On the Parity of the Order of Appearance in the Fibonacci Sequence

Abstract

Let (F-n)(n >= 0) be the Fibonacci sequence. The order of appearance function (in the Fibonacci sequence) z : Z(>= 1)-> Z(>= 1) is defined as z(n) := min{k >= 1 : F-k equivalent to 0 (mod n)}. In this paper, among other things, we prove that z(n) is an even number for almost all positive integers n (i.e., the set of such n has natural density equal to 1).