The continued-fractions tag has no usage guidance.

**28**

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**5**answers

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### 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 ...

**25**

votes

**4**answers

2k views

### 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 ...

**23**

votes

**0**answers

1k 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 ...

**22**

votes

**17**answers

6k views

### Applications of finite continued fractions

I know some applications of finite continued fractions. Probably you know more. Can you add anything? (For for Applications of periodic continued fractions I have made a special topic.)
1) (Trivial) ...

**18**

votes

**3**answers

2k 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 ...

**18**

votes

**1**answer

621 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 ...

**17**

votes

**6**answers

3k 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,...]?

**16**

votes

**5**answers

3k 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 ...

**15**

votes

**8**answers

1k 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?

**15**

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 ...

**15**

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 ...

**14**

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 ...

**14**

votes

**1**answer

1k 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 ...

**12**

votes

**3**answers

3k 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 ...

**12**

votes

**1**answer

548 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 , ...

**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( ...

**11**

votes

**1**answer

587 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 ...

**11**

votes

**0**answers

459 views

### Connection between Infinite continued fractions 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 + ...

**10**

votes

**3**answers

449 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 ...

**10**

votes

**5**answers

976 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 ...

**10**

votes

**2**answers

810 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,
...

**10**

votes

**6**answers

788 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.

**10**

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**1**answer

586 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 ...

**9**

votes

**3**answers

1k 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 ...

**9**

votes

**2**answers

797 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 $k\in \lbrace 1,2,\dots, n-1\rbrace$ ...

**9**

votes

**1**answer

1k views

### The Riemann's Zeta Function represented as a continued fraction and a question of convergence.

The Riemann's 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 ...

**9**

votes

**1**answer

336 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 ...

**9**

votes

**1**answer

296 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 ...

**8**

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**2**answers

374 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 ...

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votes

**2**answers

323 views

### Coefficients in the periodic part of continued fractions for real quadratic algebraic numbers

Given a strictly positive integer $A$, let $D(A)$ denote the set of all
real quadratic algebraic numbers with a continued fraction having almost all coefficients
$\leq A$.
Consider the field $Q_A$ ...

**8**

votes

**1**answer

364 views

### Do simple, multi-dimensional generalizations of this continued fraction formula exist?

Background
Let $x\in [0,1)\setminus \mathbb{Q}$ have regular continued fraction expansion given by
$$
x = [a_1, a_2, a_3, \dots] = \cfrac{1}{a_1+\cfrac{1}{a_2+\dots}}, \qquad a_i \in \mathbb{N}.
$$
...

**7**

votes

**2**answers

839 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 $\Vert x \Vert$ ...

**7**

votes

**1**answer

591 views

### Continued fraction representation of Zeta

A question at math.SE is asking for references. The fraction is quite nice! Check it out and post some references if you know of any.
I found this at arxiv, but it doesn't apply to Zeta.

**6**

votes

**4**answers

663 views

### Orbits of the projective special linear group on $\mathbf{Q} \cup \{\infty\}$

The group $\mathrm{PSL}_2(\mathbf{Q})$ of fractional linear transformations $x \mapsto (ax+b)/(cx+d)$ such that $a,b,c,d \in \mathbf{Q}$ and $ad-bc = 1$ acts on $\mathbf{Q} \cup \lbrace ...

**6**

votes

**3**answers

731 views

### English translation of Voronoi's dissertation

I am looking for an English translation of Voronoi's doctoral dissertation, "On a generalization of the Algorithm of Continued Fractions." I can only find it in the original Russian.

**6**

votes

**2**answers

951 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 ...

**6**

votes

**2**answers

379 views

### How Symmetric is Diophantine Approximation using Fractions with Square Denominators?

Let $S$ be an infinite set of positive integers.
Let us say that a "best S-approximation" to a real irrational $r$ is a rational number
$p/q$, with $p$ and $q$ integers and $q \in S$, such that for ...

**6**

votes

**2**answers

1k views

### Applications of periodic continued fractions

Some answers from Applications of finite continued fractions in fact are Applications of periodic continued fractions. I think that it should be separate question.
What can you add to the following ...

**6**

votes

**1**answer

143 views

### Young-Fibonacci lattice and purely periodic continued fractions

The Fibonacci lattice $\mathcal{F}$ is the poset of all finite words consisting of 1's and 2's where a word $v$ covers a word $u$ if $v$ is obtained from $v$ by either (a) inserting a 1 in $u$ prior ...

**6**

votes

**1**answer

355 views

### Previous work on this generalization of continued fractions?

The 2x2 matrix representation of a continued fraction makes it clear that we're multiplying together a bunch of group elements. Inversion is essentially freely adjoining a generator to a Coxeter ...

**6**

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**0**answers

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### A conjectured q-continued fraction related to the Göllnitz-Gordon partition identities

Given $q=e^{2i\pi\tau}$ with $|q|\lt1$, define
the well-known Göllnitz-Gordon identities
$$A(q)=\sum_{n=0}^\infty \frac{q^{n(n+1)}(-q;q^2)_n}{(q^2;q^2)_n}=\prod_{n=1}^\infty ...

**6**

votes

**1**answer

273 views

### Constructing a family of convergents from continued fractions formed by a set of prime partial quotients

For a given real number $x$, the continued fraction representation $x = [a_0; a_1, a_2, \cdots]$ where $(a_n)_{n \geq 0}$is defined by setting $x = \alpha_0$, then $a_i = \lfloor \alpha_i \rfloor$, ...

**6**

votes

**0**answers

334 views

### Why does the Rogers-Ramanujan continued fraction $R(q)$ appear in Emma Lehmer's quintic?

Define the Ramanujan theta function $f(a,b)$ as,
$$f(a,b) = \sum_{n=-\infty}^{\infty} a^{n(n+1)/2}\,b^{n(n-1)/2}$$
and the Dedekind eta function,
$$\eta(\tau) = q^{1/24}\prod_{n=1}^{\infty} ...

**6**

votes

**0**answers

290 views

### Birth-Death Process associated with Orthogonal Polynomials

I have read in various places the following objects are related:
orthogonal polynomials
birth-death processes
Lattice paths
continued fractions
After a lot of searching online, I found sketches ...

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**0**answers

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### Lower bounds for period length of continued fraction of square root

EDIT: Dror answered this to my complete satisfaction. The simple continued fraction for $$ \sqrt{n^2 + 1}$$ has length exactly the same, no dependence on $n$!!!!! No lower bound of the type I wanted ...

**5**

votes

**3**answers

399 views

### Proto-Euclidean algorithm

Consider the Euclidean algorithm (EA) as a way to measure the relative length $b/a$ of a shorter stick $b$ compared to a longer one $a$ by recursively determining
$$q_i = \left\lfloor ...

**5**

votes

**2**answers

588 views

### About two 'negative' continued fractions whose sum equals $1$

Letting $a_1,a_2,\cdots,a_r$ be integers which are larger than or equal to $2$, let us define
$$[a_1,a_2,\cdots,a_r]=\cfrac{1}{a_1-\cfrac{1}{a_2-\cfrac{1}{\ddots-\cfrac{1}{a_r}}}}$$
(Note that the ...

**5**

votes

**1**answer

186 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 $\ell$.
...

**5**

votes

**2**answers

624 views

### Lower bounds (or less) for the period of \sqrt(D) and related sequences.

This is a continuation of
Lower bounds for period length of continued fraction of square root
which is a continuation of
Upper bound of period length of continued fraction representation of very ...

**5**

votes

**1**answer

998 views

### Unsolved Problem from AmMathMonthly

Here is a simply described but fiendishly diophanterrorizing problem
I asked on AMM eons ago. Maybe you can shed some light upon it.
0.2 (base 4) = 0.2 (continued fraction)
0.24 (base 6) = 0.24 ...