Abstract
A numerical semigroup S is a subset of the set of nonnegative integers closed under addition, containing the zero element and with finite complement in \({\mathbb {N}}_{0}\) (this finite cardinality is named the genus of S). It is well-known that every numerical semigroup S is finitely generated and there are many works concerning the properties of numerical semigroups with a particular type of generators. For instance, Song (Bull Korean Math Soc 57:623–647, 2020) worked on these semigroups whose generators are Thabit numbers of the first, second kind base b and Cunningham numbers. A classical result of Sylvester ensures that if \(\gcd (a,b) = 1\), then the numerical semigroup \(\langle a, b \rangle \) has genus \(\frac{(a-1)(b-1)}{2}\). In this paper, we search for two-generator numerical semigroups whose generators and/or the genus are related to Fibonacci numbers. Our propose is fixing the sets A, B and G and looking for triples \((a, b, g) \in A\times B\times G\), where at least one of the sets is related to the Fibonacci numbers.
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Notes
On-Line Encyclopedia of Integer Sequences.
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Acknowledgements
The authors thank to the anonymous referees for their careful corrections and their comments that helped to improve the quality of the paper.
Funding
Matheus Bernardini was supported by University of Brasilia, Edital DPI/DIRPE 03/2020. Diego Marques is supported by CNPq—Brazil. Pavel Trojovský was supported by the Project of Excellence PrF UHK no. 2214/2021, University of Hradec Králové, Czech Republic.
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Bernardini, M., Marques, D. & Trojovský, P. On two-generator Fibonacci numerical semigroups with a prescribed genus. RACSAM 115, 149 (2021). https://doi.org/10.1007/s13398-021-01091-7
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DOI: https://doi.org/10.1007/s13398-021-01091-7