Generalization of Jarden's Theorem

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

and

is odd and positive, then

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

where

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

He went on to show that if

and

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

which are odd and square-free to the Lucas sequence $(V_n)_{n\geq 0}$ .

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

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

E. L. Roettger
Department of General Education
Mount Royal University
4825 Mount Royal Gate SW
Calgary, AB T3E 6K6
Canada
H. C. Williams
Department of Mathematics and Statistics
University of Calgary
2500 University Drive NW
Calgary, AB T2N 1N4
Canada