On the Lucas sequence equations $V_{n}=kV_{m}$ and $U_{n}=kU_{m}$

Volume 130 / 2013

Refik Keskin, Zafer Şiar Colloquium Mathematicum 130 (2013), 27-38 MSC: Primary 11B37; Secondary 11B39. DOI: 10.4064/cm130-1-3

Abstract

Let $P$ and $Q$ be nonzero integers. The sequences of generalized Fibonacci and Lucas numbers are defined by $U_{0}=0$, $U_{1}=1$ and $ U_{n+1}=PU_{n}-QU_{n-1}$ for $n\geq 1$, and $V_{0}=2$, $V_{1}=P$ and $ V_{n+1}=PV_{n}-QV_{n-1}$ for $n\geq 1$, respectively. In this paper, we assume that $P\geq 1$, $Q$ is odd, $(P,Q)=1$, $V_{m}\ne 1$, and $V_{r}\ne 1$. We show that there is no integer $x$ such that $V_{n}=V_{r}V_{m}x^{2}$ when $m\geq 1$ and $r$ is an even integer. Also we completely solve the equation $V_{n}=V_{m}V_{r}x^{2}$ for $m\geq 1$ and $r\geq 1$ when $Q\equiv 7\pmod{8}$ and $x$ is an even integer. Then we show that when $P\equiv 3\pmod{4}$ and $Q\equiv 1\pmod{4}$, the equation $V_{n}=V_{m}V_{r}x^{2}$ has no solutions for $% m\geq 1$ and $r\geq 1$. Moreover, we show that when $P>1$ and $Q=\pm 1$, there is no generalized Lucas number $V_{n}$ such that $V_{n}=V_{m}V_{r}$ for $m>1$ and $r>1$. Lastly, we show that there is no generalized Fibonacci number $U_{n}$ such that $U_{n}=U_{m}U_{r}$ for $Q=\pm 1$ and $1< r< m$.

Authors

  • Refik KeskinDepartment of Mathematics
    Sakarya University
    TR54187 Sakarya, Turkey
    e-mail
  • Zafer ŞiarDepartment of Mathematics
    Bilecik Şeyh Edebali University
    TR11030 Bilecik, Turkey
    e-mail

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