Using Double-Loop digraphs for solving Frobenius' Problems

Abstract

Given a set $A=\{a_1,\dots ,a_k\}$ with $1⩽a_1<\dots <a_k$ and $\text{gcd}(a_1,\dots ,a_k)=1$, let us denote

$$\mathcal{R}(A)=\{m\in \mathbb{N}|\exists x_1,\dots ,x_k\in \mathbb{N}:m=\sum _{i=1}^k{x}_i{a}_i\}$$

and $\overline{\mathcal{R}}(A)=\mathbb{N}\backslash \mathcal{R}(A)$. The classical study of the Frobenius' Problem for a given set A is the computation of the number $f(A)=\max \overline{\mathcal{R}}(A)$ (also called the Frobenius Number) and $|\overline{\mathcal{R}}(A)|$.

In this work we propose a method to explicitly find the set $\overline{\mathcal{R}}(A)$ in a closed form when $k=3$. As far as we know, this is the first proposed method to find a set $\overline{\mathcal{R}}(A)$.