Marques, Diego; Togbé, Alain On the sum of powers of two consecutive Fibonacci numbers. (English) Zbl 1222.11024 Proc. Japan Acad., Ser. A 86, No. 10, 174-176 (2010). 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 \(2\). Reviewer: Florin Nicolae (Berlin) Cited in 1 ReviewCited in 13 Documents MSC: 11B39 Fibonacci and Lucas numbers and polynomials and generalizations Keywords:Fibonacci numbers PDF BibTeX XML Cite \textit{D. Marques} and \textit{A. Togbé}, Proc. Japan Acad., Ser. A 86, No. 10, 174--176 (2010; Zbl 1222.11024) Full Text: DOI OpenURL 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 [4] D. Marques and A. Togbé, Perfect powers among Fibonomial coefficients, C. R. Acad. Sci. Paris, Ser. I 348 (2010) 717-720. · Zbl 1219.11032 [5] A. S. Posamentier and I. Lehmann, The (fabulous) Fibonacci numbers , Prometheus Books, Amherst, NY, 2007. · Zbl 1123.11008 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.