# Farey series

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of order $n$

The increasing sequence of non-negative irreducible fractions not exceeding 1 with denominators not exceeding $n$. For example, the Farey series of order 5 is the sequence

$$\frac01,\frac15,\frac14,\frac13,\frac25,\frac12,\frac35,\frac23,\frac34,\frac45,\frac11.$$

The following assertions hold.

1) If $a/b$ and $a'/b'$ are two consecutive terms of the Farey series of order $n$, then

$$ba'-ab'=1.$$

2) If $a/b$, $a'/b'$, $a''/b''$ are three consecutive terms of the Farey series of order $n$, then the mediant

$$\frac{a''}{b''}=\frac{a+a'}{b+b'}.$$

3) The number of terms in the Farey series of order $n$ is equal to

$$1+\sum_{x=1}^n\phi(x).\label{*}\tag{*}$$

Farey series were investigated by J. Farey (1816).

How to Cite This Entry:
Farey series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Farey_series&oldid=51169
This article was adapted from an original article by V.I. Nechaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article