The Diophantine equation \(F_{n}^{y} + F_{n+1}^{x} = F_{m}^{x}\) (Q2855611)

Let \((F_n)_{n \geq 0}\) be the Fibonacci sequence given by \(F_0 = 0, \;F_1 = 1, \;F_{n+2} = F_{n+1} + F_n\) for all \(n \geq 0\). In their previous paper [Proc. Japan Acad., Ser. A 87, No. 4, 45--50 (2011; Zbl 1253.11046)], the authors studied the Diophantine equation \(F^x_n + F^x_{n+1} = F_m\) in positive integers \(m, n, x\). In the present paper, they prove that the only positive integer solution \((m, n, x, y)\) of one of the equations \(F^x_n + F^y_{n+1} = F^x_m\) or \(F^y_n + F^x_{n+1} = F^x_m\) with \(n \geq 3\) and \(x \neq y\) is \((5, 3, 2, 4)\).