Markoff-Rosenberger triples and generalized Lucas sequences

Hayder Hashim; László Szalay; Szabolcs Tengely

paper

Markoff-Rosenberger triples and generalized Lucas sequences

arXiv:2005.13935

Abstract

We consider the Markoff-Rosenberger equation $a x^{2} + b y^{2} + c z^{2} = d x y z$ 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.