The continued-fractions tag has no usage guidance.

**6**

votes

**4**answers

675 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 ...

**2**

votes

**2**answers

691 views

### Liouville's Theorem in Diophantine Approximation

Liouville's Theorem states that for any algebraic $\alpha \in \mathbb{R}$ of degree $n$, there exists a positive constant $c:=c(\alpha)$ such that $$|\alpha-\frac{p}{q}|>\frac{c}{q^n}$$ for any $p ...

**4**

votes

**2**answers

193 views

### Generators of a 2D lattice

Dear MO_World,
I'm hoping someone can point me towards a reference for something. I have an invertible $2\times 2$ matrix, $A$, with real entries such that for both of the rows, the entries are ...

**8**

votes

**2**answers

328 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 $Q_A$ ...

**5**

votes

**3**answers

408 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 ...

**1**

vote

**1**answer

354 views

### Simple and general relation between continuant polynomials

Continued fraction $[a_0,a_1,...,a_n]$ may be expressed as quotient of two polynomials of $(a_0,a_1,...,a_n)$, named continuants (see http://en.wikipedia.org/wiki/Continuant_%28mathematics%29 )
...

**12**

votes

**1**answer

563 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 , ...

**10**

votes

**5**answers

987 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 ...

**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 ...

**2**

votes

**1**answer

923 views

### How to find the region of convergence of this series using the theory of continued fractions?

Please, consider the following series
\begin{equation}
f(z)=1+\sum_{n=1}^{\infty}2^{-\sum_{k=1}^{n}\frac{2s}{k}} =1+ \sum_{n=1}^{\infty}\left( \prod_{k=1}^{n}2^{-\frac{2z}{k}} \right)
\end{equation}
...

**1**

vote

**2**answers

703 views

### Composition of circle inversions

I would like to understand the map of $\mathbb{C}$ to $\mathbb{C}$
that results by iterating inversion in a unit circle.
Let $f(z)$ for $z \in \mathbb{C}$ invert $z$ in a unit circle
centered on ...

**6**

votes

**2**answers

987 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 ...

**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 ...

**2**

votes

**0**answers

684 views

### Applications of the length of the continued fraction

The fractions $n/q$ and $n/(n-q)$ can be representated by Hirzebruch continued fractions (also called Jung-Hirzebruch CF), the length of each we denote by $s$ and $t$. Is there any bound for $s+t$?

**4**

votes

**1**answer

809 views

### Searching for an inhomogeneous diophantine approximation algorithm

Given two nonzero real numbers $x$ and $y$ such that $y/x$ is irrational, a real number $z$ to be approximated, and a tolerance $\epsilon$, what is an algorithm that will provide coprime integers $a$ ...

**5**

votes

**1**answer

297 views

### Algorithm to determine sign of a polynomial

I've been working with a collaborator (Arek Goetz) on a dynamics problem involving
piecewise isometries (a map $T$ on a domain $X$ (say a subset of the plane)
such that $X$ is divided into a finite ...

**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 ...

**5**

votes

**1**answer

1k 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 ...

**4**

votes

**0**answers

302 views

### A question on M. Mignotte's Paper: “Petho's Cubics”

I have been reading the paper "Petho's Cubics" by M. Mignotte (appeared in Publ. Math. Debrecen, 56/3-4 (2000)) unfortunately I don't understand section 4 of the paper. I am hoping someone could help ...

**2**

votes

**1**answer

316 views

### Best rational approximation in a special sense

Let $\alpha$ be an irrational number, $n\geq 1$ and
$ X_n=\lbrace (x,y) \in {\mathbb Z}^2 | |y| \leq n, \ x+y\alpha >0 \rbrace$
Now let $(x_n,y_n)$ minimize the quantity $x+y\alpha$ on $X_n$. ...

**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 ...

**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 ...

**23**

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) ...

**6**

votes

**3**answers

742 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.

**11**

votes

**1**answer

597 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 ...

**10**

votes

**1**answer

345 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 ...

**25**

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 ...

**7**

votes

**2**answers

392 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 ...

**7**

votes

**2**answers

842 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 $\Vert x \Vert$ ...

**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 ...

**2**

votes

**1**answer

866 views

### Periods of Continued Fractions

Does the period length $l(pq)$ of the continued fraction of $\sqrt{pq}$, for $p$ and $q$ primes, follow some type of divisibility property, say
$$
l(pq) = c\frac{l(p)}{l(q)} \quad\text{or}\quad ...

**5**

votes

**2**answers

634 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 ...

**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( ...

**1**

vote

**1**answer

708 views

### Is it possile for all real algebraic numbers to have continued fractions with bounded partial quotients ?

Roth's theorem states that for every real algebraic $\alpha$ and $\epsilon>0$, there is a $c>0$ such that
$$|\alpha -\frac{p}{q}| > \frac{c}{q^{2+\epsilon}}.$$
Lang's conjecture strengthened ...

**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,...]?

**2**

votes

**1**answer

239 views

### Is this related to the j-function?

I was browsing in Plouffe's inverter and found that $\frac{\exp(\pi)-\ln(3)}{\ln(2)}$ is very nearly $\frac{159}{5}.$ The continued fraction is
[31, 1, 4, 12029125, ...].
Is this the same magic as ...

**3**

votes

**0**answers

193 views

### Cassels' algorithm vs. “divided cells” algorithm

Cassels' algorithm mentioned in link text looks similar to Delone's “divided cells” algorithm. Are there any differences in these algorithms?

**6**

votes

**0**answers

1k views

### 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 ...

**11**

votes

**1**answer

600 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 ...

**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 ...

**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 ...

**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 ...

**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?

**16**

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 ...