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Divisibility properties of the Fibonacci entry point


Authors: Paul Cubre and Jeremy Rouse
Journal: Proc. Amer. Math. Soc. 142 (2014), 3771-3785
MSC (2010): Primary 11B39; Secondary 11R32, 14G25
Published electronically: July 28, 2014
MathSciNet review: 3251719
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Abstract: For a prime $ p$, let $ Z(p)$ be the smallest positive integer $ n$ so that $ p$ divides $ F_{n}$, the $ n$th term in the Fibonacci sequence. Paul Bruckman and Peter Anderson conjectured a formula for $ \zeta (m)$, the density of primes $ p$ for which $ m \vert Z(p)$ on the basis of numerical evidence. We prove Bruckman and Anderson's conjecture by studying the algebraic group $ G : x^{2} - 5y^{2} = 1$ and relating $ Z(p)$ to the order of $ \alpha = (3/2,1/2) \in G(\mathbb{F}_{p})$. We are then able to use Galois theory and the Chebotarev density theorem to compute $ \zeta (m)$.


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Additional Information

Paul Cubre
Affiliation: Department of Mathematics, Clemson University, Clemson, South Carolina 29634
Email: pcubre@gmail.com

Jeremy Rouse
Affiliation: Department of Mathematics, Wake Forest University, Winston-Salem, North Caro- lina 27109
Email: rouseja@wfu.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12269-6
Received by editor(s): December 26, 2012
Published electronically: July 28, 2014
Additional Notes: The first author was partially supported by the Wake Forest University Graduate School.
The second author was supported by NSF grant DMS-0901090
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.