On the Multiplicity of a Proportionally Modular Numerical Semigroup. (8th September 2021)

• Record Type: Journal Article
• Authors: Gu, Ze
• Other Names: Peris Alfred Academic Editor.
• Is Part Of: Discrete dynamics in nature and society. Volume 2021(2021)
• Publication Date: 2021-09-08
• Subjects: System analysis -- Periodicals; Dynamics -- Periodicals; Chaotic behavior in systems -- Periodicals; Differentiable dynamical systems -- Periodicals; 003.05
• DOI: 10.1155/2021/3982297
• Languages: English
• ISSNs: 1026-0226

Abstract

Abstract : A proportionally modular numerical semigroup is the set S a, b, c of nonnegative integer solutions to a Diophantine inequality of the form a x mod b ≤ c x, where a, b, and c are positive integers. A formula for the multiplicity of S a, b, c, that is, m S a, b, c = k b / a for some positive integer k, is given by A. Moscariello. In this paper, we give a new proof of the formula and determine a better bound for k . Furthermore, we obtain k = 1 for various cases and a formula for the number of the triples a, b, c such that k ≠ 1 when the number a − c is fixed.