Markoff-Rosenberger triples and generalized Lucas sequences

TL;DR

This paper investigates solutions to the Markoff-Rosenberger equation where variables are generalized Lucas numbers, providing bounds on indices and explicitly solving cases involving balancing and Jacobsthal numbers.

Contribution

It offers new bounds on indices and fully solves specific equations involving balancing and Jacobsthal numbers within the Markoff-Rosenberger framework.

Findings

Established upper bounds for indices in solutions

Solved equations with balancing numbers

Solved equations with Jacobsthal numbers

Abstract

We consider the Markoff-Rosenberger equation $$ax^2+by^2+cz^2=dxyz$$ with $(x,y,z)=(U_i,U_j,U_k),$ where $U_i$ denotes the $i$-th generalized Lucas number of first/second kind. We provide upper bound for the minimum of the indices and we apply the result to completely resolve concrete equations, e.g. we determine solutions containing only balancing numbers and Jacobsthal numbers, respectively.