On concatenations of two $k$-generalized Lucas numbers

For an integer $k\ge 2$, the sequence of $k$-generalized Lucas numbers is defined by the recurrence relation $L_n^{(k)}=L_{n-1}^{(k)}+\cdots +L_{n-k}^{(k)}$ for all $n\ge 2$, with initial conditions $L_0^{(k)}=2$, $L_1^{(k)}=1$ for all $k\ge 2$, and $L_{2-k}^{(k)}=\cdots =L_{-1}^{(k)}=0$ for $k\ge 3$. In this paper, we determine all $k$-generalized Lucas numbers that are concatenations of two terms of the same sequence and completely solve this problem for $k\ge 3$. Our approach combines nonzero lower bounds for linear forms in logarithms, reduction techniques based on the Baker–Davenport method and the LLL-algorithm, together with continued fraction analysis and computational verification using SageMath.