Abstract
The order of appearance (in the Fibonacci sequence) function z:Z≥1→Z≥1 is an arithmetic function defined for a positive integer n as z(n)=min{k≥1:Fk≡0(modn)}. A topic of great interest is to study the Diophantine properties of this function. In 1992, Sun and Sun showed that Fermat’s Last Theorem is related to the solubility of the functional equation z(n)=z(n2), where n is a prime number. In addition, in 2014, Luca and Pomerance proved that z(n)=z(n+1) has infinitely many solutions. In this paper, we provide some results related to these facts. In particular, we prove that limsupn→∞(z(n+1)−z(n))/(logn)2−ϵ=∞, for all ϵ∈(0,2).
Highlights
The arithmetic function z : Z≥1 → Z≥1 defined by z(n) = min{k ≥ 1 : n | Fk } is known as the order of appearance in the Fibonacci sequence
Its sharpest upper bound is z(n) ≤ 2n as proved by Sallé [2] (the sharpness follows from z(6 · 5k ) = 12 · 5k, for all k ≥ 0)
Νp (r ) denotes the p-adic valuation of r, that is, the largest non-negative integer k for which pk divides r. We remark that this conjecture was verified for all prime numbers p < 3 × 1017 (PrimeGrid—December 2020)
Summary
Let ( Fn )n be the Fibonacci sequence. The arithmetic function z : Z≥1 → Z≥1 defined by z(n) = min{k ≥ 1 : n | Fk } is known as the order of appearance (or rank of apparition) in the Fibonacci sequence. Νp (r ) denotes the p-adic valuation (or order) of r, that is, the largest non-negative integer k for which pk divides r (see [13,14,15] for more facts on p-adic valuation of the Fibonacci sequence and its generalizations). We remark that this conjecture was verified for all prime numbers p < 3 × 1017 (PrimeGrid—December 2020).
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.
