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

On perfect powers that are sum of two balancing numbers

“…Also, perfect powers that are sums of two Pell numbers has been studied (see [1]). Recently, in [4] Bhoi et al, study the Diophantine equation U n + U m = x q in integers n ≥ m ≥ 0, x ≥ 2, and q ≥ 2, where (U k ) ≥0k is Lucas sequence of first kind. In particular, they proved that there are only finitely many of them for a fixed x using linear forms in logarithms and that there are only finitely many solutions in (n, m, x, q) with q, x ≥ 2 under the assumption of the abc conjecture.…”