Farey sequence F(n): all reduced fractions 0/1 ≤ p/q ≤ 1/1 with q ≤ n, in ascending order. F(5) = 0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1. |F(n)| = 1 + Σφ(k) for k=1..n. Mediant property: if a/b, c/d are neighbors, (a+c)/(b+d) appears in F(b+d).
How to Use
Enter order n:
- F(n): All fractions
- Count: |F(n)|
- Neighbors: Mediant property
Mediant Property
For consecutive fractions a/b, c/d in F(n): |bc−ad|=1 (determinant). The mediant (a+c)/(b+d) appears in F(b+d). This is how fractions 'insert' as n increases.
Applications
- Best rational approximations
- Ford circles (tangent circles)
- Riemann hypothesis connection
- Number theory proofs
Step-by-Step Instructions
- 1Enter order n.
- 2View F(n).
- 3Count fractions.
- 4Check mediants.
- 5Explore neighbors.