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Generalization of Jarden's Theorem E. L. Roettger
Department of General Education
Mount Royal University
4825 Mount Royal Gate SW
Calgary, AB T3E 6K6
Canada
eroettger@mtroyal.ca

H. C. Williams
Department of Mathematics and Statistics
University of Calgary
2500 University Drive NW
Calgary, AB T2N 1N4
Canada
hwilliam@ucalgary.ca

in

Abstract:

Let $(F_n)_{n\geq 0}$ and $(L_n)_{n\geq 0}$ denote the sequences of Fibonacci numbers and Lucas numbers, respectively. In 1950 Dov Jarden showed that if $m=5$ and $n$ is odd and positive, then

$$L_{mn}/L_n=A_nB_n,$$

where

$$A_n=5F_n^2-5F_n+1, \quad B_n=5F_n^2+5F_n+1.$$

He went on to show that if $n$ and $k$ are both odd and positive and $\eta$ is the value of the Legendre symbol $(k\vert 5)$, then $A_n \mid A_{kn}$, $B_n \mid B_{kn}$ when $\eta=1$ and $A_n \mid B_{kn}$, $B_n\mid A_{kn}$ when $\eta=-1$. In this paper we show how to generalize these results for values of $m$ which are odd and square-free to the Lucas sequence $(V_n)_{n\geq 0}$.