The Proof of a Conjecture Related to Divisibility Properties of z(n)

Abstract

The order of appearance of n (in the Fibonacci sequence) z(n) is defined as the smallest positive integer k for which n divides the k-the Fibonacci number F-k. Very recently, TrojovskATIN SMALL LETTER Y WITH ACUTE proved that z(n) is an even number for almost all positive integers n (in the natural density sense). Moreover, he conjectured that the same is valid for the set of integers n & GE;1 for which the integer 4 divides z(n). In this paper, among other things, we prove that for any k & GE;1, the number z(n) is divisible by 2(k) for almost all positive integers n (in particular, we confirm TrojovskATIN SMALL LETTER Y WITH ACUTE's conjecture).