For questions on continued fractions.

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Help understanding a geometric proof of the ergodicity of the Gauss measure for continued fractions

Any $x\in(0,1)$ can be written as a (regular) continued fraction $$ x = \cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cdots}}} = [a_1,a_2,a_3,\dotsc] $$ An irrational number has a unique expansion, ...
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89 views

Semi-convergent of continued fractions

I have read this from here The simple continued fraction for $x$ generates all of the best rational approximations for $x$ according to three rules: Truncate the continued fraction, and ...
2
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1answer
58 views

Decomposition of modular group elements

The modular group $PSL_2(\mathbb{Z})$ acts on the hyperbolic half-space $H$ by $$h\cdot z=\frac{az+b}{cz+d},\;z\in H,\;h=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in PSL_2(\mathbb{Z})$$ with ...
3
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2answers
55 views

Continued fraction and classification of real numbers.

I would be grateful if anyone can tell if there are any methods to classify real numbers using continued fraction. eg: Suppose $[a_0;a_1,a_2,\ldots,a_n]$ is the representation of some real number ...
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1answer
85 views

Convergent's numerators of the continued fraction for $\pi$

Call $C_{\pi} = \{ 1,3,22,333,355,… \}$, it´s the sequence of the numerators of convergents of the continued fraction for $\pi$, its OEIS' A002485, http://oeis.org/A002485. Let $n \in C_{\pi}$, such ...
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25 views

a periodic continued fraction $c_{n+2} = 1 - \frac{c_{n+1}}{1 - c_{n}}$

Howvever, start with two numbers $c_0, c_1$ I read in a paper the following sequence has period 5: $$ c_{n+1} = 1 - \frac{c_n}{1 - c_{n-1}} $$ Example if we have $c_0=1, c_1=2$ the sequence ...
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1answer
12 views

A weak inequality than Hurwitz

How can i prove that among any two consecuent convergents to x, al least one of the satisfy $|x-h_{n}/k_{n}|$ $< 1/2k_{n}^2$ I know, by the Thoerem of Hurwitz, that among any three consecutive ...
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29 views

About an rational aproximation to an irrational

How to show that if $x$ is an irrational number, then $x= a_{0} + \sum_{n=0}^{\infty} \frac{(-1)^n}{k_{n}k_{n+1}}$ where the $k_{n}$ are the denominators of the $n$th convergents to $x$? Maybe a ...
19
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1answer
2k views

What was Ramanujan's solution?

The wikipedia entry on Ramanujan contains the following passage: One of his remarkable capabilities was the rapid solution for problems. He was sharing a room with P. C. Mahalanobis who had a ...
7
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1answer
243 views

Chinese estimate for $\pi$. Were they lucky?

The famous chinese estimate $\pi\approx\frac{355}{113}$ is good. I think that is too good. As a continued fraction: $$\pi=[3:7,15,1,292,\ldots]$$ That $292$ is a bit too big. Is there a reason for a ...
2
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1answer
30 views

How to find the terms of the continued fraction representation for $e^\pi$

The question is - Find the first ten terms of the continued fraction representation for $e^ \pi $
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1answer
79 views

Summing up numbers from the continued fraction of $e ^ \pi$ and $\pi ^e$

I don't remember it well ,but it was around 5-6 years ago , I was 8 and I had found this new interest - continued fractions .I used to play with their terms sum them up and thought of getting ...
0
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1answer
90 views

Continued fraction proof

I am really confused about the proof of this theorem: For any continued fraction, $$q_n\alpha - p_n = \frac{(-1)^n}{\alpha_{n+1}q_n + q_{n-1}}$$ I got the base induction case for $n=0$ but I can't ...
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1answer
60 views

Proof of continued fraction convergence theorem

How does one prove that $$|\alpha - \frac{p_n}{q_n}| < |\alpha - \frac{p_m}{q_m}|$$ for all $n>m$? I know that the left side is less than $\frac{1}{2q_n^2}$ and the right side is less than ...
21
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1answer
554 views

Direct proof that for a prime $p$ if $p\equiv 1 \bmod 4$ then $l(\sqrt{p})$ is odd.

Definition: Assume $p$ is a prime. $l(\sqrt{p})=$ length of period in simple continued fraction expansion of $\sqrt{p}$. The standard proof of this uses the following: $p$ is a prime implies $p ...
3
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1answer
92 views

Rational aproximations of golden ratio

I read a blogpost that mentions that for golden ratio, the sets of best rational approximations of the first kind and the second kind are the same. Is this true? If so, why? Are there other numbers ...
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0answers
25 views

PDF describing nth term in continued fraction

For a real number r chosen uniformly at random in the range (0,1), what's the marginal Probability Density Function that describes the nth term in the continued fraction representation of r? What ...
0
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1answer
38 views

How to write continued fraction as polynomial?

I have \begin{align} r(x)= 1 + \frac{x}{\frac{1}{2}+\frac{x-1}{-1+\frac{x+1}{1+\frac{x-1}{-1}}}} \end{align} for an interpolation problem, and I need to write $r(x)$ such that nominator and ...
3
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2answers
75 views

Evaluating $\frac{2}{3-\frac{2}{3-\frac{2}{3-\ldots} } }$

What is the value of the fraction $\frac{2}{3-\frac{2}{3-\frac{2}{3-\ldots} } } $? I let $x=\frac{2}{3-\frac{2}{3-\frac{2}{3-\ldots} } } $ and hence $x=\frac{2}{3-x} $ which gives $x=1$ or $x=2$. ...
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1answer
41 views

uniform spanning tree of $2 \times n$ graph

In Probability on Trees and Networks Chapter 1 study the uniform spanning tree on the ladder graph: _ |_| |_| |_| ... |_| |_| The probability the bottom rung ...
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3answers
2k views

How to find continued fraction of pi

I have always been amazed by the continued fractions for $\pi$. For example some continued fractions for pi are: $\pi=[3:7,15,1,292,.....]$ and many others given here. Similarly some nice continued ...
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Updated:Sum of entries in continued fraction of $\sqrt d$ and $\sqrt{d}-\lfloor \sqrt{d}\rfloor$ equals (divides) $d$.

(1)I noted as a joke in class, for $\sqrt{13}$ which has continued fraction expansion $[3;\overline{1,1,1,1,6}]$ that $3+1+1+1+1+6=13$. Another eg. $\sqrt{22}=[4;\overline{{1,2,4,2,1,8}}]$, as ...
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2answers
30 views

functions on a continued fraction expansion

Let $x$ be an irrational number with continued fraction expansion $[a_0;a_1,a_2,\ldots]$. Is there an $x$ and a non-identity function $f$ such that $f(x)=[f(a_0);f(a_1),f(a_2),\ldots]$. Given that I ...
2
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1answer
136 views

How long should you descent in Stern-Brocot-Tree to get a fixed approximation guarantee?

I've read in Wikipedia: By stopping once the desired precision is reached, floating-point numbers can be approximated to arbitrary precision. If a real number x is approximated by any rational ...
4
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0answers
112 views

Cube root of two continued fraction

I know there is a nice way of getting the continued fraction expansion of quadratic irrationals mainly because they recur after a point, and if they recur after a point they are quadratic irrationals. ...
2
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2answers
170 views

Question in fraction (not simple )

I have a question and its answer but I don't know how can i solve $$\frac {37}{13} = 2+ \frac {1}{x+\frac{1}{5+\frac{1}{y}}} $$ the answer $ x =1, y=2$ Could any one explain how to solve this ?? ...
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4answers
677 views

How to detect when continued fractions period terminates

I'm doing continued fractions arithmetic. Is there a method to detect when a continued fractions period terminates? Let me give you an example: $\sqrt{2} = [1; \overline{2}]$, $\sqrt{7} = [2; ...
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Showing iterates of a complex function on the upper half plane converges uniformly on compact sets

The following is an irksome problem that my complex analysis class is having trouble solving: Let $*$ be an operator that takes a function $F:\mathcal{H}\to\mathcal{H}$ to a function ...
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2answers
635 views

Riemann's zeta as a continued fraction over prime numbers.

Riemann's zeta function is a function with many faces, I mean representations. I recently derived this one, bellow, as a continued fraction over prime numbers. $$ \zeta(s)=1 ...
2
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1answer
61 views

Proving something about $|f(x)|$ when the lim of $f(x)/x^2$ is known

I've been trying to crack this issue for 2 days and I got pretty much nothing Given that $f$ is a continuous function and the following limits exists and are finite: $$ (1) ...
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1answer
53 views

Math and Taxes - Items versus Sum Total differs

First, The question is a bit confusing as I am not really sure how to word this problem as a question. The math problem I encountered which is a bit of an anomaly is this : Suppose you are ...
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1answer
36 views

Convergence of two unusual “nested” sums

I was contemplating convergent sums, trying to think of very unusual or unorthodox sums that might be treatable recursively. Eventually, the following sum occurred to me: $$ \xi = 1 + \frac{ ...
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3answers
203 views

Convergents of square root of 2

On wikipedia I read about the continued fraction of the square root of 2: $$1+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{...}}}}$$ The first convergents are ...
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1answer
60 views

Is there a bijection between real numbers and continued fractions?

It is known that there is a bijection between rational numbers and finite continued fractions, so every rational number is uniquelly identified by a finite continued fractions and vice versa. It is ...
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40 views

Champernowne constant - summation and behavior of terms in continued fraction expansion

Is there an infinite summation that gives the Champernowne constant? Wikipedia has one, and so does Wolfram MathWorld. Are they valid? If so, could someone explain why, i.e how do they work? Also, ...
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3answers
1k views

Computing Infinite Continued Fractions

I am looking for "tricks" used to compute infinite continued fractions. For example, $$1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\ddots}}}$$ is the golden ratio since if we denote it by $x$, then we have ...
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1answer
99 views

Infinite Continued Fraction Notation

I can't find anywhere via googling; is there some sort of $\sum$ like notation for infinite continued fractions? In other words, for a sum we do this: $$ 1+x+x^2+x^3+... = \sum_{n=0}^\infty x^n $$ ...
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2answers
323 views

Taking the negative of a continued fraction

If I have a continued fraction for an irrational number $z= \langle a_0;a_1,a_2,a_3,\ldots\rangle$ it seems that $(-1)*z = \langle-a_0;-a_1,-a_2,-a_3,\ldots\rangle$. Is this true? In general, if you ...
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2answers
186 views

Reconstruct a quadratic irrational from its periodic continued fraction

How can one find a quadratic irrational when knowing its periodic continued fraction? For example(using Wikipedia notion), how can one find the quadratic irrational that its continued fraction is ...
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1answer
73 views

Solve the equation $285x \equiv 177 \pmod{924}$ using continued fraction

Solve the equation $285x \equiv 177 \pmod{924}$ using continued fraction My attempt(using Wikipedia notion): Continued fraction form for $\frac{924}{285}$ is $[3;4,6,1,9]=[q_1;q_2,q_3,q_4,q_5]$ ...
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2answers
95 views

Closed form for this continued fraction [closed]

Is there a closed form for this continued fraction? $$x+\frac{1}{x+\frac{1}{x+\frac{1}{...}}}$$
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1answer
88 views

Nested square root of continued fraction references

$$\sqrt {a_0 + \cfrac{b_1}{\sqrt{a_1 + \cfrac{b_2}{\sqrt{a_2 + \cfrac{b_3}{ \ddots }}}}}}$$ Are there any texts that explain how to deal with expressions like this?
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2answers
79 views

Find value of unending continued fraction

I am trying to find the limit if it exists for the following unending continued fraction: $$1+{1\over{2+{1\over2+{1\over{2+...}}}}}$$ I have discovered this is the continued fraction for $\sqrt2$, ...
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0answers
76 views

Is this a bounded sequence ? (about continued fraction)

Represent $\sqrt{2}$ in the form $$\sqrt{2}=1+\frac{8}{A_1+\displaystyle\frac{8}{A_2+\displaystyle\frac{8}{A_3+\ddots}}},$$ where $A_n$ is a positive integer and $A_n \geq 8$ for all $n$. So we have ...
3
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2answers
72 views

Why is the coefficient in front of $\sqrt n$ always 1 in the intermediate terms for finding the continued fraction expansion of $\sqrt n$?

After playing around on paper for a bit, I came up with a short python generator to find the continued fraction expansion of $\sqrt n$. I understand why it gets the right answer when it gets an ...
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5answers
739 views

Is π unusually close to 7920/2521?

EDIT: One can look at a particular type of approximation to $\pi$ based on comparing radians to degrees. If you try to approximate $\pi$ by fractions of the form $180n/(360k+1)$, you can find that ...
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107 views

Different ways of operating an infinite continued fraction

Given the continued fraction below, $$ \cfrac{1}{\cfrac{1}{\cfrac{1}{\cdots}+\cfrac{1}{\cdots}}+\cfrac{1}{\cfrac{1}{\cdots}+\cfrac{1}{\cdots}}} $$ I wanted to know to which number it converged, so I ...
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2answers
129 views

Reference for relation between class number of $\Bbb Q[\sqrt{-p}]$ and partial quotients of $\sqrt p$

So in Ireland and Rosen's, "Classical Introduction to Modern Number Theory", they mention the following incredible fact at the end of Chapter 13, section 1. Suppose $p \neq 3$ and $p \equiv 3 \pmod 4$ ...
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33 views

Explanations of the Euler's continued fractions to compute exponential

After looking for explanations of the Euler's continued fractions to compute exponential on internet and after reading Euler's explanations about, I still don't understand how Euler find this ...
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1answer
69 views

Number made from the first digits of $2^n$

Consider the number c made from the first digits of $2^n$. To be more precise, the n-th decimal digit of c is the first digit of $2^n$. The first digits from c are : ...