The Diophantine equation \(F_{n}^{y} + F_{n+1}^{x} = F_{m}^{x}\). (English) Zbl 1301.11013

Luca, Florian; Oyono, Roger

Integers 13, Paper A33, 17 p. (2013).

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

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\(F^x_n + F^x_{n+1} = F_m\) 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 \((5, 3, 2, 4)\).

Reviewer: Krassimir Atanassov (Sofia)

MSC:

11B39	Fibonacci and Lucas numbers and polynomials and generalizations
11D45	Counting solutions of Diophantine equations
11D41	Higher degree equations; Fermat’s equation

Keywords:

Fibonacci number; diophantine equation

Citations:

Zbl 1253.11046