Generalized Lucas numbers of the form $11x^{2}\mp 1$

Authors : Refik KESKİN, Ümmügülsüm ÖĞÜT

Pages : 1035-1045

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Publication Date : 2019-08-08

Article Type : Research Paper

Abstract :Let $P\geq3$ be an integer and $insert ignore into journalissuearticles values(V_{n});$ denote generalized Lucas sequence defined by $V_{0}=2,V_{1}=P,$ and $V_{n+1}=PV_{n}-V_{n-1}$ for $n\geq1.$ In this study, we solve the equation $V_{n}=11x^{2}\mp1.$ We show that the equation $V_{n}=11x^{2}+1$ has a solution only when $n=1$ and $P\equiv 1insert ignore into journalissuearticles values({mod}11);$. Moreover, we show that if the equation $V_{n}=11x^{2}-1$ has a solution, then $P\equiv2insert ignore into journalissuearticles values({mod}8);$ and $P\equiv-1insert ignore into journalissuearticles values({mod}11);.$
Keywords : generalized Lucas numbers, congruences, diophantine equation