On Wagstaff primes in the $k$-Lucas number sequence

Abstract

A Wagstaff prime is a prime number of the form $(2^{\mathfrak{p}}+1)/3$, where $\mathfrak{p}$ is an odd prime. Let $(L_n^{(k)})_{n\geq 2-k}$ be the $k$-Lucas number sequence defined by the recurrence relation $ L_n^{(k)} = L_{n-1}^{(k)} + \cdots + L_{n-k}^{(k)}$, for all $n \ge 2$, with initial terms \( L_0^{(k)} = 2 \) and \( L_1^{(k)} = 1 \) for all \( k \ge 2 \), and \( L_{2-k}^{(k)} = \cdots = L_{-1}^{(k)} = 0 \) for \( k \ge 3 \). In this paper, we show that the only solutions to the Diophantine equation $L_n^{(k)} = (2^{\mathfrak{p}}+1)/3$ are $(n,k,\mathfrak{p})\in\{(5,2,5),(6,4,7)\}\cup \{(2,k,3):k\ge 2\}$. We use linear forms in logarithms and the LLL reduction method to prove our result.