Abstract
Let (Fn)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≥1:n∣Fk}. Very recently, Trojovská and Venkatachalam proved that, for any k≥1, the number z(n) is divisible by 2k, for almost all integers n≥1 (in the sense of natural density). Moreover, they posed a conjecture that implies that the same is true upon replacing 2k by any integer m≥1. In this paper, in particular, we prove this conjecture.
Highlights
Let (Fn)n be the sequence of Fibonacci numbers that is defined by the second-order recurrenceFn+2 = Fn+1 + Fn, with Fi = i, for i ∈ {0, 1}
We recall the theoretical result of Spearman and Williams [19], which is very useful for estimating the order of counting functions
We proved a conjecture related to the order of appearance in the Fibonacci sequence function z : Z≥1 → Z≥1, defined as z(n) := min{k ≥ 1 : n | Fk}
Summary
Let (Fn)n be the sequence of Fibonacci numbers that is defined by the second-order recurrence. 300) and [2] for the sharpest upper bound for z(n), namely, z(n) ≤ 2n) This function is known as the order of appearance in the Fibonacci sequence. Trojovský ([13], Theorem 2) proved that the natural density of Ez(2) is equal to 1. Trojovská and Venkatachalam [14] generalized the previous result by showing that δ(Ez(2k)) = 1, for all k ≥ 1. They posed the following conjecture: Conjecture 1. The proof of the theorem combines Diophantine properties of z(n) with analytical tools concerning the average growth rate of multiplicative functions
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
