We use the notation

$$\displaystyle [a_0; a_1, \cdots, a_n] = a_0 + \cfrac{1}{a_1 + \cfrac{1}{\ddots + \cfrac{1}{a_n}}}$$

to denote a finite continued fraction, and for a given real number $\alpha$, we attach the (possibly infinite) sequence of partial quotients $[a_0; a_1, a_2, \cdots] = \alpha$.

It is well-known by classic theorems of Euler and Lagrange that a real number has eventually periodic continued fraction expansion if and only if $\alpha$ is a quadratic irrational. Moreover, it is known that if $\alpha$ is an irrational number and $p/q$ is a reduced fraction with $q > 0$ such that $|\alpha - p/q| < 1/(2q^2)$, then $p/q$ is in fact a *convergent* of $\alpha$; i.e., there is a positive integer $k$ such that $p/q = [a_0; a_1, \cdots, a_k]$.

Suppose that $\theta$ is a *purely periodic* quadratic irrational, so that its continued fraction is of the form $[\overline{a_0; a_1, \cdots, a_n}]$. Let $f(x) = ax^2 + bx + c$ be the minimal polynomial of $\theta$. What do we know about the evaluation of the quadratic form $F(x,y) = y^2 f(x/y)$ at the pairs $(p_0, q_0), \cdots, (p_n, q_n)$ corresponding to the convergents? In particular, what are the sizes of these values compared to the discriminant $\Delta(F)$ of $F$?