The Frobenius Problem for the Proth Numbers

2023/11/21 by Srivastava, Pranjal, Thakkar, Dhara
#11D07 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2311.12462

Abstract

Let n be a positive integer greater than 2. We define the Proth numerical semigroup, P_k(n), generated by \k 2ⁿ⁺ⁱ+1 | i ∈ ℕ\, where k is an odd positive number and k < 2ⁿ. 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.

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