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

A Wagstaff prime is a prime number of the form $(2^{\mathfrak{p}}+1)\mathrm{/}3$, where $\mathfrak{p}$ is an odd prime. Let $(L_n^{(k)}{)}_{n\ge 2-k}$ 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 $L_n^{(k)}=(2^{\mathfrak{p}}+1)\mathrm{/}3$ 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.