Periodic continued fractions and Kronecker symbols

TL;DR

This paper investigates the behavior of Kronecker symbols for convergents of purely periodic continued fractions, revealing conditions under which the sequence is periodic or aperiodic, unlike the always periodic Jacobi symbols.

Contribution

It provides a detailed characterization of the period length of Kronecker symbols and identifies when the sequence becomes aperiodic, extending understanding of continued fraction symbol sequences.

Findings

Kronecker symbol sequences can be aperiodic, unlike Jacobi sequences.

A necessary and sufficient condition for aperiodicity is established.

The period length relates to the period of Jacobi symbol sequences.

Abstract

We study the Kronecker symbol $\left(\frac st\right)$ 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.