Journal of Integer Sequences, Vol. 25 (2022), Article 22.7.2

Generalization of Jarden's Theorem

E. L. Roettger
Department of General Education
Mount Royal University
4825 Mount Royal Gate SW
Calgary, AB T3E 6K6

H. C. Williams
Department of Mathematics and Statistics
University of Calgary
2500 University Drive NW
Calgary, AB T2N 1N4


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

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


$\displaystyle 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 $(V_n)_{n\geq 0}$.

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Received May 30 2022; revised version received August 8 2022. Published in Journal of Integer Sequences, August 8 2022.

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