Pythagorean triples

A Pythagorean triple consists of three positive integers $a$ , $b$ , and $c$ , such that $a 2 + b 2 = c 2$ . Such a triple is commonly written $(a, b, c)$ , and a well-known example is $(3, 4, 5)$ . If $(a, b, c)$ is a Pythagorean triple, then so is $(ka, kb, kc)$ for any positive integer $k$ . A primitive Pythagorean triple is one in which $a$ , $b$ and $c$ are coprime. A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle.

The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula $a 2 + b 2 = c 2$ ; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. For instance, the triangle with sides $a = b = 1$ and $c = \sqrt 2$ is right, but $(1, 1, \sqrt 2)$ is not a Pythagorean triple because $\sqrt 2$ is not an integer. Moreover, $1$ and $\sqrt 2$ do not have an integer common multiple because $\sqrt 2$ is irrational.

Pythagorean triples have been known since ancient times. The oldest known record comes from Plimpton 322, a Babylonian clay tablet from about 1800 BC, written in a sexagesimal number system. It was discovered by Edgar James Banks shortly after 1900, and sold to George Arthur Plimpton in 1922, for $10.

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