Questions tagged [continued-fractions]

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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$k$ is a fixed integer $> 2$. Prove that there're infinitely many natural number $n$ s.t. the continued fraction of $\sqrt{n}=[a, \overline{k, b}]$ [closed]

$k$ is a fixed integer greater than $2$. Prove that there exist infinitely many natural number $n$ such that the continued fraction of $\sqrt{n}=[a, \overline{k, b}]$ for some integer $a, b$, and ...
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Rewrite $2+\frac{3}{2+\frac{1}{2+\frac{1}{2+\cdots}}}$ in the form of $ a+b\sqrt{c}$ [closed]

Write the expression $2+\frac{3}{2+\frac{1}{2+\frac{1}{2+\cdots}}}$ in the form of $a+b\sqrt{c}$, where $a,b,c$ are integers Um, I’m not really sure on how to start; can anyone give me a hint please? ...
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102 views

Rationality of Euler-type Generalised Continued Fraction

My question concerns the following: If we have a convergent series $S$ (in some field) equivalent to a Euler-type GCF: $$S = a_0 + a_0 a_1 + a_0 a_1 a_2 + \cdots = \cfrac{a_0}{1-\cfrac{a_1}{1+a_1 - ...
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Determining the second row (and first column of subsequent rows) of a continued-fraction conversion table

I wish to use continued fractions to solve a second-order nonlinear ordinary differential equation, by expressing the associated power series solution ($\phi(r)= \sum a_n r^n $, for $n$ even, as ...
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Finding the optimal $\frac pq$ approximation for a real number given upper limits on $p$ and $q$

When answering a question on Stackoverflow I got curious about how to find the optimal $\frac pq$ approximation for a real number, $r$, where $p$ and $q$ are integers that are limited by the integer ...
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What is meant by “the convergent just preceding $\frac{a}{b}$” in continued fractions?

I’m reading about continued fractions and integer solutions of a linear equation. In Higher Algebra by Hall and Knight, article 347 we have To find the general solution in positive integers of $$...
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Convergence of generalised continued fractions (with positive partial numerators)

Suppose that we have a sequence of positive numbers $(x_n)_{n \in \mathbb N}: x_n>0$ which are not necessarily integers. Q1 Can you give some examples of necessary/sufficient conditions for the ...
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Why does this continued fraction factorization magic trick work?

I discovered this by accident when I first learned about continued fractions. It's hardly foolproof, but maybe half the time, you can instantly factor semiprimes if you have one in continued fraction ...
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How to write $x= \sqrt{n^2+2}$ as a continued fraction and prove that x is irrational?

I have to write $x= \sqrt{n^2+2}$ as a continued fraction, where $n \in N^*$. I tried something like this: $$n< \sqrt{n^2+2}<n+1 \text{ so } [a_{0}]=n\\ x_1= \frac{1}{x-a_0}=\frac{1}{2}(\sqrt{...
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If $N = y^2+1$ then the continued fraction of $\sqrt(N)$ has period 1.

I am reading a book that says that if $N = y^2+1$ then the continued fraction of $\sqrt{N}$ has period 1, i.e $\sqrt{N} = [q_0;\overline{q_1}]$ or similarly $$x=q_0+\cfrac{1}{q_1+\cfrac{1}{q_1+\cfrac{...
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Is the Sum of Coefficients of a Continued Fraction unique?

Let $a$ be a rational number and $$ a = a_0+\frac{1}{a_1+\frac{1}{a_2+\ldots}} \iff a = [a_0,a_1,\ldots,a_i] $$ a corresponding continued fraction. Now, the coefficients of $a$, i.e $a_0,a_1,\ldots$ ...
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Does the set of real numbers with bounded partial quotients have positive measure?

We say a real number $x$ has bounded partial quotients if its continued fraction expansion $[a_0; a_1, a_2 \cdots]$ is bounded by some constant $M=M(x)$. The set $A$ consisting of those numbers whose ...
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151 views

How do we prove this 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 continued fraction holds $$\frac{\displaystyle4\Gamma\left(\...
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How is a continued fraction for $f(x/y)$ derived from the continued fraction for $f(z)$?

Given the continued fraction expansion for $f(z)$, how can we derive a continued fraction for $f(x/y)$ (with partial numerators and denominators limited to integers), where $x$ and $y$ are integers? ...
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What is the simple continued fraction of $τ$ ($2π$)?

I cannot find any information on Google or Wolfram Mathworld to answer this question. I also don't have the skills to calculate it myself so I thought it would be good if someone with this knowledge ...
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Proof that $n$-th root of $n$-th partial convergent's denominator almost always tends to $e^{\frac{\pi^2}{12\ln{2}}}$

While reading Steven Finch's book Mathematical constants (I believe), I once came across and wrote down the following theorem: For almost all real numbers $x$, if $\frac{P_n}{Q_n}$ is the $n^{\text{th}...
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Conversion of definite integral to continued fraction:

Consider integral of the form : $$\int_a^b f(x)dx$$ $f(x)$ is analytic and real valued for real domain. Now fix $a$ and $b$ ( most likely $[0,1]$ and $[0,\infty]$ ) . Can we construct a continued ...
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Why Is $\ln 23+\cfrac{1}{\color{red}{163}+\cfrac{1}{1+\cfrac{1}{\color{red}{41}}}}\approx\pi$

I know from reading that the Heegner number 163 yields the prime generating or Euler Lucky Number 41. Now apparently $\ln23<\pi$ and this can be shown without calculators. I noticed that $$ \pi-\...
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Proving continued fraction for $\frac{I_{\frac{p}{q}}\left(\frac{2}{q}\right)}{I_{1+\frac{p}{q}}\left(\frac{2}{q}\right)}$

I have a question about the following well-known continued fraction:$$p+q+\cfrac1{p+2q+\cfrac1{p+3q+\cfrac1{p+4q+\ddots}}}=\frac{I_{\frac pq}\left(\frac 2q\right)}{I_{1+\frac pq}\left(\frac 2q\right)}\...
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Expressing continued fractions through $e$

The following are some conjectures of mine that I have discovered empirically. The last three conjectures are true if the first four are true, and vice versa. i. $$e=3-\cfrac{1}{4-\cfrac{2}{5-\...
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Continued Fraction using all Perfect Squares

What is known about the infinite continued fraction $$1 + \cfrac{1}{4 + \cfrac{1}{9 + \cfrac{1}{16 + \cdots}}} $$ whose terms include all perfect squares in order? Do we have a closed form ...
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Convergence of a finite continued fraction

Given a sequence of natural numbers $(a_n)_{n\geq 1}$, we know that the finite continued fraction $$[a_1,a_2,\ldots,a_n]:=\cfrac{1}{a_1+\cfrac{1}{\ddots+\cfrac{\ddots}{a_{n-1}+\cfrac{1}{a_n}}}}$$ is ...
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Does this equivalence relation characterize continued fractions?

Consider a pair of integer-valued sequences $(a(n), b(n))$ such that the continued fraction $$a(0) + \cfrac{b(1)}{a(1) + \cfrac{b(2)}{a(2) + \cdots}},$$ exists. Given a strictly-nonzero sequence $c(...
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Raising a continued fraction to a power

Is it possible to give a generalized formula for raising an continued fraction to a power? This is assuming that the continued fraction is in the form of $$a_0+\frac{b_1}{a_1+\frac{b_2}{a_2+\ddots}}$$ ...
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Evaluating $1+\frac{1}{2-\frac{1}{3+\cdots}}$

I was messing around on desmos and accidentally mistyped and wrote "$1+1/(2-1/3+)$" instead of "$1+(1/2)-(1/3)+$". Anyways, I thought I might continue that trend and try to see what that infinite ...
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Not proving the irrationality of $e^e$

How would one go about proving that there does not exist such a product that satisfies $$ e^k=\prod_{n=1}^{k}\frac{a_n}{b_n} $$ for $k$, $a_n$, $b_n\in\mathbb{N}$ and $\lim_{n\to\infty}\frac{a_n}{b_n}=...
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Conjectured continued fraction formula for Catalan's constant

Yesterday I posted this conjecture, but then deleted it thinking it was false. Turns out Python doesn't define $a^b$ as a^b, but rather as ...
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Transcendence of $e(T)$

It is well known that Euler have proved that $e=[2, 1, 4, 1, 1, 6, \ldots, 2n, 1,1, \ldots]$ and that $e$ is a transcendental number by Hermite's evidence. Let us consider the function $e(T)$ ...
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Continued fraction of $π$ using sums of cubes

Recently I came across this identity: $$\pi=3+\cfrac1{6+\cfrac{1^3+2^3}{6+\cfrac{1^3+2^3+3^3+4^3}{6+\cfrac{1^3+2^3+3^3+4^3+5^3+6^3}{6+\ddots}}}},$$ thus $$\pi=3+\cfrac{1}{6+\cfrac{(1\cdot3)^2}{6+\...
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Strictly increasing function from $\mathbb{R}$ into $\mathbb{R}\backslash\mathbb{Q}$

It seems this question is duplicated...Still I would be grateful for any hint to my second question. Does there exist a strictly increasing function $f:\mathbb{R}\rightarrow\mathbb{R}\backslash\...
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Looking for proof: bases without any 'Rotate-Left-Double numbers' is the same sequence as A056469

I'm very active on the codegolf stackexchange, where the goal of codegolf is to complete a certain task/challenge in as few bytes as possible. Although the challenge isn't live yet, someone proposed ...
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Calculating the continued fraction

I need to calculate the following continued fraction: $$1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\ddots}}}$$ While I was searching for something similar, I found this one But I'm still wondering (...
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Let $[a_0; a_1, a_2, \cdots ]$ be a positive irrational number. Show that $a_n > 0$ for all $n \geq 1.$

Question: Let $[a_0; a_1, a_2, ...]$, an infinite simple continued fraction, be a positive irrational number. Show that $a_n > 0$ for all $n\geq 1$ (every integer past $a_0$ in the continued ...
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Verifying a solution to the continued fraction representation of rational numbers

I am reading through these notes and have come across Exercise 3.5.12. I tried to basically brute force the claim with algebra using the two given equations for $\alpha$ and the recurrence relations $...
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1answer
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Simple Continued Fraction of Square Root using Integer Operations Only

I am trying to find out a way to compute the simple continued fraction of a square root. Simple means that the numerators of the expansion is always one. I have an ...
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Comparing $2$ infinite continued fractions

$A = 1 +\dfrac{1}{1 + \frac{1}{1 + \frac{1}{\ddots}}} \\ B = 2 +\dfrac{1}{2 + \frac{1}{2 + \frac{1}{\ddots}}}$ Given the two infinite continued fractions $A$ and $B$ above, which is larger, $2A$ ...
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Verifying a continued fraction related to $\logφ$.

The continued fraction is the following, $${1+\cfrac{1\cdot 2}{3φ+\cfrac{1\cdot 2}{5+\cfrac{3\cdot 4}{7φ+\cfrac{3\cdot 4}{9+\ddots}}}}}=\frac{2}{3\logφ}\tag{1}$$ Where, $$φ=\frac{1+\sqrt{5}}{2}$$ ...
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Show convexity of a continued fraction

Prove convexity of function $$f(x_1, x_2 \dots x_n):=\dfrac{1}{x_1 - \dfrac{1}{x_2 - \dfrac{1}{\ddots - \dfrac{1}{x_n}}}}$$ defined on subset of $\mathbb{R}^n$, where every denominator is strictly ...
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109 views

Continued fractions for $\sin 1$ and $\cos 1$ (radians)

I am aware of Euler's continued fraction: $$a_0+a_0a_1 + a_0a_1a_2 + a_0a_1a_2a_3 +\cdots = \cfrac{a_0}{1-\cfrac{a_1}{1+a_1-\cfrac{a_2}{1+a_2-\cfrac{a_3}{1+a_3-\ddots}}}}$$ (inductive proof) I am ...
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Is there a general way of finding the value of a continued fraction whose terms form a geometric sequence?

So something like $$\cfrac{1}{2+\cfrac{4}{8+\cfrac{16}{32+\cfrac{64}{\ddots}}}}.$$
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A relation between $3-\sqrt 5$ and $\sqrt 5-1$

Let $a=\sqrt{1+\cfrac{\sqrt{5}+1}{2}}$ and let $b=\sqrt{1+\cfrac{\color{red}{\sqrt{5}-1}}{2}}$. Then $$\bigg(a+\frac 1a + b + \frac 1b\bigg)\bigg(a-\frac 1a+b-\cfrac 1b\bigg)=\color{blue}{3-\sqrt 5}.$$...
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my question is about recursion of continued fraction

1 is not equals 2, then whats wrong with the following... 1=2/(3-1) and if we replace (can we or cannot?) 1 on right hand side by 1=2/(3-1) that is 1=2/(3-2/(3-1)) and if we continue replacing 1 on ...
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On expressing $\frac{\pi^n}{4\cdot 3^{n-1}}$ as a continued fraction.

It is a celebrated equation that $$\frac{\pi}{4}=\cfrac{1}{1+\cfrac{1^2}{3+\cfrac{2^2}{5+\cfrac{3^2}{7+\ddots}}}}$$ However, there are two other conjectured equations that I found which, if true (...
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135 views

On conjectured continued fractions and $e$

Playing around with numbers, I conjectured three incredibly interesting things: $$9+\cfrac{1}{18+0\times 12\cfrac{1}{18+1\times 12+\cfrac{1}{18+2\times 12+\cfrac{1}{18+3\times 12+\ddots}}}}=\frac{4e^{...
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Infinite rationals approaching irrational within denominator squared [duplicate]

We have been asked to prove that given an irrational $\alpha$ there exist infinitely many relatively prime pairs $(p,q)$ such that $|p/q-\alpha|<1/q^2$. I have so far found a constructive solution ...
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158 views

Formula for Feigenbaum’s constant?

I have conjectured a formula to calculate the Feigenbaum constant $\delta \approx 4.66920$. $\delta\stackrel{?}{=}$ $$4+\cfrac{1\times 2 -1}{1+\cfrac{2\times 3 -1}{2^2+\cfrac{3\times 4 -1}{1+\cfrac{...
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1answer
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Negative solution for continued fraction 1 + 1/(1+(1/… [duplicate]

I am interested in the continued fraction $$1 + \dfrac{1}{1 + \dfrac{1}{1 + ...}}$$ You can solve this by letting $$y = 1 + \dfrac{1}{1 + \dfrac{1}{1 + ...}}$$ Then since it is infinite $$y = 1 + ...
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To which $\alpha\in\mathbb{R}$ can $\frac{p}{q}$ be a convergent of its (alpha's) continued fractions?

I read that for any to consecutive convergents of a number $\alpha$, at least one of them must be distance at most $\frac{1}{2q^2}$ from $\alpha$. I don't see how this helps me into determining in ...
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Diophantine properties of $\log(3)/\log(2)$ (and other questions)

Let's call $\gamma \in \mathbb R$ $\alpha$-badly approximable if there is some $d>0$ such that for all pairs $(k,l)\in \mathbb Z^2$ such that $ l\neq 0,$ $$ \left| \gamma - \frac kl \right| > \...
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27 views

Continued fraction convergence

Any idea how to show that the following 2 continued fraction representations have the same limit as $n \mapsto \infty$ and $x<0<a<b$? $$ \frac{xa}{b-x+} \frac{x(a+1)}{b+1-x+}\cdots\frac{x(a+...

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