The continued-fractions tag has no usage guidance.

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $s+t$?

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### 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 $a$ ...

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### 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 ...

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### Some unpublished notes of Hofstadter

I'm looking for some unpublished notes called "Eta Lore," which are apparently related to a talk Douglas Hofstadter first gave at the Stanford Math Club in 1963.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 ...

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### 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 ...

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### 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 ...

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### 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 $X_n$. ...

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### 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 ...

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### 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 ...

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### 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) ...

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### 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.

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $\Vert x \Vert$ ...

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### 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 ...

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**1**answer

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### 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 ...

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### 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 ...

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### 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( ...

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### 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 ...

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### 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,...]?

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### 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 ...

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### 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?

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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 ...

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### 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 ...

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### 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 ...

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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 ...

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### 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 ...

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### 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?

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### 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 ...