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

Year 2019, Volume: 48 Issue: 4, 1035 - 1045, 08.08.2019

Refik Keskin , Ümmügülsüm Öğüt

Abstract

Let $P\geq3$ be an integer and $(V_{n})$ 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 $P\equiv 1({mod}11)$. Moreover, we show that if the equation $V_{n}=11x^{2}-1$ has a solution, then $P\equiv2({mod}8)$ and $P\equiv-1({mod}11).$

Keywords

generalized Lucas numbers, congruences, diophantine equation

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