On the sum of powers of two consecutive Fibonacci numbers

Marques, Diego; Togbé, Alain

Proc. Japan Acad. Ser. A Math. Sci., Tome 86 (2010) no. 1, p. 174-176 / Harvested from Project Euclid

Résumé

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.

Publié le : 2010-10-15
Classification: Fibonacci numbers, linear forms in logarithms, 11B39, 11J86

@article{1291644508,
     author = {Marques, Diego and Togb\'e, Alain},
     title = {On the sum of powers of two consecutive Fibonacci numbers},
     journal = {Proc. Japan Acad. Ser. A Math. Sci.},
     volume = {86},
     number = {1},
     year = {2010},
     pages = { 174-176},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1291644508}
}

Marques, Diego; Togbé, Alain. On the sum of powers of two consecutive Fibonacci numbers. Proc. Japan Acad. Ser. A Math. Sci., Tome 86 (2010) no. 1, pp.  174-176. http://gdmltest.u-ga.fr/item/1291644508/