On the nonnegative integer solutions to the equation F ± F = y

Sums of Fibonacci numbers that are perfect powers

“…In view of the results due to Kebli et.al. [6] we obtain the following result as a direct consequence from Theorem 1: Corollary 2. Assume that y = 2, 3, 4, 6, 10.…”

“…In this section we give a recap of the method of Bravo and Luca [4] and bring their result in a form which is suitable for us. In particular, the following proposition is essentially a slightly improved version of Theorem 1 in [6].…”

“…Using the method due to Bravo and Luca [4] we are not restricted to powers of 2 and we can handle by the same method the Diophantine equation (1) F n + F m = y a , n > m > 0, a > 0 for any fixed integer y. For instance in the case that y = 3 we find the solutions (n, m, a) = (3, 1, 1), (3, 2, 1), (6, 1, 2), (6,2,2). In this paper we want to show that Diophantine Equation (1) has in almost all cases at most one solution for fixed integer y.…”

“…Furthermore, Kebli et.al. [6] considered a similar but slightly different problem. They considered the Diophantine equation ( 4)…”

On solutions of the Diophantine equation $F_{n_{1}}+F_{n_{2}}+F_{n_{3}}+F_{n_{4}}=2^a$

“…We should admit that recent work suggests that this will be diffcult and may not happen any time soon. When solving F n ± F m = y a in [7], the authors were compelled to use the abc−conjecture to show that the equation admits finitely-many integer solutions (n, m, y, a) with min{y, a} 2.…”