An exponential Diophantine equation related to powers of two consecutive Fibonacci numbers. (English) Zbl 1253.11046

Luca, Florian; Oyono, Roger

Proc. Japan Acad., Ser. A 87, No. 4, 45-50 (2011).

The authors prove that there is no integer \(s\geq 3\) such that the sum of \(s\)-powers of two consecutive Fibonacci numbers is a Fibonacci number. The main tool of the proof is a lower bound for a linear form in three logarithms of algebraic numbers combined with the use of the Baker-Davenport Lemma.

Reviewer: Maurice Mignotte (Strasbourg)

MSC:

11D61	Exponential Diophantine equations
11B39	Fibonacci and Lucas numbers and polynomials and generalizations
11J86	Linear forms in logarithms; Baker’s method

Keywords:

Fibonacci numbers; applications of linear forms in logarithms

References:

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