# Definition:Bézout Numbers

## Definition

Let $a, b \in \Z$ such that $a \ne 0$ or $b \ne 0$.

Let $d$ be the greatest common divisor of $a$ and $b$.

By Bézout's Lemma:

- $\exists x, y \in \Z: a x + b y = d$

The numbers $x$ and $y$ are known as **Bézout numbers** of $a$ and $b$.

## Complete Set of Bézout Numbers

These numbers, are not unique for a given $a, b \in \Z$.

For a given $a, b \in \Z$ there is a countably infinite number of **Bézout numbers**.

From Solution of Linear Diophantine Equation, if $x_0$ and $y_0$ are **Bézout numbers**, then:

- $\displaystyle \forall k \in \Z: x = x_0 + \frac {kb} {\gcd \left\{{a,b}\right\}}, y = y_0 - \frac {ka} {\gcd \left\{{a,b}\right\}}$

are also **Bézout numbers**.

## Also known as

**Bézout numbers** are also known as **Bézout coefficients**.

## Source of Name

This entry was named for Étienne Bézout.