# Tagged Questions

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### What is the effect of adding 1/2 to a continued fraction?

Is there a simple answer to the question "what happens to the continued fraction expansion of an irrational number when you add 1/2?" A closely related question is "what happens to such an expansion ...
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### How does this relationship between the Catalan numbers and SU(2) generalize?

This is a question, or really more like a cloud of questions, I wanted to ask awhile ago based on this SBS post and this post I wrote inspired by it, except that Math Overflow didn't exist then. As ...
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### Continued fractions using all natural integers

What can one say about the set of continued fractions $[0;a_1,a_2,\ldots]$, where $a_1,a_2,\ldots$ are a permutation of the set of natural numbers?
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### Open problems in continued fractions theory

I propose to collect here open problems from the theory of continued fractions. Any types of continued fractions are welcome.
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### Poles from the Continued Fraction Expansion of the Tangent Function?

Consider the well known continued fraction expansion $$z \tan z = \frac{z^2}{1 - \cfrac{z^2}{3- \cfrac{z^2}{5 - \ldots}}}$$ of the tangent function going back to Euler and Lambert (Lambert used it ...
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### Repetend digit graphs for $1/n$ in base $b$

Here is a decimal expansion of $\frac{1}{34}$: $$(1/34)_{10}=0.02941176470588235\overline{2941176470588235}\ldots$$ And here is a graphical representation of the 16-digit "repetend," as a directed ...
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### Confusion with practically implementing rational approximations

Writing a program visualizing Ford circles I've encountered a seemingly purely programmatic puzzle but then gradually realized there are some mathematical aspects of it which I don't understand. Let ...
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### distance formula in Farey graph?

Consider the usual "Farey graph", i.e. the 1-skeleton of the (essentially unique) triangulation of the disk by ideal triangles. If one insists that 1/0, 0/1, and 1/1 are vertices of a triangle then ...
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### Length of Hirzebruch continued fractions

Suppose $a,b$ are two natural numbers relatively prime to $n$ and to each other. Assume $n\geq ab+1$. Suppose further that $\frac{a}{b}\equiv k \pmod{n}$ for some $k\in \lbrace 1,2,\dots, n-1\rbrace$ ...
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### The Riemann's Zeta Function represented as a continued fraction and a question of convergence.

The Riemann's zeta function can be expressed as a continued fraction as follows \begin{align*} \zeta(z)=\newcommand{\bigk}{\mathop{\Huge\vcenter{\hbox{K}}}}\left(1-\bigk_{k=1}^{\infty }\frac{-e^{-2\...
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### Constructing a family of convergents from continued fractions formed by a set of prime partial quotients

For a given real number $x$, the continued fraction representation $x = [a_0; a_1, a_2, \cdots]$ where $(a_n)_{n \geq 0}$is defined by setting $x = \alpha_0$, then $a_i = \lfloor \alpha_i \rfloor$, ...
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### Last term of repeating continued fraction expansion

Once again, working with stable vector bundles on $\mathbb{P}^2$ I have run into a question that is really out of my area. (Thanks to everybody who helped out with my last question!) Let $D>9$ be ...
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### Coefficients in the periodic part of continued fractions for real quadratic algebraic numbers

Given a strictly positive integer $A$, let $D(A)$ denote the set of all real quadratic algebraic numbers with a continued fraction having almost all coefficients $\leq A$. Consider the field $Q_A$ ...
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### Orbits of the projective special linear group on $\mathbf{Q} \cup \{\infty\}$

The group $\mathrm{PSL}_2(\mathbf{Q})$ of fractional linear transformations $x \mapsto (ax+b)/(cx+d)$ such that $a,b,c,d \in \mathbf{Q}$ and $ad-bc = 1$ acts on $\mathbf{Q} \cup \lbrace \infty\rbrace$....
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### English translation of Voronoi's dissertation

I am looking for an English translation of Voronoi's doctoral dissertation, "On a generalization of the Algorithm of Continued Fractions." I can only find it in the original Russian.
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### Gauss-Kuzmin Theorem (continued fractions) - why is important?

As we know, Gauss wrote that $$\lim_{n \rightarrow \infty} \lambda \left(\tau^n \leq x\right) = \frac{\log(1+x)}{\log2}, \quad 0 \leq x < 1,$$ with $\lambda$ is Lebesgue ...
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### An upper bound for the length of the continued fraction expansion of $\sqrt d$

Let $d\ge 2$ and let $$\sqrt d =[a_0; \overline{a_1,\dots, a_\ell, 2a_0}]$$ be its continued fraction expansion. Clearly, if $d=n^2+1$, then $\ell=0$, which gives the lower bound for $\ell$. ...
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### Motivation for Hirzebruch-Jung Modified Euclidean Algorithm

Let $a,b \in \mathbb{N} \ \ s.t. \ \ a > b$ have $\gcd(a,b) =1$. We can define the Hirzebruch-Jung modified euclidean algorithm as follows: Let $e_i \in \mathbb{N} >2$, and $r_k \in \mathbb{N}$...