Pythagorean Triple

A Pythagorean triple consists of three positive integers a, b, and c, such that a² + b² = c². 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 (that is, they have no common divisor larger than 1). For example, (3, 4, 5) is a primitive Pythagorean triple whereas (6, 8, 10) is not. A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle.

Animation demonstrating the simplest Pythagorean triple, 3² + 4² = 5².

The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula ${\displaystyle a^{2}+b^{2}=c^{2}}$; 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 ${\displaystyle a=b=1}$ and ${\displaystyle c={\sqrt {2}}}$ is a right triangle, but ${\displaystyle (1,1,{\sqrt {2}})}$ is not a Pythagorean triple because ${\displaystyle {\sqrt {2}}}$ is not an integer. Moreover, ${\displaystyle 1}$ and ${\displaystyle {\sqrt {2}}}$ do not have an integer common multiple because ${\displaystyle {\sqrt {2}}}$ 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.

When searching for integer solutions, the equation a² + b² = c² is a Diophantine equation. Thus Pythagorean triples are among the oldest known solutions of a nonlinear Diophantine equation.