Exercise 7.7.1 (When $(\lfloor \sqrt{d} \rfloor + \sqrt{d})/c$ has a purely periodic expansion?)

Question

For what positive integers $c$ does the quadratic irrational $(\lfloor \sqrt{d} \rfloor + \sqrt{d})/c$ have a purely periodic expansion?

Richard Ganaye · Accepted Answer

ewcommand{\Q}{\mathbb{Q}} ewcommand{\Z}{\mathbb{Z}} ewcommand{\N}{\mathbb{N}} \begin{proof} Let $c \in \N^*$. By Theorem 7.20, the quadratic irrational $(\lfloor \sqrt{d} floor + \sqrt{d})/c$ has a purely periodic expansion if and only if \begin{align} &\frac{\lfloor \sqrt{d} floor + \sqrt{d}}{c} >1\quad ext{ and} \quad \quad -1 < \frac{\lfloor \sqrt{d} floor - \sqrt{d}}{c} <0. \end{align} Since $\frac{\lfloor \sqrt{d} floor - \sqrt{d}}{c} <0$ is always true, (1) is equivalent for $c>0$ to \begin{align} \sqrt{d} - \lfloor \sqrt{d} floor < c < \sqrt{d} + \lfloor \sqrt{d} floor. \end{align} Since $(\lfloor \sqrt{d} floor + \sqrt{d})/c$ is a quadratic irrational, $d$ is not a perfect square and $\sqrt{d}$ is not rational. Then \begin{align*} c < \sqrt{d} + \lfloor \sqrt{d} floor &\iff c \leq 2 \sqrt{d},\ & ext{and}\ \sqrt{d} - \lfloor \sqrt{d} floor < c &\iff 1 \leq c. \end{align*} Therefore (2) is equivalent to $$1 \leq c \leq 2 \sqrt{d}.$$ The quadratic irrational $(\lfloor \sqrt{d} floor + \sqrt{d})/c$ has a purely periodic expansion if and only if $1 \leq c \leq 2 \lfloor \sqrt{d} floor$. \end{proof}

Exercise 7.7.1 (When $(\lfloor \sqrt{d} \rfloor + \sqrt{d})/c$ has a purely periodic expansion?)

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Exercise 7.7.1 (When $(\lfloor \sqrt{d} \rfloor + \sqrt{d})/c$ has a purely periodic expansion?)

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