Farey sequence
E656694
The Farey sequence is an ordered list of completely reduced fractions between 0 and 1 with denominators up to a given integer, widely studied in number theory for its connections to fractions, mediants, and modular forms.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical sequence
ⓘ
object of number theory ⓘ |
| adjacencyProperty |
if a/b and c/d are neighbors then bc − ad = 1
ⓘ
if a/b and c/d are neighbors then their mediant (a+c)/(b+d) appears in higher-order sequences between them ⓘ |
| alsoStudiedBy | Cauchy NERFINISHED ⓘ |
| application |
analysis of gaps between fractions
ⓘ
approximation of real numbers by rationals ⓘ study of modular symbols ⓘ visualization via Ford circles ⓘ |
| cardinalityFormula |
|F_n| = 1 + sum of Euler totient function up to n
ⓘ
|F_n| = 1 + sum_{m=1}^n φ(m) ⓘ |
| connection |
related to Farey graph
ⓘ
related to tessellations of the hyperbolic plane ⓘ |
| constraint |
denominator is a positive integer
ⓘ
denominator ≤ n for order n ⓘ |
| constructionRule | start with 0/1 and 1/1 and repeatedly insert mediants with bounded denominators ⓘ |
| definition | for a positive integer n, the Farey sequence of order n is the ascending sequence of completely reduced fractions between 0 and 1 whose denominators do not exceed n NERFINISHED ⓘ |
| domain | rational numbers ⓘ |
| elementType | reduced fractions ⓘ |
| endpointInclusion | includes both 0 and 1 ⓘ |
| field | number theory ⓘ |
| firstTerms |
F_1 = {0/1, 1/1}
ⓘ
F_2 = {0/1, 1/2, 1/1} ⓘ F_3 = {0/1, 1/3, 1/2, 2/3, 1/1} ⓘ |
| historicalNote | properties of the sequence were rigorously proved by Cauchy after Farey ⓘ |
| includesEndpoint |
0/1
ⓘ
1/1 ⓘ |
| interval | [0,1] ⓘ |
| monotonicity | F_n is a subsequence of F_{n+1} ⓘ |
| namedAfter | John Farey Sr. NERFINISHED ⓘ |
| notation | F_n ⓘ |
| orderingType | total order on rationals in [0,1] with bounded denominator ⓘ |
| property |
denominators are less than or equal to the order n
ⓘ
fractions are in lowest terms ⓘ fractions are ordered by increasing value ⓘ |
| relatedTo |
Diophantine approximation
NERFINISHED
ⓘ
Ford circles ⓘ Riemann hypothesis NERFINISHED ⓘ Stern–Brocot tree NERFINISHED ⓘ continued fractions ⓘ distribution of fractions ⓘ mediant operation ⓘ modular forms ⓘ modular group PSL(2,Z) NERFINISHED ⓘ |
| symmetryProperty | sequence is symmetric around 1/2 except for endpoints ⓘ |
| usedIn |
analytic number theory
ⓘ
geometry of the modular surface ⓘ metric number theory ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.