On Numerical Experiments with Symmetric Semigroups Generated by Three Elements and Their Generalization

2009/03/22 by Leonid G. Fel, Fel, Leonid G.
Computer Science · Mathematics · #20M14 #FOS: Mathematics #Matrix Theory and Algorithms #Number Theory (math.NT) #Spectral Theory in Mathematical Physics #math.NT #msc:20M14

paper · pdf · doi:10.48550/arxiv.0903.3756

17 pages

arxiv created 2009/03/22 · openalex publication_date 2009/03/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a simple explanation of numerical experiments of V. Arnold with two sequences of symmetric numerical semigroups, S(4,6+4k,87-4k) and S(9,3+9k,85-9k) generated by three elements. We present a generalization of these sequences by numerical semigroups S(r₁²,r₁r₂+r₁²k,r₃-r₁²k), k∈\mathbb Z, r₁,r₂,r₃∈\mathbb Z⁺, r₁≥ 2 and gcd(r₁,r₂)=gcd(r₁,r₃)=1, and calculate their universal Frobenius number Phi(r₁,r₂,r₃) for the wide range of k providing semigroups be symmetric. We show that this kind of semigroups admit also nonsymmetric representatives. We describe the reduction of the minimal generating sets of these semigroups up to r₁²,r₃-r₁²k for sporadic values of k and find these values by solving the quadratic Diophantine equation.

Related

• Genus of numerical semigroups generated by three elements
• Almost symmetric numerical semigroups with given Frobenius number and type
• p-numerical semigroups with p-symmetric properties
• Numerical semigroups generated by squares, cubes and quartics of three consecutive integers
• Almost symmetric numerical semigroups with odd generators