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

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 \(U_{n}(P,1)=7\square \) has only the solution \((n,P)=(2,7\square )\) when 7|P and the equation \(V_{n}(P,1)=7\square \) has only the solution \((n,P)=(1,7\square )\) when 7|P or \((n,P)=(4,1)\) when \(P^{2}\equiv 1({\mathrm{mod }}\, 7)\). In addition, we show that the equation \(V_{n}(P,1)=7V_{m}(P,1) \square \) has a solution if and only if \(P^{2}=-3+7\square \) and \((n,m)=(3,1)\). Moreover, we show that the equation \(U_{n}(P,1)=7U_{m}(P,1) \square \) has only the solution \((n,m,P,\square )=(8,4,1,1)\) when P is odd.