The Frobenius Problem for the Proth Numbers

Abstract

Let $n$ be a positive integer greater than $2$. We define \textit{the Proth numerical semigroup}, $P_k(n)$, generated by $\{k{2}^{n+i}+1\mid i\in \mathbb{N}\}$, where $k$ is an odd positive number and $k<2^n$. In this paper, we introduce the Frobenius problem for the Proth numerical semigroup $P_k(n)$ and give formulas for the embedding dimension of $P_k(n)$. We solve the Frobenius problem for $P_k(n)$ by giving a closed formula for the Frobenius number. Moreover, we show that $P_k(n)$ has an interesting property such as being Wilf.

A preliminary version of this paper will appear in CALDAM 2024