Pifagorovka trojka - the ordered set from three natural numbers satisfying the following homogeneous quadratic equation :
The numbers that make up the Pythagorean triple are called Pythagorean numbers . Named in honor of Pythagoras of Samos , although open long before him.
Primitive threes
Since the equation uniformly when multiplied , and on the same natural number we get another Pythagorean triple. Pythagorean triple is called primitive if it cannot be obtained in this way from some other Pythagorean triple, that is, if are mutually prime numbers. In other words, the greatest common divisor of a primitive Pythagorean triple equals 1.
In the primitive top three numbers and have different parity , the even divided by 4, and - always odd.
Any primitive Pythagorean triple where - odd, but - evenly, clearly represented in the form for some natural mutually simple numbers different parity.
These numbers can be calculated by the formulas:
On the contrary, any such pair of numbers sets the primitive Pythagorean trio [1] .
Examples
There are 16 primitive Pythagorean triples with :
(3, 4, 5) | (5, 12, 13) | (8, 15, 17) | (7, 24, 25) |
(20, 21, 29) | (12, 35, 37) | (9, 40, 41) | (28, 45, 53) |
(11, 60, 61) | (16, 63, 65) | (33, 56, 65) | (48, 55, 73) |
(13, 84, 85) | (36, 77, 85) | (39, 80, 89) | (65, 72, 97) |
Not all threes with primitive, for example, (6, 8, 10) is obtained by multiplying by two triples (3, 4, 5). Each of the triples with a small hypotenuse forms a well distinguishable radial line of multiples of triples in the scatter diagram.
Primitive threes with :
(20, 99, 101) | (60, 91, 109) | (15, 112, 113) | (44, 117, 125) |
(88, 105, 137) | (17, 144, 145) | (24, 143, 145) | (51, 140, 149) |
(85, 132, 157) | (119, 120, 169) | (52, 165, 173) | (19, 180, 181) |
(57, 176, 185) | (104, 153, 185) | (95, 168, 193) | (28, 195, 197) |
(84, 187, 205) | (133, 156, 205) | (21, 220, 221) | (140, 171, 221) |
(60, 221, 229) | (105, 208, 233) | (120, 209, 241) | (32, 255, 257) |
(23, 264, 265) | (96, 247, 265) | (69, 260, 269) | (115, 252, 277) |
(160, 231, 281) | (161, 240, 289) | (68, 285, 293) |
Possible values in Pythagorean triples form a sequence (sequence A009003 in OEIS )
- 5, 10, 13, 15, 17, 20, 25, 26, 29, 30, 34, 35, 37, 39, 40, 41, 45, 50, ...
Based on the properties of Fibonacci numbers , it is possible to form from these numbers, for example, such Pythagorean triples:
History
The most famous in developed ancient cultures was the troika (3, 4, 5), which allowed the ancients to build right angles. Vitruvius considered this top three to be the highest achievement of mathematics, and Plato - a symbol of matrimony, which indicates the great importance that the ancient three gave (3, 4, 5).
In the architecture of ancient Mesopotamian tombstones there is an isosceles triangle composed of two rectangular ones with sides of 9, 12, and 15 cubits. The pyramids of Pharaoh Snofru (XXVII century BC. E.) Are built using triangles with sides 20, 21 and 29, as well as 18, 24 and 30 dozens of Egyptian cubits.
Babylonian mathematicians were able to calculate the Pythagorean triples. The Babylonian clay tablet , called Plimpton 322 , contains fifteen Pythagorean triples (more precisely, fifteen pairs of numbers such that ). It is believed that this tablet was created around 1800 BC. er [2]
Generating triples
The Euclidean formula [3] is the main tool for constructing Pythagorean triples. According to her for any pair of natural numbers and ( ) whole numbers
form the Pythagorean trio. The triples formed by the Euclidean formula are primitive if and only if and mutually simple and is odd. If and odd then , and will be even and the troika is not primitive. However division , and 2 gives a primitive top three if and mutually simple [4] .
Any primitive triple is obtained from a single pair of mutually simple numbers. and one of which is even. It follows that there are infinitely many primitive Pythagorean triples.
Despite the fact that the Euclidean formula generates all primitive triples, it does not generate all triples. When adding an additional parameter it turns out a formula that generates all Pythagorean triangles in a unique way:
Where , and - integers, , odd and mutually simple.
The fact that these formulas form Pythagorean triples can be verified by substitutions in and checking that the result matches . Since any Pythagorean trio can be divided into some to get a primitive triple, any triple can be formed in a unique way using and to create a primitive triple and then it multiplies by .
Since the days of Euclid, many formulas have been found for generating triples.
Proof of Euclidean Formulas
The fact that the numbers , , that satisfy the Euclidean formula, always constitute a Pythagorean triangle, obvious for positive integers and , because after substitution into formulas , and will be positive numbers, and also from what is performed
Reverse statement that , , are expressed by the Euclidean formula for any Pythagorean triple, it follows from the following [5] . All such triples can be written as ( , , ) where and , , are mutually simple as well and have the opposite parity (one of them is even, the other is odd). (If a has the same parity with both legs, in the case of their parity they will not be mutually simple, and in the case of oddness will give an even number, and it can not be equal to the odd .) Of we get , and consequently, . Then . Insofar as is rational, we will present it in the form of an irreducible fraction . From here we get that fraction equals . Solving equations
regarding and get
Insofar as and irreducible by assumption, numerators and denominators will be equal if and only if the right sides of each equality are irreducible. As we agreed, the fraction also irreducible, whence it follows that and mutually simple. The right parts will be irreducible if and only if and have the opposite parity, so the numerator is not divisible by 2. (A and must have opposite parity - both cannot be even due to irreducibility, and in the case of oddness of both numbers, the division by 2 will give a fraction, in the numerator and denominator of which will be odd numbers, but this fraction is equal to , in which the numerator and the denominator will have different parity, which contradicts the assumption.) Now, equating the numerators and denominators, we obtain the Euclidean formula , , with and mutually simple and having different parity.
Longer, but more generally accepted evidence is given in the books of Maor (Maor, 2007) [6] and Sierpinski [7] .
Interpretation of parameters in the Euclidean formula
Let the sides of the Pythagorean triangle be equal , and . Denote the angle between the leg and hypotenuse letter . Then [8]
Elementary properties of primitive Pythagorean triples
Properties of a primitive Pythagorean triple ( a , b , c ) , where a < b < c (without specifying the parity of numbers a or b ):
- always a complete square [9] . This is especially useful for checking whether a given triple of numbers is Pythagorean, although this is not a sufficient condition. The triple (6, 12, 18) passes this test, because ( c - a ) ( c - b ) / 2 is a complete square, but this triple is not Pythagorean. If a triple of numbers a , b, and c forms a Pythagorean triple, then the number ( c minus even katet) and half the number ( c minus odd katet) are full squares, but this is not a sufficient condition, and the triple (1, 8, 9) is counterexample, since 1 2 + 8 2 ≠ 9 2 .
- A maximum of one of the numbers a , b, and c is a square [10] .
- The area of the Pythagorean triangle cannot be a square [11] or a double square [12] of a natural number.
- Exactly one of the numbers a and b is odd , c is always odd. [13] .
- Exactly one of the numbers a and b is divisible by 3. [14]
- Exactly one of the numbers a and b is divisible by 4. [7]
- Exactly one of the numbers a , b, and c is divisible by 5. [7]
- The maximum number that abc always divides is sixty. [15]
- All prime factors c are of the form 4 n + 1 [16] . Thus, c has the form 4 n + 1 .
- The number ( b - a ) is a product of primes of the form 8 n ± 1 , that is, it does not have such factors as 2, 3, 5, 11, 13, 19, 29, 37, 43, 53, 59, 61, 67, 83, 101, ...
- The area ( K = ab / 2 ) is an even congruent number [17] .
- In any Pythagorean triple, the radius of the inscribed circle and the radii of three extra-circumscribed circles are natural numbers. In particular, for a primitive triple, the radius of the inscribed circle is r = n ( m - n ) , and the radii of non-inscribed circles touching the legs m 2 - n 2 , 2 mn , and the hypotenuse m 2 + n 2 are respectively m ( m - n ) , n ( m + n ) and m ( m + n ) [18] .
- As with any right triangle, the converse assertion to the Thales theorem says that the diameter of the circumcircle is equal to the hypotenuse. Since for primitive triples the diameter is √ m 2 + n 2 , the radius of the circumscribed circle is half of this number and this number is rational, but not integer (since m and n have different parity).
- If the area of the Pythagorean triangle is multiplied by the curvatures of the inscribed circle and the three non-written ones, we end up with four positive integers w > x > y > z, respectively. These numbers w , x , y , z satisfy the equation of Cartesian circles [19] . Equivalently, the radius of the any right-angled triangle is equal to its half-meter. The outer center of Soddy is located at point D , where ACBD is a rectangle, ACB is a right triangle, and AB is its hypotenuse. [20]
- There are no Pythagorean triples for which the hypotenuse and one of the legs are the legs of the other Pythagorean triple. This is one of the formulations of Fermat's right-angled triangle theorem [21] .
- Each primitive Pythagorean triangle has a unique ratio of the area to the square of the semi-perimeter (that is, the relations for different primitive triangles are different), and this ratio is [22]
- In no primitive Pythagorean triangle, the height based on the hypotenuse is not expressed as an integer, and therefore it cannot be divided into two Pythagorean triangles. [23]
In addition, there may be special Pythagorean triples with some additional properties:
- Any integer greater than 2 which is not (in other words, if it is greater than 2 and does not have the form 4 n + 2) is part of a primitive Pythagorean triple.
- Any integer greater than 2 is included in a primitive or non-primitive Pythagorean triple. For example, the numbers 6, 10, 14, and 18 are not contained in any primitive triplet, but are included in triples 6, 8, 10; 14, 48, 50 and 18, 80, 82.
- There are infinitely many Pythagorean triples in which the hypotenuse and the larger of the legs differ exactly by one (such triples are obviously primitive). One of the ways to obtain such triples is the equality (2 n + 1) 2 + [2 n ( n + 1)] 2 = [2 n ( n + 1) + 1] 2 , leading to triples (3, 4, 5) , (5, 12, 13) , (7, 24, 25) , etc. A more general statement: for any odd integer j, there are infinitely many primitive Pythagorean triples, in which the hypotenuse and even katet differ by j 2 .
- There are infinitely many primitive Pythagorean triples in which the hypotenuse and longer in length of the leg differs by exactly two. Generalization: For any integer k > 0 , there are infinitely many primitive Pythagorean triples in which the hypotenuse and the odd leg differ by 2 k 2 .
- There are infinitely many Pythagorean triples, in which two legs differ exactly by one. For example, 20 2 + 21 2 = 29 2 .
- For any natural n, there are n Pythagorean triples with different hypotenuse and the same area.
- For any natural n, there are at least n different Pythagorean triples with the same cat a , where a is some natural number
- For any natural n, there are at least n different Pythagorean triples with the same hypotenuse. [24]
- There are infinitely many Pythagorean triples, in which the squares are the hypotenuse c and the sum of the legs a + b . In the smallest such triple [25], a = 4,565,486,027,761 ; b = 1,061,652,293,520 ; c = 4,687,298,610,289 . Here a + b = 2 372 159 2 and c = 2 165 017 2 . In the Euclidean formula, these values correspond to m = 2 150 905 and n = 246 792 .
- There are Pythagorean triangles with a full height, based on the hypotenuse . Such triangles are known as breakables, since they can be divided by this height into two smaller Pythagorean triangles. None of the triangles to be broken is formed by a primitive triple [26] .
- The set of all primitive Pythagorean triangles forms the root natural way, see. Tree of primitive Pythagorean triples .
It is not known whether there are two different Pythagorean triples with the same product of the numbers in them [27] .
Euclidean Formula Geometry
Euclidean formula for the Pythagorean triple
can be understood in terms of the geometry of rational points on the unit circle [28] . Suppose there is a triangle with legs a and b and hypotenuse c , where a , b and c are positive integers. By the Pythagorean theorem a 2 + b 2 = c 2 , and after dividing both sides by c 2
Geometrically, a point on a Cartesian plane with coordinates
lies on the unit circle x 2 + y 2 = 1 . In this equation, the x and y coordinates are given by rational numbers. And vice versa, any point on a circle with rational x and y coordinates gives a primitive Pythagorean triple. Indeed, we write x and y as irreducible fractions :
where the greatest common divisor of a , b and c is 1. Since the point with x and y coordinates lies on the unit circle,
Q.E.D.
Thus, there is a correspondence between and primitive Pythagorean triangles. On this basis, Euclidean formulas can be obtained by trigonometry methods or using stereographic projection .
To apply the stereographic approach, we assume that P ′ is a point on the x axis with rational coordinates
Then, using algebraic calculations, it can be shown that the point P has coordinates
Thus, we find that any the x axis corresponds to a rational point of the unit circle. And back, let P ( x , y ) be a point of the unit circle with rational coordinates x and y . Then the stereographic projection of P ′ on the x axis has rational coordinates
In terms of algebraic geometry, an algebraic variety of rational points of the unit circle is birational to an affine line over rational numbers. The unit circle is then called a rational curve . The correspondence of rational points of a line and a circle makes it possible to give explicit parametrization of (rational) points on a circle using rational functions.
Pythagorean threes group
Any rational point on the unit circle corresponds to the Pythagorean triple ( a , b , c ) , more precisely, a generalized Pythagorean triple, since a and b can be zero and negative.
Let two Pythagorean triangles ( a 1 , b 1 , c 1 ) and ( a 2 , b 2 , c 2 ) be given with angles α and β . You can build triangles with angles α ± β , using the formulas for the addition of angles:
These right triangles will also be integral, that is, Pythagorean. You can enter an operation on triples using the formulas above. This operation will be commutative and associative, that is, generalized Pythagorean triples form an Abelian group [29] .
Pythagorean triples on a two-dimensional lattice
A two-dimensional lattice is a set of isolated points in which, if you select one point as the origin (0, 0), all other points have coordinates ( x , y ) , where x and y run through all positive and negative integers. Any Pythagorean triple ( a , b , c ) can be drawn on a two-dimensional lattice as points with coordinates ( a , 0) and (0, b ) . By the Pick theorem, the number of lattice points lying strictly inside a triangle is given by the formula [30] . For primitive Pythagorean triples, the number of lattice points is equal to and it is comparable to the area of a triangle
Interestingly, the first case of coincidence of the areas of primitive Pythagorean triples appears on triples (20, 21, 29), (12, 35, 37) with an area of 210 [31] . The very first appearance of primitive Pythagorean triples with the same number of lattice points appears only on (18,108, 252,685, 253,333), (28,077, 162,964, 165,365) with the number of points 2,287,674,594 [32] . Found three primitive Pythagorean triples with the same areas (4485, 5852, 7373), (3059, 8580, 9109), (1380, 19 019, 19 069) and an area of 13 123 110. Nevertheless, not a single triples of primitive Pythagorean triples with the same the number of lattice points has not yet been found.
Spinors and modular group
Pythagorean triples can be represented as matrices of the form
The matrix of this type is symmetric . In addition, its determinant
equals zero exactly when ( a , b , c ) is a Pythagorean triple. If X corresponds to the Pythagorean triple, then it must have rank 1.
Since X is symmetric, it is known from linear algebra that there is a vector ξ = [ m n ] T , such that
- (one)
where T means transpose . The vector ξ is called a spinor (for the Lorentz group SO (1, 2). In abstract terms, the Euclidean formula means that each primitive Pythagorean triple can be written as an outer product on itself of a spinor with integer elements, as in formula (1).
The Γ modular group is a set of 2 × 2 matrices with integer elements.
and a determinant equal to one: αδ - βγ = 1 . This set forms a group , since the inverse of a matrix from Γ is again a matrix from Γ , like the product of two matrices from Γ . The modular group acts on the set of all integer spinors. Moreover, the group is transitive on the set of whole spinors with mutually simple elements. If [ m n ] T contains mutually simple elements, then
where u and v are chosen (using the Euclidean algorithm ) so that mu + nv = 1 .
Acting on the spinor ξ in (1), the action in Γ goes into action over the Pythagorean triples, while allowing triples with negative values. If A is a matrix in Γ , then
- (2)
gives rise to actions on the matrix X in (1). This does not give a well-defined effect on primitive triples, since it can translate a primitive triples into non-primitive ones. In this place it is accepted (following Trautman [28] ) to call a triple ( a , b , c ) standard , if c > 0 and either ( a , b , c ) are mutually simple, or ( a / 2, b / 2, c / 2) are mutually simple and a / 2 is odd. If the spinor [ m n ] T has mutually simple elements, then the connected triple ( a , b , c ) , given by formula (1), is a standard triple. It follows that the action of the modular group is transitive on the set of standard triples.
Alternatively, we restrict ourselves to those values of m and n , for which m is odd and n is even. Let the subgroup Γ (2) of the group Γ be the kernel of the homomorphism
where SL (2, Z 2 ) is a special linear group over a finite field Z 2 integer modulo 2 . Then Γ (2) is a group of unimodular transformations that preserves the parity of each element. Thus, if an element of the vector ξ is odd and the second is even, then the same is true for Aξ for all A ∈ Γ (2) . In fact, under the action of (2), the group Γ (2) acts transitively on a set of primitive Pythagorean triples [33] .
The group Γ (2) is a free group whose matrices are generated by
Therefore, any primitive Pythagorean triple can be uniquely obtained as the product of copies of the matrices U and L.
Parent-child relationship
As shown by Berggren [34] , all primitive Pythagorean triples can be obtained from a triangle (3, 4, 5) using three linear transformations T1, T2, T3, where a , b , c are sides of a triple:
new side a | new side b | new side c | |
T1: | a - 2 b + 2 c | 2 a - b + 2 c | 2 a - 2 b + 3 c |
T2: | a + 2 b + 2 c | 2 a + b + 2 c | 2 a + 2 b + 3 c |
T3: | - a + 2 b + 2 c | −2 a + b + 2 c | −2 a + 2 b + 3 c |
If we start with 3, 4, 5, then all other primitive triples will eventually be obtained. In other words, any primitive triple will be a “parent” of 3 additional primitive triples. If we start with a = 3, b = 4 and c = 5, then the next generation of triples will be
new side a | new side b | new side c |
3 - (2 × 4) + (2 × 5) = 5 | (2 × 3) - 4 + (2 × 5) = 12 | (2 × 3) - (2 × 4) + (3 × 5) = 13 |
3 + (2 × 4) + (2 × 5) = 21 | (2 × 3) + 4 + (2 × 5) = 20 | (2 × 3) + (2 × 4) + (3 × 5) = 29 |
−3 + (2 × 4) + (2 × 5) = 15 | - (2 × 3) + 4 + (2 × 5) = 8 | - (2 × 3) + (2 × 4) + (3 × 5) = 17 |
The linear transformations T1, T2 and T3 have a geometric interpretation in the language of quadratic forms. They are closely related (but not equivalent) to reflections generated by the orthogonal group x 2 + y 2 - z 2 over the integers. Another set of three linear transformations is discussed in the article [35] .
Relationship with Gaussian integers
Euclidean formulas can be analyzed and proved using Gaussian integers [36] . Gaussian integers are complex numbers of the form α = u + vi , where u and v are ordinary integers , and i is the root of minus ones . The units of Gaussian integers are ± 1 and ± i. Regular integers are called integers and are denoted by Z. Gaussian integers are denoted by Z [ i ]. The right side of the Pythagorean theorem can be decomposed into Gaussian integers:
The primitive Pythagorean triple is a triple, in which a and b are mutually simple , that is, they have no common prime dividers. For such triples, either a or b is even and the second is odd. It follows that c is also odd.
Each of the two factors z = a + bi and z * = a - bi of a primitive Pythagorean triple is equal to the square of a Gaussian integer. This can be proved with the help of the property that any Gaussian integer can be uniquely decomposed into Gaussian simple ones with an accuracy of one [37] . (The uniqueness of the expansion, roughly speaking, follows from the fact that for them it is possible to determine the version of the Euclidean algorithm ) The proof has three steps. It is first proved that if a and b do not have primes in integers, then they do not have simple common factors in Gaussian integers. It follows that z and z * do not have common prime factors in Gaussian integers. Finally, since c 2 is a square, any Gaussian prime in the decomposition is repeated twice. Since z and z * do not have common prime factors, this doubling is also true for them. Therefore, z and z * are squares.
Thus, the first factor can be written as
The real and imaginary parts of this equation give two formulas:
For any primitive Pythagorean triple there must exist integer m and n such that these two equalities hold. From here, any Pythagorean triple can be obtained by choosing these integers.
Like a full square of Gaussian integers
If we take the square of a Gaussian integer, we obtain the following interpretation of Euclidean formulas as a representation of the full square of Gaussian integers.
If we use the fact that Gaussian integers are a Euclidean region and that for Gaussian integers p is the square of the module always a complete square, it can be shown that the Pythagorean triples correspond to the squares of simple Gaussian integers if the hypotenuse is a prime number.
Distribution of triples
There are many results regarding the distribution of Pythagorean triples. In the scatter diagram some obvious patterns are manifested. If the legs ( a , b ) of a primitive triple appear on the diagram, then all the works by an integer of these legs must also be on the diagram, and this property explains the appearance on the diagram of the radial lines from the origin.
The diagram shows sets of parabolas with a high density of points having foci at the origin. Parabolas are reflected from the axes with an angle of 45 degrees, and at the same point the third parabola approaches the axis perpendicularly.
These patterns can be explained as follows. If a natural number, then ( a , , ) is a Pythagorean triple. (In fact, any Pythagorean triple ( a , b , c ) can be written in this way with integer n , possibly after exchanging a and b places, because and a , b cannot be odd at the same time.) Pythagorean triples then lie on the curves given by the equations . Thus, the parabolas are reflected from the a axis, and the corresponding curves with a and b change places. Если a меняется при заданном n (то есть на выбранной параболе), целые значения b появляются относительно часто, если n является квадратом или произведением квадрата на небольшое число. Если некоторые такие значения лежат близко друг от друга, соответствующие параболы почти совпадают и тройки образуют узкую параболическую ленту. Например, 38 2 = 1444, 2 × 27 2 = 1458, 3 × 22 2 = 1452, 5 × 17 2 = 1445 и 10 × 12 2 = 1440. Соответствующая параболическая лента около n ≈ 1450 чётко видна на диаграмме рассеяния.
Угловые свойства, описанные выше следуют немедленно из функционального вида парабол. Параболы отражаются от оси a в точке a = 2 n и производная b по a в этой точке равна −1. Таким образом, угол наклона равен 45°. Поскольку кластеры, как и треугольники, повторяются при умножении на целую константу, значение 2 n тоже принадлежит кластеру. Соответствующая парабола пересекает ось b под прямым углом в точке b = 2 n , а потому является симметричным отражением параболы, которая получается обменом переменных a и b и которая пересекает ось a под прямым углом в точке a = 2 n .
Альберт Фесслер ( Albert Fässler ) и др. показали значимость этих парабол в контексте конформных отображений [38] [39] .
Специальные случаи
Последовательность Платона
Случай n = 1 общей конструкции пифагоровых троек известен давно. Прокл , в своём комментарии к 47-му утверждению в первой книге Начал Евклида , описывает это следующим образом:
Некоторые методы получения таких треугольников этого вида легко получить, один из них принадлежит Платону , другой — Пифагору . (Последний) начал с нечётных чисел. Для этого он выбрал нечётное число в качестве меньшего из катетов. Затем он возвёл его в квадрат, вычел единицу и половину этой разницы использовал как второй катет. Наконец, он добавил единицу к этому катету и получил гипотенузу.
…Метод Платона работает с чётными числами. Он использует заданное чётное число в качестве одного из катетов. Половина этого числа возводится в квадрат и добавляется единица, что даёт гипотенузу, а вычитание единицы даёт второй катет. … И это даёт тот же треугольник, что и другой метод.
В виде уравнений:
- a нечётно (Пифагор, 540 до н. э.):
- a чётно (Платон, 380 до н. э.):
Можно показать, что все пифагоровы тройки получаются из последовательности Платона ( x , y , z ) = p , ( p 2 − 1)/2 и ( p 2 + 1)/2, если позволить p принимать нецелые (рациональные) значения. Если в этой последовательности p заменить рациональной дробью m / n , получим 'стандартный' генератор троек 2 mn , m 2 − n 2 и m 2 + n 2 . Отсюда следует, что любой тройке соответствует рациональное значение p , которое можно использовать для получения подобного треугольника с рациональными сторонами, пропорциональными сторонам исходного треугольника. Например, платоновым эквивалентом тройке (6, 8, 10) будет (3/2; 2, 5/2).
Уравнение Якоби — Маддена
The equation
эквивалентно специальной диофантовой тройке
Существует бесконечное число решений этого уравнения, которые можно получить используя эллиптическую кривую . Два из этих решений:
Равные суммы двух квадратов
Один из способов генерации решений для — параметризовать a , b , c , d в терминах натуральных чисел m , n , p , q следующим образом: [40]
Равные суммы двух четвёртых степеней
Если даны два набора пифагоровых троек:
то задача поиска равных произведений катета и гипотенузы
как легко видеть, эквивалентна уравнению
для которого Эйлер получил решение . Поскольку он показал, что эта точка является рациональной точкой эллиптической кривой , то существует бесконечное число решений. Фактически, он также нашёл полиномиальную параметризацию 7-й степени.
Теорема Декарта об окружностях
В случае , когда все переменные являются квадратами,
Эйлер показал, что это эквивалентно трём пифагоровым тройкам:
Здесь тоже существует бесконечное число решений, а для специального случая уравнение упрощается до
которое имеет решение с небольшими числами и может быть решено как .
Почти равнобедренные пифагоровы тройки
Имеются с целыми сторонами, у которых длины катеты отличающиеся на единицу, например:
и бесконечное число других. Для них можно вывести общую формулу
где ( x , y ) являются решениями уравнения Пелля .
В случае, когда катет и гипотенуза отличаются на единицу, как в случаях
общим решением будет
откуда видно, что все нечётные числа (большие 1) появляются в примитивных пифагоровых тройках.
Обобщения
Имеется несколько вариантов обобщения концепции пифагоровых троек.
Пифагоровы четвёрки
Множество из четырёх натуральных чисел a , b , c и d , таких, что a 2 + b 2 + c 2 = d 2 называется пифагоровой четвёркой . Простейший пример — (1, 2, 2, 3), поскольку 1 2 + 2 2 + 2 2 = 3 2 . Следующий (примитивный) простейший пример — (2, 3, 6, 7), поскольку 2 2 + 3 2 + 6 2 = 7 2 .
Все четвёрки задаются формулой
Пифагоровы n -наборы
Используя простое алгебраическое тождество
для произвольных x 0 , x 1 , просто доказать, что квадрат суммы n квадратов сам является суммой n квадратов, для чего положим x 0 = x 2 2 + x 3 2 + … + x n 2 и раскроем скобки [41] . Можно легко видеть, что пифагоровы тройки и четвёрки являются просто частными случаями x 0 = x 2 2 и x 0 = x 2 2 + x 3 2 соответственно, что можно продолжать для других n , используя формулу для пятёрки квадратов
Поскольку сумма F ( k , m ) k последовательных квадратов, начиная с m 2 , задаётся формулой [42]
можно найти значения ( k , m ) такие, что F ( k , m ) является квадратом. Так, Хиршхорн нашёл формулу для последовательностей, в которых число членов само является квадратом [43] ,
и v ⩾ 5 есть любое натуральное число, не делящееся на 2 или 3. Наименьшее значение v = 5, откуда k = 25, что даёт хорошо известное значение из задачи Люка складирования пушечных ядер:
факт, который связан с решёткой Лича .
Кроме того, если в пифагоровом n -наборе ( n ⩾ 4) все слагаемые являются последовательными натуральными числами, за исключением последнего, можно использовать равенство [44]
Поскольку вторая степень p сокращается, остаётся линейное уравнение, которое легко решается , хотя k и m следует выбрать так, чтобы p был целым, и пример получаем при k = 5 и m = 1:
Таким образом, получаем метод генерации пифагоровых n -наборов путём подбора x [45] :
где q = n − 2 и
Fermat's great theorem
Обобщением концепции пифагоровых троек служит поиск троек натуральных чисел a , b и c , таких, что a n + b n = c n для некоторого n , большего 2. Пьер Ферма в 1637 году высказал утверждение, что таких троек не существует, и это утверждение стало известно как Великая теорема Ферма , поскольку её доказательство или опровержение отняло много больше времени, чем любая другая гипотеза Ферма. Первое доказательство было дано Уайлсом в 1994 году.
n — 1 или n n -х степеней как n -я степень
Другим обобщением является поиск последовательностей из n + 1 натуральных чисел, для которых n -я степень последнего члена последовательности равна сумме n -х степеней предыдущих членов. Наименьшие последовательности для известных значений n :
- n = 3: {3, 4, 5; 6}.
- n = 4: {30, 120, 272, 315; 353}
- n = 5: {19, 43, 46, 47, 67; 72}
- n = 7: {127, 258, 266, 413, 430, 439, 525; 568}
- n = 8: {90, 223, 478, 524, 748, 1088, 1190, 1324; 1409}
В слегка отличном обобщении сумма ( k + 1) n -х степеней приравнивается сумме ( n − k ) n -х степеней. For example:
- ( n = 3): 1 3 + 12 3 = 9 3 + 10 3 . Пример стал известным после воспоминаний Харди о разговоре с Рамануджаном о числе 1729, которое является наименьшим числом, которое можно представить в виде суммы двух кубов двумя различными способами.
Может существовать также n − 1 n -х степеней натуральных чисел, дающих в сумме n -ю степень натурального числа (хотя, согласно великой теореме Ферма , не для n = 3). Эти последовательности являются контрпримерами гипотезе Эйлера . Наименьшие известные контрпримеры [46] [15]
- n = 4: (95800, 217519, 414560; 422481)
- n = 5: (27, 84, 110, 133; 144)
Тройки треугольника Герона
Треугольник Герона обычно определяется как треугольник с целыми сторонами, площадь которого тоже целое число, и мы будем полагать, что стороны треугольника различны . Длины сторон такого треугольника образуют тройку Герона ( a, b, c ), где a < b < c . Ясно, что пифагоровы тройки являются тройками Герона, поскольку в пифагоровой тройке по меньшей мере один из катетов a и b является чётным числом, так что площадь треугольника ab /2 будет целым числом. Не всякая тройка Герона является пифагоровой, поскольку, например, тройка (4, 13, 15) с площадью 24 не пифагорова.
Если ( a , b , c ) является тройкой Герона, то таковой будет и ( ma , mb , mc ) при любом натуральном m , большим единицы. Тройка Герона ( a , b , c ) примитивна , если a , b и c попарно взаимно просты (как и в случае пифагоровых троек). Ниже приведено несколько троек Герона, не являющихся пифагоровыми:
- (4, 13, 15) с площадью 24,
- (3, 25, 26) с площадью 36,
- (7, 15, 20) с площадью 42,
- (6, 25, 29) с площадью 60,
- (11, 13, 20) с площадью 66,
- (13, 14, 15) с площадью 84,
- (13, 20, 21) с площадью 126.
- (3, 25, 26) с площадью 36,
По формуле Герона , чтобы тройка натуральных чисел ( a , b , c ) с a < b < c была тройкой Герона, необходимо, чтобы
- ( a 2 + b 2 + c 2 ) 2 − 2 ( a 4 + b 4 + c 4 )
или, что то же самое,
- 2 ( a 2 b 2 + a 2 c 2 + b 2 c 2 ) − ( a 4 + b 4 + c 4 )
было ненулевым полным квадратом, делящимся на 16.
Use
Примитивные пифагоровы тройки используются в криптографии в качестве случайных последовательностей и для генерации ключей [47] .
See also
- Тройка Эйзенштейна
- Совершенный кубоид
- Геронов треугольник
- Теорема Гильберта 90
- Целочисленный треугольник
- Modulo comparison
- Негипотенузное число
- Простое число Пифагора
- Пифагорова четвёрка
- Формула тангенса половинного угла
- Тригонометрические тождества
Notes
- ↑ В. Серпинский . Пифагоровы треугольники. — М. : Учпедгиз, 1959. — 111 с.
- ↑ Robson, Eleanor (February 2002), " Words and pictures: new light on Plimpton 322 ", American Mathematical Monthly (Mathematical Association of America) . — Т. 109 (2): 105–120, doi : 10.2307/2695324 , < http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Robson105-120.pdf >
- ↑ DE Joyce. Euclid's Elements. — Clark University, June 1997. — С. Book X, Proposition XXIX .
- ↑ Douglas W. Mitchell. An Alternative Characterisation of All Primitive Pythagorean Triples // The Mathematical Gazette. — July 2001. — Т. 85 , вып. 503 . — С. 273–5 .
- ↑ Raymond A. Beauregard, ER Suryanarayan. Proofs Without Words: More Exercises in Visual Thinking / Roger B. Nelsen. — Mathematical Association of America , 2000. — Т. II . — С. 120 . — ISBN 978-0-88385-721-2 .
- ↑ Eli Maor. The Pythagorean Theorem. — Princeton University Press, 2007. — С. Appendix B.
- ↑ 1 2 3 Sierpinski, 2003 .
- ↑ Houston, 1993 , с. 141.
- ↑ Posamentier, 2010 , с. 156.
- ↑ Несуществование решения, в котором и a , и b являются квадратами, первоначально доказано Пьером Ферма . Для других случаев, в которых c является одним из квадратов, см. в книге Стиллвела.
- ↑ Carmichael, 1959 , с. 17
- ↑ Carmichael, 1959 , с. 21.
- ↑ Sierpinski, 2003 , с. 4-6.
- ↑ Sierpinski, 2003 , с. 23–25.
- ↑ 1 2 MacHale, Bosch, 2012 , с. 91-96.
- ↑ Sally, 2007 , с. 74–75.
- ↑ Это следует из факта, что одно из чисел a или b делится на четыре, и из определения конгруэнтных чисел как площадей прямоугольных треугольников с рациональными сторонами
- ↑ Baragar, 2001 , с. 301, упражнение 15.3.
- ↑ Bernhart, Price, 2005 .
- ↑ Bernhart, Price, 2005 , с. 6
- ↑ Carmichael, 1959 , с. 14.
- ↑ Rosenberg, Spillane, Wulf, May 2008 , с. 656–663.
- ↑ Paul Yiu, 2008 .
- ↑ Sierpinski, 2003 , с. 31.
- ↑ Pickover, 2009 , с. 40
- ↑ Paul Yiu, 2008 , с. 17
- ↑ Weisstein, Eric W. Пифагорова тройка (англ.) на сайте Wolfram MathWorld .
- ↑ 1 2 Trautman, 1998 .
- ↑ Eckert, 1984 .
- ↑ Paul Yiu, 2003 .
- ↑ The sequence A093536 in OEIS .
- ↑ Sequence A225760 in OEIS .
- ↑ Alperin, 2005 .
- ↑ Berggren, 1934 .
- ↑ Further discussion of the parent-child relationship - Pythagorean triple (Wolfram) , Alperin, 2005 .
- ↑ Stillwell, 2002 , p. 110–2 Chapter 6.6 Pythagorean Triples.
- ↑ Gauss, 1832 See also Werke , 2 : 67-148.
- ↑ 1988 Preprint See Figure 2 on p. 3. It was later printed in ( Fässler 1991 )
- ↑ Benito, Varona, 2002 , p. 117–126.
- ↑ Nahin, Paul. An Imaginary Tale: The Story of p. 25-26.
- ↑ A Collection of Algebraic Identities: Sums of n Squares .
- ↑ Sum of consecutive cubes equal a cube (Unavailable (inaccessible link) . Archived May 15, 2008.
- ↑ Michael Hirschhorn. When is the sum of consecutive squares a square? // The Mathematical Gazette. - November 2011. - Vol . 95 . - p . 511–2 . - ISSN 0025-5572 .
- ↑ John F. Jr. Goehl. Reader reflections // Mathematics Teacher. - May 2005. - Vol. 98 , no. 9 - p . 580 .
- ↑ John F. Goehl, Jr. Triples, quartets, pentads // Mathematics Teacher. - May 2005. - Vol . 98 . - p . 580 .
- ↑ Scott Kim. Bogglers // Discover . - May 2002. - p . 82 .
The equation more difficult, only in 1988 after 200 years of unsuccessful attempts by mathematicians to prove the impossibility of solving the equation Noam Elkis from Harvard found a counterexample - 2.682.440 4 + 15.365.639 4 + 18.796.760 4 = 20.615.673 4 :- Noam Elkies. On A 4 + B 4 + C 4 = D 4 // Mathematics of Computation. - 1988. - T. 51 . - p . 825–835 .
- ↑ S. Kak, M. Prabhu. Cryptographic applications of primitive Pythagorean triples // Cryptologia. - 2014. - V. 38 , no. 3 - p . 215-222 .
Literature
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- John Stillwell. Numbers and Geometry. - Springer, 1998. - p. 133. - (Undergraduate Texts in Mathematics). - ISBN 9780387982892 .
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