Definition:Pythagorean Triple

A Pythagorean triple is a triple of positive integers $$\left({x, y, z}\right)$$ such that $$x^2 + y^2 = z^2$$.

That is, a Pythagorean triple is a solution to the Pythagorean equation.

Primitive Pythagorean Triple
If in addition $$x \perp y$$ (that is, $$x$$ and $$y$$ are coprime) then $$\left({x, y, z}\right)$$ is said to be primitive.

Also note, from All Elements of Primitive Pythagorean Triple are Coprime, that $$y \perp z$$ and $$x \perp z$$.

Canonical Form
From Parity of Elements of Primitive Pythagorean Triple we have that $$x$$ and $$y$$ can not both be odd or both be even.

Hence $$z$$ must also be odd.

The convention for representing $$\left({x, y, z}\right)$$ as a Pythagorean triple is that $$x$$ is the even element, while $$y$$ and $$z$$ are both odd.

This is the canonical form of a Pythagorean triple.

Also see

 * Pythagoras's Theorem
 * Pythagorean equation