Summary: 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 $(n,m)$.

11B39, 11B50, 11D99

Fibonacci numbers