On the Frobenius Number and Genus of a Collection of Semigroups Generalizing Repunit Numerical Semigroups

2023/06/06 by Feihu Liu, Guoce Xin, Liu, Feihu +5 · 1 citation
Computer Science · Mathematics · #Combinatorics (math.CO) #Commutative Algebra and Its Applications #Computational Drug Discovery Methods #FOS: Mathematics #Number Theory (math.NT) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2306.03459

openalex publication_date 2023/06/06 · openalex created_date 2023/06/09 · openalex updated_date 2026/07/28

Abstract

Let A=(a₁, a₂, …, a_n) be a sequence of relative prime positive integers with a_i≥ 2. The Frobenius number F(A) is the largest integer not belonging to the numerical semigroup ⟨ A⟩ generated by A. The genus g(A) is the number of positive integer elements not in ⟨ A⟩. The Frobenius problem is to determine F(A) and g(A) for a given sequence A. In this paper, we study the Frobenius problem of A=(a,h₁a+b₁d,h₂a+b₂d,…,h_ka+b_kd) with some restrictions. An innovation is that d can be a negative integer. In particular, when A=(a,ba+d,b²a+(b²-1)/(b-1)d,…,b^ka+(b^k-1)/(b-1)d), we obtain formulas for F(A) and g(A) when a≥ k-1-(d-1)/(b-1). Our formulas simplify further for some special cases, such as Mersenne, Thabit, and repunit numerical semigroups. Finally, we partially solve an open problem for the Proth numerical semigroup.

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• On the Frobenius Number of Quotients of Numerical Semigroups

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