Geodesic
Some remarks on the equation \(F[n]=kF[m]\) in Fibonacci numbers
Journal of integer sequences, Tome 10 (2007) no. 5
Let ${F_{n}}_{n= 1}^{\infty } = {1,1,2,3,\dots }$ be the sequence of Fibonacci numbers. In this paper we give some sufficient conditions on a natural number $k$ such that the equation $F_{n} = kF_{m}$ is solvable with respect to the unknowns $n$ and $m$. We also show that for $k > 1$ the equation $F_{n} = kF_{m}$ has at most one solution $(n,m)$.
Classification : 11B39, 11B50, 11D99
Keywords: Fibonacci numbers 
@article{JIS_2007__10_5_a0,
     author = {Farrokhi D.G.,  M.},
     title = {Some remarks on the equation {\(F[n]=kF[m]\)} in {Fibonacci} numbers},
     journal = {Journal of integer sequences},
     year = {2007},
     volume = {10},
     number = {5},
     zbl = {1140.11005},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2007__10_5_a0/}
}
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Farrokhi D.G.,  M. Some remarks on the equation \(F[n]=kF[m]\) in Fibonacci numbers. Journal of integer sequences, Tome 10 (2007) no. 5. http://geodesic.mathdoc.fr/item/JIS_2007__10_5_a0/