Questions tagged [continued-fractions]

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Are these continued fractions for the "tail" of $\zeta(3)$ 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{...
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3 votes
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69 views

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 ...
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2 votes
1 answer
207 views

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, $\ell=\ell(d)$. ...
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1 vote
0 answers
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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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9 votes
2 answers
569 views

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 ...
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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 ...
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31 votes
2 answers
2k views

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 ...
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8 votes
1 answer
450 views

$\text{SL}_2(\mathbb{Z})$ 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 \...
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1 vote
1 answer
155 views

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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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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  • 213
4 votes
2 answers
217 views

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 ...
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2 votes
1 answer
150 views

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=\...
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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 $$\frac{a}{b}=\sum_{j=0}^\infty x_jp^j,\ x_j\in\{0,\ldots,p-1\}\ ?$$ ...
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15 votes
1 answer
1k views

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)=...
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2 votes
0 answers
79 views

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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5 votes
1 answer
222 views

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 ...
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3 votes
0 answers
279 views

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 ...
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4 votes
0 answers
285 views

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{...
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2 votes
1 answer
165 views

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 ...
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2 votes
1 answer
143 views

(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 $T$, ...
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7 votes
1 answer
386 views

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 -...
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1 vote
1 answer
146 views

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 $\...
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1 vote
0 answers
150 views

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 ...
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1 vote
1 answer
336 views

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{\...
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5 votes
1 answer
291 views

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 }}}}}}}}}}...
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6 votes
1 answer
567 views

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 ...
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8 votes
1 answer
565 views

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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1 vote
0 answers
58 views

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 ...
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17 votes
0 answers
583 views

Picture of Lambert's proof that $\pi$ is irrational?

With a suitably generous notion of "picture proof" or "proof without words" or "geometric proof," there do exist such proofs of the irrationality of square roots and even ...
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3 votes
0 answers
207 views

Asymptotic expansions for the continued fraction $[1,x,x^2,x^3,\cdots]$

The $n$-th convergent is defined as $$R_n(x) = \frac{P_n(x)}{Q_n(x)}=[1;x,x^2,\cdots,x^n]=1+\frac{1}{x+}\frac{1}{x^2+}\frac{1}{x^3+\cdots}\frac{1}{x^n}$$ where $P_n(x), Q(x)$ are polynomials ...
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2 votes
1 answer
99 views

Continued fractions, Chebyshev and non-homogenous approximation

In Khinchin's book, "Continued Fractions," he considers the question, given an irrational, $\alpha$, and a real number, $\beta$, how to find integral $x$ and $y$ such that $$\alpha x - y \...
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6 votes
0 answers
194 views

Continued fractions and class groups

Let $d$ be a positive integer. It is well-known (due to Lagrange) that the continued fraction of $\sqrt{d}$ is eventually periodic. Moreover, it is known that the equation $$\displaystyle x^2 - dy^2 = ...
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4 votes
0 answers
238 views

Why is Haven's discovery important?

Today my attention was caught by one of those little stories that appear when you open a certain browser: an inmate achieved a number theoretic breakthrough It is about continued fractions and I would ...
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2 votes
1 answer
124 views

On a transcendental number defined as a variation involving the Lambert $W$ function in the nested square root representation of the golden ratio

Define the real number $\xi$ satisfying $$\xi=\sqrt{1+W\left(1+\sqrt{1+W(1+\ldots)}\right)}\tag{1}$$ where $W(x)$ denotes the main branch of the Lambert $W$ function, as reference I add that Wikipedia ...
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1 vote
0 answers
154 views

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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3 votes
1 answer
285 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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7 votes
0 answers
170 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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3 votes
1 answer
233 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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1 vote
0 answers
39 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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6 votes
1 answer
495 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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  • 427
7 votes
2 answers
418 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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  • 413
7 votes
2 answers
180 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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  • 9,201
5 votes
2 answers
697 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 for ...
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3 votes
0 answers
137 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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9 votes
3 answers
1k 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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18 votes
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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  • 354
8 votes
2 answers
347 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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  • 9,201
32 votes
2 answers
1k 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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5 votes
0 answers
89 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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8 votes
1 answer
170 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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