On the Structure of the Period of the continued fraction for $\sqrt{d}$

TL;DR

This paper investigates the structure of the periodic continued fraction expansion of square roots, providing formulas for the central term and the repeated partial quotients in its period.

Contribution

It introduces explicit formulas for the central term and the repeated partial quotients in the continued fraction expansion of .

Findings

Formulas for the central term of the period

Explicit description of repeated partial quotients

Deeper understanding of the structure of continued fractions for 

Abstract

The paper examines the structure of the periodic continued fraction for $\sqrt{d}$ and gives formulae for the central term as well as the repeated partial quotients occurring in its period.