Malaysian Journal of Mathematical Sciences, September 2025, Vol. 19, No. 3
Solutions of the Jin-Schmidt Equation in Lucas Numbers
Alabrahimi, Q. N. and Hashim, H. R.
Corresponding Email: hayderr.almuswi@uokufa.edu.iq
Received date: 2 January 2025
Accepted date: 6 April 2025
Abstract:
In this paper, we study the integer solutions of the following equation that is so called the Jin-Schmidt equation,
\begin{align*}
AX^2 + BY^2 +CZ^2 = DXYZ+1,
\end{align*}
where $(X,Y,Z)=(L_i,L_j,L_k)$, with $i,j,k\geq 1$ such that $L_i$, $L_j$ and $L_k$ represent terms in the Lucas sequence that is defined by the relation $L_0=2$, $L_1=1$, $L_{n+1}=L_{n}+L_{n-1}$ with $n\geq 1$.
Keywords: Diophantine equations; Lucas sequences; Jin-Schmidt equation; Magma software; elliptic curve equation
