An inductive proof of the Frobenius coin problem of two denominations

TL;DR

This paper presents an inductive proof of the Frobenius coin problem for two denominations, establishing the existence of solutions for all sufficiently large amounts and providing a recursive algorithm for finding these solutions.

Contribution

It offers a new inductive proof and a constructive recursive algorithm for the Frobenius coin problem with two denominations.

Findings

Proves solutions exist for all amounts greater than ab - a - b

Provides a recursive algorithm to find solutions

Establishes a constructive approach for the problem

Abstract

Let $a,b$ be positive, relatively prime, integers. We prove, using induction, that for every $d > ab-a-b$ there exist $x,y\in\mathbb{Z}_{\geq 0}$, such that $d=ax+by$. As a byproduct, we obtain a constructive recursive algorithm for identifying appropriate $x,y$ as above.