On the local behavior of the order of appearance in the Fibonacci sequence-Reference-Cited by-同舟云学术

On the local behavior of the order of appearance in the Fibonacci sequence

Published:2014-05-21 Issue:04 Volume:10 Page:915-933
ISSN:1793-0421
Container-title:International Journal of Number Theory
language:en
Short-container-title:Int. J. Number Theory

Author:

Luca Florian¹²,Pomerance Carl³

Affiliation:

1. Mathematical Institute, UNAM Juriquilla, Juriquilla, 76230 Santiago de Querétaro, Querétaro de Arteaga, Mexico

2. School of Mathematics, University of the Witwatersrand, P. O. Wits 2050, South Africa

3. Mathematics Department, Dartmouth College, Hanover, NH 03755, USA

Abstract

Let z(N) be the order of appearance of N in the Fibonacci sequence. This is the smallest positive integer k such that N divides the k th Fibonacci number. We show that each of the six total possible orderings among z(N), z(N + 1), z(N + 2) appears infinitely often. We also show that for each nonzero even integer c and many odd integers c the equation z(N) = z(N + c) has infinitely many solutions N, but the set of solutions has asymptotic density zero. The proofs use a result of Corvaja and Zannier on the height of a rational function at 𝒮-unit points as well as sieve methods.

Publisher

World Scientific Pub Co Pte Lt

Subject

Algebra and Number Theory

Link

https://www.worldscientific.com/doi/pdf/10.1142/S1793042114500079

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