Farey Sequence Generator

About this calculator

The Farey sequence of order n, F_n, lists every fraction a/b in lowest terms with 0 ≤ a ≤ b ≤ n, arranged in increasing order between 0/1 and 1/1 — F_3, for instance, is 0/1, 1/3, 1/2, 2/3, 1/1. Consecutive fractions in the sequence always satisfy the property that their cross-multiplied difference (bc − ad for neighbors a/b and c/d) equals exactly 1.

Number theory students and enthusiasts use Farey sequences to study rational approximation, the Stern-Brocot tree, and the distribution of fractions with bounded denominators, and they show up in problems involving continued fractions, the Riemann hypothesis's Franel-Landau reformulation, and generating all reduced fractions up to a given denominator without duplicates.