The terms of the form $7kx^{2}$ in the generalized Lucas sequence with parameters $P$ and $Q$
Volume 173 / 2016
Acta Arithmetica 173 (2016), 81-95
MSC: 11B37, 11B39, 11B50.
DOI: 10.4064/aa8247-2-2016
Published online: 14 April 2016
Abstract
Let $V_{n}(P,Q)$ denote the generalized Lucas sequence with parameters $P$ and $Q.$ For all odd relatively prime values of $P$ and $Q$ such that $P^{2}+4Q \gt 0,$ we determine all indices $n$ such that $V_{n}(P,Q)=7kx^{2}$ when $k\, |\, P.$ As an application, we determine all indices $n$ such that the equation $V_{n}=21x^{2}$ has solutions.
