On the resolution of the Diophantine equation $U_n + U_m = x^q$

Abstract

Suppose that $(U_{n})_{n \geq 0}$ is a binary recurrence sequence and has a dominant root $\alpha$ with $\alpha>1$ and the discriminant $D$ is square-free. In this paper, we study the Diophantine equation $U_n + U_m = x^q$ in integers $n \geq m \geq 0$, $x \geq 2$, and $q \geq 2$. Firstly, we show that there are only finitely many of them for a fixed $x$ using linear forms in logarithms. Secondly, we show that there are only finitely many solutions in $(n, m, x, q)$ with $q, x\geq 2$ under the assumption of the {\em abc-conjecture}. To prove this, we use several classical results like Schmidt subspace theorem, a fundamental theorem on linear equations in $S$-units and Siegel's theorem concerning the finiteness of the number of solutions of a hyperelliptic equation.