Markoff-Rosenberger triples and generalized Lucas sequences
Abstract
We consider the Markoff-Rosenberger equation ax2+by2+cz2=dxyz with (x,y,z)=(Ui,Uj,Uk), where Ui 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.
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