Arf Numerical Semigroups With Prime Multiplicity

Abstract

In the literature, parametrizations are given for Arf numerical semigroups with small multiplicity and arbitrary conductor. From those parametrizations, formulas are obtained for the number of such Arf numerical semigroups. These formulas show that the number of Arf numerical semigroups with multiplicity 3, 5 or 7 and arbitrary conductor depends only on the congruence class of the conductor modulo the multiplicity. In a recent work with S. Ilhan and M. Suer, observing that the same is true for Arf numerical semigroups with multiplicity 11 and 13, we asked if that was true for Arf numerical semigroups with arbitrary prime multiplicity. In the present work this question is answered affirmatively.