# 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).

#### 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)