Suppose that \((U_{n})_{n \ge 0}\) 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 \(n \ge m \ge 0\) , \(x \ge 2\) , and \(q \ge 2\) . 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 \(q, x\ge 2\) under the assumption of the 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 superelliptic equation.
