# 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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### 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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### 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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### 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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### 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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### 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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### 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 ...