0
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0answers
25 views

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 ...
0
votes
0answers
20 views

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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0answers
23 views

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 ...
1
vote
1answer
50 views

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 ...
1
vote
2answers
57 views

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 $1 \leq k \leq n$. $C_o=a_0$. ...
1
vote
1answer
59 views

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 ...
1
vote
1answer
23 views

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 ...
1
vote
2answers
142 views

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 ...
1
vote
1answer
241 views

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 ...
5
votes
1answer
82 views

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 ...
1
vote
1answer
116 views

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.
0
votes
1answer
41 views

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 ...
7
votes
1answer
91 views

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 ...
0
votes
0answers
88 views

Continued fraction of $\gamma+1$ using recursion

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} ...
2
votes
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 ...
4
votes
1answer
132 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 ...
0
votes
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 ...
2
votes
1answer
97 views

A trigonometric identity for special angles

Prove that for a natural number $n$, $$\prod_{k=1}^n \tan\left(\frac{k\pi}{2n+1}\right) = 2^n \prod_{k=1}^n \sin\left(\frac{k\pi}{2n+1}\right)=\sqrt{2n+1}.$$
5
votes
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 ...
5
votes
5answers
296 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+ ...
1
vote
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. ...
1
vote
0answers
140 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 ...
5
votes
3answers
390 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, ...
4
votes
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 ...
18
votes
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 ...
1
vote
2answers
99 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 ...
0
votes
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}\\ ...
3
votes
3answers
450 views

Equivalence of Two Different Irrational Numbers

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 $\beta=\frac{a\alpha+b}{c\alpha+d}$. ...
0
votes
1answer
47 views

Continued Fractions Convergents $C_k-C_{k-3}$

Derive a formula for $C_k-C_{k-3}$ in terms of partial quotients $a_k$ and nominal denominators $q_k$. Recall that $$C_k=\frac{p_k}{q_k}$$ where $$\begin{matrix} \begin{align} p_0&=a_0 & ...
2
votes
2answers
110 views

All Even-Numbered Convergents of a Finite Continued Fraction Are Less Than the Value

Let $x=[a_0;a_1,a_2]$ be shorthand notation for the continued fraction $$x=a_0+\frac{1}{a_1+\frac{1}{a_2}}.$$ Then every $x\in\mathbb{Q}$ can be represented as a finite continued fraction ...
1
vote
1answer
90 views

Calculate the quadratic irrational number given by a certain periodic cont. fraction

Calculate the quadratic irrational number $\alpha$ given by the periodic continued fraction $\alpha = \overline{ [1,2,1] } $. To be honest I am not sure how to tackle this one. I know the algorithm ...
5
votes
2answers
175 views

A continued fraction involving composite numbers

What is the limit of the continued fraction whose partial denominators are the composites?
2
votes
1answer
462 views

Convergent of continued fractions the best rational approximation of a number? [duplicate]

Possible Duplicate: A nicer proof of Lagrange's 'best approximations' law? I was reading through the wikipedia article on continued fractions, and they state, essentially, that ...
1
vote
3answers
210 views

Continued fraction form for rational numbers less than $1$

How could we convert a rational number (less than $1$) to the continued fraction form? This is probably an extension of this question. After reading Bill Dubuque's answer here and here, I got ...
0
votes
3answers
1k views

Extract a Pattern of Iterated continued fractions from convergents

I have been working on an article at https://oeis.org/wiki/Table_of_convergents_constants where I posted a table of "convergents constants" (defined at https://oeis.org/wiki/Convergents_constant) ...
7
votes
1answer
238 views

What causes the convergence of Iterated continued fractions from convergents?

Here is a small discovery I stumbled across a few weeks ago. I hope at least one person will find it interesting enough to help me. The iterated continued fractions from convergents (or convergents ...
5
votes
1answer
240 views

How to find the number of continued fraction from a periodic representation?

Problem Find the number that represented by $[2,2,2 \ldots]$ I know it wasn't difficult, but I was absent the last two classes. So I just want to make sure that I got it right. My attempt was, ...
3
votes
1answer
143 views

How to find continued fraction of the form $a\sqrt{b}$?

For the form $\sqrt{b}$, I could just apply the recursive quadratic formula: $$P_{k+1} = a_kQ_k - P_k$$ $$Q_{k+1} = \dfrac{d - P^2_{k+1}}{Q_k}$$ $$\alpha_k = \dfrac{P_k + \sqrt{d}}{Q_k}$$ ...