Pythagorean triple

[edit] Definition

A Pythagorean triple is a solution $(x, y, z)$ of the equation $x 2 + y 2 = z 2$ , in which $x$ , $y$ and $z$ are positive integers.

A Pythagorean triple is called primitive, if $gcd(x, y, z) = 1$ .

All primitive Pythagorean triples are given by:

$x=2mn, \quad y=m^2-n^2, \quad z=m^2+n^2$

where $m$ and $n$ are relatively-prime positive integers, and $m > n$ .

Every non-primitive Pythagorean triple is some positive integer multiple of a primive triple.