On $k$-Pell numbers that are Palindromes formed by two distinct Repdigits

TL;DR

The paper addresses which terms of the $k$-generalized Pell sequence $P_n^{(k)}$ can equal palindromes formed by two distinct repdigits. It fuses a Binet-type expansion and dominant-root analysis with explicit lower bounds for linear forms in logarithms via Matveev's theorem, together with LLL-based lattice reduction, to derive finite bounds on the involved parameters and perform a targeted search. The authors prove that only two instances occur: $P_{8}^{(3)}=545$ and $P_{7}^{(5)}=232$. This result highlights the rarity of repdigit-palindromic representations within generalized Pell sequences and demonstrates a powerful combination of transcendence methods and lattice reduction for Diophantine problems in linear recurrences.

Abstract

Let $k \ge 2$ and consider the sequence $\{P_n^{(k)}\}_{n \ge 2-k}$ of $k$-generalized Pell numbers, which begins with the first $k$ terms as $0, \ldots, 0, 0, 1$, and satisfies the recurrence relation $P_n^{(k)} = 2P_{n-1}^{(k)} + P_{n-2}^{(k)} + \cdots + P_{n-k}^{(k)}$ for all $n \ge 2$. In this work, we identify all terms in the $k$-Pell sequence that can be expressed as palindromes formed by concatenating two distinct repdigits.

Theorems & Definitions (12)

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