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2024 | OriginalPaper | Chapter

p-Numerical Semigroups of the Triples of the Sequence \((a^n-b^n)/(a-b)\)

Authors : Takao Komatsu, Ruze Yin

Published in: Mathematical Methods for Engineering Applications

Publisher: Springer Nature Switzerland

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Abstract

For a non-negative integer p, we give explicit formulas for the p-Frobenius number and the p-genus of numerical semigroups of \((\nu _n,\nu _{n+1},\nu _{n+2})\), where \(\nu _n=(a^n-b^n)/(a-b)\) with \(\gcd (a,b)=1\) and \(a>b>1\). Here, the p-numerical semigroup \(S_p\) is the set of integers whose non-negative integral linear combinations of given positive integers are expressed more than p ways. When \(p=0\), \(S_0\) with the 0-Frobenius number and the 0-genus is the original numerical semigroup \(S_0\) with the Frobenius number and the genus. Symmetric properties of numerical semigroups are important to characterize the numerical semigroup. In recent works, some closed formulas of p-Frobenius numbers have been successfully given, but no symmetric property has been found. We give a symmetric property of this numerical semigroup.

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Title: p-Numerical Semigroups of the Triples of the Sequence
Authors: Takao Komatsu
Ruze Yin
Publisher: Springer Nature Switzerland
Book: Mathematical Methods for Engineering Applications
Print ISBN: 978-3-031-49217-4

Electronic ISBN: 978-3-031-49218-1

Copyright Year: 2024
DOI: https://doi.org/10.1007/978-3-031-49218-1_2

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