◀ ▲ ▶Branches / Number-theory / Proof

Proof

(related to Proposition: A Linear Term for 1 Using Two Co-prime Coefficients)

• The linear equation $ax+by=1$ is solvable, which follows immediately from the existence of solutions of an LDE with more variables for $r=2$ and the coefficients $a,b$ being co-prime.
• If $x_0,y_0$ are solutions of the above equation, then it follows from all solutions given a solution of an LDE with two variables that all solutions have the form $x=x_0+hb$ and $y=y_0-hb$ for all integers $h\in\mathbb Z.$
  ∎

Thank you to the contributors under CC BY-SA 4.0!

Github:

References

Bibliography

1. Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927