## Divisibility properties of the Fibonacci entry point

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- by Paul Cubre and Jeremy Rouse PDF
- Proc. Amer. Math. Soc.
**142**(2014), 3771-3785 Request permission

## 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 $m | Z(p)$ 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 $\zeta (m)$.## References

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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
- MR Author ID: 741123
- Email: rouseja@wfu.edu
- 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
- © Copyright 2014
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**142**(2014), 3771-3785 - MSC (2010): Primary 11B39; Secondary 11R32, 14G25
- DOI: https://doi.org/10.1090/S0002-9939-2014-12269-6
- MathSciNet review: 3251719