Pythagorean triple

In number theory, a Pythagorean triple or Pythagorean number of three natural numbers is taken, which may occur as lengths of the sides of a right triangle.

They are found on Babylonian clay tablets already, which are dated to the time of Hammurabi dynasty ( 1829-1530 BC ). The cuneiform tablet Plimpton 322 contains 15 different Pythagorean triples, including, , and, which suggests that more than 3500 years ago, a method for calculating such triples must have been known.

The Indian - Sulbasutra Baudhayana exacts from the 6th century BC, contains five Pythagorean triples.

Pythagorean triples were also treated by Diophantus. Because of the Pythagorean theorem, they are exactly the positive integer solutions of the Diophantine equation

If x, y and z have no common factor, we speak of a primitive Pythagorean triple. For each primitive triple z is odd, and the number of X and Y is even and the other uneven.

Examples

• The smallest Pythagorean triple is (3,4,5). It is primitive. It is used in the twelve- node line for the production of a right angle.
• ( 5,12,13 )
• ( 15,20,25 ) and ( 15,36,39 ) are not primitive.

Generation of Pythagorean triples

The formulas

Provide for any a Pythagorean triple. It is precisely then primitive if and are relatively prime is odd.

Conversely, every primitive triple with the help of these formulas of prime produce. This result already appeared on in Euclid's Elements.

Pythagorean each triplet ( X, Y, Z) of a primitive triple ( x, y, z ) and (x = n, y = ny, nz = Z ) by an integer n greater than zero are calculated. The natural number n is the greatest common divisor of X, Y, Z, and thus uniquely determined.

Examples:

• 2.1 provides the triple ( 3,4,5)
• 3.1 provides the triple ( 6,8,10 ) which is not a primitive as 3 and 1 are both odd. It is the multiplied with 2 triple ( 3,4,5).
• 3.2 provides the triple ( 5,12,13 )
• Multiplication by 7 provides ( 35,84,91 )

Derivation of the formula for the formation of Pythagorean triples

Is a Pythagorean triple, it follows the division of the corresponding equation by

The numbers and are rational and positive and satisfy the Cartesian equation of the unit circle

Thus, (x, y) is a point with rational coordinate on the unit circle. The line through the points ( -1,0) and (x, y) intersects the y axis at the point ( 0, t). Especially this is true for the slope

Where t is a rational number, and u, v are natural numbers relatively prime.

Substituting this equation into the equation of a unit circle, we obtain:

So

Since the first solution x = -1 because not interested, follows:

The coordinates ( x, y) are therefore:

Or

This yields the Pythagorean triples:

It may happen that possess, and a common divisor D. For example, would

The only way to do this, however, D = 2 is concerned, because, assuming shared a prime number as well as, the following applies

From which one

Can close. Due to the coprimality from and from which would arise, not possible, and it stays together with the only possible solution.

It is those that are coprime and both odd, sort out, however, without losing primitive Pythagorean triples, because if and give the triples, we give and the triplets are relatively prime, and they are not both odd.

The first primitive Pythagorean triples

Following these rules is obtained as a primitive Pythagorean triple, for example (in order of u v):

Two consequences of Pythagorean triples are still remarkable:

• For v = u -1: (3, 4, 5 ), (5, 12, 13 ), (7, 24, 25 ), (9, 40, 41 ), (11, 60, 61 ), (13, 84, 85 ), ..., (2n 1, 2n 2 ² n, 2n 2 ² n 1 ), ..., that is, for any odd number 2n 1 (except for 1 ), a triple, wherein the number 2N 1 is the smallest number, and the two other figures differ by exactly 1: x = 2n 1; y, z = x ² / 2 ± ½. This is related to that of the first binomial formula (m 1) ² = m² 2 m 1, and therefore every odd number 2m 1 is the difference of two square numbers. Since the square of an odd number 2n 1 is odd, there is every odd number 2n 1 is a Pythagorean triple ( 2m 1 = (2n 1) ²).
• For v = 1 (and straight u): (3, 4, 5 ), (15, 8, 17 ), ( 35, 12, 37 ), ( 63, 16, 65 ), ( 99, 20, 101), (143, 24, 145 ), ..., (4n ² -1, 4n, 4n ² 1), ... Thus, for each natural number n, a triple, containing the number 4n, and in which the other two numbers differ by exactly 2: x = 4n; y, z = x ² / 4 ± 1 Also this results from the first binomial formula: (m 2) ² = m² 4 m 4. Each square number is divisible by 16 can be written as 4m 4, so that to each number 4n a Pythagorean triple exists (4m 4 = ( 4n) ²).

And for each even number x is greater than 2 which is not a multiple of 4, it is possible with the first sequence forming a Pythagorean triple ( 2n 1 = x / 2) and then doubling the number. Thus, one can at any natural number x, which is greater than 2, a pair of numbers y and z find that at odd x is 1, wherein a straight x differs by 2:

* Non-primitive triples. The cases for x = 4n 2 are redundant, since they represent a doubling of x = 2n 1.

Connection with the American hero triangles

Each corresponding to a Pythagorean triple triangle is a triangle African hero, i.e. both the side length and the area are rational numbers. Each hero African triangle can be divided into two right triangles, which are given by Pythagorean triples of rational numbers.

The Fermat equation

A generalization of the Pythagorean triples are obtained when the exponent 2 is replaced by a natural number n. So you studied the Diophantine equation

And look for solutions with natural (or whole ) numbers x, y, z excluding the trivial solutions in which one of the three digits is equal to zero.

Pierre de Fermat introduced around the year 1637 on the assertion that there is no such triple. Although he gave no proof this conjecture is called Fermat's great theorem. For centuries, no evidence can be found. However, the search for it has led to many interesting results, particularly in number theory. 1995 could finally prove Fermat's theorem of the mathematician Andrew Wiles.

Algorithm

A possible algorithm in the programming language Haskell might look like this:

PythTripels n = [( k * x k * y * z k ) | ( x, y, z) <- primitive, k < - [ 1, n ' div ' z]. ] WHERE     primitive = [(p ^ 2 -q ^ 2, 2 * p * q, p ^ 2 q ^ 2 ) |. p < - take while ( \ p -> p ^ 2 1 < = n) [ 1 ], q < - take while ( \ q -> p ^ 2 q ^ 2 <= n ). [ 1 p], odd (p q ) && pq gcd == 1 ] It creates a natural number n all possible triples whose hypotenuse does not exceed n.