Generalization of a theorem of Jarden (Q2883408)

For the Fibonacci numbers \(F_n\) \textit{D. Jarden} [Am. Math. Mon. 53, 425--427 (1946; Zbl 0060.08512)] proved that for positive integers \(m,n\) \(F_mF_n\mid F_{mn}\) if and only if \((m,n)=1,2\), or 5. Here this is generalized to nondegenerate Lucas sequences \(u(a,b)\) by showing that \(u_mu_n\mid u_{mn}\) only if \((m,n)\) takes on a limited number of values. Conditions are given for \((m,n)=2,3,4,5,7,13\).NEWLINENEWLINEThe case \((m,n)=1\) was proved in \textit{Y. Bilu, G. Hanrot} and \textit{P. M. Voutier} [J. Reine Angew. Math. 539, 75--122 (2001; Zbl 0995.11010)].