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1. CMB 2015 (vol 58 pp. 704)

Benamar, H.; Chandoul, A.; Mkaouar, M.
On the Continued Fraction Expansion of Fixed Period in Finite Fields
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
Categories:11A55, 13J05

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