# Questions tagged [continued-fractions]

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160
questions

**3**

votes

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194 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 ...

**4**

votes

**0**answers

273 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{...

**2**

votes

**1**answer

133 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 ...

**2**

votes

**1**answer

63 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 $T$, ...

**6**

votes

**1**answer

329 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 -...

**1**

vote

**1**answer

106 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 $\...

**1**

vote

**0**answers

137 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 ...

**1**

vote

**1**answer

235 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{\...

**5**

votes

**1**answer

266 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 }}}}}}}}}}...

**6**

votes

**1**answer

528 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 ...

**8**

votes

**1**answer

493 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, ...

**1**

vote

**0**answers

51 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 ...

**17**

votes

**0**answers

505 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 ...

**0**

votes

**0**answers

90 views

### Generating function to continued fraction?

Assume $f_i \ne 0$. I want to convert a sequence:
$$F(z) = 1+f_1z+f_2z^2+f_3z^3+\ldots$$
to a Stieltjes continued fraction (S-fraction):
$$\frac{1}{1+\frac{g_1z}{1+ \cdots}}$$
See page 230 of these ...

**3**

votes

**0**answers

199 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 ...

**2**

votes

**1**answer

95 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 \...

**0**

votes

**0**answers

48 views

### Function near lines mod 1

While thinking about some problems, I came across the following:
Does there exist a function $f: \mathbb N \to \mathbb R$, and some $c \in \mathbb R$, such that for any $n$, any block of $cn$ ...

**6**

votes

**0**answers

180 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 = ...

**4**

votes

**0**answers

227 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 ...

**2**

votes

**1**answer

104 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 ...

**1**

vote

**0**answers

137 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^...

**3**

votes

**1**answer

236 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-\...

**6**

votes

**0**answers

127 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 $\alpha$, ...

**3**

votes

**1**answer

203 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 ...

**1**

vote

**0**answers

30 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 ...

**6**

votes

**1**answer

487 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^{...

**7**

votes

**2**answers

357 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 ...

**7**

votes

**2**answers

173 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 $d=0$, $a=1$ ...

**2**

votes

**1**answer

481 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 ...

**3**

votes

**0**answers

129 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 ...

**7**

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, ...

**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 ...

**8**

votes

**2**answers

303 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 ...

**21**

votes

**0**answers

493 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 ...

**5**

votes

**0**answers

86 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 ...

**8**

votes

**1**answer

164 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 \\...

**1**

vote

**0**answers

51 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....

**1**

vote

**1**answer

100 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 ...

**0**

votes

**0**answers

51 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+\...

**6**

votes

**0**answers

324 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]...

**4**

votes

**2**answers

274 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 ...

**5**

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**0**answers

242 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|\...

**0**

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**0**answers

77 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{...

**3**

votes

**1**answer

209 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 ...

**1**

vote

**2**answers

178 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,\...

**11**

votes

**0**answers

372 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 ...

**1**

vote

**1**answer

131 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}/\...

**0**

votes

**0**answers

166 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 ...

**10**

votes

**1**answer

209 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 ...

**3**

votes

**1**answer

258 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 ...