Farey Sequence

The Farey sequence of order n is the increasing sequence consisting of the fractions 0/1 and 1/1 and all the irreducible proper fractions whose numerator and denominator are greater than 0 and do not exceed n. For example, 0/1,1/3, 1/2, 2/3,1/1 is the Farey sequence of order 3.

If a/b and a’/b’ are two consecutive terms in a Farey sequence, then a’b – ab’ = 1. If a/b, a’/b’, and a”/b” are three consecutive terms in a Farey sequence, then a’/b’ = (a + a”)/(b + b”). The uses of Farey sequences include the approximation of irrational numbers by rational numbers and the reduction of binary quadratic forms.

The Farey sequence is named for the British scientist J. Farey, who reported some of its properties without proof in 1816.