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


The following assertions hold.

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


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


3) The number of terms in the Farey series of order $n$ 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 \eqref{*} $\phi$ denotes the Euler function.

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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