On the sum of powers of two consecutive Fibonacci numbers. (English) Zbl 1222.11024

Let \(F_n\) be the Fibonacci numbers, \(s\) a positive integer. The authors prove that if \(F_n^s+F_{n+1}^s \) is a Fibonacci number for all suficiently large \(n\), then \(s=1\) or \(2\).


11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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