The Frobenius problem for numerical semigroups generated by sequences\n that satisfy a linear recurrence relation

2021/11/08 by Fabián Arias, Arias, Fabián, Jerson Borja +1
Computer Science · Mathematics · #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2111.04899

openalex publication_date 2021/11/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider a sequence of positive integers of the form caⁿ-d, n\≥ 1,\nwhere a, c and d are positive integers, a>1. For each n\≥ 1, let\nS_n be the submonoid of mathbb N generated by mathbf s_j=ca^n+j-d,\nwith j\∈ mathbb N. We obtain a numerical semigroup (1/e)S_n by dividing\nevery element of S_n by e=\gcd(S_n).\n We characterize the embedding dimension of S_n and describe a method to\nfind the minimal generating set of S_n. We also show how to find the maximum\nelement of the Ap 'ery set rm Ap(S_n, mathbf s₀), characterize the\nelements of rm Ap(S_n, mathbf s₀), and use these results to compute the\nFrobenius number of the numerical semigroup (1/e)S_n, where e=\gcd(S_n).\n

Related

• The Frobenius problem for generalized repunit numerical semigroups
• On the Frobenius Number and Genus of a Collection of Semigroups Generalizing Repunit Numerical Semigroups
• Numerical Semigroups Generated by Quadratic Sequences
• A Note on Generalized Repunit Numerical Semigroups
• The Frobenius problem for four numerical semigroups