Malaysian Journal of Mathematical Sciences, June 2024, Vol. 18, No. 2
On the Solutions of the Lucas Sequence Equation $\pm\frac{1}{V_n(P_2,Q_2)}=\sum_{k=1}^\infty\frac{U_{k-1}(P_1,Q_1)}{x^k}$
Abdulzahra, A. A. and Hashim, H. R.
Corresponding Email: hayderr.almuswi@uokufa.edu.iq
Received date: 1 January 2024
Accepted date: 22 February 2024
Abstract:
Suppose that $\{U_n(P,Q)\}$ and $\{V_n(P,Q)\}$ are respectively the Lucas sequences of the first and second kinds with $P\neq 0$, $Q\neq 0$ and $gcd(P,Q)=1$. In this paper, we introduce an approach for studying the solutions $(x,n)$ of the diophantine equation
$$
\pm\frac{1}{V_n(P_2,Q_2)}=\sum_{k=1}^\infty\frac{U_{k-1}(P_1,Q_1)}{x^k},
$$
in the cases of $(P_1,Q_1)\neq(P_2,Q_2)$ and $(P_1,Q_1)=(P_2,Q_2)$. Moreover, we apply the procedure of this approach with which $-3\leq P_1,P_2\leq3$, $-2\leq Q_1\leq2$ and $ -1\leq Q_2\leq1$. Our approach is mainly based on transferring this equation into either an elliptic curve equation that can be solved easily using the Magma software, or a quadratic equation that can be solved using the quadratic formula.
Keywords: Lucas sequences, diophantine equations, elliptic curves, quadratic equation
