# Definition:Pythagorean Triple/Primitive

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## 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.

## 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**