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 Identity, or Bézout's Lemma:
 * $$\exists x, y \in \Z: a x + b y = d$$

The numbers $$x$$ and $$y$$ are known as Bézout numbers or Bézout coefficients 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:
 * $$\forall k \in \Z: x = x_0 + \frac {kb} {\gcd \left\{{a,b}\right\}}, y = y_0 - \frac {ka} {\gcd \left\{{a,b}\right\}}$$

are also Bézout numbers.