Solutions of Pythagorean Equation/Primitive

Theorem
The set of all primitive Pythagorean triples is generated by:
 * $\left({2 m n, m^2 - n^2, m^2 + n^2}\right)$

where:
 * $m, n \in \Z$ are positive integers
 * $m \perp n$, that is, $m$ and $n$ are coprime
 * $m$ and $n$ are of opposite parity
 * $m > n$.

Historical Note
This solution was known to.