# Farey series

From Encyclopedia of Mathematics

*of order *

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

The following assertions hold.

1) If and are two consecutive terms of the Farey series of order , then

2) If , , are three consecutive terms of the Farey series of order , then

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

(*) |

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

#### References

[1] | A.A. Bukhshtab, "Number theory" , Moscow (1966) (In Russian) |

[2] | R.R. Hall, "A note on Farey series" J. London Math. Soc. , 2 (1970) pp. 139–148 |

[3] | G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) |

#### Comments

Of course, in (*) denotes the Euler function.

**How to Cite This Entry:**

Farey series.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Farey_series&oldid=16556

This article was adapted from an original article by V.I. Nechaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article