On the Multiplicity of a Proportionally Modular Numerical Semigroup

Abstract

A proportionally modular numerical semigroup is the set $S\left(a,b,c\right)$ of nonnegative integer solutions to a Diophantine inequality of the form $ax\mathrm{mod}b\le cx$ , where $a,b$ , and $c$ are positive integers. A formula for the multiplicity of $S\left(a,b,c\right)$ , that is, $m\left(S\left(a,b,c\right)\right)=⌈kb/a⌉$ for some positive integer $k$ , is given by A. Moscariello. In this paper, we give a new proof of the formula and determine a better bound for $k$ . Furthermore, we obtain $k=1$ for various cases and a formula for the number of the triples $\left(a,b,c\right)$ such that $k\ne 1$ when the number $a-c$ is fixed.