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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 \(2\).

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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References:

[1] Y. Bugeaud, M. Mignotte and S. Siksek, Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers, Ann. of Math. (2) 163 (2006), no. 3, 969-1018. · Zbl 1113.11021
[2] D. Kalman and R. Mena, The Fibonacci numbers exposed, Math. Mag. 76 (2003), no. 3, 167-181. · Zbl 1048.11014
[3] M. Laurent, Linear forms in two logarithms and interpolation determinants. II, Acta Arith. 133 (2008), no. 4, 325-348. · Zbl 1215.11074
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