# Questions tagged [farey-sequences]

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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### Diophantine approximation and Farey sequences

I am self studying Apostol Dirichlet Series and Modular Functions in Number Theory and could not solve this question given in Chapter 7. Please help. Question is – Let $\Theta$ be an irrational ...
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### Regarding properties of Farey sequences related to lattice points.

I am self studying Apostol's Dirichlet series and Modular Functions in Number Theory and need help in this question. Question is – let $n \ge 1$ and $T_n$ denotes the set of lattice points $(x, y)$ ...
1answer
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### Regarding property of Farey sequence

I am studying apostol Dirichlet series and modular functions book and struck upon this question. Question is - Two reduced fractions $a/ b$ and $c/d$ are called similarly ordered if $(c-a)(d-b) \ge 0$...
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### Determine if a number is the length of a farey sequence

Given a number $j$, is there a way to determine that it is the length of a valid Farey sequence. The length is given by: $$|F_n| = \frac{1}{2}(n+3)n - \sum_{d=2}^n |F_\frac{n}{d}|$$ For example, 2,...
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### Find the Farey triple given the base vertex of any isosceles triangle in Farey diagram.

There is a visualization of the circular Farey diagram where all triangles are isosceles. Observe that any rational $\frac{a}{b}$ distinct from $\frac{0}{1}$ and $\frac{1}{0}$ is always the vertex ...
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### Is there a heuristic reason behind this numerical coincidence?

Write $N(m, n; c)$ for the number of $m\times n$ zero-one matrices where each zero is adjacent to precisely $c$ others, where by "adjacent" I mean up/down/left/right but not diagonally. (Notice that ...
1answer
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### Interesting approximation of distribution of numbers in a Farey sequence

I was investigating the distribution of the numbers in a Farey sequence and found some pattern. It is known that the number of elements in Farey sequence can be found using Euler totient function. So ...
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1answer
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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 $a/b\in\mathcal{F}_n$ ...
2answers
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### What is the sum of the squares of the differences of consecutive element of a Farey Sequence

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/...
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### How to compute next/previous representable rational number?

An (approximate) non-negative rational number representation is a pair of natural numbers each not greater than some fixed limit M (and of course denominator being non-zero). With this condition ...