Questions tagged [continued-fractions]

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4 answers
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How does this relationship between the Catalan numbers and SU(2) generalize?

This is a question, or really more like a cloud of questions, I wanted to ask awhile ago based on this SBS post and this post I wrote inspired by it, except that Math Overflow didn't exist then. As ...
Qiaochu Yuan's user avatar
34 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 ...
Wolfgang's user avatar
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34 votes
1 answer
2k views

Ramanujan's $\tau(n)$ and continued fractions

In D.H. Lehmer's paper "Ramanujan's function $\tau(n)$, (Duke J. Math v. 10 1943, pp. 483-492), Lehmer states the Ramanujan conjecture $|\tau( p )|< 2p^{11/2}$, so that $p^{-11/2}\tau( p )=2\cos(\...
Stopple's user avatar
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33 votes
21 answers
11k views

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) ...
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
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30 votes
5 answers
3k views

What is the effect of adding 1/2 to a continued fraction?

Is there a simple answer to the question "what happens to the continued fraction expansion of an irrational number when you add 1/2?" A closely related question is "what happens to such an expansion ...
Justin Moore's user avatar
  • 3,617
28 votes
3 answers
4k views

Is any particular algebraic number known to have unbounded continued fraction coefficients?

The continued fraction $$[1;1,2,3,4,5,\dots]=1+\cfrac{1}{1+\cfrac{1}{2+\cdots}}, $$ for instance, is known explicitly as a ratio of Bessel function values and is (I believe - SS) known to be ...
XL _At_Here_There's user avatar
28 votes
3 answers
5k views

Is there any pattern to the continued fraction of $\sqrt[3]{2}$? [closed]

Is there any pattern to the continued fraction of $\sqrt[3]{2}$ ? Wolfram Alpha returns for cube root of 2: $\sqrt[3]{2}=$ [1; 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, 2, 1, 3, ...
john mangual's user avatar
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28 votes
1 answer
2k views

Can we use the Rogers-Ramanujan cfrac to parameterize the Fermat quintic $x^5+y^5=1$?

Define $\color{blue}{q=e^{2\pi i \tau}}$ and Dedekind eta function $\eta(\tau)$. Note: I found these relations empirically, but their consistent forms suggest they can be rigorously proven. I. $p=2$...
Tito Piezas III's user avatar
23 votes
6 answers
5k views

Showing e is transcendental using its continued fraction expansion

Can the transcendence of e be shown using its continued fraction expansion e = [2;1,2,1,1,4,1,1,6,...]?
paarshad's user avatar
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19 votes
5 answers
5k views

Which Diophantine equations can be solved using continued fractions?

Pell equations can be solved using continued fractions. I have heard that some elliptic curves can be "solved" using continued fractions. Is this true? Which Diophantine equations other than Pell ...
Samuel Hambleton's user avatar
19 votes
1 answer
843 views

Is anything known about which numbers appear in the continued fraction expansion of $\pi$?

This question is mostly idle curiosity, and certainly is not related to any research activities of my own. The motivation and background are as follows. I am currently teaching a Freshman Seminar in ...
Theo Johnson-Freyd's user avatar
19 votes
2 answers
759 views

"Fractally self-similar" numbers

This is another question about visualization of Ford circles, the previous one being Confusion with practically implementing rational approximations. Here is an output of zooming into Ford circles at $...
მამუკა ჯიბლაძე's user avatar
19 votes
2 answers
978 views

Poles from the Continued Fraction Expansion of the Tangent Function?

Consider the well known continued fraction expansion $$ z \tan z = \frac{z^2}{1 - \cfrac{z^2}{3- \cfrac{z^2}{5 - \ldots}}} $$ of the tangent function going back to Euler and Lambert (Lambert used it ...
Franz Lemmermeyer's user avatar
18 votes
8 answers
3k views

Open problems in continued fractions theory

I propose to collect here open problems from the theory of continued fractions. Any types of continued fractions are welcome.
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 ...
Marcus's user avatar
  • 396
18 votes
0 answers
697 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
17 votes
5 answers
1k views

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 ...
GM Hurley's user avatar
  • 171
17 votes
2 answers
2k views

Some unpublished notes of Hofstadter

I'm looking for some unpublished notes called "Eta Lore," which are apparently related to a talk Douglas Hofstadter first gave at the Stanford Math Club in 1963. I know these notes exist because they'...
17 votes
0 answers
702 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 ...
Timothy Chow's user avatar
16 votes
3 answers
5k views

Upper bound of period length of continued fraction representation of very composite number square root

Given natural numbers of special very composite form, like primorials or factorials, how to give some useful upper bound limit of continued fraction period length of their square roots? I'm not a ...
user5539's user avatar
  • 263
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)=...
KStarGamer's user avatar
16 votes
2 answers
1k views

Is there an explicit example of such a real number with the following property?

In Diophantine approximation, for a given positive real number $\alpha$ let $[a_0, a_1, \cdots]$ denote its continued fraction expansion and let $p_n/q_n = [a_1, \cdots, a_n]$. Then it is known that $...
Stanley Yao Xiao's user avatar
15 votes
8 answers
2k views

Continued fractions using all natural integers

What can one say about the set of continued fractions $[0;a_1,a_2,\ldots]$, where $a_1,a_2,\ldots$ are a permutation of the set of natural numbers?
timur's user avatar
  • 3,292
15 votes
2 answers
1k views

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) =...
Tito Piezas III's user avatar
15 votes
2 answers
2k views

Is there an elementary proof of a result about the parity of the period of the repeating block in the continued fraction expansion of square roots

It is a known fact that for a Prime $P$, $P\equiv 1$ mod $4$ iff the length of the period in the repeating block for the continued fraction expansion of $\sqrt{P}$ is odd. I have an elementary proof ...
Jason Smith's user avatar
14 votes
6 answers
1k views

Euler's divergent series sum n!*(-1)^n: what is known about the resulting constant?

Much of the theory of continued fractions has been developped by Euler in the 18th century. The little survey "Euler: continued fractions and divergent series (and Nicholas Bernoulli)", mentions ...
Wolfgang's user avatar
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14 votes
1 answer
3k views

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 ...
A.Neves's user avatar
  • 536
14 votes
1 answer
784 views

Continued fractions and projective resolutions

Hello, This question might be vague and not thought-through enough. If we have a real positive number $x$, we can start to write it as a continued fraction: $x = a_0 + \frac{1}{x_1} , \ldots , x_n=...
Sasha's user avatar
  • 5,492
14 votes
0 answers
344 views

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 ...
F Flicker's user avatar
  • 141
13 votes
1 answer
1k views

Connection between Infinite continued fractions, elliptic integrals and AGM

It is known that at $x=1$, the following continued fraction represents $\frac{4}{\pi}$ and can be approximated rapidly using Gauss' Arithmetic Geometric mean. $$C(x) = x + \frac{1^{2}}{2x + \frac{3^{...
Turbo's user avatar
  • 13.6k
12 votes
3 answers
2k views

distance formula in Farey graph?

Consider the usual "Farey graph", i.e. the 1-skeleton of the (essentially unique) triangulation of the disk by ideal triangles. If one insists that 1/0, 0/1, and 1/1 are vertices of a triangle then ...
Kevin M Pilgrim's user avatar
12 votes
1 answer
813 views

Applications of pattern-free continued fractions

Questions about continued fractions reminded me about a related diophantine problem. I am not quite sure that diophantine equations are still in fashion but $$ 1^k+2^k+\dots+(m-1)^k=m^k, $$ the Erd&#...
Wadim Zudilin's user avatar
12 votes
1 answer
398 views

Lengths of continued fractions for the numbers with fixed ratio

Let $s(x)$ is the length of continued fraction expansion of $x$, and let $l(x)$ be the sum of partial quotients. I can prove that for any rational $\alpha$ ratios $\frac{s(\alpha x)}{s(x)}$ and $\frac{...
Alexey Ustinov's user avatar
11 votes
5 answers
2k views

Relation between indefinite quadratic forms and continued fractions

Let $D$ be a positive square free integer; for simplicity let's take $D$ to be $2$ or $3$ modulo $4$. Then ideal classes in $\mathbb{Z}[\sqrt{D}]$ are in bijection with matrices $\left( \begin{...
David E Speyer's user avatar
11 votes
0 answers
398 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 ...
Johann Cigler's user avatar
10 votes
3 answers
556 views

Continued Fractions from Digit Streams

let $x=\sum_{i=1}^{\infty}\delta_i2^{-i},\ \delta_i\in\{0,1\}$. Is there an algorithm that converts the sequence $(\delta_0,\ \delta_1,\ ...)$ of the binary digits of $x$ to the sequence $[a_0;a_1,\...
Manfred Weis's user avatar
  • 12.6k
10 votes
3 answers
617 views

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 ...
Jack Huizenga's user avatar
10 votes
2 answers
648 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
10 votes
2 answers
1k views

Length of Hirzebruch continued fractions

Suppose $a,b$ are two natural numbers relatively prime to $n$ and to each other. Assume $n\geq ab+1$. Suppose further that $\frac{a}{b}\equiv k \pmod{n}$ for some $k\in \lbrace 1,2,\dots, n-1\rbrace$ ...
Gjergji Zaimi's user avatar
10 votes
1 answer
496 views

An upper bound for the length of the continued fraction expansion of $\sqrt d$

Let $d\ge 2$ and let $$ \sqrt d =[a_0; \overline{a_1,\dots, a_\ell, 2a_0}] $$ be its continued fraction expansion. Clearly, if $d=n^2+1$, then $\ell=0$, which gives the lower bound for $\ell$. ...
Nikita Sidorov's user avatar
10 votes
3 answers
2k views

Gauss-Kuzmin Theorem (continued fractions) - why is important?

As we know, Gauss wrote that \begin{equation} \lim_{n \rightarrow \infty} \lambda \left(\tau^n \leq x\right) = \frac{\log(1+x)}{\log2}, \quad 0 \leq x < 1, \end{equation} with $\lambda$ is Lebesgue ...
Dan L's user avatar
  • 101
10 votes
2 answers
1k views

Simultaneous rational approximation of two reals using their continued fractions

Littlewood's well-known conjecture about simultaneous rational approximation is that for all $x, y \in \mathbb{R}$, $\liminf_{n \to \infty} n \Vert nx \Vert \Vert ny \Vert = 0$ (where $\Vert x \Vert$ ...
Alec Edgington's user avatar
10 votes
1 answer
385 views

Repetend digit graphs for $1/n$ in base $b$

Here is a decimal expansion of $\frac{1}{34}$: $$(1/34)_{10}=0.02941176470588235\overline{2941176470588235}\ldots$$ And here is a graphical representation of the 16-digit "repetend," as a directed ...
Joseph O'Rourke's user avatar
10 votes
1 answer
820 views

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 $|ab|<1$ ...
Tito Piezas III's user avatar
10 votes
1 answer
288 views

Confusion with practically implementing rational approximations

Writing a program visualizing Ford circles I've encountered a seemingly purely programmatic puzzle but then gradually realized there are some mathematical aspects of it which I don't understand. Let ...
მამუკა ჯიბლაძე's user avatar
10 votes
1 answer
226 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 ...
David E Speyer's user avatar
10 votes
1 answer
267 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
9 votes
2 answers
621 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
  • 4,762
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, ...
Andreas Rüdinger's user avatar

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