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

TL;DR

This work proves that for every $k\ge3$, no term of the $k$-generalized Lucas sequence $L_n^{(k)}$ equals a palindromic concatenation of two distinct repdigits, extending the known $k=2$ result. The authors combine a Binet-like representation with linear forms in logarithms (Matveev) and aggressive LLL-based lattice reductions to bound the growth parameters $n$, $\ell$, and $m$, then perform targeted computations to exhaust all possibilities. The analysis yields explicit, sharp reductions: initial digit-count bounds, then case-by-case eliminations (including exhaustive searches for small $\ell,m$ and $n$ ranges, and large-$k$ asymptotics), ultimately ruling out any solutions. The results advance our understanding of Diophantine properties of recurrence sequences and demonstrate the effectiveness of Baker theory plus lattice reduction in handling repdigit-palindrome questions in generalized Lucas 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.

Theorems & Definitions (13)

• Theorem 1.1
• Lemma 2.1: Lemma 2, gomez
• Definition 3.1
• Theorem 3.1: Matveev, see Theorem 9.4 in Matveev
• Definition 3.2
• Lemma 3.1
• Lemma 3.2: Lemma VI.1 in SMA
• Lemma 4.1
• proof
• Lemma 4.2