The Frobenius problem for numerical semigroups generated by the Thabit numbers of the first, second kind base $b$ and the Cunningham numbers
Bull. Korean Math. Soc. 2020 Vol. 57, No. 3, 623-647
https://doi.org/10.4134/BKMS.b190399
Published online May 31, 2020
Kyunghwan Song
Ewha Womans University
Abstract : The greatest integer that does not belong to a numerical semigroup $S$ is called the Frobenius number of $S$. The Frobenius problem, which is also called the coin problem or the money changing problem, is a mathematical problem of finding the Frobenius number. In this paper, we introduce the Frobenius problem for two kinds of numerical semigroups generated by the Thabit numbers of the first kind, and the second kind base $b$, and by the Cunningham numbers. We provide detailed proofs for the Thabit numbers of the second kind base $b$ and omit the proofs for the Thabit numbers of the first kind base $b$ and Cunningham numbers.
Keywords : Frobenius problem, Thabit numerical semigroups base $b$, Ap\'{e}ry set, genus, type
MSC numbers : Primary 11A67, 20M30
Supported by : This research was supported by the National Research Foundation of Korea grant funded by the Korea government (Grant Number: NRF2018R1A2A1A05079095).
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