Green–Tao theorem

E286292

mathematical theorem result in number theory

The Green–Tao theorem is a landmark result in number theory proving that the sequence of prime numbers contains arbitrarily long arithmetic progressions.

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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

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 ⓘ

Referenced by (5)

Full triples — surface form annotated when it differs from this entity's canonical label.

Terence Tao → notableWork → Green–Tao theorem ⓘ

Green–Tao theorem → hasGeneralization → Green–Tao theorem self-linksurface differs ⓘ

this entity surface form: Green–Tao–Ziegler theorem

Ben Green → knownFor → Green–Tao theorem ⓘ

Ben Green → notableWork → Green–Tao theorem ⓘ

this entity surface form: Green–Tao theorem on arithmetic progressions in the primes

Ben Green → notableTheorem → Green–Tao theorem ⓘ