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.
Categories: math.NT