Farey sequence

Named after British geologist John Farey Sr., whose letter about the sequences was published in the Philosophical Magazine in 1816.

Farey sequence (plural Farey sequences)

1. (number theory) For a given positive integer n, the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size.
   • 2002, Alfred S. Posamentier, Jay Stepelman, Teaching Secondary Mathematics, Merrill, page 403:

     Students should then see the number of fractions ${\displaystyle N}$, in the Farey sequence is equal to ${\displaystyle \phi (2)+\phi (3)+\phi (4)+\dots +\phi (n)}$, where ${\displaystyle \phi (n)}$ is the number of positive integers less than or equal to ${\displaystyle n}$ that are relatively prime to ${\displaystyle n}$.

   • 2007, Jakub Pawlewicz, Order Statistics in the Farey Sequences in Sublinear Time, Lars Arge, Michael Hoffmann, Emo Welzl (editors), Algorithms - ESA 2007: 15th Annual European Symposium, Proceedings, Springer, LNCS 4698, page 218,

     The Farey sequence of order ${\displaystyle n}$ (denoted ${\displaystyle {\mathcal {F}}_{n}}$) is the increasing sequence of all irreducible fractions from interval ${\displaystyle \textstyle [0,1]}$ with denominators less than or equal to ${\displaystyle \textstyle n}$. The Farey sequences have numerous interesting properties and they are well known in the number theory and in the combinatorics.

   • 2009, Michel Weber, Dynamical Systems and Processes, European Mathematical Society, page 549:

     Riemann sums have also important connections with various problems from number theory, among them the Riemann Hypothesis, through their link with Farey sequences.

• The sequence for given ${\displaystyle n}$ may be called the Farey sequence of order ${\displaystyle n}$, and is often denoted ${\displaystyle F_{n}}$.
• The sequences are cumulative: each ${\displaystyle F_{n}}$ is contained in ${\displaystyle F_{n+1}}$. The added elements are those fractions ${\displaystyle \textstyle {\frac {m}{n+1}}}$ for which ${\displaystyle m}$ and ${\displaystyle n+1}$ are coprime.
• The restriction that the fraction be in the range ${\displaystyle [0,1]}$ (i.e., ${\displaystyle 0\leq }$ numerator ${\displaystyle \leq }$ denominator) is sometimes omitted.
  • With the restriction in place, every Farey sequence begins with ${\displaystyle \textstyle {\frac {0}{1}}}$ and ends with ${\displaystyle \textstyle {\frac {1}{1}}}$.

• Farey neighbour
• Farey pair