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8 votes
2 answers
166 views

Distribution of traces and max entries of words of fixed length in $\operatorname{SL}_2(\mathbb{N})$

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\tr{\mathsf{tr}}$$\SL_2(\mathbb{N})$ is a free monoid on the generators $$ L=\begin{pmatrix}1&0\\1&1\end{pmatrix},\quad R=\begin{pmatrix}1&1\...
yoyo's user avatar
  • 599
12 votes
3 answers
685 views

When does $2$ arise when using the Euclidean algorithm to compute greatest common divisors?

When using the standard Euclidean algorithm to compute the greatest common divisor of a pair of relatively prime positive integers, the integer $2$ sometimes arises and sometimes does not. For example,...
Joel Louwsma's user avatar
0 votes
0 answers
101 views

q-factor in the Rogers-Ramanujan continued fraction

The Rogers-Ramanujan continued fraction is defined by $$R(q)=\cfrac{q^{1/5}}{1+\cfrac{q}{1+\cfrac{q^2}{1+\cfrac{q^3}\ddots}}}.$$ I would like to know role played by the factor $q^{1/5}$ and why not ...
Sangama's user avatar
2 votes
1 answer
241 views

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. $abc$ ...
joro's user avatar
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2 votes
0 answers
56 views

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}{...
Sidharth Ghoshal's user avatar
4 votes
1 answer
124 views

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}...
Vessel's user avatar
  • 145
7 votes
0 answers
255 views

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 ...
D.R.'s user avatar
  • 771
3 votes
1 answer
77 views

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 ...
Monk's user avatar
  • 125
1 vote
1 answer
200 views

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 $$y:=a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cfrac{1}{\begin{align}\ddots& \\ &a_{n-1}+\cfrac{1}{a_n+y}\end{align}}}}}$$ $$...
Manfred Weis's user avatar
  • 12.8k
2 votes
0 answers
63 views

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 $(x,y)$-...
Notamathematician's user avatar
0 votes
1 answer
272 views

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 $1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}$ , ...
mathoverflowUser's user avatar
4 votes
2 answers
302 views

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 ...
mtheorylord's user avatar
3 votes
1 answer
203 views

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 ...
blancket's user avatar
  • 189
4 votes
0 answers
177 views

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 ...
Jesse Elliott's user avatar
2 votes
0 answers
72 views

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 ...
Lucian Ionescu's user avatar
6 votes
1 answer
121 views

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 $\pi/4$: $$ \...
Nanhui Lee's user avatar
0 votes
0 answers
83 views

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 ...
0x11111's user avatar
  • 493
2 votes
1 answer
145 views

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_{&...
Roland Bacher's user avatar
6 votes
0 answers
120 views

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 ...
Oleksandr  Kulkov's user avatar
1 vote
0 answers
206 views

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 ...
Rodrigo's user avatar
  • 51
0 votes
1 answer
132 views

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 ...
Notamathematician's user avatar
18 votes
0 answers
710 views

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 $$f(s)=\dfrac1{1+\dfrac s{1+\dfrac{s^2/2!}{1+\dfrac{s^3/3!}{1+\cdots}}}}$$ ...
TheSimpliFire's user avatar
5 votes
1 answer
176 views

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 ...
Tito Piezas III's user avatar
7 votes
0 answers
433 views

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, ...
Tito Piezas III's user avatar
2 votes
0 answers
154 views

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}\...
Tito Piezas III's user avatar
7 votes
0 answers
264 views

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. ...
Gerald Edgar's user avatar
  • 40.6k
8 votes
2 answers
590 views

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 $...
Tito Piezas III's user avatar
6 votes
1 answer
256 views

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-...
Tito Piezas III's user avatar
10 votes
2 answers
686 views

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\...
Tito Piezas III's user avatar
-1 votes
2 answers
115 views

How to solve the following infinite ladder fraction? ( Through pen & paper ) [closed]

The fraction continues till infinity as shown in the image :
Chandra Prakash Bairagi's user avatar
0 votes
0 answers
111 views

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 ...
RaffaeleScandone's user avatar
3 votes
1 answer
177 views

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 $(a,b)$ ...
Joshua Stucky's user avatar
0 votes
2 answers
226 views

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 ...
Nikita Sidorov's user avatar
10 votes
1 answer
294 views

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 $$e(x)=\log(1+x)+\frac{1+\lbrace x\rbrace}{1+x}\left(e(1/\lbrace x\rbrace)-\log(1+\lbrace x\rbrace)\right)$$ ...
Roland Bacher's user avatar
5 votes
0 answers
209 views

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 $...
Henri Cohen's user avatar
  • 12.6k
2 votes
0 answers
84 views

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 $[...
Roland Bacher's user avatar
6 votes
1 answer
640 views

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{...
Wolfgang's user avatar
  • 13.3k
3 votes
0 answers
94 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 ...
Wolfgang's user avatar
  • 13.3k
3 votes
1 answer
461 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, $\ell=\ell(d)$. ...
Nikita Sidorov's user avatar
1 vote
0 answers
85 views

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 = ...
Станислав Крымский's user avatar
9 votes
2 answers
635 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 ...
Ivan Meir's user avatar
  • 4,822
2 votes
0 answers
79 views

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 ...
Ivan Meir's user avatar
  • 4,822
33 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 ...
Ivan Meir's user avatar
  • 4,822
9 votes
1 answer
742 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 \...
Sprotte's user avatar
  • 1,065
1 vote
1 answer
182 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 $\alpha_n= \frac{1}{(n+1)^2}$...
VSP's user avatar
  • 233
0 votes
0 answers
137 views

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-...
VSP's user avatar
  • 233
4 votes
2 answers
258 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 ...
Johann Cigler's user avatar
3 votes
1 answer
254 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=\...
Marcus's user avatar
  • 406
3 votes
0 answers
163 views

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 $$\frac{a}{b}=\sum_{j=0}^\infty x_jp^j,\ x_j\in\{0,\ldots,p-1\}\ ?$$ ...
Roland Bacher's user avatar
16 votes
1 answer
2k 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)=...
KStar's user avatar
  • 533

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