On the Multiplicity of a Proportionally Modular Numerical Semigroup
A proportionally modular numerical semigroup is the set Sa,b,c of nonnegative integer solutions to a Diophantine inequality of the form ax mod b≤cx, where a,b, and c are positive integers. A formula for the multiplicity of Sa,b,c, that is, mSa,b,c=kb/a for some positive integer k, is given by A. Mos...
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| Format: | Article |
| Language: | English |
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Wiley
2021-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2021/3982297 |
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| author | Ze Gu |
| author_facet | Ze Gu |
| author_sort | Ze Gu |
| collection | DOAJ |
| description | A proportionally modular numerical semigroup is the set Sa,b,c of nonnegative integer solutions to a Diophantine inequality of the form ax mod b≤cx, where a,b, and c are positive integers. A formula for the multiplicity of Sa,b,c, that is, mSa,b,c=kb/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. |
| format | Article |
| id | doaj-art-a2f65dda39c64215a23b458480ab33ed |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-a2f65dda39c64215a23b458480ab33ed2025-02-03T01:24:41ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2021-01-01202110.1155/2021/39822973982297On the Multiplicity of a Proportionally Modular Numerical SemigroupZe Gu0School of Mathematics and Statistics, Zhaoqing University, Zhaoqing, Guangdong 526061, ChinaA proportionally modular numerical semigroup is the set Sa,b,c of nonnegative integer solutions to a Diophantine inequality of the form ax mod b≤cx, where a,b, and c are positive integers. A formula for the multiplicity of Sa,b,c, that is, mSa,b,c=kb/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.http://dx.doi.org/10.1155/2021/3982297 |
| spellingShingle | Ze Gu On the Multiplicity of a Proportionally Modular Numerical Semigroup Discrete Dynamics in Nature and Society |
| title | On the Multiplicity of a Proportionally Modular Numerical Semigroup |
| title_full | On the Multiplicity of a Proportionally Modular Numerical Semigroup |
| title_fullStr | On the Multiplicity of a Proportionally Modular Numerical Semigroup |
| title_full_unstemmed | On the Multiplicity of a Proportionally Modular Numerical Semigroup |
| title_short | On the Multiplicity of a Proportionally Modular Numerical Semigroup |
| title_sort | on the multiplicity of a proportionally modular numerical semigroup |
| url | http://dx.doi.org/10.1155/2021/3982297 |
| work_keys_str_mv | AT zegu onthemultiplicityofaproportionallymodularnumericalsemigroup |