primitive Pythagorean triple

A primitive Pythagorean triple consists of three positive integers a, b, and c, such that a² + b² = c², and these integers have no common factors. These triples represent the side lengths of a right triangle.1

The ancient Babylonians were aware of these triples, as evidenced by the Plimpton 322 tablet dating back to around 1700 BCE, which lists number triples satisfying the Pythagorean theorem. While the Greeks, particularly Pythagoras and Euclid, are credited with proving the theorem, the Babylonians had practical knowledge of these number relationships much earlier.234

A formula exists to generate all primitive Pythagorean triples using two relatively prime integers, p and q, where one is even and the other is odd, and p > q. For example, the triple (3, 4, 5) is the only primitive Pythagorean triple composed of consecutive integers.1