A Conjecture Connected with Units of Quadratic Fields

In this article, we consider the order $$\mathcal{O}${f}={x+yf\sqrt{d}:x,\ y \in \Z}$withconductor$f\in\N$inarealquadraticfield$K=$\mathbb{Q}$(\sqrt{d})$where$d>0$issquare-freeand$d\equiv2,3\pmod 4$. We obtain numerical information about $ n(f)=n(p)=min{\nu\in\N : \varepsilon^{\nu}\in $\mathcal{O}${p}}$where$\varepsilon>1$ is the fundamental unit of $K$ and $p$ is an odd prime. Our numerical results suggest that the frequencies of $\frac{p\pm 1}{2n(p)}$ or $\frac{p\pm 1}{n(p)}$ should have a limit as the ranges of $d$ and $p$ go to infinity.