Unit of the form $n\alpha + m \beta \sqrt{d}$ in real quadratic fields

Question

Let $d$ be a positive square-free integer equal to $2$ or $3\mod 4$. The unit group of $\mathbb{Z}[\sqrt{d}]$, denoted $\mathbb{Z}[\sqrt{d}]^\times$, is generated by $\pm 1$ and $p+q\sqrt{d}$, where $p/q$ is the first convergent in the recurring fraction expansion of $\sqrt{d}$. Let's take $\alpha, \beta \in \mathbb{Z}$ such that $\gcd(\alpha, \beta)=\gcd(\alpha,d)=1$. My question is, for every coprime pair $(\alpha,\beta)$ does there exist $n,m \in \mathbb{Z}$ (also necessarily coprime and $n$ coprime to $d$) such that

$$ (n\alpha) + \sqrt{d} (m\beta) \in \mathbb{Z}[\sqrt{d}]^\times? $$

Answer 1

Here's a counterexample . . .

Let $U$ be the set of units of $\mathbb{Z}{\large{[}}\sqrt{2}\,{\large{]}}$.

Then $U=\{\pm x_n\pm y_n\sqrt{2}\mid n\in\mathbb{N}_0\}$, where $\mathbb{N}_0$ is the set of nonnegative integers, and $x_n,y_n$ are defined by the recursion $$ \begin{cases} (x_0,y_0)=(1,0)\\[4pt] (x_{n+1},y_{n+1})=(x_n+2y_n,x_n+y_n)\\ \end{cases} $$ Applying the recursion mod $5$, one can verify that

• The sequence $(x_n,y_n)$ repeats mod $5$ in a cycle of length $12$.$\\[4pt]$
• $x_n\not\equiv 0\;(\text{mod}\;5)$ for $0\le n\le 11$.

It follows that $x_n\not\equiv 0\;(\text{mod}\;5)$ for all $n$.

Thus, letting $d=2$ and $\alpha=5$, we have a counterexample.