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

1
vote
0answers
43 views
+50

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 ...
1
vote
1answer
25 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 ...
2
votes
1answer
84 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 ...
2
votes
1answer
39 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 ...
1
vote
1answer
25 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 ...
3
votes
0answers
58 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 ...
5
votes
1answer
89 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 ...
0
votes
1answer
30 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.
1
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) = ...
2
votes
1answer
83 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 ...
1
vote
1answer
56 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 ...
0
votes
0answers
57 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
35 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. ...
-1
votes
2answers
60 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 ...
8
votes
1answer
74 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 ...
0
votes
0answers
15 views

Let $\theta(z) = \sum q^{n^2}$, is $\theta(-1/z)$ also a theta function?

I am learning about theta functions. Let $q = e^{2\pi i \, z}$ and $\theta(z) = \sum q^{n^2}$. How does it behave under $\mathrm{SL}_2(\mathbb{Z})$ ? In general we have: $$ \theta\left( - ...
1
vote
0answers
27 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 ...
2
votes
0answers
50 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 ...
0
votes
1answer
38 views

Approximation of a continued fraction

I'm new to continuous fractions and since I haven't dabbled in mathematics for several years I'm finding it quite difficult to get back on the horse. I'm trying to find e given: $$e = 2 + \frac{1}{1 ...
3
votes
2answers
80 views

Infinite nested radical and infinite continued fractions

If $$a = \sqrt{k_0+\sqrt{k_1+\sqrt{k_2+\sqrt{k_3+\sqrt{\cdots}}}}}$$ and $$b = \cfrac{1}{k_0+\cfrac{1}{k_1+\cfrac{1}{k_2+\cfrac{1}{\cdots}}}}$$ what is the relation between $a$ and $b$. What function ...
9
votes
0answers
111 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 $(2,3,4,5,6)$. ...
13
votes
2answers
231 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

Show that $p_n=q_{n-1}=\frac{\alpha^n-\beta^n}{\alpha-\beta}$

If $\frac{p_n}{q_n}$ be the nth convergent of the simple continued fraction $$\cfrac{1}{a+\cfrac{1}{a+\cfrac{1}{a+\ddots}}}$$ show that $$p_n=q_{n-1}=\frac{\alpha^n-\beta^n}{\alpha-\beta}$$ where, ...
377
votes
6answers
26k views

Why is $1 - \frac{1}{1 - \frac{1}{1 - \ldots}}$ not real?

So we all know that the continued fraction containing all $1$s... $$ x = 1 + \frac{1}{1 + \frac{1}{1 + \ldots}} $$ yields the golden ratio $x = \phi$, which can easily be proven by rewriting it as ...
2
votes
1answer
37 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 ...
0
votes
1answer
57 views

Problem on continued fraction of $\frac{\sqrt{5}+1}{2}$ [closed]

If $\frac{p_r}{q_r}$ be the $r^{\text{th}}$ convergent of the continued fraction of $\frac{\sqrt{5}+1}{2}$ then prove that $p_{n+1}=p_{n}+p_{n-1}$ and $p_{2n}=p_{2n-1}+p_{2n-2}$. Attempt: I have ...
2
votes
3answers
65 views

Compare five ways of solving cubic equation by iterations (nested expressions)

Say we have a depressed cubic equation in the general form: $$x^3-bx-c=0$$ There are basically five ways of solving it by iterations. Let's consider them in no particular order (the names are my ...
6
votes
1answer
105 views

Liouville numbers and continued fractions

First, let me summarize continued fractions and Liouville numbers. Continued fractions. We can represent each irrational number as a (simple) continued fraction by $$[a_0;a_1,a_2,\cdots\ ...
3
votes
1answer
43 views

How to prove that this infinite product of continued fractions converges to $1-\frac{1}{z}$?

$$\cfrac{z}{1+z} \cdot \cfrac{z}{1+z-\cfrac{z}{1+z}} \cdot \cfrac{z}{1+z-\cfrac{z}{1+z-\cfrac{z}{1+z}}} \cdots= 1-\frac{1}{z}$$ I propose that this works for any $z \in C$ if and only if $|z|>1$. ...
5
votes
1answer
61 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 ...
1
vote
0answers
50 views

How to rigorously prove the following continued fraction identity?

The following identity seems to be satisfied for any $z \in C$ $$\cfrac{z^2}{1+z^2-\cfrac{z^2}{1+z^2-\cfrac{z^2}{1+z^2-\cfrac{z^2}{1+z^2-\cdots}}}}=\begin{cases}1 & |z| \geq 1\\z^2 & |z| \leq ...
19
votes
0answers
197 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 ...
3
votes
0answers
56 views

Euler exponential continued fraction to compute the trigonometric functions and the golden ratio

Using the Euler continued fraction for the exponent, which is convergent everywhere on the complex plane: $$e^{-z}=1-\cfrac{z}{1+z-\cfrac{z}{2+z-\cfrac{2z}{3+z-\cfrac{3z}{4+z-\cdots}}}}$$ We can ...
0
votes
0answers
19 views

Computation of general continued fractions by $2 \times 2$ matrix multiplication - is it the best way?

There are two main ways to compute a continued fraction (or its $n$th convergent). Let's say we have a general fraction: $$ x= a_0 + ...
1
vote
0answers
24 views

Stern-Brocot Tree and sum of coefficients of continued fraction

Suppose we are given a continued fraction $$\frac{p}{q}=a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+\frac{1}{a_{4}+\cdots}}}$$ I am trying to find an expression, possibly asymptotic, for the sum of the ...
1
vote
1answer
47 views

Definite integral of a continued fraction function

I came up with this function written as the following continued fraction (please correct me if my notation is incorrect): for $n\in\mathbb{N}$, let $$f(x;n)=x+\operatorname*{\LARGE ...
1
vote
0answers
37 views

Recommended reading for continued fractions? And some results

First of all, I apologize for my amateurness and inexperience. Although I always enjoyed math, only two years ago I started experimenting with continued fractions and gained a deep reverence for them. ...
1
vote
5answers
94 views

Limit of the sequence defined by a recurrence

Given a recurrence formula for an arithmetic sequence, $$U_{n} = \frac{1}{2+U_{n-1}}$$ Show that$$\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+ ...}}}} = (SomeGivenValue)$$ How can we solve questions ...
1
vote
0answers
80 views

Continued fraction and order of a real number

If $\alpha$ is a irrational number prove that $\operatorname{ord}\alpha \geq 1+\exp(\limsup \log(\log(a_n+1))/n)$ I tried the simple things like use the well known formula $\operatorname {ord}\alpha ...
1
vote
0answers
15 views

Continued fraction expansion of $1-x$ for $x\in [0,1[$.

If $\alpha$ is a real number in $[0,1[$ that can be written as $[a_0;a_1,a_2,\ldots]$, what can I say about the continued fraction expansion of $1-\alpha$?
3
votes
0answers
33 views

How to write this function in a “well-formed” way

Given an input $0 \lt x \lt 1$, find $x$'s Nearest Integer Continued Fraction with structure $$x = a_0 \pm \cfrac{1}{a_1 \pm \cfrac{1}{a_2 \pm \cdots}}.$$ Then $$f(c) = a_0 + 1 \mp \cfrac{1}{a_1 + ...
0
votes
2answers
33 views

Sum and Product of continued fraction expansion?

Give the continued fraction expansion of two real numbers $a,b \in \mathbb R$, is there an "easy" way to get the continued fraction expansion of $a+b$ or $a\cdot b$? If $a,b$ are rational it is easy ...
6
votes
2answers
75 views

Specification of Hurwitz's Theorem

Hurwitz's Theorem in Number Theory states that for every irrational number $\xi$, the equation $$\left|\xi-\frac{p}{q}\right|<\frac{1}{\sqrt{5}q^2}$$ has infinitely many solutions $(p,q)$ ...
0
votes
1answer
60 views

determine the continued fraction of $\sqrt{n^2 + 2}$ for $n \in \mathbb{N}$

determine the continued fraction of $\sqrt{n^2 + 2}$ for $n \in \mathbb{N}$. For rationals it is rather easy to do this and i know the algorithm, i only get stuck a lot by irrational number such as ...
2
votes
2answers
82 views

Which continued fraction for $e$ is the most computationally efficient?

I know that famous numbers like $\pi$ and $e$ have multiple representations as continued fractions and I'm fascinated with the variety of representations. My question: What continued fraction for ...
1
vote
0answers
22 views

Convergents of continued fractions

Let $d$ and $m$ be positive integers such that $d$ is not a square and such that $m\leq\sqrt{d}$. I want to prove that if $x$ and $y$ are positive integers stafisfying $x^2-dy^2=m$ then $x/y$ is a ...
1
vote
1answer
33 views

How to calculate remainder value of a fraction

Question: Four brothers split a sum of money between them. The first brother received 50% of the total, the second received 25% of the total, the third received 20% of the total, and the fourth ...
-1
votes
5answers
56 views

Find $x$ defined as a continued fraction [duplicate]

I have solved the above using the below method. $$x= 12 + \frac{1}{2+\left(\frac{1}{2}+x\right)}$$ After solving for $x$, I got the answer as $11.7515$ and $-1.41824$ So what is the real value of ...
-1
votes
1answer
59 views

Find the real number $x$ represented by continued fraction $[12;2,2,12,2,2,12,2,2,12\dots]$

I need to fins the real vlaue of x for the continued fraction (Image attached) I have tried partial coefficient method, but didn't get the exact answer. I there any way where we can identify the ...
2
votes
1answer
35 views

Find the sum of the integers in the continued fraction

Find the sum of integers $a,b,c,d,$ and $e$ if $\dfrac{2011}{1990} = a+\dfrac{1}{b+\dfrac{1}{c+\dfrac{1}{d+\dfrac{1}{e}}}}$. I could simplify the big fraction on the RHS, but I don't see how that ...