On Periodic Points of the Order of Appearance in the Fibonacci Sequence

Abstract

Let ( F n ) n ≥ 0 be the Fibonacci sequence. The order of appearance z ( n ) of an integer n ≥ 1 is defined by z ( n ) = min { k ≥ 1 : n ∣ F k } . Marques, and Somer and Křížek proved that all fixed points of the function z ( n ) have the form n = 5 k or 12 · 5 k . In this paper, we shall prove that z ( n ) does not have any k-periodic points, for k ≥ 2 .