A Complete List of all Numbers not of the Form $ax+by$

TL;DR

This paper provides an explicit complete list of all numbers that cannot be expressed as a linear combination of two coprime positive integers, advancing understanding of the Frobenius coin problem.

Contribution

It offers a simple explicit description of all nonrepresentable numbers and presents two different proofs, including one using Eisenstein's lattice point counting techniques.

Findings

Complete list of nonrepresentable numbers provided

Two proofs of the main result, including a geometric approach

Enhanced understanding of Frobenius coin problem solutions

Abstract

For given coprime positive integers $a$ and $b$, the classical Frobenius coin problem asked to find the largest number that cannot be expressed in the form $ax+by$ for nonnegative integers $x$ and $y$, also known as the Frobenius number. Sylvester answered this question and also discovered the number of these nonrepresentable numbers. In recent times, a lot of progress has been made regarding the sums of a fixed power of these nonrepresentable numbers, commonly known as the Sylvester sums. However, the actual list of nonrepresentable numbers has remained mysterious so far. In this note, we obtain a complete list, that is a simple explicit description of these nonrepresentable numbers. We give two different proofs of our result. The first one enables us to use Eisenstein's lattice point counting techniques to study the set of nonrepresentable numbers, whereas the second one is a short direct proof.