On The Lucas Sequence Equations $V_{n}=7\square$ and $V_{n}=7V_{m}\square$

Abstract

Let P be a nonzero integer and let $(U_{n})$ and $(V_{n})$ denote Lucas sequences of first and second kind defined by $U_{0}=0, U_{1}=1; V_{0}=2, V_{1}=P;$ and $U_{n+1}=PU_{n}+U_{n-1},V_{n+1}=PV_{n}+V_{n-1}$ for $n\ge 1.$ In this study, when P is odd, we show that the equation $U_{n} =7\square$ has only the solution $(n,P)=(2,7\square )$ when 7|P and the equation $V_{n}=7\square$ has only the solution $(n,P)=(1,7\square )$ when 7|P or $(n,P)=(4,1)$ when $P^{2}\equiv 1( \text{ mod } 7).$ In addition, we show that the equation $V_{n}=7V_{m}\square$ has a solution if and only if $P^{2}=-3+7\square$ and $(n,m)=(3,1).$ Moreover, we show that the equation $U_{n}=7U_{m} \square$ has only the solution $(n,m,P,\square )=(8,4,1,1)$ when P is odd.