A is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number.

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51 views

Is there an advantage in using continued fractions for Catalan or Fibonacci-Lucas primality tests?

I am studying the basic theory about continued fractions and also reviewed here at MSE former questions to learn more. While reviewing the questions and answers, I found references to the Fibonacci ...
1
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0answers
42 views

a q-continued fraction related to the octahedral group

Let $q=e^{2\pi i\tau}$. If $u(\tau)$ is Ramanujan's octic continued fraction, $$u(\tau)=\cfrac{\sqrt{2}\,q^{1/8}}{1+\cfrac{q}{1+q+\cfrac{q^2}{1+q^2+\cfrac{q^3}{1+q^3+\ddots}}}}$$ is it true that ...
10
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1answer
78 views

Continued fraction estimation of error in Leibniz series for $\pi$.

The following arctan formula for $\pi$ is quite well known $$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\tag{1}$$ and bears the name of Madhava-Gregory-Leibniz series after ...
3
votes
0answers
90 views

Continued fraction for $\int_{0}^{\infty}(e^{-xt}/\cosh t)\,dt$

In one of the comments to a question I posted on MSE, I got this wonderful continued fraction $$\int_{0}^{\infty}\frac{e^{-xt}}{\cosh t}\,dt = \frac{1}{x +}\frac{1^{2}}{x +}\frac{2^{2}}{x ...
2
votes
0answers
84 views

The ratio of jacobi theta functions

Let $q=e^{2\pi i\tau}$. If $\theta_2$ and $\theta_3$ are jacobi theta functions , is it true that the ratio of the two functions can be expressed as a continued fraction of the form $$ ...
7
votes
1answer
83 views

Are the unit partial quotients of $\pi, \log(2), \zeta(3) $ and other constants $all$ governed by $H=0.415\dots$?

Khinchin showed that given the simple continued fraction of a real number, $$r = a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1} {\ddots}}}$$ then it is almost always true that the partial quotients $a_i$ ...
5
votes
2answers
284 views

What is the geometric average of 2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10…?

It is known that the Khinchin constant is not the geometric mean of the first $n$ coeffecients, as $n$ approaches infinity, of the continued fraction of e, which is $$[2; 1, 2, 1, 1, 4, 1,1, 6, 1, 1, ...
0
votes
2answers
53 views

Visualisation of the reciprocal of an continued fraction?

If: $$a=\cfrac{l}{m+\cfrac{n}{o+\cfrac{p}{q+\cdots}}}$$ Then could you help me visualize $1/a$? I really don't understand it. Thank you so much!
3
votes
1answer
122 views

A new $q$-continued fraction of order $12$

I think I may have discovered a $q$-continued fraction of order $12$ with a form different from that established by Mahadeva Naika. Let $q=e^{2i \pi \tau}=\exp(2i \pi \tau)$, then, $$\begin{aligned} ...
4
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0answers
38 views

The $q$-continued fraction for tribonacci constant and others

Let $q = e^{-2\pi}$. We are familiar with Ramanujan's beautiful continued fraction, $$\cfrac{q^{1/5}}{1 + \cfrac{q} {1 + \cfrac{q^2} {1 + \cfrac{q^3} {1+\ddots}}}} = {\sqrt{5+\sqrt{5}\over ...
1
vote
1answer
36 views

Continued fraction manipulation

I have the following continued fraction $$ \frac{1}{a_1x+}\;\;\frac{1}{b_1+}\;\;\frac{1}{a_2x+}\;\;\frac{1}{b_2} $$ The paper I am reading then converts this to the following continued z-fraction ...
3
votes
1answer
115 views

A real number is rational $\iff$ its continued fraction expansion is finite.

I know that if this expansion is finite, then I can go to the lowest denominator in the whole fraction and turn it into a fraction and keep doing so until I get a fraction which means the number is ...
0
votes
2answers
48 views

How can I approximate a decimal with two fractions where denominator is less or equal to $d$

I was looking for a way to approximate a decimal number with a fraction, whose denominator is less or equal to $d$. Basically, having a decimal $X$, I want to find two fractions such that ...
11
votes
1answer
526 views

Deriving a trivial continued fraction for the exponential

Lately, I learned about the following continued fraction for the exponential function: $$\exp(x)=1+\cfrac{x}{1-\cfrac{x/2}{1+x/2-\cfrac{x/3}{1+x/3-\cfrac{x/4}{1+x/4-\dots}}}}$$ I thought it was ...
15
votes
2answers
883 views

Continued fraction for $\frac{1}{e-2}$

A couple of years ago I found the following continued fraction for $\frac1{e-2}$: $$\frac{1}{e-2} = 1+\cfrac1{2 + \cfrac2{3 + \cfrac3{4 + \cfrac4{5 + \cfrac5{6 + \cfrac6{7 + \cfrac7{\cdots}}}}}}}$$ ...
0
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0answers
12 views

continued fractions-decreasing series

I am dealing with this problem. Let $\theta=[a_0,...]$ be irrational number. Prove that the following series are strictly decreasing: \begin{eqnarray} \left| \theta-\frac{p_n}{q_n} \right|\\ \left| ...
10
votes
1answer
914 views

How do I prove the partial denominators formula of the Bauer-Muir transformation of a generalized continued fraction?

Notation: $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ is the Gauss Notation for generalized continued fractions. Description of the Bauer-Muir transformation ...
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0answers
38 views

For which prime numbers $p$ there exist $x,y\in \Bbb{Z}$ such that $p=x^2+2y^2$? [duplicate]

For which prime numbers $p$ there exists $x,y\in \Bbb{Z}$ such that $p=x^2+2y^2$? I guess I am to use continued fraction, but I am not sure how. I know how to find solutions for defined numbers but I ...
2
votes
1answer
36 views

Computing periodic continued fractions.

Compute $[1,2,3,\overline{1,4}]$ where $\overline{1,4}$ is the periodic part. I looked into explanations about that, but haven't come by an actual algorithm of computing such a thing. I know it is ...
6
votes
1answer
80 views

Find the derivative of $f$ if it exists, else, prove it doesn't exist

Let $f: \mathbb R^+ \to \mathbb R $ $$f=\mathop{\vcenter{\LARGE\mathrm K}}\limits_{j=1}^{+\infty}\frac{x}{x^j}=\cfrac{x}{x^1+\cfrac{x}{x^2+\cfrac{x}{x^3+\ddots}}}$$ I saw this problem in a math ...
7
votes
3answers
3k 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 ...
6
votes
1answer
91 views

Even Fibonacci Numbers and $\sqrt{5}$

My question is simple, but a mystery to me. Skip to the last paragraph if you're not interested in the story of my exploration. EDIT: I seem to have misinterpreted a key detail regarding how the ...
8
votes
1answer
69 views

Multiply all terms in continued fraction by a constant

I noticed that continued the fraction for $\sqrt{12}$ is $3;2,6,2,6,2,\ldots$ and the continued fraction for $\sqrt{7\times12}$ is $9;6,18,6,18,6,\ldots$ all the terms in the continued fraction are ...
0
votes
1answer
115 views

The radius of image of a circle under mobius transformation

A Mobius transformation of the plane takes $z \mapsto \frac{az+b}{cz+d}$. These are known to take circles to circles, but given an explicit circle, how do we compute the radius. Let's parameterize ...
5
votes
1answer
418 views

Continued fraction for $\tan(nx)$

I found this beautiful continued fraction expansion of $\tan(nx)$, $n$ being a positive integer, online but I don't remember the source now: $\displaystyle \tan(nx) = \cfrac{n\tan x}{1 -\cfrac{(n^{2} ...
1
vote
2answers
91 views

Finding two solutions to $x^2 - 6y^2 = 1$ using continued fractions [closed]

Can anyone show me how to find the solutions to $x^2-6y^2=1$ by using continued fractions? I know how to find the fractions for $\sqrt6$ but do not know how to proceed. THANK YOU!!!
6
votes
2answers
232 views

“Bizarre” continued fraction of Ramanujan! But where's the proof?

$$\frac{e^\pi-1}{e^\pi+1}=\cfrac\pi{2+\cfrac{\pi^2}{6+\cfrac{\pi^2}{10+\cfrac{\pi^2}{14+...}}}}$$ "Bizarre" continued fraction of Ramanujan! But where's the proof? i have no training in continued ...
3
votes
0answers
83 views

Integral formulas involving continued fractions

Ramanujan posed the following formulas as questions in the Journal of Indian Mathematical Society: $$\int_{0}^{\infty}\dfrac{\sin nx\,\,dx}{{\displaystyle x + \dfrac{1}{x +}\dfrac{2}{x +}\dfrac{3}{x ...
0
votes
1answer
21 views

Farey Sequence implemenatation

I'm trying to use the Farey sequence to get the next lowest reduced fraction in a list. For example, for $n = 8$, we have $\dots, \frac13, \frac38, \frac25, \frac37, \frac12, \dots$ So let's take ...
-1
votes
1answer
33 views

Continued Fraction Expansions Confusion

Let $\theta$ be an irrational number with continued fraction expansion $[a_0; a_1, a_2, \cdots]$. Suppose $P_n/Q_n = [a_0; a_1, \cdots , a_n]$ is the $n^{th}$ convergent. Then how do I show that ...
8
votes
3answers
109 views

Minimal $ab$ for Rational Number $a/b$ in an Interval

Given rational numbers $L$ and $U$, $0<L<U<1$, find rational number $M=a/b$ such that $L \le M<U$ and $(a\times b)$ is as small as possible---$a$ and $b$ are integers. For example, If ...
0
votes
2answers
42 views

Theorem 1 in Khinchin's “Continued Fractions”

I'm reading an English translation of Khinchin's Continued Fractions and I may have found an error in Theorem 1, page 4. Khinchin observes that if we simplify a finite continued fraction $[a_0; a_1, ...
50
votes
3answers
580 views

Closed form solution for $\sum_{n=1}^\infty\frac{1}{1+\frac{n^2}{1+\frac{1}{\stackrel{\ddots}{1+\frac{1}{1+n^2}}}}}$.

Using a symbolic computation software (Mathematica), I got the following interesting results: $$ \begin{align} \sum_{n=1}^\infty\frac{1}{1+\frac{n^2}{1+\frac{1}{1+\frac{1}{1+n^2}}}} &= ...
29
votes
10answers
1k views

What are the applications of continued fractions?

What is the most motivating way to introduce continued fractions? Are there any real life applications of continued fractions?
6
votes
2answers
564 views

Baire space homeomorphic to irrationals

I try to show that the Baire space $\Bbb N^{\Bbb N}$, with regular product metric, is homeomorphic to the unit interval of irrationals $(0,1)\setminus\Bbb Q$. I already know that the needed function ...
2
votes
1answer
31 views

Why can't we get a better diophantine approximation to the golden ratio?

Essentially, my question is why $|\frac{1 + \sqrt{5}}{2} - \frac{a}{b}| < 1/b^c$ (for $c>2$) is satisfied by only a finite number of $\frac{a}{b}$. This is intrinsically related to Hurwitz's ...
5
votes
0answers
39 views

A conjecture about “equiharmonic numbers” of Flajolet via Doron Zeilberger

While semi-randomly browsing, I came across this conjecture which Philippe Flajolet sent to Doron Zeilberger as a "gift" (the "gift" is here, so you can check to see if I have typeset it correctly): ...
0
votes
1answer
35 views

modification of Dedekind cuts

Dedekind defining real numbers as equivalence classes of Cauchy sequences of rational numbers. $x=y$ means $x-y=0$ ie $x_n - y_n \to 0$. addition and multiplication are defined for each coordinate. ...
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2answers
94 views

Can we say that $\sqrt{2}=2/(2/(2/(2/\ldots)))$?

Can we say that $\sqrt{2}= \cfrac{2}{\cfrac{2}{\cfrac{2}{\cfrac{2}{\ldots}}}}$? We have ...
8
votes
4answers
731 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; ...
0
votes
1answer
48 views

Continued Fraction

I am working on the following question "Use the continued fraction $[1;0,1,1,2,1,1,4,1,1,6,1,1]$ to get an estimate for $e$." But I got stuck when I tried to compute $q_i$, since $a_1=0$ , $q_1 =0$. ...
2
votes
0answers
31 views

Continued fraction for $[1,2,3,4,5,6,\dots]$ [duplicate]

Any continued fraction that does not terminate or repeat can't be rational or a quadratic irrational. It is not hard to write something that does not fit these two categories. Can we still get a ...
16
votes
1answer
343 views

An infinite series plus a continued fraction by Ramanujan

Here is a famous problem posed by Ramanujan Show that $$\left(1 + \frac{1}{1\cdot 3} + \frac{1}{1\cdot 3\cdot 5} + \cdots\right) + ...
2
votes
2answers
151 views

Fundamental unit in the ring of integers $\mathbb Z[\frac{1+\sqrt{141}}{2}]$

Find a fundamental unit in the ring of integers $\mathbb Z[\frac{1+\sqrt{141}}{2}]$ of $\mathbb Q(\sqrt{141})$ I have different corollaries for different numbers, the most appropriate for $141$ ...
2
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0answers
262 views

nth-root of continued fraction with Raney transducers

There are some algorithms for doing basic arithmetic by using regular continued fraction expansions. These algorithms are mainly due to Gosper (1972) and Raney (1973). These two approaches use ...
3
votes
2answers
180 views

Continued fraction to irrational number

Let $[1;\overline{2,1}$] be a continued fraction. I want to find the corresponding number. I know how to transform fractions of the form $[a;\overline{b}]$ but I am having a hard time here. Thank you. ...
6
votes
2answers
242 views

What are the reduced elements of $\mathbb Q(\sqrt{30})$?

What are the reduced elements of $\mathbb Q(\sqrt{30})$ ? From the definition here(on page $32$); An element $\beta\in\mathbb Q(\sqrt{d})$ is said to be reduced, if $\beta>1$ and ...
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1answer
33 views

How to find periodic continued fraction expansion of $\frac{\sqrt{7}}3$

How to find periodic continued fraction expansion of $\frac{\sqrt{7}}3$ Using this formula here (it begins in the middle of the page), I obtained $\frac{\sqrt{7}}3=[0;1,\overline{7,2}]$ but ...
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vote
2answers
49 views

Continued Fraction Algorithm for 113/50

The numbers $a_k$ can be found for $\frac{113}{50}$ by using a continued fraction algorithm. Note that $\frac{113}{50}$ is rational, and as a result it will have to terminate. Can anyone help me ...
1
vote
2answers
65 views

Continued Fraction for Root 5 [duplicate]

How can I find the continued fraction expansion for the square root of 5. Do this without the use of a calculator and show all the steps.