# Tagged Questions

The Farey sequence of order $n$ is the sequence of all lowest-terms fractions between 0 and 1 whose denominators do not exceed $n$, in increasing order.

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### How does this proof concerning the farey sequence work exacty?

In Hardy and Wright, sixth edition we have theorem 28 which states that if $\dfrac{h}{k}$ and $\dfrac{h'}{k'}$ are two consecutive terms in a Farey sequence: $$kh'-hk'=1$$ and theorem 29 states that, ...
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### Stern-Brocot Tree and sum of coefficients of continued fraction

Suppose we are given a continued fraction $$\frac{p}{q}=a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+\frac{1}{a_{4}+\cdots}}}$$ I am trying to find an expression, possibly asymptotic, for the sum of the $a_i$'...
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### Find a sequence (Sn) that for any a between [0,1] there is a subsequence of Sn that converges to a.

Find a sequence (Sn) that for any a between [0,1] there is a subsequence of Sn that converges to a. I've been stumped for days, my guess is that it is an addition of sequences each expressing its ...
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### Ford circles and possible algebraic mistakes

For each Farey fraction $\frac{a}{b}$ let $C(\frac{a}{b})$ denote the circle in the plane of radius $\frac{1}{2b^2}$ and center $(a/b,\frac{1}{2b^2})$. These circles, called the Ford circles, lie in ...
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### Product of the first n cyclotomic polynomials.

Let $$F_n(\alpha) = \prod_{k = 1}^n \Phi_k(e(\alpha))$$ where $e(\alpha) = e^{2\pi i\alpha}$ It is clear that $F_n(\alpha) = 0$ iff $\alpha = \frac{a}{q}$ for relatively prime $a, q$ s.t. $q \le n$. ...
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### Find $x, y$ such that $\left | \frac ab -\frac xy \right |$ is minimal

Given positive integers $a, b, D$. How to find $x, y \in \mathbb{Z^+}$ such that $$M =\left | \frac ab -\frac xy \right |$$ is minimal and $x + y \le D$? For a solution, I can get it by brute-...
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### Computational Complexity of Finding Adjacent Terms in Farey Sequence

The Farey sequence $\mathcal{F}_n$ is the list of all fractions in increasing order (in lowest terms) from $0$ to $1$, having denominator at most $n$. My question is, given some $a/b\in\mathcal{F}_n$ ...
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Farey sequence of order $n+1$ ($F_{n+1}$) can be construct by adding mediant value (${a+c \over b+d}$) into $F_{n}$, where ${a \over b}$ and ${c \over d}$ are consecutive term in $F_{n}$, and $b+d = ... 1answer 164 views ### Farey sequences for polynomials? Does a notion of Farey sequence (or something equivalent) exist for polynomials over finite fields? 3answers 571 views ### Constructing Farey sequences inductively Objective: I'd like to prove that$F_{n+1}$(the Farey sequence of order$n+1$) is obtained form the Farey sequence$F_n$of order$n$by adding all fractions of the form$\frac{a+c}{b+d}$when$\...
A Farey sequence of order $n$ is a list of the rational numbers between 0 and 1 inclusive whose denominator is less than or equal to $n$. For example \$F_6= \{0,1/6,1/5,1/4,1/3,2/5,1/2,3/5,2/3,3/4,4/...