Michel Mendès France's “Folding Lemma” for continued fraction expansions provides an unusual explanation for the well known symmetry in the expansion of a quadratic irrational integer.

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...

We give the relationship between regular continued fractions and Lehner fractions, using a procedure known as insertion}. Starting from the regular continued fraction expansion of any real irrational x, when the maximal number of insertions is applied one obtains the Lehner fraction of x. Insertions (and singularizations) show how these (and other) continued fraction expansions are related. We also investigate the relation between Lehner fractions and the Farey expansion (also known as the full...

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...