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The Square Terms in Generalized Lucas Sequence with Parameters $P$ And $Q$

Zafer Şi̇ar, Refi̇k Keski̇n

Abstract


Let $P$ and $Q$ be nonzero integers. Generalized Lucas sequence is defined as follows: $V_{0}=2$, $V_{1}=P$, and $V_{n+1}=PV_{n}+QV_{n-1}$ for $n\geq 1$. We assume that $P$ and $Q$ are odd relatively prime integers. Firstly, we determine all indices $n$ such that $V_{n}=kx^{2}$ and $V_{n}=2kx^{2}$ when $k|P$. Then, as an application of our these results, we find all solutions of the equations $V_{n}=3x^{2}$ and $V_{n}=6x^{2}$. Moreover, we find integer solutions of some Diophantine equations.

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DOI: http://dx.doi.org/10.7146/math.scand.a-23292

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ISSN 0025-5521 (print) ISSN 1903-1807 (online)

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