In this paper we give a lower bound with respect to block length for the trace of non-elliptic conjugacy classes of the Hecke groups. One consequence of our bound is that there are finitely many conjugacy classes of a given trace in any Hecke group. We show that another consequence of our bound is that class numbers are finite for related hyperbolic \( \mathbb{Z}[\lambda] \)-binary quadratic forms. We give canonical class representatives and calculate class numbers for some classes of hyperbolic \( \mathbb{Z}[\lambda] \)-binary quadratic forms.

Dufresnoy and Pisot characterized the smallest Pisot number of degree $n \geq 3$ by giving explicitly its minimal polynomial. In this paper, we translate Dufresnoy and Pisot's result to the Laurent series case. The aim of this paper is to prove that the minimal polynomial of the smallest Pisot element (SPE) of degree $n$ in the field of formal power series over a finite field is given by $P(Y)=Y^{n}-\alpha XY^{n-1}-\alpha^n,$ where $\alpha$ is the least element of the finite field $\mathbb{F}_{q}\backslash\{0\}$ (as a finite total ordered set). We prove that the sequence of SPEs of degree $n$ is decreasing and converges to $\alpha X.$ Finally, we show how to obtain explicit continued fraction expansion of the smallest Pisot element over a finite field.

We provide a criterion for the central norm to be any value in the simple continued fraction expansion of $\sqrt{D}$ for any non-square integer $D>1$. We also provide a simple criterion for the solvability of the Pell equation $x^2-Dy^2=-1$ in terms of congruence conditions modulo $D$.

We look at the simple continued fraction expansion of $\sqrt{D}$ for any $D=2^hc $ where $c>1$ is odd with a goal of determining necessary and sufficient conditions for the central norm (as determined by the infrastructure of the underlying real quadratic order therein) to be $2^h$. At the end of the paper, we also address the case where $D=c$ is odd and the central norm of $\sqrt{D}$ is equal to $2$.

In 1988 Rieger exhibited a differentiable function having a zero at the golden ratio\break $(-1+\sqrt5)/2$ for which when Newton's method for approximating roots is applied with an initial value $x_0=0$, all approximates are so-called ``best rational approximates''---in this case, of the form $F_{2n}/F_{2n+1}$, where $F_n$ denotes the $n$-th Fibonacci number. Recently this observation was extended by Komatsu to the class of all quadratic irrationals whose continued fraction expansions have period length $2$. Here we generalize these observations by producing an analogous result for all quadratic irrationals and thus provide an explanation for these phenomena.

The purpose of this article is to provide criteria for the simultaneous solvability of the Diophantine equations $X^2 - DY^2 = c$ and $x^2 - Dy^2 = -c$ when $c \in \mathbb{Z}$, and $D \in \mathbb{N}$ is not a perfect square. This continues work in \cite{me}--\cite{alfnme}.

We discuss the $q$-exponential functions $e_q$, $E_q$ for $q$ on the unit circle, especially their continuity in $q$, and analogues of the limit relation $ \lim_{q\rightarrow 1}e_q((1-q)z)=e^z$.