On Palindromic forms in the $k$-Lucas sequence composed of two distinct Repdigits

Abstract

For integers $k\ge 2$, the $k$-generalized Lucas sequence $\{L_n^{(k)}{\}}_{n\ge 2-k}$ is defined by the recurrence relation \[ L_n^{(k)} = L_{n-1}^{(k)} + \cdots + L_{n-k}^{(k)} \quad \text{for } n \geq 2, \] with initial terms given by $L_0^{(k)}=2$, $L_1^{(k)}=1$, and $L_{2-k}^{(k)}=\cdots =L_{-1}^{(k)}=0$. In this paper, we extend work in \cite{Lucas} and show that the result in \cite{Lucas} still holds for $k\ge 3$, that is, we show that for $k\ge 3$, there is no $k$-generalized Lucas number appearing as a palindrome formed by concatenating two distinct repdigits.

arXiv comment14 pages. arXiv admin note: text overlap with arXiv:2504.14261, arXiv:2504.10138