Mathematics Stack Exchange
Mathematics Stack Exchange

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.

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.

63 questions
13
votes
1 answer

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 $F_6= \{0,1/6,1/5,1/4,1/3,2/5,1/2,3/5,2/3,3/4,4/5,5/6,1\}$. The consecutive differences of…
deinst
  • 5,716
12
votes
2 answers

How the recursive structure of Apollonian gaskets can be described in order to be able to reproduce them?

The classical Descartes-Soddy relationship between the signed curvatures $b_k$ ("b" for "bend") of 4 mutually tangent circles (Apollonian configuration): $$\sum_{k=1}^4 b_k^2=\tfrac12 \left(\sum_{k=1}^4 b_k\right)^2\tag{1}$$ allows to obtain the…
Jean Marie
  • 88,997
11
votes
3 answers

Minimal $ab$ for Rational Number $a/b$ in an Interval

Given rational numbers $L$ and $U$, $0
mhp
  • 510
7
votes
0 answers

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. $q \le n$. Now, let $\frac{p_1}{q_1}, \frac{p_2}{q_2}$ be…
Mayank Pandey
  • 2,047
7
votes
1 answer

Is there an elegant Stern Brocot like way to generate all coprime triples?

As one might know, the Stern Brocot tree elegantly and compactly models all rational numbers. I am now left wondering if a process like this tree modeling could be done not only for pairs but for triples and even beyond. To be more specific, what…
Eric sidenis
  • 585
7
votes
1 answer

Is the link between Stern's diatomic sequence and binary subsequences genuine or a coincidence with exceptions?

In a sister stack, Martin Ender raised a question about the following function: Let's define a function $f(N)$ on the integers via the following algorithm. We'll use $N = 38$ as an example: Get the binary representation of $N$:…
Peter Taylor
  • 13,701
6
votes
1 answer

Why do the Gaussian rationals make these patterns?

I was looking for a cool thing to visualize, when I found this picture on Wolfram MathWorld. I obtained the image below by taking all integers $a, b, c, d$ in a given range (between $-15$ and $15$ here). I then calculated $\frac{a+bi}{c+di}$ and…
zenzicubic
  • 453
6
votes
6 answers

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 there is finite number of representable rational…
Adam Badura
  • 163
5
votes
2 answers

Finding good rational approximations

This question was asked in INMO 2005 and here is the problem: Let $(a,b)$ $\in$ $\mathbb{N}$ $s.t$ $\cfrac{43}{197}$ $\lt$ $\cfrac{a}{b}$ $\lt$ $\cfrac{17}{77}$. $\text{Find}$ $\min(b)$. My first approach to this question was to take reciprocal of…
p_square
  • 1,037
5
votes
2 answers

How to prove that construction of Farey sequence by mediant is coverage?

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 $b+d = n+1$. I've already prove that ${a \over b} <…
neizod
  • 194
4
votes
0 answers

Unitary Farey Sequence Matrices

Take the Farey sequence $\mathcal{F}_n$ with values $a_m\in \mathcal{F}_n$ and put them into a vector $$ \vec v_k=\frac1{\sqrt{|\mathcal{F}_n|}}\biggr(\exp(2\pi i k a_m)\biggr)_m $$ The dimension of this vector is $|\mathcal{F}_n| = 1 +…
draks ...
  • 18,755
4
votes
0 answers

Alternative Proof for "Roots of Mertens Function-Farey Sequence-Cosines Relations"

You can write Merten's function as $$ M(n)= \sum_{a\in \mathcal{F}_n} e^{2\pi i a} , $$ where $\mathcal{F}_n$ is the Farey sequence of order $n$. The sum may be split into imaginary and real parts, due to $e^{2\pi i a}=\cos(2\pi a)+i\sin(2\pi…
draks ...
  • 18,755
4
votes
2 answers

When are two neighbouring fractions in Farey sequence are similarly ordered

I am trying exercises from Tom M Apostol and I could not think about this problem in Chapter 5. Problem is - Two reduced fractions $a/b$ and $c/d$ are said to be similarly ordered if $(c-a)\times(d-b)\ge0$. Prove that any two neighboring fractions…
user775699
4
votes
3 answers

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 $\frac{a}{b}<\frac{c}{d}$ are neighbours in $F_n$ and…
sssuuusssuccc
  • 43
3
votes
1 answer

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 $a/b\in\mathcal{F}_n$ for known $n$, what is the most efficient method…
tc1729
  • 3,217
1
2 3 4 5