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Proposition: A Linear Term for 1 Using Two Co-prime Coefficients

Given two co-prime integers $a,b$, there exist integers $x,y$ such that the linear equation $ax+by=1$ holds. If $x_0,y_0$ is any solution, then all solutions have the form $$x=x_0+hb\wedge y=y_0-hb\quad\forall h\in\mathbb Z.$$

Notes

• see also extended greatest common divisor algorithm.

Table of Contents

Proofs: 1

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References

Bibliography

1. Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927
2. Jones G., Jones M.: "Elementary Number Theory (Undergraduate Series)", Springer, 1998