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.
All labels observed (5)
How this entity was disambiguated
This entity first appeared as the object of triple T7338654 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Farey sequence Context triple: [Farey tessellation, relatedTo, Farey sequence]
-
A.
Farey tessellation
The Farey tessellation is a geometric partition of the hyperbolic plane into ideal triangles whose vertices correspond to rational numbers, closely linked to number theory and modular group actions.
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B.
Sylvester sequence
The Sylvester sequence is an integer sequence defined recursively where each term is one more than the product of all previous terms, yielding rapidly growing, pairwise coprime numbers closely related to Egyptian fraction representations.
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C.
Continued Fractions
Continued Fractions is a classic mathematical monograph by Aleksandr Khinchin that systematically develops the theory and applications of continued fraction expansions in number theory and analysis.
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D.
Ulam sequence
The Ulam sequence is an integer sequence starting with 1 and 2 in which each subsequent term is the smallest integer that can be written uniquely as the sum of two distinct earlier terms.
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E.
Zeckendorf
Zeckendorf is a surname most notably associated with American real estate developer William Zeckendorf and his influential role in mid-20th-century urban development.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Farey sequence Target entity description: 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.
-
A.
Farey tessellation
The Farey tessellation is a geometric partition of the hyperbolic plane into ideal triangles whose vertices correspond to rational numbers, closely linked to number theory and modular group actions.
-
B.
Sylvester sequence
The Sylvester sequence is an integer sequence defined recursively where each term is one more than the product of all previous terms, yielding rapidly growing, pairwise coprime numbers closely related to Egyptian fraction representations.
-
C.
Continued Fractions
Continued Fractions is a classic mathematical monograph by Aleksandr Khinchin that systematically develops the theory and applications of continued fraction expansions in number theory and analysis.
-
D.
Ulam sequence
The Ulam sequence is an integer sequence starting with 1 and 2 in which each subsequent term is the smallest integer that can be written uniquely as the sum of two distinct earlier terms.
-
E.
Zeckendorf
Zeckendorf is a surname most notably associated with American real estate developer William Zeckendorf and his influential role in mid-20th-century urban development.
- F. None of above. chosen
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
ⓘ
linked to:
Augustin-Louis Cauchy
|
| 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
ⓘ
linked to:
Farey sequence
|
| 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. ⓘ |
| 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
ⓘ
Ford circles ⓘ Riemann hypothesis ⓘ Stern–Brocot tree ⓘ continued fractions ⓘ distribution of fractions ⓘ mediant operation ⓘ modular forms ⓘ modular group PSL(2,Z) ⓘ |
| symmetryProperty | sequence is symmetric around 1/2 except for endpoints ⓘ |
| usedIn |
analytic number theory
ⓘ
geometry of the modular surface ⓘ metric number theory ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Farey sequence Description of subject: 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.
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Farey sequence
linked to: Farey sequence
Farey sequence
→
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
ⓘ
linked to: Farey sequence
linked to: Farey sequence