# Questions tagged [continued-fractions]

The continued-fractions tag has no usage guidance.

140
questions

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### What is the nearest Ford circle for any point in $\mathbb R^2$

I want to draw Ford circles within a "distance Estimated system" (ray marching). Therefore, given a point $(x,y)$ from $\mathbb R^2$, I need the shortest distance to any circle with center $(p/q,1/2q^...

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90 views

### Proof of continued fraction identity of subfactorial

This question is part of a wider conjecture I have formed with someone which has its roots in Raayoni et al. (2019). The subfactorial function can be written as $$!n=\frac{n!}e+\frac{(-1)^n}{n+2-\...

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98 views

### Possible Birkhoff spectra for irrational rotations

Let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ be the unit circle (think of it as of the interval $[0,1)$ with endpoints identified). Assume that $\alpha$ is irrational and consider the rotation by $\alpha$, ...

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**1**answer

166 views

### Distribution of partial quotients of $\sqrt{d}$ with $X < d \leq 2X$

Recall that for a real number $\alpha$, it has a continued fraction expansion usually written as
$$\displaystyle \alpha = [a_0; a_1, a_2, \cdots].$$
Moreover, $\alpha$ is rational if and only if its ...

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27 views

### Maximal orders in Clifford algebras

Let
$$
\mathcal{C}_n(R)=R\langle e_1,\ldots,e_n\rangle/(\{e_i^2+1\}, \{e_ie_j+e_je_i:i\neq j\})
$$
be the Clifford algebra for the negative definite quadratic form $-\sum_ix_i^2$ obtained by adjoining ...

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**1**answer

481 views

### Are the coefficients of certain product of Rogers-Ramanujan Continued Fraction non-negative?

Let $$R(q) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^{2}}{1 + \cfrac{q^{3}}{1 + \cdots}}}}$$
The following equality is famous:
$$\cfrac{q^{1/5}}{R(q)} = \prod_{k>0} \cfrac{(1-q^{5k-2})(1-q^{...

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305 views

### Average number of iterations for the Euclidean algorithm to terminate

Let $N$ be a positive integer and $0 \leq s < N$.
We try to divide $s$ into $N$ using the Euclidean algorithm:
$N = q_1 s + r_1 $
$r = q_2 r_1 + r_2 $
$\vdots$
$r_{K-1} = q_{K-1} r_K$
If we ...

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**2**answers

160 views

### Evaluation of hypergeometric type continued fraction

Is there a (possibly hypergeometric-type) explicit evaluation of the
continued fraction
$$a-\dfrac{1.(c+d)}{2a-\dfrac{2.(2c+d)}{3a-\dfrac{3.(3c+d)}{4a-\ddots}}}$$
Even the special case $d=0$, $a=1$ ...

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**1**answer

461 views

### Does using continued fractions work to give a homeomorphism $\mathbb{Q}^+ \rightarrow (\mathbb{Q}^+)^2$?

Let $\mathbb{Q}$ be the topological space of rational numbers (with topology induced by inclusion in the real line) and let $\mathbb{Q}^+$ be the set of positive ($x>0$) rationals.
I'm looking ...

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116 views

### Why does division parallelize but not continued fractions and is there an analog of multiplication to continued fractions?

All the basic arithmetic operations $\times,+,/,-$ can be parallelized. However continued fraction representation of a rational number is not parallelized. The process of Euclid's algorithm looks ...

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637 views

### Have new conjectures generated by the Ramanujan machine been proven?

Recently the following preprint was published with new automatically generated conjectures on generalizes continuous fractions, e.g., for the Euler constant: Raayoni, Pisha, Manor, Mendlovic, Haviv, ...

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**1**answer

1k views

### Divergent Series & Continued Fraction (from Gauss' Mathematical Diary)

I've asked that question before on History of Science and Mathematics but haven't received an answer
Does someone have a reference or further explanation on Gauß' entry from May 24, 1796 in his ...

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**1**answer

184 views

### Riemann-Hilbert and Orthogonal polynomials

Sorry for perhaps naive questions, I am not at all a specialist in the subject
but I need it for my research.
I know that there are close relations between Riemann-Hilbert problems and
orthogonal ...

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446 views

### Representations of $\zeta(3)$ as continued fractions involving cubic polynomials

$\zeta(3)$ has at least two well-known representations of the form $$\zeta(3)=\cfrac{k}{p(1) - \cfrac{1^6}{p(2)- \cfrac{2^6}{ p(3)- \cfrac{3^6}{p(4)-\ddots } }}},$$
where $k\in\mathbb Q$ and $p$ is a ...

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82 views

### “middle” partial denominator in continued fraction expansion of square roots

Suppose $d$ is a positive integer that is not a perfect square such that the negative Pell equation, $x^{2}-dy^{2}=-1$ has no solution. Then we know the minimal period of the continued fraction ...

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**1**answer

158 views

### Limit of quotients of elements of special Fibonacci matrices

Let $F_n$ be the $n$-th Fibonacci number, started with $F_0=0,F_1=1$, and consider the matrices
$$M_n=\pmatrix{F_{n+3} & F_{n+1} \\ F_{n+2} & F_{n}}.$$
Let
$$\pmatrix{\alpha_n & \beta_n \\...

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48 views

### Bound for truncation error of continued fraction for $E_1(z)$

Let $z \in \mathbb C \setminus(-\infty,0)$. It is known that
$$E_1(z) = \cfrac{e^{-z}}{z+\cfrac{1}{1+\cfrac{1}{z+\cfrac{2}{1+\cfrac{2}{z+\cfrac{3}{1+\cdots}}}}}}.$$
For example, see http://functions....

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**1**answer

94 views

### Distinctness of quadratic surd continued fraction convergent ratio limit

In this question on math.stackexchange.com I have made two conjectures the first of which I have proved. The second has not been settled. I post it here to seek a proof.
Given a quadratic surd $\sqrt ...

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43 views

### Reduction of a Jacobi-type continued fraction

I am trying to reduce the following Jacobi-like continued fraction(or J-fraction):
$$f(z)=z+K_{n=1}^{\infty}\frac{R_{n}k^2}{z+Q_{n}}$$
where,
$$R_{n}=n\left(n-\frac{1}{2}\right),\; Q_n=n\left(n+m+\...

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252 views

### Theory of Irrational Tangles?

According to one possible definition, an $n$-tangle $T$ is a subset $T \subseteq \Bbb{R}^2\times [0,1] =: X$ that is homeomorphic to a disjoint union $[0,1] \times n := [0,1] \amalg \ldots \amalg [0,1]...

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234 views

### Periods of the continued fraction expansions of Galois-conjugate quadratic-irrationals

Question: Given a quadratic irrational $x = a + b\sqrt{D}$ ($a,b \in \Bbb{Q}$, $D \in \Bbb{N}_{> 0}$ square-free) and its Galois conjugate $x' = a - b\sqrt{D}$, is it true that the continued ...

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215 views

### Transcendental Continued Fractions

Liouville famously showed the existence of a transcendental number by considering $\alpha =\sum\limits_{n=0}^\infty10^{-n!}$ and showing that it did not satisfy 'Liouville's criterion' $|\alpha-p/q|\...

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75 views

### Different characterizations of Liouville numbers

Usually, Liouville numbers are defined as follows:
$x$ is Liouville if for ever $i\in\mathbb N$ there exist $n,m\in\mathbb Z$ such that
\begin{equation}
\left|x-\frac nm\right|<\frac1{m^i}.
\end{...

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**1**answer

178 views

### Matrix continued fractions

I am aware of the classical continued fraction in the field of real numbers, but recently I have come across the term matrix continued fraction and when I checked on the internet there are varieties ...

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165 views

### Does the set of Diophantine $m$-tuples has full measure?

We say that an $m$-tuple $\omega=(\omega_1,\ldots,\omega_m)$ satisfies the Diophantine condition of order $\nu \geq 0$ if there is a constant $C>0$ such that for all natural $q$ and integer $p_1,\...

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345 views

### Relation between a continued fraction and partitions

I am interested in the continued fraction
$$\sum\limits_k {{z^{{2^k} - 1}}} = \frac{1}{{1 - \frac{{{T_0}z}}{{1 - \frac{{{T_1}z}}{{1 - \frac{{{T_2}z}}{{1 -{ \ddots }}}}}}}}}.$$
OEIS A104977 states ...

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119 views

### Simultaneous Diophantine Condition and Growth Rate of Convergents Denominators

Let $\omega=(\omega_1,\ldots,\omega_{m})$ be an $m$-tuple of real numbers. Let $|\omega|_{m}:=\sup\limits_{1 \leq j \leq m}|\omega_j|_{1}$ be a metric on flat torus $\mathbb{T}^{m}=\mathbb{R}^{m}/\...

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151 views

### whether the quotient of continued fraction of algebraic irrational number is bounded or not is similar or equivalent to Collatz conjecture?

I vaguely recall that whether the quotient of continued fraction of algebraic irrational number is bounded or not is similar or equivalent to Collatz conjecture, could any one give the reference? or ...

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201 views

### Distribution of good diophantine approximations

Let $\langle x \rangle: \mathbb{R} \to (-1/2,1/2]$ be the periodic function with period $1$ which is $x$ for $x \in (-1/2,1/2]$. Is there some function $D(a,b)$ of real numbers $a<b$ such that, for ...

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**1**answer

235 views

### Continued Fraction of Random Variables

So this is my first post in mathoverflow. I posted this problem in Mathstack, an I've also
put a bounty on it, but did not get any response. If anyone can at least point out a reference on this ...

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2k views

### Distribution of distances of successive zeros of $f(x)={\rm cos}(x)+{\rm cos}(\alpha x)+{\rm cos}(\beta x)$

Let $\alpha$ and $\beta$ be incommensurate real numbers.
Consider the function
$f(x)={\rm cos}(x)+{\rm cos}(\alpha x)+{\rm cos}(\beta x)$
and its positive zeros $x_k(\alpha,\beta)$.
Fix $\alpha$ and ...

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214 views

### quasiperiodic continued fractions

Is anything known about continued fractions in which the sequence of integers is quasiperiodic?
Quasiperiodic is meant here in the sense of 1D quasicrystals. For example, draw an irrationally-sloped ...

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325 views

### Irrationality of generalized continued fractions

An infinite simple continued fraction
$$\frac{1}{b_1 + \frac{1}{b_2 + \frac{1}{b_3+\dots}}} (b_i\in\mathbb Z)$$
is irrational. Now for a generalized continued fraction:
$$\frac{a_1}{b_1 + \frac{a_2}...

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141 views

### Has anybody studied continued fractions in function spaces?

For the text below, define $f^\infty(x) = \lim_{n\to\infty} f^n(x)$ where $f^n = \underbrace{f \circ \ldots \circ f}_{n}$.
Usually 'continued fraction' means continued fraction in $\mathbb{R}$. For ...

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**1**answer

275 views

### Some nice functional equations for $q$-continued fractions

Given $\large q=e^{2\pi i \tau}$. Define,
$$\alpha(\tau) = \sqrt2\,q^{1/8}\prod_{n=1}^\infty\frac{ (1-q^{4n-1})(1-q^{4n-3})}{(1-q^{4n-2})(1-q^{4n-2})}$$
$$\beta(\tau) = q^{1/5}\prod_{n=1}^\infty\frac{ ...

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141 views

### The Heine $q$-continued fraction

Let $q=e^{2\pi i \tau}$. The Heine continued fraction is $$H_2(\tau)=\frac1{q^{1/24}}\frac{\eta(2\tau)}{\eta(\tau)} =1+\cfrac{q}{1-q+\cfrac{q^3-q^2}{1+\cfrac{q^5-q^3}{1+\cfrac{q^7-q^4}{1+\ddots}}}}$$
...

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175 views

### limit of denominator in continued fraction expansion algebraic?

I asked this on MathStackExchange but didn't get an answer, so I'm trying it here:
Let $\alpha$ be an algebraic number and denote with $\frac{p_n}{q_n}$ the $n$-th convergent of $\alpha$ that we get ...

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**1**answer

1k views

### Can we use the Rogers-Ramanujan cfrac to parameterize the Fermat quintic $x^5+y^5=1$?

Define $\color{blue}{q=e^{2\pi i \tau}}$ and Dedekind eta function $\eta(\tau)$. Note: I found these relations empirically, but their consistent forms suggest they can be rigorously proven.
I. $p=2$...

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77 views

### Stern-Stolz in $p$-adic case

I'm trying to figure out if the following statement is trivial or not:
For $b_i \in \mathbb{C}_p$ (the complete $p$-adic field), if $\sum |b_i|_p < \infty$, then the continued fraction $b_0+\...

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245 views

### when is the Brun continued fraction periodic?

I was hoping to figure this one out on my own. There's this nice paper by Avila on various "subtractive" Euclidean algorithms. Here is one he attributes to Viggo Brun:
$$ (x,y,z) \mapsto \text{sort}...

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237 views

### Growth rate for the average of the entries in the fundamental period of the continued fraction expansion of $\sqrt{n}$

(Cross-posted from stackexchange: https://math.stackexchange.com/questions/1976296/what-is-known-about-the-average-of-the-partial-quotients-in-the-fundamental-peri)
I'd like references concerning ...

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378 views

### Constant related to continued fraction of quadratic irrationals

Let $d$ be a positive, non-square integer, and define $c_d$ to be the smallest positive number with the following property: for all pairs of co-prime integers $(p,q)$ with $q > 0$, the inequality
$...

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77 views

### Criterion for irrational numbers of constant type 2

From Kuiper's and Niederreiter's book Uniform distribution of sequences, Ch.2, § 3, I learn that an irrational number $\alpha\in \mathbf{R}\smallsetminus \mathbf{Q}$ is of constant type $\eta$ if ...

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630 views

### Rogers-Ramanujan continued fraction $R(e^{-2 \pi \sqrt 5})$

Let $$R(q) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^{2}}{1 + \cfrac{q^{3}}{1 + \cdots}}}}$$
It is easy to evaluate $R(e^{-2 \pi/ \sqrt 5})$ using the Dedekind eta function identity $\eta(-\frac{...

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**3**answers

443 views

### Combinatorial aspects of continued fractions

Recently, I got interested in the study of the combinatorial aspects of continued fractions. Precisely, I read of the following lemma of Flajolet (see here):
Lemma. It holds
$$\sum_{\omega} \nu(\...

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258 views

### Identity with Ramanujan's generalized continued fraction

Let $F(x,q)=\sum_{n\geq 0}x^n\dfrac{q^{n^2}}{(q)_n}$, where $(q)_n=(1-q)(1-q^2)\dots(1-q^n)$. Then:
$$H(x,q)=\frac{F(-xq,q)}{F(-x,q)}=\dfrac{1}{1-\dfrac{qx}{1-\dfrac{q^2x}{1-\dots}}}$$ is the ...

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**1**answer

134 views

### Mean value of a function associated with continued fractions

Suppose that an irrational $x$ in $(0,1)$ has convergents $c(k,x)$, and let
$$d(x) = \sum_{k=0}^{\infty} \mid x - c(k,x)\mid.$$
What is the mean value of $d$?

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75 views

### identity which include continued fraction

In this encyclopedia,
http://encyclopedia-of-equation.webnode.jp/including-continued-fraction/
I found this identity.
$$\sum_{n=1}^{k-1} \frac{2n}{(n^2+r^2)^2}+ \frac{1}{ (k-\frac{1}{2})^2+\frac{1+...

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**2**answers

652 views

### “Fractally self-similar” numbers

This is another question about visualization of Ford circles, the previous one being Confusion with practically implementing rational approximations. Here is an output of zooming into Ford circles at $...

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**1**answer

264 views

### 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 ...