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.

learn more… | top users | synonyms

3
votes
0answers
126 views
+50

conjectured arithmetic properties of some continued fraction

Given the continued fraction found in this post,bearing a striking resemblance to the one in this post $$G(q)=\cfrac{1}{1-q+\cfrac{q(1-q)^2}{1-q^3+\cfrac{q(1-q^2)^2}{1-q^5+\cfrac{q(1-q^3)^2}{1-q^7+\...
6
votes
3answers
100 views

Convergence of a Harmonic Continued Fraction

Does this continued fraction converge? $$\large\frac { 1 }{ 1+\frac { 1 }{ 2+\frac { 1 }{ 3+\frac { 1 }{ 4+\dots } } } } $$ ($[0;1,2,3,4, \dots]$) I tried approximating a few values but I ...
4
votes
2answers
90 views

Writing continued fractions of irrational numbers as infinite series

Infinite sums have been formulated for famous irrational numbers, such as $\pi, \phi,e,\sqrt2$ and a few others that can be listed here and here: Here are some examples: (There are more examples ...
-1
votes
0answers
32 views

How to remap continued fractions from $\mathbb{R}$ to a discrete set of integers [on hold]

Assuming that I have a continuous fraction \begin{equation} x = a_0 + k_1 \cfrac{x_1}{a_1 + k_2 \cfrac{x_2}{a_2 + k_3 \cfrac{x_3}{a_3 + k_4 \cfrac{x_4}{a_4 \ddots } } } } \end{...
1
vote
0answers
59 views

Is this stronger Kintchine theorem true?

Let $\phi(n)$ be an increasing real valued function on the positive integers. Suppose that almost every $x \in (0,1)$ has $a_n \geq \phi(n)$ for infinitely many $n$, where $a_n$ is the n'th integer ...
2
votes
1answer
19 views

Prove that these continued fractions are equal.

The wikipedia page on Generalized Continued Fractions gives this equation. $$ \log \left( 1+\frac{x}{y} \right) = \cfrac{x} {y+\cfrac{1x} {2+\cfrac{1x} {3y+\cfrac{2x} {2+\cfrac{2x} {5y+\cfrac{3x} {2+\...
0
votes
0answers
23 views

Is the inverse of Minkowski's question mark function continuous on the dyadic fractions?

I'm looking for a continuous function from the dyadic fractions between 0 and 1 to the rational numbers between 0 and 1. The inverse of Minkowski's question mark (also known as Conway's box function) ...
7
votes
1answer
479 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 ...
0
votes
1answer
256 views

Using Maple for continued fraction expansions

I can find the continued fraction expansion of a value using Maple. Is there a simple way for finding the sequence of convergents (approximants) of the continued fraction expansion in Maple? Currently ...
2
votes
3answers
646 views

Can every transcendental number be expressed as an infinite continued fraction?

Every infinite continued fraction is irrational. But can every number, in particular those that are not the root of a polynomial with rational coefficients, be expressed as a continued fraction?
0
votes
1answer
12 views

Estimation of numerators and denominators of convergents of continued fractions

I was going through C.Odd's textbook on continued fractions and in the introductory chapter it introduced the formula for the numerator and denominator of the $\ k$ convergent in terms of the ...
0
votes
0answers
32 views

Almost everywhere growth of continued fraction partial quotients

What is an upper bound for the growth of the largest partial quotient (i.e. the 'digit') among the first $n$ partial quotients in the continued fraction expansion of almost all real numbers as $n$ ...
2
votes
1answer
54 views

Proving a particular infinite continued-fraction identity

By iterating the basic relation $$ \forall z \in \mathbb{C} \setminus \{ -1 \}: \quad z = 1 + \frac{z^{2} - 1}{z + 1}, $$ one obtains the following finite continued-fraction identities: \begin{alignat}...
3
votes
0answers
88 views

Continued fraction $1 + \frac 2{3 + \frac 4 {5 + \cdots}} = \frac 1 {\sqrt{e} - 1}$?

I saw this link (written in Japanese) and found an interesting problem: Calculate $1 + \frac 2{3 + \frac 4 {5 + \cdots}}$. The link provides the answer ($\frac 1 {\sqrt e - 1}$) and a hint that one ...
4
votes
1answer
695 views

Pell's Equation $x^2-11y^2=1$ through Continued Fractions

Use continued fractions to find the minimal solution to $x^2-11y^2=1$. I know that $\sqrt{11}=3+\frac{1}{3+\frac{1}{6+\frac{1}{3+...}}}$ I took $\sqrt{11}=3+\frac{1}{3+\frac{1}{6+\sqrt{11}}}$ and I ...
0
votes
1answer
56 views

Proof of $\frac{q_n}{q_{n-1}} = [a_n,a_{n-1},a_{n-2},…,a_2,a_1]$?

Proof of continued fractions axiom. Let $c=[a_0,a_1,a_2,\dots,a_n,\dots] = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \ddots}}$ be a continued fraction which could be finite or infinite. By $\frac{p_n}{...
0
votes
0answers
21 views

Interpretation of $ \tau $ in the Stephen Astels paper '' Cantor set and numbers with restricted partial quotients?

I am trying to read Stephen Astels paper 'Cantor sets and numbers with restricted partial quotients'. Visit http://www.ams.org/journals/tran/2000-352-01/S0002-9947-99-02272-2 In this he directly ...
8
votes
3answers
87 views

Continued fraction for $c= \sum_{k=0}^\infty \frac 1{2^{2^k}} $ - is there a systematic expression?

I want to use the convergents of the continued fraction for $$c= \sum_{k=0}^\infty \frac 1{2^{2^k}} $$ - but of course a numeric software is very limited here, so I hope there exists a systematic ...
5
votes
1answer
133 views

Does this integral $\int_0^\infty \frac{dx}{(1+e^x)(a+x)}$ have a closed form?

Note that $a>0$, thus I'm not sure if we can apply residues here. (For $a=0$ the integral doesn't converge). $$\int_0^\infty \frac{dx}{(1+e^x)(a+x)}$$ Despite the simple expression under the ...
0
votes
0answers
31 views

Recurrence relation connected with the continued fraction of the Exponential Integral.

I've been trying to solve this non-homogeneous recurrence relation with no luck. $$k_n=a_nk_{n-1}+k_{n-2}$$ $$a_n= \begin{cases} x, & \text{if $n$ is even}\\ -2/(n+1), & \text{if $n$ is odd} \...
1
vote
0answers
20 views

Transformation of this log-type continued fraction

I've learned that $$\ln\left(1+\cfrac{x}{y}\right)=\frac{x}{y+\cfrac{1x}{2+\cfrac{1x}{3y+\cfrac{2x}{2+\cfrac{2x}{5y+\frac{3x}{2+\cdots}}}}}}=\cfrac{2x}{2y+x-\cfrac{(1x)^2}{3(2y+x)-\cfrac{(2x)^2}{5(2y+...
0
votes
0answers
73 views

a continued fraction related to the exponential function $e^x$

Given a natural number $n$,with $|x|\lt1$ define the following conjectured identity $$G(n,x)=\begin{aligned}\cfrac{-n}{1-x-\cfrac{(1+n)(1-x^2)}{1-x^3-\cfrac{x^2(1-x)(1-x^3)}{1-x^5-\cfrac{x^3(1-x^2)(1-...
1
vote
1answer
106 views

continued fraction $F(x)$ that is a generating function of central binomial coefficients

Given the following continued fraction $$F(x) =\cfrac{1}{x+\cfrac{2^2(2^2-1)}{6x+\cfrac{3^2(3^2-1)}{12x+\cfrac{4^2(4^2-1)}{20x+\cfrac{5^2(5^2-1)}{30x+\ddots}}}}}=\frac{1}{\sqrt{x^2+4}}$$ Then $$\...
4
votes
1answer
220 views

conjectured general continued fraction for the quotient of gamma functions

Given complex numbers $a=x+iy$, $b=m+in$ and a gamma function $\Gamma(z)$ with $x\gt0$ and $m\gt0$, it is conjectured that the following general continued fraction which is symmetric on $a$ and $b$ is ...
1
vote
1answer
48 views

Estimation on the accuracy of the convergents of $\sqrt{n}$

I have noticed that the accuracy of the best rational approximations to $\sqrt{n}$ given by his continued fraction expansion, when the numerator and deniminator are large numbers, is approximately ...
1
vote
0answers
30 views

On a problem of Erdős about continued fractions and Liouville numbers

In 1938, Erdős and Mahler raised the following question: Let $\xi$ be a real number such that $(p_n/q_n)_n$ are convergents of its continued fraction. If there exists a subsequence $(p_{n_j}/q_{...
26
votes
1answer
567 views

Curious about an empirically found continued fraction for tanh

First of all, and since this is my first question in this forum, I would like to specify that I am not a professional mathematician (but a philosophy teacher); I apologize by advance if something is ...
3
votes
2answers
225 views

a general continued fraction satisfying $\frac{(i+\Theta\sqrt{z})^m}{(i-\Theta\sqrt{z})^m}=\frac{(ik+\sqrt{z})^{m+1}}{(ik-\sqrt{z})^{m+1}}$

Given any natural number $m\gt2$, let $z$,$k$ be complex numbers, where $i=\sqrt{-1}$ and consider the general continued fraction $$\Theta(k,z,m)=\cfrac{(m+1)}{km+\cfrac{z(0m-1)(2m+1)} {3km+\cfrac{z(...
21
votes
1answer
233 views

Continued fraction analog to zeta function - how to properly define it and find its properties?

I do not mean the continued fraction representation of zeta function; I mean the function which has the form: $$f(s)=\cfrac{1}{1^s+\cfrac{1}{2^s+\cfrac{1}{3^s+\cfrac{1}{4^s+\cdots}}}}$$ For some ...
4
votes
1answer
55 views

How to prove this continued fraction connection between $\gamma$ and $e$?

There is apparently a curious connection between Euler-Mascheroni constant $\gamma$ and $e$ in the form of an infinite series and continued fraction: $$e \gamma=e \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}...
1
vote
0answers
43 views

On a particular continued fraction related to $\pi$.

Is this continued fraction well studied? $$\Theta(m)=m+\cfrac{1^2}{2m+\cfrac{3^2} {2m+\cfrac{5^2}{2m +\cfrac{7^2}{2m+\cfrac{9^2}{2m+\ddots}}}}}$$ Note $\Theta(1)=\frac4\pi$. Denote $\Theta(m)=\...
3
votes
2answers
198 views

A pair of continued fractions that are algebraic numbers and related to $a^2+b^2=c^m$

Similar to the cfracs in this post, define the two complementary continued fractions, $$x=\cfrac{-(m+1)}{km\color{blue}+\cfrac{(-1)(2m+1)} {3km\color{blue}+\cfrac{(m-1)(3m+1)}{5km\color{blue} +\cfrac{...
4
votes
4answers
330 views

Relationship between degrees of continued fractions

I'm trying to compute the values of differing degrees of continued fractions like $\sqrt 2$, $e$ and other similar fractions. My theory was to take the reduced fraction at an arbitrary depth and the ...
6
votes
2answers
181 views

A $q$-continued fraction connected to the divisor function?

In this post, the following two continued fractions discussed by Nicco are given, $$A(q)= \left(\frac{\vartheta_2(0,q)}{2\,q^{1/4}}\right)^2= \cfrac{1}{1-q+\cfrac{q(1\color{red}-q)^2}{1-q^3+\cfrac{q^...
13
votes
1answer
1k views

a conjectured continued-fraction for $\displaystyle\cot\left(\frac{z\pi}{4z+2n}\right)$ that leads to a new limit for $\pi$

Given a complex number $\begin{aligned}\frac{z}{n}=x+iy\end{aligned}$ and a gamma function $\Gamma(z)$ with $x\gt0$, it is conjectured that the following continued fraction for $\displaystyle\cot\left(...
3
votes
0answers
94 views

Two complementary continued fractions that are algebraic numbers

Define the two similar continued fractions, $$x=\cfrac{1}{km\color{blue}+\cfrac{(m-1)(m+1)} {3km\color{blue}+\cfrac{(2m-1)(2m+1)}{5km\color{blue}+\cfrac{(3m-1)(3m+1)}{7km\color{blue}+\ddots}}}}\tag1$$...
2
votes
1answer
183 views

Use the simple continued fraction of $\sqrt{27323}$ to factor $27323$…

Use the simple continued fraction of $\sqrt{27323}$ to factor $27323$. So far I have: $\sqrt{27323} = 1 + (\sqrt{27323} - 1)$ which gives... $= 1 + \frac{1}{(\frac{1}{164.2967029})}$ I'm ...
7
votes
2answers
352 views

Ramanujan theta function and its continued fraction

I believe Ramanujan would have loved this kind of identity. After deriving the identity, I wanted to share it with the mathematical community. If it's well known, please inform me and give me some ...
4
votes
1answer
211 views

a continued fraction related to pythagoras theorem $a^2+b^2=c^2$

For our purpose,let $a,b,c$ and $x\gt2$ be natural numbers such that the positive integers $a,b$ and $c$ form a special pythagorean triple $(a,b,c)$,then it is conjectured that the following is true $...
0
votes
0answers
28 views

Periodicity of the continued fraction of a square root

Writing $\sqrt{n}=[a_0; a_1, a_2, \dots ]$, at which $a_i$ does the period start? Is it $a_1$? I just put "for some $n\ge 1$, where $a_{n-1}=a_i$", is that a good enough answer?
0
votes
2answers
54 views

Can I make an infinte sum using rational numbers that makes an irrational but not transcendental number?

I looked a lot on the internet for examples and I tried to do it myself, but I haven't seen any infinite sums of rational numbers that equal for example something like square root of 10 or cube root ...
0
votes
0answers
45 views

Is there a Similar type for Fibonacci numbers?

The sequence of Lucas numbers is: $2,1,3,4,7,11,18,\ldots$ $L_0=2$, $L_1=1$; $L_{n+1}=L_n+L_{n-1}$ $\phi=\frac{1+\sqrt5}{2}$ is the golden ratio Ramanujan's continued fraction $$\frac{5}{1+[5^{0....
1
vote
3answers
72 views

Only valid for Pythagoraean triples $\sqrt2+\frac{b}{\sqrt2+\frac{b}{\sqrt2+\frac{b}{\sqrt2\cdots}}}=\sqrt{c+a}$?

$$\sqrt2+\frac{b}{\sqrt2+\frac{b}{\sqrt2+\frac{b}{\sqrt2\cdots}}}=\sqrt{c+a}$$ Where (a,b,c) are the Pythagoraean Triples and are satisfy by the Pythagoras theorem $a^2+b^2=c^2$ An example of ...
0
votes
2answers
29 views

Calculate a quadratic irrational from its periodic continued fraction

I have a periodic continued fraction [2; 1, 3] and I want to convert it into a quadratic irrational. Any helps?
2
votes
1answer
97 views

Is $ 2.7182818281828…$ a semiconvergent of e?

Euler's number $e=2.71828 18284 59045... $ can be approximated by the rational number: $$ x=\frac{271,828-27}{100,000-10}= \frac{271,801}{99,990} =2.7182818281... $$ Also, $e$ has the well-known ...
1
vote
1answer
32 views

Find the infinite simple continued fractions for …

Find the infin ite simple continued fractions for $\sqrt{2};\sqrt{5};\sqrt{6};\sqrt{7};\sqrt{8}$. I have solved similar equations for continued fractions but only using a fraction, if someone could ...
1
vote
1answer
37 views

Find the simple continued fractions for both $\pm \frac{39}{25}$…

Find the simple continued fractions for both $\pm \frac{39}{25}$? So far for $\frac{39}{25}$ I have: $39 = 1 \times 25 + 14 $ $ 25 = 1\times 14 + 11 $ $14 = 1 \times 11 + 3$ $11 = 3 \times 3 +...
2
votes
1answer
48 views

Which irrational number represents the infinite simple continued fraction [0;7]?

Which irrational number represents the infinite simple continued fraction [0;7]? -So from my current understanding [o;7] can be represented as the following: $ = \frac{1}{7 + \frac{1}{7 + \frac{1}{7 ...
3
votes
0answers
64 views

Is this well known? [duplicate]

How to prove $$1+\cfrac{1}{1+\cfrac{e^{-2\pi}}{1+\cfrac{e^{-4\pi}}{1+\cfrac{e^{-6\pi}}{\cdots}}}} = \left(\sqrt{5\phi}-\phi\right) e^{2\pi/5}.$$ i dont know how to do it. like if there were repeating ...
2
votes
1answer
38 views

Consecutive smooth number generator recovery

The numbers $n=811150370266636218705704$ and $n+1$ have highest factors 173 and 167, and they happen to be the largest consecutive 173-smooth numbers. They were found via Størmer's theorem and the ...