The Frobenius Problem for the Proth Numbers
Source
Lecture Notes in Computer Science Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics
ISSN
03029743
Date Issued
2024-01-01
Author(s)
Srivastava, Pranjal
Thakkar, Dhara
Abstract
Let n be a positive integer greater than 2. We define the Proth numerical semigroup, P<inf>k</inf>(n), generated by {k2n+i+1∣i∈N}, where k is an odd positive number and k< 2 <sup>n</sup>. In this paper, we introduce the Frobenius problem for the Proth numerical semigroup P<inf>k</inf>(n) and give formulas for the embedding dimension of P<inf>k</inf>(n). We solve the Frobenius problem for P<inf>k</inf>(n) by giving a closed formula for the Frobenius number. Moreover, we show that P<inf>k</inf>(n) has an interesting property such as being Wilf.
Subjects
Apéry Set | Combinatorial techniques | Frobenius problem | Numerical semigroup | Proth Number | pseudo-Frobenius number | type | Wilf’s conjecture