A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is one in which , and are coprime (that is, they have no common divisor larger than 1). For example, is a primitive Pythagorean triple whereas is not. 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 ; 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 and is a right triangle, but is not a Pythagorean triple because is not an integer. Moreover, and do not have an integer common multiple because 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 is a Diophantine equation. Thus Pythagorean triples are among the oldest known solutions of a nonlinear Diophantine equation. Examples There are 16 primitive Pythagorean triples of numbers up to 100: Other small Pythagorean triples such as (6, 8, 10) are not listed because they are not primitive; for instance (6, 8, 10) is a multiple of (3, 4, 5). Each of these points (with their multiples) forms a radiating line in the scatter plot to the right. Additionally, these are the remaining primitive Pythagorean triples of numbers up to 300: Generating a triple Euclid's formula is a fundamental formula for generating Pythagorean triples given an arbitrary pair of integers and with . The formula states that the integers form a Pythagorean triple. The triple generated by Euclid's formula is primitive if and only if and are coprime and one of them is even. When both and are odd, then , , and will be even, and the triple will not be primitive; however, dividing , , and by 2 will yield a primitive