Green–Tao theorem
E286292
The Green–Tao theorem is a landmark result in number theory proving that the sequence of prime numbers contains arbitrarily long arithmetic progressions.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Green–Tao theorem canonical | 4 |
| Green–Tao theorem for primes | 1 |
| Green–Tao theorem on arithmetic progressions in the primes | 1 |
| Green–Tao–Ziegler theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2648104 — 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: Green–Tao theorem Context triple: [Terence Tao, notableWork, Green–Tao theorem]
-
A.
Hardy–Littlewood conjectures
The Hardy–Littlewood conjectures are a collection of influential unproven hypotheses in analytic number theory that generalize the prime number theorem to describe the distribution of prime numbers and prime constellations.
-
B.
Deuring–Heilbronn phenomenon
The Deuring–Heilbronn phenomenon is a result in analytic number theory describing how the presence of an exceptional (Siegel) zero of a Dirichlet L-function forces other zeros away from the real axis, sharpening zero-free regions and affecting the distribution of primes in arithmetic progressions.
-
C.
Fermat polygonal number theorem
The Fermat polygonal number theorem is a result in number theory stating that every positive integer can be expressed as a sum of a fixed number of polygonal numbers of a given order.
-
D.
Hardy–Littlewood circle method
The Hardy–Littlewood circle method is a powerful analytic number theory technique that uses complex analysis and Fourier series to study additive problems such as Waring’s problem and the Goldbach conjecture.
-
E.
Chebotarev density theorem
The Chebotarev density theorem is a fundamental result in algebraic number theory that generalizes the prime number theorem to describe how often primes in a number field have a given Frobenius conjugacy class in its Galois group.
- 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: Green–Tao theorem Target entity description: The Green–Tao theorem is a landmark result in number theory proving that the sequence of prime numbers contains arbitrarily long arithmetic progressions.
-
A.
Hardy–Littlewood conjectures
The Hardy–Littlewood conjectures are a collection of influential unproven hypotheses in analytic number theory that generalize the prime number theorem to describe the distribution of prime numbers and prime constellations.
-
B.
Deuring–Heilbronn phenomenon
The Deuring–Heilbronn phenomenon is a result in analytic number theory describing how the presence of an exceptional (Siegel) zero of a Dirichlet L-function forces other zeros away from the real axis, sharpening zero-free regions and affecting the distribution of primes in arithmetic progressions.
-
C.
Fermat polygonal number theorem
The Fermat polygonal number theorem is a result in number theory stating that every positive integer can be expressed as a sum of a fixed number of polygonal numbers of a given order.
-
D.
Hardy–Littlewood circle method
The Hardy–Littlewood circle method is a powerful analytic number theory technique that uses complex analysis and Fourier series to study additive problems such as Waring’s problem and the Goldbach conjecture.
-
E.
Chebotarev density theorem
The Chebotarev density theorem is a fundamental result in algebraic number theory that generalizes the prime number theorem to describe how often primes in a number field have a given Frobenius conjugacy class in its Galois group.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in number theory ⓘ |
| author |
Ben Green
ⓘ
Terence Tao ⓘ |
| concerns |
arithmetic progressions
ⓘ
prime numbers ⓘ |
| difficulty | high ⓘ |
| field |
additive combinatorics
ⓘ
analytic number theory ⓘ number theory ⓘ |
| firstPreprint | arXiv:math/0404188 ⓘ |
| generalizesFrom | Szemerédi's theorem ⓘ |
| hasConsequence |
existence of 3-term arithmetic progressions of primes
ⓘ
existence of 4-term arithmetic progressions of primes ⓘ existence of k-term arithmetic progressions of primes for all k ⓘ |
| hasGeneralization |
Green–Tao–Ziegler theorem
ⓘ
linked to:
Green–Tao theorem
|
| hasMSCClassification |
11B25
ⓘ
11N13 ⓘ 11P32 ⓘ |
| hasProofType | non-constructive proof ⓘ |
| hasStatus | proven ⓘ |
| implies | for every positive integer k there exists an arithmetic progression of length k consisting entirely of prime numbers ⓘ |
| influenced |
development of higher-order Fourier analysis
ⓘ
research on relative Szemerédi theorems ⓘ subsequent work on polynomial progressions in the primes ⓘ |
| inspiredBy | Szemerédi's theorem ⓘ |
| isLandmarkResultIn |
additive number theory
ⓘ
prime number theory ⓘ |
| mainStatement | the sequence of prime numbers contains arbitrarily long arithmetic progressions ⓘ |
| namedAfter |
Ben Green
ⓘ
Terence Tao ⓘ |
| publicationYear | 2004 ⓘ |
| publishedIn | Annals of Mathematics ⓘ |
| relatedTo |
Erdős–Turán conjecture on arithmetic progressions
ⓘ
linked to:
Erdős–Turán conjecture
Hardy–Littlewood prime tuples conjecture ⓘ
linked to:
Hardy–Littlewood conjectures
Szemerédi's theorem ⓘ |
| shows |
primes are not randomly distributed with respect to long arithmetic progressions
ⓘ
primes contain arbitrarily long linear patterns of the form a+nd ⓘ |
| topic |
patterns in the primes
ⓘ
structure of the primes ⓘ |
| usesMethod |
Hardy–Littlewood circle method ideas
ⓘ
Szemerédi-type regularity methods ⓘ transference principle ⓘ |
| yearAnnounced | 2004 ⓘ |
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: Green–Tao theorem Description of subject: The Green–Tao theorem is a landmark result in number theory proving that the sequence of prime numbers contains arbitrarily long arithmetic progressions.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Green–Tao theorem
linked to: Green–Tao theorem
linked to: Green–Tao theorem