Geodesic
On the Continued Fraction Expansion of Fixed Period in Finite Fields
Canadian mathematical bulletin, Tome 58 (2015) no. 4, pp. 704-712

Voir la notice de l'article provenant de la source Cambridge University Press

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 $Per\sqrt{Q}\,=\,t$ .
DOI : 10.4153/CMB-2015-055-9
Mots-clés : 11A55, 13J05, continued fractions, polynomials, formal power series 
Benamar, Hela; Chandoul, Amara; Mkaouar, M. On the Continued Fraction Expansion of Fixed Period in Finite Fields. Canadian mathematical bulletin, Tome 58 (2015) no. 4, pp. 704-712. doi: 10.4153/CMB-2015-055-9
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