rope tensioners", who, during the solemn ceremony of laying temples and pyramids, marked right angles with a rope having 12 (= 3 + 4 + 5) equally spaced knots. The method of constructing a right angle with harpedonapts is obvious from Figure 36.

It must be said that another connoisseur of ancient mathematics, van der Waerden, categorically disagrees with Cantor, although the very proportions of ancient Egyptian architecture testify in favor of Cantor. Be that as it may, today a right triangle with an aspect ratio is called Egyptian.

As noted on p. 76, a clay tablet dating back to the ancient Babylonian era and containing 15 lines of Pythagorean triplets has been preserved. In addition to the trivial triple obtained from the Egyptian (3, 4, 5) by multiplying by 15 (45, 60, 75), there are also very complex Pythagorean triples, such as (3367, 3456, 4825) and even (12709, 13500, 18541)! There is no doubt that these numbers were found not by simple enumeration, but by some uniform rules.

Nevertheless, the question of the general solution of equation (1.2.1) in natural numbers was raised and solved only by the Pythagoreans. The general formulation of any mathematical problem was alien to both the ancient Egyptians and the ancient Babylonians. Only with Pythagoras does the formation of mathematics as a deductive science begin, and one of the first steps along this path was the solution of the problem of Pythagorean triples. The ancient tradition associates the first solutions of equation (1.2.1) with the names of Pythagoras and Plato. Let's try to reconstruct these solutions.

It is clear that Pythagoras thought of equation (1.2.1) not in an analytical form, but in the form of a square number , inside which it was necessary to find the square numbers and . It was natural to represent the number in the form of a square with a side y one less side z original square, i.e. . Then, as it is easy to see from Figure 37 (just see!), for the remaining square number, the equality must be satisfied. Thus, we arrive at a system of linear equations

Adding and subtracting these equations, we find the solution of equation (1.2.1):

It is easy to see that the resulting solution gives natural numbers only for odd . Thus, we finally have

And so on. Tradition connects this decision with the name of Pythagoras.

Note that system (1.2.2) can also be obtained formally from equation (1.2.1). Indeed,

whence, assuming , we arrive at (1.2.2).

It is clear that the Pythagorean solution was found under a rather rigid constraint () and contains far from all Pythagorean triples. The next step is to put , then , since only in this case will be a square number. So the system arises will also be a Pythagorean triple. Now the main

Theorem. If p And q coprime numbers of different parity, then all primitive Pythagorean triples are found by the formulas