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).
|||A.A. Bukhshtab, "Number theory" , Moscow (1966) (In Russian)|
|||R.R. Hall, "A note on Farey series" J. London Math. Soc. , 2 (1970) pp. 139–148|
|||G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979)|
Of course, in (*) denotes the Euler function.
Farey series. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Farey_series&oldid=16556