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AI Summary & Key Takeaways
AI Insight⚡ TL;DR(Too Long; Didn't Read)
In this paper we give formulas, in terms of a, b, c, d, for the genus, the Frobenius number, and the set of pseudo-Frobenius numbers of ⟨a, b, cb − da⟩ in the case in which the interval $[a/c, b/d]$ contains some integer.
📌 3 Key Takeaways
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Objective & Hypothesis:If S is a numerical semigroup with embedding dimension equal to three whose minimal generators are pairwise relatively prime numbers, then S = ⟨a, b, cb − da⟩ with a, b, c, d positive integers such that gcd(a, b) = gcd(a,c) = gcd(b,d) = 1, c ∈ {2,...,a−1}, and a < b < cb−da.
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Key Findings & Data:If S is a numerical semigroup with embedding dimension equal to three whose minimal generators are pairwise relatively prime numbers, then S = ⟨a, b, cb − da⟩ with a, b, c, d positive integers such that gcd(a, b) = gcd(a,c) = gcd(b,d) = 1, c ∈ {2,...,a−1}, and a < b < cb−da.
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Conclusion & Impact:In this paper we give formulas, in terms of a, b, c, d, for the genus, the Frobenius number, and the set of pseudo-Frobenius numbers of ⟨a, b, cb − da⟩ in the case in which the interval $[a/c, b/d]$ contains some integer.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Frobenius problem · numerical semigroups · Apéry set
Abstract
If S is a numerical semigroup with embedding dimension equal to three whose minimal generators are pairwise relatively prime numbers, then S = ⟨a, b, cb − da⟩ with a, b, c, d positive integers such that gcd(a, b) = gcd(a,c) = gcd(b,d) = 1, c ∈ {2,...,a−1}, and a < b < cb−da. In this paper we give formulas, in terms of a, b, c, d, for the genus, the Frobenius number, and the set of pseudo-Frobenius numbers of ⟨a, b, cb − da⟩ in the case in which the interval $[a/c, b/d]$ contains some integer.
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Cite This Article
ROBLES-PÉREZ, A. M., ROSALES, J. C. (2022). The frobenius problem for some numerical semigroups with embedding dimension equal to three. Hacettepe Journal of Mathematics and Statistics, Cilt: 44 - Sayı: 4, 901–908. https://ajindex.com/articles/530260