Farey Sequences

Definition. The Farey sequence $F_n$ is the sequence of all rational numbers in $[0,1]$ with denominator at most $n$, arranged in increasing order. Each fraction is written in lowest terms.

$$\begin{aligned} F_1 &= \left\{\frac{0}{1},\,\frac{1}{1}\right\}, & F_2 &= \left\{\frac{0}{1},\,\frac{1}{2},\,\frac{1}{1}\right\}, & F_3 &= \left\{\frac{0}{1},\,\frac{1}{3},\,\frac{1}{2},\,\frac{2}{3},\,\frac{1}{1}\right\}, \\[0.4em] F_4 &= \left\{\frac{0}{1},\,\frac{1}{4},\,\frac{1}{3},\,\frac{1}{2},\,\frac{2}{3},\,\frac{3}{4},\,\frac{1}{1}\right\}, & F_5 &= \left\{\frac{0}{1},\,\frac{1}{5},\,\frac{1}{4},\,\frac{1}{3},\,\frac{2}{5},\,\frac{1}{2},\,\frac{3}{5},\,\frac{2}{3},\,\frac{3}{4},\,\frac{4}{5},\,\frac{1}{1}\right\}. \end{aligned}$$

Remark. The phrase “the fraction $\frac{x}{y}$ is in the Farey sequence $F_n$” means that the fraction is in lowest terms.

Two consecutive terms of $F_n$ are called adjacent (or neighbouring).

Definition. The fraction $\frac{a+c}{b+d}$ is called the mediant of $\frac{a}{b}$ and $\frac{c}{d}$.

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Problems in this set

The nine problems build the classical theory of Farey sequences in order.

1. Same denominator — distinct reduced fractions $\frac{a}{n}$ and $\frac{b}{n}$ cannot be adjacent.
2. Mediant — the mediant of two positive fractions lies strictly between them.
3. Denominator sum — if $\frac{a}{b}$ and $\frac{c}{d}$ are adjacent in $F_n$, then $b+d>n$.
4. Distance lower bound — $|\frac{a}{b}-\frac{c}{d}|\ge\frac{1}{bd}$.
5. Three-part determinant chain — consecutive triples, the determinant condition $bc-ad=1$, and when determinant one forces adjacency.
6. Extremal distances — maximum and minimum gaps between neighbours in $F_n$.
7. Odd positions — a counting argument for denominators $p$, $q$, and $p+q$.

Work through the problems in order; later proofs refer to earlier results.