# Questions tagged [continued-fractions]

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### Is the continued fraction of a constructible number special in some way?

Rationals have finite CF and quadratic have periodic CF. CF in turn can be represented in terms of the modular group SL2(Z), e.g. using the standard generators S(z)=-1/z and T(z)=z+1. On the other ...
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### On four Ramanujan-type "Legendrian" sequences with a 3-term recurrence?

I. Recurrences In a previous post, it was mentioned how Almkvist-Zudilin did a computer search for solutions to the recurrence relation, $$(n+1)^3s_{n+1}=(2n+1)(an^2+an+b)s_n+c\,n^3s_{n-1}$$ within a ...
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### On H. Cohen's four continued fractions for $\zeta(3), \zeta(5), \zeta(7)$?

After 6 years from this old MO post, I finally find in the literature polynomials of deg-$5$ for the continued fraction of $\zeta(5)$. I. Recurrences involving $\zeta(5)$ In Cohen's 2022 paper, ...
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### Finding similar Zudilin-Cohen recurrence relations and cfracs for $\frac{\zeta(4)}{13}$?

I. Two recurrence relations The first one was also discussed in this MO post. We have the similar, \begin{align} (n+1)^5 u_{n+1} &= (2n + 1)(9n^2 + 9n + 3)(15n^2 + 15n + 4)u_n +3n^3(9n^2-1)u_{n-1}\...
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### Can you identify this irrational number?

There is a certain number, say $v$. I can prove it is irrational. That would be more interesting if it is expressible in terms of known values ... zeta functions, Catalan's number, L-functions, etc. ...
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### Is $\mathbb{Q}$ the orbit of a rational function under iteration?

In this previous post I asked for the smallest set of continuous real functions that could generate $\mathbb Q$ by iteration starting from $0$. Surprisingly one continuous function suffices. In the ...
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### Is there a finite set of polynomials generating all rational numbers by iteration?

In this previous post I asked for the smallest set of continuous real functions that could generate $\mathbb{Q}$ by iteration starting from 0. Surprisingly one continuous function suffices. The ...
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### What is the smallest set of real continuous functions generating all rational numbers by iteration?

I recently came across this problem from USAMO 2005: "A calculator is broken so that the only keys that still work are the $\sin$, $\cos$, $\tan$, $\arcsin$, $\arccos$ and $\arctan$ buttons. The ...
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### $\text{SL}_2(\mathbb{Z})$ and continued fractions?

I know the following facts: $\text{SL}_2(\mathbb{Z})$ is generated by everyone's favorite matrices \begin{equation*} S = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \end{equation*} and \...
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### A continued J fraction for $a_n = \frac{1}{(n+1)^2}$?

The following is called a J continued fraction: $$\cfrac{\alpha_0}{1+a_0x-\cfrac{b_1x^2}{1+a_1x-\cfrac{b_2x^2}{1+a_2x-\cdots}}}$$ where the constants are real numbers. Let $\alpha_n= \frac{1}{(n+1)^2}$...
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### Reconstructing elements of $\mathbb Q$ in $\mathbb Z_p$

Can a rational number $a/b$ (with $b$ coprime to a prime number $p$) be recovered efficiently from a $p$-adic expansion of the form $$\frac{a}{b}=\sum_{j=0}^\infty x_jp^j,\ x_j\in\{0,\ldots,p-1\}\ ?$$ ...
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### Continued fraction associated to KdV solitons

Background (may be skipped by those interested only in the basic question and not important associations): “An essay on continued fractions” by Euler (translated by Myra and Bostwick Wyman) contains ...
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### Fundamental Units in $\mathbb{Z}[\sqrt{d}]$ with $d \equiv 1 \mod 4$

It is well known and often cited how the fundamental units for the number ring of $\mathbb{Q}[\sqrt{d}]$ look like. In the case of $d\equiv 2,3 \mod 4$ the number ring is $\mathbb{Z}[\sqrt{d}]$ and in ...