Introduction

This report focuses on the square roots of positive integers whose continued fraction expansions have centers. Let $d$ be such an integer. Then the continued fraction expansion of $\sqrt{d}$ is given by

$$\sqrt{d} = [a_0;\overline{a_1,a_2,\ldots,a_n,b,a_n,\ldots,a_2,a_1,2a_0}]$$ (1)

We are interested in the factorization of $[a_1,a_2,\ldots,a_n]$ into $[r]\overline{[2r]}$ where $[r]$ is the continued fraction expansion of a rational number $r$ and $\overline{[2r]}$ represents the continued fraction expansion of $2r$in reverse order.

For a thorough introduction to continued fractions and concepts used in this report, see Justin Miller's report: Families of Continued Fractions.