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

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.