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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Abstract

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

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

Dates
First available in Project Euclid: 6 December 2010

Permanent link to this document
https://projecteuclid.org/euclid.pja/1291644508

Digital Object Identifier
doi:10.3792/pjaa.86.174

Mathematical Reviews number (MathSciNet)
MR2779831

Zentralblatt MATH identifier
1222.11024

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

Keywords
Fibonacci numbers linear forms in logarithms

Citation

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. https://projecteuclid.org/euclid.pja/1291644508


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References

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