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1answer
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Brun's algorithm

Does anyone have an exact reference for the weak convergence (convergence in angle) of Brun's subtractive multi-dimensional continued fractions algorithm (in all dimensions)? I have been given ...
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1answer
99 views

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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1answer
522 views

Is anything known about which numbers appear in the continued fraction expansion of $\pi$?

This question is mostly idle curiosity, and certainly is not related to any research activities of my own. The motivation and background are as follows. I am currently teaching a Freshman Seminar in ...
7
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1answer
528 views

Continued fraction representation of Zeta

A question at math.SE is asking for references. The fraction is quite nice! Check it out and post some references if you know of any. I found this at arxiv, but it doesn't apply to Zeta.
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1answer
210 views

How to prove that the odd continued fraction approximants of ln(1+X) are upper bounds?

The odd order continued fraction approximants for $\ln(1+X)$ are $$X,\quad \frac{X^2+6X}{4X+6,}\quad \frac{X^3+21X^2+30X}{9X^2+36X+30,}\quad \dots.$$ In "Some bounds for the logarithmic function", ...
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1answer
63 views

Expressions in “continued” monotone functions

Recall continued fractions: http://en.wikipedia.org/wiki/Continued_fraction Now take a look at this question: ...
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214 views

Why does the Rogers-Ramanujan continued fraction $R(q)$ appear in Emma Lehmer's quintic?

Define the Ramanujan theta function $f(a,b)$ as, $$f(a,b) = \sum_{n=-\infty}^{\infty} a^{n(n+1)/2}\,b^{n(n-1)/2}$$ and the Dedekind eta function, $$\eta(\tau) = q^{1/24}\prod_{n=1}^{\infty} ...
5
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2answers
540 views

About two 'negative' continued fractions whose sum equals $1$

Letting $a_1,a_2,\cdots,a_r$ be integers which are larger than or equal to $2$, let us define $$[a_1,a_2,\cdots,a_r]=\cfrac{1}{a_1-\cfrac{1}{a_2-\cfrac{1}{\ddots-\cfrac{1}{a_r}}}}$$ (Note that the ...
7
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1answer
265 views

Do simple, multi-dimensional generalizations of this continued fraction formula exist?

Background Let $x\in [0,1)\setminus \mathbb{Q}$ have regular continued fraction expansion given by $$ x = [a_1, a_2, a_3, \dots] = \cfrac{1}{a_1+\cfrac{1}{a_2+\dots}}, \qquad a_i \in \mathbb{N}. $$ ...
2
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1answer
247 views

Complex continued fractions with given digits

For which sets of Gaussian integers $A\subseteq\mathbb{Z}[i]$ is the set of continued fractions with digits in $A$ $$ C(A)=\{a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cdots}}\mid a_i\in A\} $$ totally ...
4
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2answers
416 views

Numerical coincidence?

(Nobody's answered this one on stackexchange after several days.) My brother built a garage whose horizontal cross-section is a rectangle that measures $45$ feet by $30$ feet. To make sure the right ...
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5answers
2k views

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 ...
1
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2answers
195 views

Linkage between singularities of algebraic varieties and continued fractions

I have an impression that there is linkage or relation between singulariry of algebraic variety and continued fraction when I read some book on resolution of singularity or algebraic geometry.Could ...
4
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0answers
218 views

Expressions of $tanh$ type whose continued fractions have two shifts per period

This is a follow-up of another thread about quasi periodic continued fractions, a.k.a. Hurwitz fractions, with some linear shifts. I seem to have found the pattern of a subclass of them, as given ...
3
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0answers
225 views

quasi periodic continued fractions and powers of e, tanh, tan

It is well known that some transcendental numbers like e.g. rational multiples of $e^{2/n}$ with $n\in\mathbb N $, when written as regular continued fractions (R.C.F.), yield what can be called a ...
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364 views

Connection between Infinite continued fractions and AGM

It is known that at $x=1$, the following continued fraction represents $\frac{4}{\pi}$ and can be approximated rapidly using Gauss' Arithmetic Geometric mean. $$C(x) = x + \frac{1^{2}}{2x + ...
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1answer
339 views

Previous work on this generalization of continued fractions?

The 2x2 matrix representation of a continued fraction makes it clear that we're multiplying together a bunch of group elements. Inversion is essentially freely adjoining a generator to a Coxeter ...
5
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1answer
268 views

3-D continued fractions

Which theorems from classical theory of continued fractions have 3-(or multi-) dimesional analogs? Of cause classical one is a periodicity of Klein polyhedra. Probably there are some more...
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2answers
678 views

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 $k\in \lbrace 1,2,\dots, n-1\rbrace$ ...
10
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2answers
718 views

The complete list of continued fractions like the Rogers-Ramanujan?

I have two questions about q-continued fractions, but a little intro first. Given Ramanujan's theta function, $$f(a,b) = \sum_{n=-\infty}^{\infty}a^{n(n+1)/2}b^{n(n-1)/2}$$ then the following, ...
6
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0answers
241 views

Birth-Death Process associated with Orthogonal Polynomials

I have read in various places the following objects are related: orthogonal polynomials birth-death processes Lattice paths continued fractions After a lot of searching online, I found sketches ...
2
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2answers
310 views

Palindromic continued fraction

Here's what I hope is a final question outside of my area that I need to understand a problem about stable vector bundles on $\mathbb{P}^2$. Thank you everybody for your help so far! Suppose I have ...
6
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2answers
304 views

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 ...
6
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4answers
596 views

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 ...
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2answers
480 views

Liouville's Theorem in Diophantine Approximation

Liouville's Theorem states that for any algebraic $\alpha \in \mathbb{R}$ of degree $n$, there exists a positive constant $c:=c(\alpha)$ such that $$|\alpha-\frac{p}{q}|>\frac{c}{q^n}$$ for any $p ...
4
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2answers
179 views

Generators of a 2D lattice

Dear MO_World, I'm hoping someone can point me towards a reference for something. I have an invertible $2\times 2$ matrix, $A$, with real entries such that for both of the rows, the entries are ...
7
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2answers
294 views

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 $Q_A$ ...
5
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3answers
379 views

Proto-Euclidean algorithm

Consider the Euclidean algorithm (EA) as a way to measure the relative length $b/a$ of a shorter stick $b$ compared to a longer one $a$ by recursively determining $$q_i = \left\lfloor ...
1
vote
1answer
265 views

Simple and general relation between continuant polynomials

Continued fraction $[a_0,a_1,...,a_n]$ may be expressed as quotient of two polynomials of $(a_0,a_1,...,a_n)$, named continuants (see http://en.wikipedia.org/wiki/Continuant_%28mathematics%29 ) ...
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1answer
497 views

Continued fractions and projective resolutions

Hello, This question might be vague and not thought-through enough. If we have a real positive number $x$, we can start to write it as a continued fraction: $x = a_0 + \frac{1}{x_1} , \ldots , ...
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5answers
894 views

Euler's divergent series sum n!*(-1)^n: what is known about the resulting constant?

Much of the theory of continued fractions has been developped by Euler in the 18th century. The little survey "Euler: continued fractions and divergent series (and Nicholas Bernoulli)", mentions ...
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1answer
1k views

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 ...
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1answer
750 views

How to find the region of convergence of this series using the theory of continued fractions?

Please, consider the following series \begin{equation} f(z)=1+\sum_{n=1}^{\infty}2^{-\sum_{k=1}^{n}\frac{2s}{k}} =1+ \sum_{n=1}^{\infty}\left( \prod_{k=1}^{n}2^{-\frac{2z}{k}} \right) \end{equation} ...
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2answers
596 views

Composition of circle inversions

I would like to understand the map of $\mathbb{C}$ to $\mathbb{C}$ that results by iterating inversion in a unit circle. Let $f(z)$ for $z \in \mathbb{C}$ invert $z$ in a unit circle centered on ...
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2answers
799 views

Gauss-Kuzmin Theorem (continued fractions) - why is important?

As we know, Gauss wrote that \begin{equation} \lim_{n \rightarrow \infty} \lambda \left(\tau^n \leq x\right) = \frac{\log(1+x)}{\log2}, \quad 0 \leq x < 1, \end{equation} with $\lambda$ is Lebesgue ...
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6answers
2k views

Which Diophantine equations can be solved using continued fractions?

Pell equations can be solved using continued fractions. I have heard that some elliptic curves can be "solved" using continued fractions. Is this true? Which Diophantine equations other than Pell ...
2
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0answers
660 views

Applications of the length of the continued fraction

The fractions $n/q$ and $n/(n-q)$ can be representated by Hirzebruch continued fractions (also called Jung-Hirzebruch CF), the length of each we denote by $s$ and $t$. Is there any bound for $s+t$?
4
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1answer
778 views

Searching for an inhomogeneous diophantine approximation algorithm

Given two nonzero real numbers $x$ and $y$ such that $y/x$ is irrational, a real number $z$ to be approximated, and a tolerance $\epsilon$, what is an algorithm that will provide coprime integers $a$ ...
5
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1answer
276 views

Algorithm to determine sign of a polynomial

I've been working with a collaborator (Arek Goetz) on a dynamics problem involving piecewise isometries (a map $T$ on a domain $X$ (say a subset of the plane) such that $X$ is divided into a finite ...
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1answer
901 views

Some unpublished notes of Hofstadter

I'm looking for some unpublished notes called "Eta Lore," which are apparently related to a talk Douglas Hofstadter first gave at the Stanford Math Club in 1963. I know these notes exist because ...
5
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1answer
968 views

Unsolved Problem from AmMathMonthly

Here is a simply described but fiendishly diophanterrorizing problem I asked on AMM eons ago. Maybe you can shed some light upon it. 0.2 (base 4) = 0.2 (continued fraction) 0.24 (base 6) = 0.24 ...
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0answers
272 views

A question on M. Mignotte's Paper: “Petho's Cubics”

I have been reading the paper "Petho's Cubics" by M. Mignotte (appeared in Publ. Math. Debrecen, 56/3-4 (2000)) unfortunately I don't understand section 4 of the paper. I am hoping someone could help ...
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1answer
303 views

Best rational approximation in a special sense

Let $\alpha$ be an irrational number, $n\geq 1$ and $ X_n=\lbrace (x,y) \in {\mathbb Z}^2 | |y| \leq n, \ x+y\alpha >0 \rbrace$ Now let $(x_n,y_n)$ minimize the quantity $x+y\alpha$ on $X_n$. ...
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2answers
1k views

Is there an explicit example of such a real number with the following property?

In Diophantine approximation, for a given positive real number $\alpha$ let $[a_0, a_1, \cdots]$ denote its continued fraction expansion and let $p_n/q_n = [a_1, \cdots, a_n]$. Then it is known that ...
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2answers
857 views

Applications of periodic continued fractions

Some answers from Applications of finite continued fractions in fact are Applications of periodic continued fractions. I think that it should be separate question. What can you add to the following ...
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14answers
4k views

Applications of finite continued fractions

I know some applications of finite continued fractions. Probably you know more. Can you add anything? (For for Applications of periodic continued fractions I have made a special topic.) 1) (Trivial) ...
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3answers
688 views

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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1answer
514 views

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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1answer
313 views

Lengths of continued fractions for the numbers with fixed ratio

Let $s(x)$ is the length of continued fraction expansion of $x$, and let $l(x)$ be the sum of partial quotients. I can prove that for any rational $\alpha$ ratios $\frac{s(\alpha x)}{s(x)}$ and ...
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891 views

Ramanujan's $\tau(n)$ and continued fractions

In D.H. Lehmer's paper "Ramanujan's function $\tau(n)$, (Duke J. Math v. 10 1943, pp. 483-492), Lehmer states the Ramanujan conjecture $|\tau( p )|< 2p^{11/2}$, so that $p^{-11/2}\tau( p ...