Abstract view
On the Continued Fraction Expansion of Fixed Period in Finite Fields


Published:20150827
Printed: Dec 2015
H. Benamar,
FacultÃ© des Sciences de Sfax, BP 1171, Sfax 3000, Tunisie
A. Chandoul,
163 avenue de Luminycase 90713288 Marseille Cedex 9France
M. Mkaouar,
FacultÃ© des Sciences de Sfax, BP 1171, Sfax 3000, Tunisie
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
squarefree 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, nonsquares polynomials $Q$ such
that $\deg Q= 2d$ and $ Per \sqrt {Q}=t$.