Definition:Pythagorean Triple/Primitive

Definition

Let $\tuple {x, y, z}$ be a Pythagorean triple such that $x \perp y$ (that is, $x$ and $y$ are coprime).

Then $\tuple {x, y, z}$ is a primitive Pythagorean triple.

Canonical Form

Let $\tuple {x, y, z}$ be a primitive Pythagorean triple.

The convention for representing $\tuple {x, y, z}$ as a (primitive) 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 known as

A primitive Pythagorean triple is also known as a primitive solution of $x^2 + y^2 = z^2$.

Also see

• Parity of Smaller Elements of Primitive Pythagorean Triple which shows that $x$ and $y$ cannot both be odd or both be even.

• Elements of Primitive Pythagorean Triple are Pairwise Coprime

Source of Name

This entry was named for Pythagoras of Samos.

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Pythagorean triple