Pythagorean triple

The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula

The oldest known record comes from Plimpton 322, a Babylonian clay tablet from about 1800 BC, written in a sexagesimal number system.

The following will generate all Pythagorean triples uniquely: where m, n, and k are positive integers with m > n, and with m and n coprime and not both odd.

Conversely, every primitive Pythagorean triple arises (after the exchange of a and b, if a is even) from a unique pair m > n > 0 of coprime odd integers.

There is a correspondence between points on the unit circle with rational coordinates and primitive Pythagorean triples.

At this point, Euclid's formulae can be derived either by methods of trigonometry or equivalently by using the stereographic projection.

As yet, no set of three primitive Pythagorean triples have been found sharing the same interior lattice count.

Since X is symmetric, it follows from a result in linear algebra that there is a column vector ξ = [m n]T such that the outer product holds, where the T denotes the matrix transpose.

In abstract terms, the Euclid formula means that each primitive Pythagorean triple can be written as the outer product with itself of a spinor with integer entries, as in (1).

The modular group Γ is the set of 2×2 matrices with integer entries with determinant equal to one: αδ − βγ = 1.

In fact, under the action (2), the group Γ(2) acts transitively on the collection of primitive Pythagorean triples (Alperin 2005).

[32] (This unique factorization follows from the fact that, roughly speaking, a version of the Euclidean algorithm can be defined on them.)

From the formula c2 = zz*, that in turn would imply that c is even, contrary to the hypothesis of a primitive Pythagorean triple.

Within the scatter, there are sets of parabolic patterns with a high density of points and all their foci at the origin, opening up in all four directions.

Within this quadrant, each arc centered on the origin shows that section of the parabola that lies between its tip and its intersection with its semi-latus rectum.

If several such values happen to lie close together, the corresponding parabolas approximately coincide, and the triples cluster in a narrow parabolic strip.

Albert Fässler and others provide insights into the significance of these parabolas in the context of conformal mappings.

Proclus, in his commentary to the 47th Proposition of the first book of Euclid's Elements, describes it as follows: Certain methods for the discovery of triangles of this kind are handed down, one which they refer to Plato, and another to Pythagoras.

Thus it has formed the same triangle that which was obtained by the other method.In equation form, this becomes: a is odd (Pythagoras, c. 540 BC): a is even (Plato, c. 380 BC): It can be shown that all Pythagorean triples can be obtained, with appropriate rescaling, from the basic Platonic sequence (a, (a2 − 1)/2 and (a2 + 1)/2) by allowing a to take non-integer rational values.

It follows that every triple has a corresponding rational a value which can be used to generate a similar triangle (one with the same three angles and with sides in the same proportions as the original).

The Platonic sequence itself can be derived[clarification needed] by following the steps for 'splitting the square' described in Diophantus II.VIII.

is to parametrize a, b, c, d in terms of integers m, n, p, q as follows:[35] Given two sets of Pythagorean triples, the problem of finding equal products of a non-hypotenuse side and the hypotenuse, is easily seen to be equivalent to the equation, and was first solved by Euler as

When it is the longer non-hypotenuse side and hypotenuse that differ by one, such as in then the complete solution for the primitive Pythagorean triple a, b, c is and where integer

It shows that all odd numbers (greater than 1) appear in this type of almost-isosceles primitive Pythagorean triple.

Another property of this type of almost-isosceles primitive Pythagorean triple is that the sides are related such that for some integer

The Pythagorean n-tuple can be made primitive by dividing out by the largest common divisor of its values.

Use (m1, ..., mn) = (c + a1, a2, ..., an) to get a Pythagorean n-tuple by the above formula and divide out by the largest common integer divisor, which is 2m1 = 2(c + a1).

Dividing out by the largest common divisor of these (m1, ..., mn) values gives the same primitive Pythagorean n-tuple; and there is a one-to-one correspondence between tuples of setwise coprime positive integers (m1, ..., mn) satisfying m21 > m22 + ... + m2n and primitive Pythagorean n-tuples.

For the smallest case v = 5, hence k = 25, this yields the well-known cannonball-stacking problem of Lucas, a fact which is connected to the Leech lattice.

In addition, if in a Pythagorean n-tuple (n ≥ 4) all addends are consecutive except one, one can use the equation,[42] Since the second power of p cancels out, this is only linear and easily solved for as

Animation demonstrating the smallest Pythagorean triple, 3 2 + 4 2 = 5 2 .
Scatter plot of the legs ( a , b ) of the first Pythagorean triples with a and b less than 6000. Negative values are included to illustrate the parabolic patterns. The "rays" are a result of the fact that if ( a , b , c ) is a Pythagorean triple, then so is (2 a , 2 b , 2 c ) , (3 a , 3 b , 3 c ) and, more generally, ( ka , kb , kc ) for any positive integer k .
Primitive Pythagorean triples shown as triangles on a graph
The primitive Pythagorean triples. The odd leg a is plotted on the horizontal axis, the even leg b on the vertical. The curvilinear grid is composed of curves of constant m − n and of constant m + n in Euclid's formula.
A plot of triples generated by Euclid's formula maps out part of the z 2 = x 2 + y 2 cone. A constant m or n traces out part of a parabola on the cone.
3,4,5 maps to x,y point (4/5,3/5) on the unit circle
The rational points on a circle correspond, under stereographic projection , to the rational points of the line.
Stereographic projection of the unit circle onto the x -axis. Given a point P on the unit circle, draw a line from P to the point N = (0, 1) (the north pole ). The point P ′ where the line intersects the x -axis is the stereographic projection of P . Inversely, starting with a point P ′ on the x -axis, and drawing a line from P ′ to N , the inverse stereographic projection is the point P where the line intersects the unit circle.
A scatter plot of the legs ( a , b ) of the first Pythagorean triples with a and b less than 4500.