Products of Three $k-$Generalized Lucas Numbers as Repdigits

TL;DR

This paper determines all cases where the product of three $k$-generalized Lucas numbers results in a repdigit number, solving a specific Diophantine equation for these sequences.

Contribution

It provides a complete characterization of repdigit products of three $k$-generalized Lucas numbers, extending previous work on Lucas sequences.

Findings

Identified all solutions to the Diophantine equation involving three $k$-generalized Lucas numbers and repdigits.

Established bounds and conditions for the solutions.

Extended the understanding of multiplicative properties of generalized Lucas sequences.

Abstract

Let $ k \geq 2 $ and let $ ( L_{n}^{(k)} )_{n \geq 2-k} $ be the $k-$generalized Lucas sequence with certain initial $ k $ terms and each term afterward is the sum of the $ k $ preceding terms. In this paper, we find all repdigits which are products of arbitrary three terms of $k-$generalized Lucas sequences. Thus, we find all non negative integer solutions of Diophantine equation $L_n^{(k)}L_m^{(k)}L_l^{(k)} =a \left( \dfrac{10^{d}-1}{9} \right)$ where $n\geq m \geq l \geq 0$ and $1 \leq a \leq 9.$