Let $\xi =[a_0;a_1,a_2,\cdots ,a_i,\cdots ]$ be an irrational number in simple continued fraction expansion, $p_i/q_i=[a_0;a_1,a_2,\cdots ,a_i]$, $M_i=q_i^2|\xi -p_i/q_i|$. In this note we find a function $G(R,r)$ such that $$\begin{array}{ccc}& M_{n+1}<R\text{and}{M}_{n-1}<r\text{imply}{M}_n>G(R,r),\hfill & \hfill M_{n+1}>R\text{and}{M}_{n-1}>r\text{imply}{M}_n<G(R,r).\end{array}$$ Together with a result the author obtained, this shows that to find two best approximation functions $\tilde{H}(R,r)$ and $\tilde{L}(R,r)$ is a well-posed problem. This problem has not been solved yet.

We make more accessible a neglected simple continued fraction based algorithm due to Lagrange, for deciding the solubility of $a{x}^2+bxy+c{y}^2=N$ in relatively prime integers $x,y$, where $N\ne 0$, gcd$(a,b,c)=\text{gcd}(a,N)=1\text{et}D=b^2-4ac\&gt;0$ is not a perfect square. In the case of solubility, solutions with least positive y, from each equivalence class, are also constructed. Our paper is a generalisation of an earlier paper by the author on the equation $x^2-D{y}^2=N$. As in that paper, we use a lemma on unimodular matrices that gives a much simpler proof than Lagrange’s for...

In the 1990s, J.C. Tong gave a sharp upper bound on the minimum of $k$ consecutive approximation constants for the nearest integer continued fractions. We generalize this to the case of approximation by Rosen continued fraction expansions. The Rosen fractions are an infinite set of continued fraction algorithms, each giving expansions of real numbers in terms of certain algebraic integers. For each, we give a best possible upper bound for the minimum in appropriate consecutive blocks of approximation...