Definition:Pythagorean Triple
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Definition
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
Source of Name
This entry was named for Pythagoras of Samos.
