Proceedings of the Japan Academy, Series A, Mathematical Sciences

On the sum of powers of two consecutive Fibonacci numbers

Diego Marques and Alain Togbé

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Let $(F_{n})_{n\geq 0}$ be the Fibonacci sequence given by $F_{n+2}=F_{n+1}+F_{n}$, for $n\geq 0$, where $F_{0}=0$ and $F_{1}=1$. In this note, we prove that if $s$ is an integer number such that $F_{n}^{s}+F_{n+1}^{s}$ is a Fibonacci number for all sufficiently large integer $n$, then $s=1$ or 2.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 86, Number 10 (2010), 174-176.

First available in Project Euclid: 6 December 2010

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Zentralblatt MATH identifier

Primary: 11B39: Fibonacci and Lucas numbers and polynomials and generalizations 11J86: Linear forms in logarithms; Baker's method

Fibonacci numbers linear forms in logarithms


Marques, Diego; Togbé, Alain. On the sum of powers of two consecutive Fibonacci numbers. Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), no. 10, 174--176. doi:10.3792/pjaa.86.174.

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  • 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.
  • D. Kalman and R. Mena, The Fibonacci numbers exposed, Math. Mag. 76 (2003), no. 3, 167–181.
  • M. Laurent, Linear forms in two logarithms and interpolation determinants. II, Acta Arith. 133 (2008), no. 4, 325–348.
  • D. Marques and A. Togbé, Perfect powers among Fibonomial coefficients, C. R. Acad. Sci. Paris, Ser. I 348 (2010) 717–720.
  • A. S. Posamentier and I. Lehmann, The (fabulous) Fibonacci numbers, Prometheus Books, Amherst, NY, 2007.