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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191 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
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1answer
71 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 ...
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1answer
42 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 ...
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0answers
24 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 ...
26
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1answer
555 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
211 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)} ...
21
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1answer
224 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
51 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} ...
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0answers
39 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 ...
3
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2answers
194 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} ...
4
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4answers
327 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
1answer
450 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 ...
6
votes
2answers
180 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= ...
12
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 ...
3
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0answers
91 views
2
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1answer
179 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 ...
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2answers
334 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
202 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 ...
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0answers
26 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?
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2answers
52 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
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0answers
42 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 ...
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3answers
69 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 ...
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2answers
26 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
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1answer
92 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
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1answer
30 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
35 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
62 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 ...
5
votes
1answer
113 views

Link between the negative pell equation $x^2-dy^2=-1$ and a certain continued fraction

Consider the generalized continued fraction $$F(x)=(x-1)-\cfrac{(x+1)}{x+\cfrac{(-1)(5)} {3x+\cfrac{(1)(7)}{5x+\cfrac{(3)(9)}{7x+\cfrac{(5)(11)}{9x+\ddots}}}}}$$ I experimentally discovered that at ...
18
votes
1answer
499 views

a new continued fraction for $\sqrt{2}$

In a q-continued fraction related to the octahedral group I defined a new q-continued fraction for the square of ramanujan's octic continued fraction which I discovered using certain three term ...
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votes
1answer
32 views

Any good books for studying Continued Fraction?

Does anyone have recommendations for cheap books on Continued Fractions? I do not have much money and so it needs to be a cheap book.
8
votes
2answers
801 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 ...
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vote
2answers
18 views

Numerical algorithm: Spectral function -> Continued Fraction

I am trying to code up a numerical algorithm which takes a spectral function of the form $$c(\zeta) = w_0 +\sum_{m=1}^N \frac{w_m}{\lambda_m+\zeta}$$ into a continued fraction of the form $$c(\zeta) = ...
15
votes
2answers
255 views

What's the formula for this series for $\pi$?

These continued fractions for $\pi$ were given here, $$\small \pi = \cfrac{4} {1+\cfrac{1^2} {2+\cfrac{3^2} {2+\cfrac{5^2} {2+\ddots}}}} = \sum_{n=0}^\infty \frac{4(-1)^n}{2n+1} = \frac{4}{1} - ...
2
votes
1answer
89 views

How to use Euler's continued fraction formula?

I am trying to convert some continued fractions to series by using Euler's continued fraction formula (see the link to Wikipedia). But there is something I obviously misunderstood in it. Take for ...
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1answer
60 views

Continued Fractions : Under which branch of mathematics do they come?

I wanted to know in which branch of Mathematics do Continued Fraction come? By branch I mean for example Geometry or Differential Equation are a branch of maths so is there any particular branch of ...
15
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0answers
465 views

a conjectured continued fraction for $\displaystyle\tan\left(\frac{z\pi}{4z+2n}\right)$

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 ...
1
vote
1answer
37 views

Good book for self study of Continued Fractions

Does anyone have a recommendation for a rigorous while readable book to use for the self study of continued fractions? PS - As examples of "rigorous while readable book" for self-learning, A. ...
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2answers
66 views

Negative solution for a positive continued fraction

$$ x=1+\cfrac{1}{1+\cfrac{1}{1+...}}\implies x=1+\frac{1}{x}\implies x=\frac{1\pm \sqrt{5}}{2} $$ Can the negative solution be considered as a solution? If yes, how is it possible to have a negative ...
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0answers
171 views

Divergent continued fractions?

The solutions to $$ x^2-6x+10=0 \tag 1 $$ are $$ 3\pm i\tag2. $$ Rearranging $(1)$ just a bit, we get $$ x = 6 -\frac{10}x \tag3 $$ and then substituting the right side of $(3)$ for $x$ within the ...
8
votes
1answer
76 views

Newton's method for square roots 'jumps' through the continued fraction convergents

I know that Newton's method approximately doubles the number of the correct digits on each step, but I noticed that it also doubles the number of terms in the continued fraction, at least for square ...
8
votes
4answers
317 views

Continued fractions for $\sqrt{x} $ and beyond, valid formula?

For $x > 0$, is this trick valid? I use $$ ( \sqrt{x}-1)(\sqrt{x}+1)=x-1 $$ then $$ \sqrt{x}+1 = \frac{x-1}{\sqrt{x}+1-2} $$ so I can use iterations to get the rational approximant $$ \sqrt{x} ...
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0answers
31 views

How to make continued fractions of any number?

I recently found an continued fraction representation of $\pi$, and I wondered how can I make an continued fraction that converges into a number? The MAIN question is: how do you make a continued ...
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votes
0answers
55 views

Does the continued fraction for $e^{3/n}$ have a pattern?

Wikipedia has patterns for the simple continued fractions $e^{1/n},e^{2/n}$, which made me wonder whether there is one known for $e^{3/n}?$ (by pattern, I mean that the partial quotients $a_n$ can ...
21
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2answers
715 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 ...
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0answers
115 views

Infinitely nested radical expansions for real numbers

Conjecture. For any real number $x \in (0,1]$ there exists a unique expansion in the form $x=-2+\sqrt{a_1+\sqrt{a_2+\sqrt{a_3+\cdots}}}$ with $a_k$ being natural numbers from the set $(2,3,4,5,6)$. ...
5
votes
1answer
64 views

Nested Radicals and Continued Fractions

Is there some interconnection between these two topics? A sort of classification of the possibile types of nested radicals and maybe some way (hopefully bijective, in some sense) to pass from a ...
2
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0answers
61 views

Rogers-Ramanujan continued fraction $R(e^{-2 \pi \sqrt 5})$

Let $$R(q) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^{2}}{1 + \cfrac{q^{3}}{1 + \cdots}}}}$$ It is easy to evaluate $R(e^{-2 \pi/ \sqrt 5})$ using the Dedekind eta function identity ...
6
votes
1answer
2k views

Approximating $\arctan x$ for large $|x|$

I would like to know if there is reasonably fast converging method for computing large arguments of arctan. Until now I've came across Taylor series that converges only on interval $(-1,1)$ and for ...