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