Definition:Bézout Numbers

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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.