Farey sequence

Definition

Let $n$ be a positive integer. The Farey sequence $F_n$ of order $n$ is the increasing sequence of all fractions $\frac{a}{b}$ with $0 \le a \le b \le n$ and $\gcd(a, b) = 1$ . Each fraction is in lowest terms, and $F_n$ starts with $\frac{0}{1}$ and ends with $\frac{1}{1}$ .

Remarks

For example, $F_5 = \left( \frac{0}{1}, \frac{1}{5}, \frac{1}{4}, \frac{1}{3}, \frac{2}{5}, \frac{1}{2}, \frac{3}{5}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{1}{1} \right)$ . If $\frac{a}{b} < \frac{c}{d}$ are neighbours in $F_n$ , then $bc - ad = 1$ . The length of $F_n$ is $1 + \sum_{k=1}^{n} \varphi(k)$ , with $\varphi$ Euler's totient function. The sequences are named after John Farey, whose letter about them appeared in 1816.

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Divisibility definition
Greatest common divisor definition
Farey sequence definition

Tag	`0PN4QJE969`
URI	`m2://0PN4QJE969`
Kind	definition
Writing	W1 stated
MSC	11B57
Defines	Farey sequence
Uses	Greatest common divisor

Farey sequence

Definition

Remarks

Reading path

Further reading

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