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On the Continued Fraction Expansion of Fixed Period in Finite Fields

Published:2015-08-27
Printed: Dec 2015
• H. Benamar,
Faculté des Sciences de Sfax, BP 1171, Sfax 3000, Tunisie
• A. Chandoul,
163 avenue de Luminy-case 907-13288 Marseille Cedex 9-France
• M. Mkaouar,
Faculté des Sciences de Sfax, BP 1171, Sfax 3000, Tunisie
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Abstract

The Chowla conjecture states that, if $t$ is any given positive integer, there are infinitely many prime positive integers $N$ such that $\operatorname{Per} (\sqrt{N})=t$, where $\operatorname{Per} (\sqrt{N})$ 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 $Q\in \mathbb{F}_q[X]$ for which the continued fraction expansion of $\sqrt {Q}$ has a fixed period, also we give a lower bound of the number of monic, non-squares polynomials $Q$ such that $\deg Q= 2d$ and $Per \sqrt {Q}=t$.
 Keywords: continued fractions, polynomials, formal power series
 MSC Classifications: 11A55 - Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15] 13J05 - Power series rings [See also 13F25]

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