Periodic continued fractions and Kronecker symbols

Abstract

We study the Kronecker symbol $\left(\frac{s}{t}\right)$ for the sequence of the convergents $s/t$ of a purely periodic continued fraction expansion. Whereas the corresponding sequence of Jacobi symbols is always periodic, it turns out that the sequence of Kronecker symbols may be aperiodic. Our main result describes the period length in the periodic case in terms of the period length of the sequence of Jacobi symbols and gives a necessary and sufficient condition for the occurrence of the aperiodic case.