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
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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.
