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On the Continued Fraction Expansion of Fixed Period in Finite Fields

Published online by Cambridge University Press:  20 November 2018

Hela Benamar ,
Amara Chandoul  and
M. Mkaouar
Hela Benamar
Affiliation:
Faculté des Sciences de Sfax, BP 1171, Sfax 3000, Tunisie e-mail: [email protected]@fss.rnu.tn
Amara Chandoul
Affiliation:
163, avenue de Luminy-case, 907-13288, Marseille Cedex 9, France e-mail: [email protected]
M. Mkaouar
Affiliation:
Faculté des Sciences de Sfax, BP 1171, Sfax 3000, Tunisie e-mail: [email protected]@fss.rnu.tn
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Abstract

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The Chowla conjecture states that if $t$ is any given positive integer, there are infinitely many prime positive integers $N$ such that $\text{Per}\left( \sqrt{N} \right)\,=\,t$, where $\text{Per}\left( \sqrt{N} \right)$ is the period length of the continued fraction expansion for $\sqrt{N}$. C. Friesen proved that, for any $k\,\in \,\mathbb{N}$, there are infinitely many square-free integers $N$, where the continued fraction expansion of $\sqrt{N}$ has a fixed period. In this paper, we describe all polynomials $Q\,\in \,{{\mathbb{F}}_{q}}\left[ X \right]$ for which the continued fraction expansion of $\sqrt{Q}$ has a fixed period. We also give a lower bound of the number of monic, non-squares polynomials $Q$ such that $\deg \,Q=\,2d$ and $Per\sqrt{Q}\,=\,t$.

Keywords

11A5513J05continued fractionspolynomialsformal power series
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 4 , 01 December 2015 , pp. 704 - 712
Copyright
Copyright © Canadian Mathematical Society 2015

References

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