On the exponential diophantine equation \(U_{n}^x+U_{n+1}^x=U_m\)

Let \( \{U_n\}_{n\ge 0} \) ${\{U_n\}}_{n\ge 0}$ be the Lucas sequence. For integers x, n and m, we find all solutions to \(U_{n}^x+U_{n+1}^x=U_m\) $U_n^x+U_{n+1}^x=U_m$ . The equation was studied and claimed to be solved completely in Ddamulira and Luca (Ramanujan J 56(2):651–684, 2021) but there are some computational bugs in that publication because of the wrong statement of Mignotte’s bound from Mignotte (A kit on linear forms in three logarithms. http://irma.math.unistra.fr/~bugeaud/travaux/kit.pdf, 2008). In this paper, the main result remains the same as in Ddamulira and Luca (Ramanujan J 56(2):651–684, 2021) but we focus on correcting the computational mistakes in Ddamulira and Luca (Ramanujan J 56(2):651–684, 2021), involving the application of Theorem 2.1 from Mignotte (A kit on linear forms in three logarithms. http://irma.math.unistra.fr/~bugeaud/travaux/kit.pdf, 2008).