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 (3)
| Label | Occurrences |
|---|---|
| Green–Tao theorem canonical | 3 |
| 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 theorem
self-linksurface differs
ⓘ
surface form:
Green–Tao–Ziegler 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
ⓘ
surface form:
Erdős–Turán conjecture on arithmetic progressions
Hardy–Littlewood conjectures ⓘ
surface form:
Hardy–Littlewood prime tuples conjecture
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
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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 (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Green–Tao–Ziegler theorem
this entity surface form:
Green–Tao theorem on arithmetic progressions in the primes