Tagged Questions

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3D extension of Euclidean algorithm jigsaw method - help!

Recently I've been learning about how the Euclidean algorithm = jigsaw method (filling a rectangle with squares) = forming continued fractions. And today I'm wondering how a 3D version of the jigsaw ...
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Comparing Generalized Continued Fractions

Gosper lays out a method (under Approximations) for comparing regular (a.k.a simple) continued fractions which have all partial numerators set to 1. Continue comparing terms until they differ, then ...
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a periodic continued fraction $c_{n+2} = 1 - \frac{c_{n+1}}{1 - c_{n}}$

Howvever, start with two numbers $c_0, c_1$ I read in a paper the following sequence has period 5: $$c_{n+1} = 1 - \frac{c_n}{1 - c_{n-1}}$$ Example if we have $c_0=1, c_1=2$ the sequence ...
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Finite Continued Fraction Proof

Let $\alpha = \left[a_0, a_1, a_2,\cdots,a_n\right]$ be a finite continued fraction with $a_0 > 0$ and let $C_i = p_i/q_i$ be the convergent of $\alpha$. If $i\ge 1$, prove that ...
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Continued fraction explanation

This is about simple infinite continued fraction. I don't understand the line '...then $C_0 < x < C_1$'. $C_k$ here refers to $C_k=[a_0;a_1,a_2,...,a_k]$ where $1 \leq k \leq n$. $C_o=a_0$. ...
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Prove that for $n\ge1$, $\xi-\frac{h_n}{k_n}=(-1)^nk_n^{-2}\left(\xi_{n+1}+\langle 0,a_n,a_{n-1},…,a_2,a_1\rangle\right)^{-1}$

Prove that for $n\ge1$, $$\xi-\frac{h_n}{k_n}=(-1)^nk_n^{-2}\left(\xi_{n+1}+\langle 0,a_n,a_{n-1},...,a_2,a_1\rangle\right)^{-1}$$ In addition, show that ...
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Interesting continued fraction problem $|r_i-u_0/u_1|\le\frac1{k_ik_{i+1}}$

Let $u_0/u_1$ be a rational number in lowest terms, and write $u_0/u_1=\langle a_0, a_1,...,a_n\rangle$ in standard continued fraction notation. Show that if $0\le i<n$, then ...
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Infinite continued fraction expansion

How can we find the first six partial quotients of the infinite continued fraction expansion of $\sqrt[3]2$? I know how to do this by expanding when we have a square root function... but I"m not sure ...
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Pell's Equation 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 ...
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Is the infinite continued fraction $[0;0,0,\ldots]=0$?

Wolfram|Alpha states that the infinite continued fraction $$\cfrac{1}{0+\cfrac{1}{0+\cdots}}=0.$$ Assuming $[0;0,0,\ldots]$ exists implies that the continued fraction is $1$, since ...
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Does the continued fractions $3+\frac{1}{5+\frac{1}{7+\cdots}}$ equal $\pi$?

$3+\frac{1}{5+\frac{1}{7+\cdots}}=\pi$ Is it true? If yes, how to show it? Please help. Thank you.
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What is the rate of decay of $\min\{k\xi-\lfloor k\xi\rfloor|k\in\{1,\dots,n\}\}$, for irrational $\xi$?

I wish to establish bounds on the sequence of infima of $\{n\xi\}_{n\in\Bbb N}$, where $\{x\}=x-\lfloor x\rfloor$ is the fractional part function and $\xi$ is irrational. I can prove that ...
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On the Pell-like $Ax^2-By^2 = 1$

This is connected to the post, Mere coincidence? (prime factors). I was looking at NeuroFuzzy's dataset and noticed the line, {{{1, {4, 2}}, {1, 4, 2, 4, 2}, 23762}} It seems this could be ...
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Number $\gamma,$ the Euler-Mascheroni constant, is defined as the value of $$\gamma = \lim_{n\to\infty} \sum_{k=1}^n \frac{1}{k} - \ln(n).$$ We know that $$\lim_{n\to\infty} ... 0answers 92 views Lower bound for the length of continued fraction Define \mathscr L: \mathbb Q \mapsto \mathbb N as the minimal number of terms in the continued fraction of a rational number. Example: the continued fraction of \frac{5}{8} is ... 1answer 123 views Finding an upper bound on a fraction 0<\varepsilon <1. If n_k and a_k are positive integers for which$$n_{k+1}=a_{k+1}n_k+n_{k-1}$$Let L\in\mathbb{N}. If L>a_k \ge 3, what's the smallest upper bound I can place on ... 1answer 87 views Non-Recursive Fundamental Recurrence Formulas Is there a non-recursive version of the fundamental recurrence formulas for continued fractions? I am trying to compute A_{1000}, and it is taking me an extremely long time. By the way, I am ... 1answer 94 views A trigonometric identity for special angles Prove that for a natural number n,$$\prod_{k=1}^n \tan\left(\frac{k\pi}{2n+1}\right) = 2^n \prod_{k=1}^n \sin\left(\frac{k\pi}{2n+1}\right)=\sqrt{2n+1}.$$1answer 103 views For which a is n\lfloor a\rfloor+1\le \lfloor na\rfloor true for all sufficiently large n? Inspired by this question I ask this. For which a is n\lfloor a\rfloor+1\le \lfloor na\rfloor true for all sufficiently large n? The original question concerned a=e, the usual ... 5answers 286 views Continued Fraction [1,1,1,…] If the continued fractional representation of an irrational number \alpha is given by [1,1,1,...], I can compute that \alpha = \frac{1+\sqrt{5}}{2} by solving the equation \alpha = 1+ ... 0answers 60 views Proof of a Continued Fraction Identity using basic CF definition. Two definitions (the first is informal) of continued fraction: This is the basic Continued Fraction algorithm for real numbers. Let \alpha \in \mathbb{R}. If [\alpha]=\alpha, then we are done. ... 0answers 136 views Uniqueness of continued fraction representation of rational numbers I have problems proving the uniqueness of simple continued fraction representation of rational numbers as claimed in http://en.wikipedia.org/wiki/Continued_fraction#Finite_continued_fractions. Let ... 3answers 389 views finding the rational number which the continued fraction [1;1,2,1,1,2,\ldots] represents I'd really love your help with finding the rational number which the continued fraction [1;1,2,1,1,2,\ldots] represents. With the recursion for continued fraction ( p_0=a_0, q_0=1, p_{-1}=1, ... 0answers 112 views Finding a closed expression for a calculated value. Sometimes, when getting some numerical results when investigating number theory sequences with a computer, I find myself suspecting that a decimal value (a) I have found might be a quadratic ... 1answer 1k views What was Ramanujan's solution? The wikipedia entry on Ramanujan contains the following passage: One of his remarkable capabilities was the rapid solution for problems. He was sharing a room with P. C. Mahalanobis who had a ... 2answers 98 views finding the quadratic irratonality of simple continued fractions For instance: find the quadratic irrationality of the simple continued fraction [1;2,3]. I have a handful of these problems to do, so any walk-through of one problem should give me the general idea ... 1answer 30 views If x=[a_0;a_1,a_2,\dots], then |x-C_k|<1/a_k^{\text{}}q_k^2. How can I show that if x=[a_0;a_1,a_2,\dots], then |x-C_k|<1/a_k^{\text{}}q_k^2 using the facts that$$\begin{align} C_k-C_{k-1}&=\frac{(-1)^{k-1}}{q_kq_{k-1}}\text{, and}\\ ...
If $\alpha$ and $\beta$ are two real numbers, we say that $\beta$ is equivalent to $\alpha$ if there are integers $a$, $b$, $c$, and $d$ such that $ad-bc=\pm1$ and $\beta=\frac{a\alpha+b}{c\alpha+d}$. ...