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 Fk. Very recently, Trojovský 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≥1 for which the integer 4 divides z(n). In this paper, among other things, we prove that for any k≥1, the number z(n) is divisible by 2k for almost all positive integers n (in particular, we confirm Trojovský’s conjecture).
Highlights
Let ( Fn )n be the Fibonacci sequence which is defined by the binary recurrence
The first ingredient is related to the value of z( pk ) for a prime number p and k ≥ 1: Lemma 1 (Theorem 2.4 of [14])
We have that z(2k ) = 3 · 2k−1 for all k ≥ 2, and z(3k ) = 4 · 3k−1 for all k ≥ 1. It holds that z( pk ) = pmax{k−e( p),0} z( p), where e( p) := max{k ≥ 0 : pk | Fz( p) }
Summary
Let ( Fn )n be the Fibonacci sequence which is defined by the binary recurrence. For any integer n ≥ 1, the order of appearance of n (in the Fibonacci sequence), denoted by z(n) as z(n) := min{k ≥ 1 : n | Fk }. The arithmetic function z : Z≥1 → Z≥1 is well defined 300)) and z(n) ≤ 2n is the sharpest upper bound (as proved by Sallé [2]). We refer the reader to [3,4,5,6,7,8,9] for more (recent) results on z(n). The first few values of z(n) (for n ∈ [1, 20]) are (see sequence A001177 in OEIS [10]): Academic Editors: Iwona Wloch
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