Questions tagged [continued-fractions]
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207
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Small solutions of $x^2-a^3 y^2=\pm 1$
We are interested in small integer solutions to the Pell equation:
$$x^2-a^3 y^2=\pm 1 \qquad (1)$$
Where in $\pm 1$ you can chose either sign.
$(x^2,a^3 y^2)$ are consecutive powerful numbers.
$abc$ ...
2
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0
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36
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Are there any known Khinchin reals for which the asymptotics of "average" of their coefficients seems experimentally known?
We can define a Khinchin Real and recall the definition of Khinchin's Constant
A real number $r$ is a Khinchin real if given the simple continued fraction expansion of $r$ as
$$ r = a_0 + \cfrac{1}{...
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3
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Irrationality of generalized continued fractions
An infinite simple continued fraction
$$\frac{1}{b_1 + \frac{1}{b_2 + \frac{1}{b_3+\dots}}} (b_i\in\mathbb Z\setminus\left\{0\right\})$$
is irrational. Now for a generalized continued fraction:
$$\...
4
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1
answer
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Legendre's Irrationality Condition for Generalized Continued Fractions
This MathOverflow post cites that Legendre allegedly showed that given $a_{i}\in\mathbb{Z}\setminus\left\{0\right\}, b_{i}\in\mathbb{Z}$,
$$\cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cdots}...
10
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1
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Bibliography request: Entropy for continued fractions
Given a strictly positive real number $x$ we set $e(x)=\log(1+x)$ if $x$ is an integer and
$$e(x)=\log(1+x)+\frac{1+\lbrace x\rbrace}{1+x}\left(e(1/\lbrace x\rbrace)-\log(1+\lbrace x\rbrace)\right)$$
...
2
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0
answers
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Curious sequences associated to continuous fractions
Given a strictly positive initial rational number $x_0$ in $\mathbb Q_>$
we define a sequence $x_0,x_1,\ldots$ recursively by
setting $x_{n+1}=x_n+1/S(x_n)$
for $S(x)=a_0+a_1+\ldots+a_k$
where $[...
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5
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Algorithm generalizing continued fractions for non-quadratic algebraic numbers
The continued fraction algorithm generates an integer sequence which terminates for a rational number, is periodic for the roots of irreducible integer quadratics, and is non-periodic for other ...
2
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1
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(Exponential) Mixing property for Gauss map - going from cylinders to intervals
I'm trying to understand the proof of a mixing property of the Gauss map from the paper - 'Some metrical theorems in number theory' and I'm getting confused by the logic in a step.
The Gauss map $T$, ...
7
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0
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Simple/Elementary derivation of Ramanujan's continued fraction for Hurwitz $\zeta(3,x+1)$
I came across this MSE post discussing a certain continued fraction for $\zeta(3)$ (more specifically, the Hurwitz zeta function $\zeta(s,z)$ at $s=3$) due to Ramanujan. I asked the original poster ...
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1
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Evaluating the generalized continued fraction obtained from the factorization of a bivariate polynomial equation
Happy New Year, MO community!
We need someone expert in Generalized Continued Fractions (GCFs), with a deep knowledge of the GCFs’ convergence properties, to solve the following problem.
PROBLEM
...
3
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1
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249
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What is the sequence of badly approximable numbers to omit in Hurwitz' second Theorem for Diophantine Approximation to obtain better constants?
The well known result of Hurwitz on Diophantine approximation says that for any irrational $\alpha$ there are infinitely many integer numbers $p$ and $q$ such that
$$
|\alpha -\frac{p}{q}|<\frac{1}{...
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1
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197
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Calculating the value of periodic continued fractions with $a_i\in\lbrace 0,1\rbrace$
Question:
How can the value of continued fractions of the form
$$y:=a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cfrac{1}{\begin{align}\ddots& \\ &a_{n-1}+\cfrac{1}{a_n+y}\end{align}}}}}$$
$$...
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1
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260
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Factorization trees and (continued) fractions?
This question is inspired by trying to understand the lexicographic sorting of the natural numbers in the fractal at this question:
Is $1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}$ , ...
2
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0
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Elementary recursion for the A258173
Let $a(n)$ be A258173 i.e. sum over all Dyck paths of semilength $n$ of products over all peaks $p$ of $y_p$, where $y_p$ is the $y$-coordinate of peak $p$.
A Dyck path of semilength $n$ is a $(x,y)$-...
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698
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Are these continued fractions of integrals known?
Simplified repost of Are these continued fractions of integrals known? on MSE
EDIT: The period of the oscillations of $$f(s)=\dfrac1{1+\dfrac s{1+\dfrac{s^2/2!}{1+\dfrac{s^3/3!}{1+\cdots}}}}$$ ...
10
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3
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Last term of repeating continued fraction expansion
Once again, working with stable vector bundles on $\mathbb{P}^2$ I have run into a question that is really out of my area. (Thanks to everybody who helped out with my last question!)
Let $D>9$ be ...
4
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2
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277
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Approximating a fraction with a given denominator
Let $M$, $N$ be large natural numbers (say ~200 bits). Let $L$ be a smaller number, (say ~100 bits).
I want to approximate the fraction:
$$\frac{M}{N} \sim \frac{k}{L+r}$$
where $r$ is at most $L$. In ...
3
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1
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What can we say about the reciprocal of a reduced regular continued fraction?
For positive integers $a>b>0$, we can represent $a/b$ uniquely as $$\frac{a}{b}=a_1-\cfrac{1}{a_2-\cfrac{1}{\cdots-\cfrac{1}{a_n}}}=:[a_1,\dots,a_n]^{-}$$ with $a_i\geq 2$, and this is called ...
4
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Simple continued fraction of Freiman's constant
The quadratic irrational $\frac{2221564096+283748\sqrt{462}}{491993569}$ is known as Freiman's constant and arises in the theory of continued fractions. I'm curious as to its simple continued ...
10
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1
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Are these two $q$-continued fractions equivalent?
In this MSE post, Nicco Mnisi defined a particular $q$-continued fraction of order $12$. More generally, define the cfrac found in Ramanujan's Notebooks, Vol III, Chap. 16, page 24, where $|ab|<1$ ...
2
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0
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Is the continued fraction of a constructible number special in some way?
Rationals have finite CF and quadratic have periodic CF. CF in turn can be represented in terms of the modular group SL2(Z), e.g. using the standard generators S(z)=-1/z and T(z)=z+1. On the other ...
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Finding the continued fraction for the "tails" of $\eta(3)$
I am interested in the continued fractions for the "tails" or "correction term" of the series sum of specific constants. For example, the Madhava's correction term for $\pi/4$:
$$
\...
6
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1
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622
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Are these continued fractions for the "tails" of $\zeta(3)$ and of the Catalan constant known?
For polynomials $a=a(x)$ and $b=b(x)$, define the continued fraction $$f(a,b):=a(1)+ \lower 2pt\overset{\infty }{\underset{n=1}{\mathbb{\LARGE K}}}~\dfrac{b(n)}{a(n+1)}=a(1)+\cfrac{b(1)}{a(2) + \cfrac{...
0
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0
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Numerical detection of Cantori
It is known that as parameters vary in Hamiltonian system, KAM tori can break [1,2].
How to construct numerically the breaking tori?
The most relevant paper that I could find is [3,4].
But it uses ...
2
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1
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143
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A bijection between odd natural integers and compositions
Given an odd natural integer $2a-1$ with $a\geq 1$, associate to it recursively the composition $\psi(1)=\emptyset$ and $\psi(2^{-n}a)+(n+\delta_{>1}(m))$ if $a=2^n m$ with $m$ odd where $\delta_{&...
6
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Complexity of continued fraction arithmetic operations
Let $A = [a_0; a_1, \dots]$ and $B = [b_0; b_1, \dots]$ be continued fractions.
Let's say that we want to compute $A+B$ or $A \cdot B$ while staying in the continued fraction representation. So, for ...
1
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0
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Proving that quotient of orthogonal polynomials is a Padé approximant of Stieltjes transform
This question is reposted from Math Stack Exchange (you can see the original post here). The motivation for reposting is that I feel like the question isn't getting much attention in MSE - if there is ...
0
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1
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128
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Series reversion using something like continued fraction
Let $f(n)$ be an arbitrary function such that $f(n)\in\mathbb{Z}$.
Let
$$
F(x)=\sum\limits_{m\geqslant 0}f(m)x^m
$$
Define the operator $\operatorname{SR}$, which is associated with the series ...
14
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1
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The Riemann's Zeta Function represented as a continued fraction and a question of convergence
The Riemann zeta function can be expressed as a continued fraction as follows
\begin{align*}
\zeta(z)=\newcommand{\bigk}{\mathop{\Huge\vcenter{\hbox{K}}}}\left(1-\bigk_{k=1}^{\infty }\frac{-e^{-2\cdot ...
5
votes
1
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On four Ramanujan-type "Legendrian" sequences with a 3-term recurrence?
I. Recurrences
In a previous post, it was mentioned how Almkvist-Zudilin did a computer search for solutions to the recurrence relation,
$$(n+1)^3s_{n+1}=(2n+1)(an^2+an+b)s_n+c\,n^3s_{n-1}$$
within a ...
8
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2
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577
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On Zagier's missing continued fraction with multiple limits?
I. Zagier's continued fraction
As pointed out by Gorodetsky in his answer, Zagier evaluated the continued fractions associated with his six sporadic sequences excepting the one for $(-9,-3,-27)$. Let $...
6
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1
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251
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On the continued fractions using Cooper's sequences $s_7,\, s_{10},\, s_{18}$ and the Zudilin-Cohen sequence
In a previous MO post, H. Cohen suggested Gorodetsky's 2021 paper which discussed $6+6+3=15$ "sporadic sequences". The first 6 are Zagier's sporadic sequences, the second 6 are by Almkvist-...
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On H. Cohen's four continued fractions for $\zeta(3), \zeta(5), \zeta(7)$?
After 6 years from this old MO post, I finally find in the literature polynomials of deg-$5$ for the continued fraction of $\zeta(5)$.
I. Recurrences involving $\zeta(5)$
In Cohen's 2022 paper, ...
2
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0
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Finding similar Zudilin-Cohen recurrence relations and cfracs for $\frac{\zeta(4)}{13}$?
I. Two recurrence relations
The first one was also discussed in this MO post. We have the similar,
\begin{align}
(n+1)^5 u_{n+1} &= (2n + 1)(9n^2 + 9n + 3)(15n^2 + 15n + 4)u_n +3n^3(9n^2-1)u_{n-1}\...
7
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Can you identify this irrational number?
There is a certain number, say $v$. I can prove it is irrational. That would be more interesting if it is expressible in terms of known values ... zeta functions, Catalan's number, L-functions, etc. ...
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On 12 cfracs: for Catalan's $K$, Gieseking's $\kappa$, and $\pi^2$, $\pi^3$, plus three for $\zeta(3)$ using Zagier's "six sporadic sequences"
I. Some functions
As these will be used in the continued fraction evaluations below, recall the Riemann zeta function $\zeta(s),$ and Dirichlet beta function $\beta(s),$
$$\beta(s) = \sum_{n=1}^\infty\...
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21
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Applications of finite continued fractions
I know some applications of finite continued fractions. Probably you know more. Can you add anything? (For Applications of periodic continued fractions I have made a special topic.)
1) (Trivial) ...
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2
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106
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How to solve the following infinite ladder fraction? ( Through pen & paper ) [closed]
The fraction continues till infinity as shown in the image :
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0
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108
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Improving Diophantine approximation by rescaling
Let $\lambda\in(0,1)$ be an irrational number such that its continued fraction expansion is bounded (for example, an irrational quadratic number, whose continued fraction is periodic). It is known ...
2
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2
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1k
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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 ...
15
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2
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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,
$$A(q) =...
3
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1
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The growth of certain continued fractions
I was recently looking into an old problem of Hardy which studies the distribution of integers of the form $2^a 3^b \leq x$, where $a,b\geq 0$. Letting $N(x)$ denote the number of pairs $(a,b)$ ...
0
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2
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210
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Natural extension of the Gauss map
Let $G:(0,1)\to(0,1)$ be the Gauss map, i.e., $G(x)=\left\{\frac1{x}\right\}$, which is known to act as the shift on the space of continued fraction expansions.
Question. Is there an explicit ...
14
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0
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345
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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 ...
5
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0
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Very slow continued fraction convergence
Let $a(0)+b(0)/(a(1)+b(1)/(a(2)+b(2)/(a(3)+\dots)))$ be a continued fraction, and $p(n)/q(n)$ its $n$-th convergent. If it converges (i.e., $p(n)/q(n)$ tends to some limit
$S$ as $n\to\infty$), then $...
6
votes
2
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1k
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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}$...
34
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2
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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 } }}},$$
where $k\in\mathbb Q$ and $p$ is a ...
3
votes
0
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93
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Infinite families of continued fractions for the Catalan constant
The recent answer to an old question of mine made me aware of The Ramanujan Machine. So it seems like so far, the number of continued fraction representations for $\zeta(3)$ of this polynomial kind is ...
16
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1
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Extending Apéry's proof to Catalan's constant?
I've been looking into Apéry's irrationality proof of $\zeta (3)$, and one of the first questions I instantly had, was how did he derive the following continued fraction?
$$\begin{equation*} \zeta (3)=...
5
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2
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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 for ...