$k$-Pell-Lucas numbers as Product of Two Repdigits

TL;DR

This paper characterizes all $k$-generalized Pell-Lucas numbers that can be expressed as the product of two repdigits, extending previous results on classical Pell-Lucas numbers.

Contribution

It generalizes prior work by identifying all such numbers for the $k$-generalized Pell-Lucas sequence, covering any integer $k \\geq 2$.

Findings

Identified all $k$-generalized Pell-Lucas numbers that are products of two repdigits.

Extended previous results from classical Pell-Lucas numbers to the generalized sequence.

Provided a complete characterization for all $k \\geq 2$.

Abstract

For any integer $k \geq 2$, let $\{Q_{n}^{(k)} \}_{n \geq -(k-2)}$ denote the $k$-generalized Pell-Lucas sequence which starts with $0, \dots ,2,2$($k$ terms) where each next term is the sum of the $k$ preceding terms. In this paper, we find all the $k$-generalized Pell-Lucas numbers that are the product of two repdigits. This generalizes a result of Erduvan and Keskin \cite{Erduvan1} regarding repdigits of Pell-Lucas numbers.