Hokkaido Mathematical Journal

Generalized Lucas Numbers of the form $wx^{2}$ and $wV_{m}x^{2}$

Merve GÜNEY DUMAN, Ümmügülsüm ÖĞÜT, and Refik KESKİN

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Abstract

Let $P\geq 3$ be an integer. Let $(V_{n})$ denote generalized Lucas sequence defined by $V_{0}=2$, $V_{1}=P$, and $V_{n+1}=PV_{n}-V_{n-1}$ for $n\geq 1$. In this study, when $P$ is odd, we solve the equation $V_{n}=wx^{2}$ for some values of $w$. Moreover, when $P$ is odd, we solve the equation $V_{n}=wkx^{2}$ with $k \mid P$ and $k \gt 1$ for $w=3,11,13$. Lastly, we solve the equation $V_{n}=wV_{m}x^{2}$ for $w=7,11,13$.

Article information

Source
Hokkaido Math. J., Volume 47, Number 3 (2018), 465-480.

Dates
First available in Project Euclid: 26 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.hokmj/1537948825

Digital Object Identifier
doi:10.14492/hokmj/1537948825

Mathematical Reviews number (MathSciNet)
MR3858373

Zentralblatt MATH identifier
06959098

Subjects
Primary: 11B37: Recurrences {For applications to special functions, see 33-XX} 11B39: Fibonacci and Lucas numbers and polynomials and generalizations

Keywords
Generalized Lucas sequence Generalized Fibonacci sequence congruence square terms in Lucas sequences

Citation

GÜNEY DUMAN, Merve; ÖĞÜT, Ümmügülsüm; KESKİN, Refik. Generalized Lucas Numbers of the form $wx^{2}$ and $wV_{m}x^{2}$. Hokkaido Math. J. 47 (2018), no. 3, 465--480. doi:10.14492/hokmj/1537948825. https://projecteuclid.org/euclid.hokmj/1537948825


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