Classification of Pythagorean triples

Statement

Let $m$ and $n$ be coprime positive integers with $m > n$ and $m - n$ odd. Then $(m^2 - n^2, \, 2mn, \, m^2 + n^2)$ is a primitive Pythagorean triple. Conversely, each primitive Pythagorean triple $(a, b, c)$ with $b$ even arises in this way from exactly one such pair $(m, n)$. Each Pythagorean triple is $(ka, kb, kc)$ for a positive integer $k$ and a primitive triple $(a, b, c)$, up to the order of the first two entries.

Remarks

The formula is called Euclid's formula. For example, $(m, n) = (2, 1)$ gives $(3, 4, 5)$, and $(m, n) = (3, 2)$ gives $(5, 12, 13)$. One proof writes $b^2 = (c - a)(c + a)$ and uses unique factorization. Another uses the rational points on the unit circle.

Explanations

L1 human

Euclid's formula produces every right triangle with whole-number sides, from two numbers $m$ and $n$. The identity $(m^2 - n^2)^2 + (2mn)^2 = (m^2 + n^2)^2$ shows that the formula always works.