Some Propositions and Lemmas

Fractions" The following proposition is fundamental to the continued fractions of surds, because it can determine whether a set of partial quotients come from a given surd.

Proposition 1   Let $x = [q_0, \overline{q_1, q_2, \ldots, q_2, q_1, 2q_0}]$; then $x = \sqrt{d}$ if and only if

$$M(\sqrt{d}) = \frac{1}{\sqrt{d}}$$

where $M = \{q_0, q_1, \ldots , q_1, q_0\}$, and $M(\sqrt{d})$ is a fractional linear transformation.

Since the partial quotients that come from a surd $\sqrt{d}$ are almost palindromic, it is possible to produce more explicit forms of Proposition 1. The following lemmas reduce the amount of calculation necessary to verify that a set of partial quotients come from a surd by half. They will be used in the proof of the theorems in Section . The rest of the report contains new ideas and results, which are discussed in [5].

Lemma 1   Let D be a non-square integer, $\omega$ be the greatest integer in $\sqrt{D}$, $\alpha = \omega + \sqrt{D}$, and $A = \left[ _c^a ~ _d^b \right]$. If $A(\frac{1}{\alpha}) = \beta$ such that $\beta \overline{\beta} = -1$, then

$$\{\omega\}A^TA\{\omega\}(\sqrt{D}) = \frac{1}{\sqrt{D}} \mbox{.}$$

Lemma 2   Let D be a non-square integer, $\omega$ be the greatest integer in $\sqrt{D}$, $\alpha = \omega + \sqrt{D}$, and $A = \left[ _c^a ~ _d^b \right]$. If $A(\frac{1}{\alpha}) = \beta$ such that $\beta + \overline{\beta} = -r$, then

$$\{\omega\}A^T\{r\}A\{\omega\}(\sqrt{d}) = \frac{1}{\sqrt{D}} \mbox{.}$$

Now that these two lemmas have been stated, it is convenient to introduce and prove the theorems regarding two families of continued fractions.