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

TL;DR

This paper proves that for generalized Lucas sequences with k ≥ 3, no sequence number forms a palindromic number by concatenating two different repdigits, extending previous results for smaller k.

Contribution

It extends prior work to show that for all k ≥ 3, no Lucas sequence term is a palindrome made from two distinct repdigits.

Findings

No k-Lucas number is a palindrome formed by two distinct repdigits for k ≥ 3.

The result generalizes previous findings for smaller k.

The proof confirms the uniqueness of repdigit-based palindromes in these sequences.

Abstract

For integers $k \geq 2$, the $k$-generalized Lucas sequence $\{L_n^{(k)}\}_{n \geq 2-k}$ 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.