Farey series

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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).


[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)


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

How to Cite This Entry:
Farey series. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Nechaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article