The Frobenius problem: a geometric approach

Abstract

For the well known Frobenius problem, we present a new geometric approach, based on the use of the $n$-dimensional lattice $\mathbb{Z}^n$, where $n$ is the number of generators. Within this approach we are able to study the cases of two and three generators. The main feature of our geometric representation is that we can nicely visualize the set of {\em gaps}, i.e., the non-representable positive integers.

In the case of two generators, we give a description of the set of gaps. Moreover, for any positive integer, $m$, we derive a simple expression for the denumerant $d(m;a,b)$.

We show that we can use the $2$-dimensional lattice associated to the set of generators $\{ a,b\}$ to study the Frobenius problem with generators $\{ a,b,c\}$. In particular, we give, as for two generators, a graphical representation of the set of gaps. For a large set of possible values of $c$, this representation allows us to simplify the computation of the Frobenius number and compute the number of gaps.