# Questions tagged [continued-fractions]

The continued-fractions tag has no usage guidance.

196
questions

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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 ...

5
votes

1
answer

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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 $\pi/4$:
$$
\...

0
votes

0
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73
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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
votes

1
answer

132
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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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190
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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
votes

1
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124
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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 ...

11
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427
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### Are these continued fractions of integrals known?

Simplified repost of Are these continued fractions of integrals known? on MSE
Define the continued fraction integral transform $$\{\mathcal If(t)\}(s)=\dfrac{f(s)}{1+\dfrac{\int_0^s f(x)\,dx}{1+\...

4
votes

1
answer

146
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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 ...

6
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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
votes

0
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130
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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. ...

8
votes

2
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537
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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 $...

5
votes

1
answer

220
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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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2
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579
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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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votes

2
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84
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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 :

0
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0
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104
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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 ...

3
votes

1
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140
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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 $(a,b)$ ...

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2
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166
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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 ...

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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
$$e(x)=\log(1+x)+\frac{1+\lbrace x\rbrace}{1+x}\left(e(1/\lbrace x\rbrace)-\log(1+\lbrace x\rbrace)\right)$$
...

5
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187
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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 $...

2
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0
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70
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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 $[...

6
votes

1
answer

573
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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{...

3
votes

0
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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 ...

2
votes

1
answer

331
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### Period of continued fraction expansion and class number

Let $d>1$ be not a square. Then the continued fraction expansion of $\sqrt d$ is $[a_0; \overline{a_1,\dots,a_\ell}]$, where $a_0=\lfloor \sqrt d\rfloor$ and $a_\ell=2a_0$.
Thus, $\ell=\ell(d)$.
...

1
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0
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78
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### Quasiperiodic sequence, finite differences, recursion

Consider a sequence $\{an\}$ consisting of fractional parts of the numbers $an$ for natural numbers $n$ where $a$ is an irrational number. Its $n$th value is in the interval $(x;y)$ for numbers $n = ...

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2
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606
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### Is $\mathbb{Q}$ the orbit of a rational function under iteration?

In this previous post I asked for the smallest set of continuous real functions that could generate $\mathbb Q$ by iteration starting from $0$. Surprisingly one continuous function suffices.
In the ...

3
votes

0
answers

70
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### Is there a finite set of polynomials generating all rational numbers by iteration?

In this previous post I asked for the smallest set of continuous real functions that could generate $\mathbb{Q}$ by iteration starting from 0. Surprisingly one continuous function suffices.
The ...

33
votes

2
answers

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### What is the smallest set of real continuous functions generating all rational numbers by iteration?

I recently came across this problem from USAMO 2005:
"A calculator is broken so that the only keys that still
work are the $\sin$, $\cos$, $\tan$, $\arcsin$, $\arccos$ and $\arctan$ buttons. The ...

9
votes

1
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606
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### $\text{SL}_2(\mathbb{Z})$ and continued fractions?

I know the following facts: $\text{SL}_2(\mathbb{Z})$ is generated by everyone's favorite matrices
\begin{equation*}
S =
\begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix}
\end{equation*}
and
\...

1
vote

1
answer

176
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### A continued J fraction for $a_n = \frac{1}{(n+1)^2}$?

The following is called a J continued fraction:
$$\cfrac{\alpha_0}{1+a_0x-\cfrac{b_1x^2}{1+a_1x-\cfrac{b_2x^2}{1+a_2x-\cdots}}}$$
where the constants are real numbers. Let $\alpha_n= \frac{1}{(n+1)^2}$...

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124
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### A question on continued J-fraction

Consider the following two continued fractions $A$ and $B$:
$$\frac{\alpha_0}{1+a_0x-\frac{b_1x^2}{1+a_1x-\frac{b_2x^2}{1+a_2x-\cdots}}}$$
$$\frac{\beta_0}{1+c_0x-\frac{d_1x}{1+c_1x-\frac{d_2x}{1+c_2x-...

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241
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### A special class of weighted Motzkin paths

Consider Motzkin paths with the following weight:
All up-steps and the horizontal steps on height $0$ have weight $1$, all down-steps have weight $t$ and the horizontal steps on even heights have ...

2
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1
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211
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### Distance formula for continued fractions

In the book Neverending fractions from Borwein, van der Poorten, Shallit and Zudilin, there is the so called distance formula (Theorem 2.45, p. 43) stated:
$$\alpha_1\alpha_2\cdot...\cdot\alpha_n=\...

3
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0
answers

162
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### Reconstructing elements of $\mathbb Q$ in $\mathbb Z_p$

Can a rational number $a/b$ (with $b$ coprime to a prime number $p$) be recovered efficiently from a $p$-adic expansion of the form
$$\frac{a}{b}=\sum_{j=0}^\infty x_jp^j,\ x_j\in\{0,\ldots,p-1\}\ ?$$
...

16
votes

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)=...

3
votes

1
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195
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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}{...

5
votes

1
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### Continued fraction associated to KdV solitons

Background (may be skipped by those interested only in the basic question and not important associations):
“An essay on continued fractions” by Euler (translated by Myra and Bostwick Wyman) contains ...

3
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### Fundamental Units in $\mathbb{Z}[\sqrt{d}]$ with $d \equiv 1 \mod 4$

It is well known and often cited how the fundamental units for the number ring of $\mathbb{Q}[\sqrt{d}]$ look like. In the case of $d\equiv 2,3 \mod 4$ the number ring is $\mathbb{Z}[\sqrt{d}]$ and in ...

4
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### Is the set of approximating sequences for irrationals dominating?

Let $r\in [0,1]\setminus\mathbb{Q}$, let $\mathbb{N}$ denote the set of non-negative integers and let $\mathbb{N}_+=\mathbb{N}\setminus\{0\}$. For $n\in\mathbb{N}_+$ let $$\alpha_r(n)=\min\{|r-\frac{...

2
votes

1
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186
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### Fibonacci-Motzkin paths and J-type continued fractions

Recall that a Motzkin path is a piece-wise linear planar path
connecting points in the integer lattice quadrant
$\Bbb{Z}_{\geq 0} \times \Bbb{Z}_{\geq 0}$ beginning at the origin $(0,0)$ and
ending at ...

2
votes

1
answer

242
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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
votes

1
answer

443
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### continued fraction for logarithmic integral

Does the logarithmic integral function $\operatorname{li}(x)$ have the continued fraction expansion
$$\operatorname{li}(x) = \cfrac{x}{\log x -1 -{}} \ \cfrac{1}{\log x -3 -{}} \ \cfrac{4}{\log x -...

1
vote

1
answer

151
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### Bounded, aperiodic irrationals with bounded, aperiodic sum

If $q = [q_0;q_1 \dots]$, say $q_i$ is the $i$-th partial quotient of $q$. My question is the following:
Can one construct an explicit example of irrational $r,s > 0$ such that
$\{ 1,r,s\}$ is $\...

1
vote

0
answers

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### Theoretical invariant distribution of discrete dynamical systems, including the Riemann Zeta map

Update on 3/10/2021: I added Example 5 in the Appendix. This generic example encompasses the Riemann Zeta dynamical system. A simple version of this post, targeted to engineers, machine learning ...

2
votes

1
answer

463
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### About generalized continued fractions

Let us consider the sequences $(x_n), (a_n)$, starting with $n=0$ and $x_0\in ]0,1[$, defined by the following generalized Gaussian map:
$$x_{n+1}=\frac{\lambda_n}{x_n^{\alpha_n}}-\Big\lfloor \frac{\...

5
votes

1
answer

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### Irrationality of $e^{x/y}$

How to prove the following continued fraction of $e^{x/y}$
$${\displaystyle e^{x/y}=1+{\cfrac {2x}{2y-x+{\cfrac {x^{2}}{6y+{\cfrac {x^{2}}{10y+{\cfrac {x^{2}}{14y+{\cfrac {x^{2}}{18y+\ddots }}}}}}}}}}...

6
votes

1
answer

631
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### Algebraic and rational parts of a real number

Let $\alpha$ be a positive real number. Does it make sense to define the closest rational to $\alpha$ as the number $R(\alpha)=\frac{p_1}{p_2}$ such that $p_1,p_2$ are positive co-prime integers ...

8
votes

1
answer

668
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### An alternative to continued fraction and applications

This post is inspired by the Numberphile video 2.920050977316, advertising the paper A Prime-Representing Constant by Dylan Fridman, Juli Garbulsky, Bruno Glecer, James Grime and Massi Tron Florentin, ...

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0
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### continued fractions and cusp non-excursions

Consider the modular surface $X:=\mathbb{H^2}/PSL_2(\mathbb{Z})$.
Fix a width-of-cusp parameter $w, 0<w<<1$.
Let $B_w$ be the cusp neighborhood of width $w$. (So $w=1$ corresponds to the ...