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The Frobenius number of a family of numerical semigroups with embedding dimension 5

Kurt Herzinger, Emelia McLaughlin and Julie Trimber

Vol. 16 (2023), No. 5, 833–847
Abstract

We answer the Frobenius problem for a specific family of numerical semigroups with embedding dimension 5. The numerical semigroups in this family are called fulcrum numerical semigroups and are related to balanced and unitary numerical semigroups introduced in earlier research. We show that fulcrum numerical semigroups come in two distinct varieties and provide formulas for the Frobenius number and genus for both.

Keywords
numerical semigroup, Frobenius number, Apéry set, genus
Mathematical Subject Classification
Primary: 20M14
Milestones
Received: 6 June 2022
Revised: 23 December 2022
Accepted: 23 December 2022
Published: 9 December 2023

Communicated by Nathan Kaplan
Authors
Kurt Herzinger
Department of Mathematical Sciences
U.S. Air Force Academy
Air Force Academy, CO
United States
Emelia McLaughlin
Department of Mathematical Sciences
U.S. Air Force Academy
Air Force Academy, CO
United States
Julie Trimber
Department of Mathematical Sciences
U.S. Air Force Academy
Air Force Academy, CO
United States