The Proof of a Conjecture on the Density of Sets Related to Divisibility Properties of z(n)

Abstract

Let (F-n)(n) be the sequence of Fibonacci numbers. The order of appearance (in the Fibonacci sequence) of a positive integer n is defined as z(n)=min{k & GE;1:n divide F-k}. Very recently, Trojovska and Venkatachalam proved that, for any k & GE;1, the number z(n) is divisible by 2(k), for almost all integers n & GE;1 (in the sense of natural density). Moreover, they posed a conjecture that implies that the same is true upon replacing 2(k) by any integer m & GE;1. In this paper, in particular, we prove this conjecture.