On the Diophantine Equation $z(n)=(2-1/k)\,n$ Involving the Order of Appearance in the Fibonacci Sequence

Abstract

Let $(F_n)_{n\geq 0}$ be the sequence of the Fibonacci numbers. The order (or rank) of appearance $z(n)$ of a positive integer $n$ is defined as the smallest positive integer $m$ such that $n$ divides $F_m$. In 1975, Sall\' e proved that $z(n)\leq 2n$, for all positive integers $n$. In this paper, we shall solve the Diophantine equation $z(n)=(2-1/k)n$ for positive integers $n$ and $k$.