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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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 ...
2
votes
0answers
83 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 $$ ...
1
vote
0answers
39 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 ...
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
votes
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 ...
0
votes
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| ...
1
vote
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 ...
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 ...
3
votes
0answers
46 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 as usual after reading basic concepts I reviewed here at MSE former questions to learn more. While reviewing the questions and answers, I ...
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 ...
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 ...
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!!!
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 ...
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 ...
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, ...
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 ...
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. ...
1
vote
2answers
93 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 ...
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 ...
2
votes
0answers
260 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 ...
1
vote
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 ...
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 $141$ ...
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 ...
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 ...
1
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.
0
votes
1answer
66 views

continued fraction expansion for √7 [duplicate]

Can someone help me find the continued fraction expansion for $\sqrt{7}$ just like I did for below. For $\sqrt{3}$ I did this: I was given that $x = \sqrt{3} -1 $ $x = \frac{1}{1+\frac{1}{2+x}} $ ...
0
votes
1answer
32 views

Geometric Proof for Slopes (Contined Fractions)

I just started learning about continued fractions, and my lecture had a theorem that estimated the slope $a$ of a given line $L$. This was done in terms of the slope of the point $P$ with coordinates ...
1
vote
2answers
61 views

Why is $\sqrt{3}=[1;1,2,1,2,\dots]$?

Why is $\sqrt{3}=[1;1,2,1,2,\dots]$ ? $\displaystyle[1;1,2,1,2,\dots]=1+\frac{1}{[1;2,1,2,\dots]}=1+\frac{1}{1+\frac{1}{2+\frac{1}{[1;2,1,2,\dots]}}}$ If I set $x=[1;2,1,2,\dots]$ then; ...
0
votes
3answers
113 views

continued fraction of the roots of $x^2 - \frac{53793390359}{1088391168}x + \frac{823543}{12230590464} = 0$

I would like to find the continued fraction expansion of the roots of: $$x^2 - \frac{53793390359}{1088391168}x + \frac{823543}{12230590464} = 0$$ Eq 1.6 from [1] What makes this problem so ...
1
vote
1answer
56 views

continued fraction of $3 + 17\sqrt{3} $

On a computer, I tried to iterate the Euclidean algorithm on the number $3 + 17\sqrt{3}$ and here is what I got: \begin{array}{ccccrcrcrcr} 3 + 17\sqrt{3} &=& 32 &\cdot\;(& ...
1
vote
0answers
34 views

Integral of a Continued Fraction

How might one go about evaluating the following integral $\int_{-\infty}^{\infty}\mathrm{K}_{j=0}^{\infty}(F_{j}e^{-x^2})dx$? Where$\mathrm{K}$ denotes a continued fraction and $F_j$ is the jth ...
2
votes
1answer
84 views

Why are there no continued fraction representation for $\pi$ obeying mathematical rules?

There are several irrational numbers that can be represented with continued fraction such that a mathematical rule arises in this continued fraction. For example, the Euler number $e$ can be ...
1
vote
0answers
20 views

Continued Fraction summation representation

I have a rational fraction of the form: $$s=\frac{p_0+p_1x+p_2x^2+\cdots+p_Mx^M}{1+q_1x+q_2x^2+\cdots+q_Mx^M} $$ The paper I am reading converts this to the form: $$s = ...
7
votes
2answers
178 views

Simple Finite Continued Fraction

I am working on my senior thesis and have encountered, unexpectedly, a finite continued fraction that I would be interested in resolving. I already know the answer (by an informed guess based on where ...
5
votes
0answers
49 views

Is there any elegant formalization of fractional numbers?

The question is just what is on the title, but I'll describe the context for completion: Natural numbers can be encoded quite elegantly on the Lambda Calculus as church numbers, that is, a function ...
2
votes
0answers
14 views

On the asymptotics of a continued fraction

I know the coefficients of the continued fraction representation of a function that diverges like $O(\sqrt{x})$, where the $a_k$ depend on $x$, $ f(x)=1+a_0/\left(1+\underset{k=1}{\overset{\infty ...
4
votes
1answer
68 views

Continued Fraction, Closest Neighbours

For setting the proper divisor/multiplier registers in a phase locked loop, I use a continued fraction expansion, which I stop if either the numerator or denominator of the fraction becomes larger ...
1
vote
1answer
71 views

Good and best rational approximations

Lately I was reading a bit about continued fractions and came up with a question that I couldn't find an answer for. Here are the definitions I will use in the question: Let $x \in \mathbb{R}$. A ...
1
vote
1answer
45 views

Closest rational approximation of $\sqrt x$ with denominator having prime powers $\lt n$

I am representing denominators in rational numbers with powers of their prime factors for easy multiplication and division in lowest terms (by adding and subtracting the prime powers). I would like ...