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

Refik Keskin¹ ; Zafer Şiar²

¹ Department of Mathematics Sakarya University TR54187 Sakarya, Turkey
² Department of Mathematics Bilecik Şeyh Edebali University TR11030 Bilecik, Turkey

Colloquium Mathematicum, Tome 130 (2013) no. 1, pp. 27-38

Résumé

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 $1 r m$.

Détail

DOI : 10.4064/cm130-1-3

Keywords: nonzero integers sequences generalized fibonacci lucas numbers defined qu n geq qv n geq respectively paper assume geq odd there integer geq even integer completely solve equation geq geq equiv pmod even integer equiv pmod equiv pmod equation has solutions geq geq moreover there generalized lucas number lastly there generalized fibonacci number