The Primes Contain Arbitrarily Long Arithmetic Progressions

Ben Green, Terence Tao · Annals of Mathematics / arXiv · 2004 · mathematics · 10.4007/annals.2008.167.481 · record greentao2004-primes
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prime numbers arithmetic progression additive combinatorics Szemerédi theorem transference principle positive density positive relative density almost primes k-pseudorandom linear forms condition correlation condition uniform cover a transference principle: embed the primes inside a pseudorandom set of positive relative density, then transfer Szemeredi theorem (which holds for dense sets) to this sparse set to find long arithmetic progressions A TERM IS ADMITTED TO BE LESS ACCURATE THAN AN AVAILABLE ALTERNATIVE AND CHOSEN ANYWAY, FOR A REASON THAT HAS NOTHING TO DO WITH ACCURACY: Quotation withheld The more accurate name is named, the trade is stated, and the choice is made against accuracy in a footnote by the authors themselves A TERM IS DECLARED TO BE USED LOOSELY, IN THE SENTENCE THAT INTRODUCES IT: WE WILL USE THE TERM GOWERS UNIFORM LOOSELY to describe a function which is small in some norm. The looseness is not discovered by a reader; it is announced, so every later use inherits the warning AND A SECOND TERM IS RESERVED FOR ONE CLASS OF OBJECT, WITH AN EXPLICIT WARNING AGAINST CONFLATING THE TWO: THIS SHOULD NOT BE CONFUSED WITH THE TERM PSEUDORANDOM, WHICH WILL BE RESERVED FOR MEASURES. Two neighbouring technical words, one used loosely and one reserved, and the boundary between them is stated rather than left to context A METHOD IS GIVEN A COINED NAME AND THEN DESCRIBED AS BASICALLY ONE RESTRICTION: we get around this using something WHICH WE REFER TO AS THE W -TRICK, which BASICALLY CONSISTS OF RESTRICTING THE PRIMES TO an arithmetic progression. The name is minted for a manoeuvre, and the word `basically` marks the description as a simplification of it AND THE AUTHORS CALL THEIR OWN MANOEUVRE A TRICK AND THEN INSIST IT IS NOT ONE: Quotation withheld The word they chose for it is the word they then have to argue against, which is what a name does when it is picked for how a step looks rather than for what it does A SECTION IS INTRODUCED AS DOING NOTHING BUT FIXING WHAT A TERM MEANS: in this section WE SPECIFY EXACTLY WHAT WE MEAN BY A PSEUDORANDOM MEASURE. The definitional work is given a section of its own and announced as its whole content - and the same word, `pseudorandom`, is the one reserved elsewhere in the paper against confusion with `Gowers uniform` A PROPERTY IS DEFINED BY NAMING A CONJUNCTION OF NUMBERED HYPOTHESES, SO THE TERM IS AN ABBREVIATION FOR A LIST: WE SAY THAT ν IS k-PSEUDORANDOM IF IT SATISFIES the linear forms condition **AND ALSO** the correlation condition. What the word denotes is exactly two earlier definitions holding at once, with parameters fixed - a term whose content is entirely inherited A CROSS-FIELD ANALOGY IS DRAWN AND HEDGED TWICE IN ONE SENTENCE: VERY ROUGHLY SPEAKING, Gowers uniform functions are **ANALOGOUS TO** the notion of “weakly mixing” functions that appear in ergodic theory proofs, whereas Gowers anti-uniform functions are SOMEWHAT ANALOGOUS TO the notion of almost periodic functions. Two fields' vocabularies mapped onto each other, with the strength of the mapping qualified separately for each half A NEW METHOD IS PLACED INSIDE AN OLD TRADITION RATHER THAN AGAINST IT: the idea of using a majorant to study the primes is BY NO MEANS NEW - indeed IN SOME SENSE SIEVE THEORY IS PRECISELY THE STUDY OF SUCH OBJECTS. The novelty claim is withdrawn in the same breath as the method is introduced, and an existing field is named as having been doing it all along A CONDITION IS RESTATED IN ANOTHER FIELD'S LANGUAGE, WITH THE BORROWED WORDS IN SCARE QUOTES: Quotation withheld. The quotation marks concede that the objects are not random variables, and the translation is offered as a point of view rather than as an identity A HOPED-FOR THEORY IS NAMED IN SCARE QUOTES BEFORE IT EXISTS, AS A SUSPICION: particularly exciting is THE SUSPICION that the notion of a k-step nilsystem might be ANALOGOUS TO A KIND OF “higher order Fourier analysis” which **COULD BE USED** to deal with systems of linear equations that cannot be handled by conventional Fourier analysis. A name is coined for something that does not yet exist, and every hedge in the sentence marks that A STRUCTURE IS DEFINED AS A CONJUNCTION OF TWO CHECKABLE PROPERTIES: then we say that Φ is a UNIFORM COVER of B by A if Φ is **SURJECTIVE** and all the fibers HAVE THE SAME CARDINALITY. The term abbreviates exactly those two tests, either of which a reader can apply without knowing what the term is for primes progressions arithmetic density pseudorandom integers theorem measure transference pseudorandom measure mathematical-proof

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relation to other work	`claim_relations`	how this paper stands to another: agrees, contradicts, extends, qualifies
claim type	`claim_type`	the shape of the claim: existence, mechanism, method, critique, prediction, null
concept edge	`concept_edges`	an asserted relation between two concepts, with the sentence asserting it
concept	`concepts`	an idea the paper works with, quoted where the paper uses it
content tag	`content_tags`	a finding, definition, quantity, statistic or variable named in the paper
limitation	`limitations`	what the paper says its evidence cannot support
mechanism	`mechanisms`	how the reported effect is said to come about
method	`method`	how the paper produced its evidence
phenomenon	`phenomenon`	the effect or regularity under study
sense	`senses`	a distinct use of a term, quoted where the paper fixes it
transfer	`transfers`	a method or concept carried in from another field
unrecordable relation	`unhoused_relations`	a relation the paper asserts that this vocabulary cannot express, with the reason

The paper

The full text is not reproduced here. This article is not cleared for redistribution, so the page carries the tag layer only. Its evidence sentences are withheld; the grounding status remains with each tag, and the article itself is at https://doi.org/10.4007/annals.2008.167.481. The reading view exists for the papers whose licence permits it.

Every tag on this page

claim_relations 6 · claim_type 1 · concept_edges 2 · concepts 8 · content_tags 2 · limitations 5 · mechanisms 13 · method 1 · phenomenon 1 · senses 3 · transfers 1 · unhoused_relations 1

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facet	tag	field	conf.	whose words
claim_type	existence Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_type`		the paper's words
phenomenon	arithmetic progressions in the primes Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`phenomenon`		the paper's words
method	mathematical-proof Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: mathematical-proof	`method`		the paper's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: a transference principle: embed the primes inside a pseudorandom set of positive relative density, then transfer Szemeredi theorem (which holds for dense sets) to this sparse set to find long arithmetic progressions	`mechanisms[0]`		the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: A TERM IS ADMITTED TO BE LESS ACCURATE THAN AN AVAILABLE ALTERNATIVE AND CHOSEN ANYWAY, FOR A REASON THAT HAS NOTHING TO DO WITH ACCURACY: Quotation withheld The more accurate name is named, the trade is stated, and the choice is made against accuracy in a footnote by the authors themselves	`mechanisms[1]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: A TERM IS DECLARED TO BE USED LOOSELY, IN THE SENTENCE THAT INTRODUCES IT: WE WILL USE THE TERM GOWERS UNIFORM LOOSELY to describe a function which is small in some norm. The looseness is not discovered by a reader; it is announced, so every later use inherits the warning	`mechanisms[2]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: AND A SECOND TERM IS RESERVED FOR ONE CLASS OF OBJECT, WITH AN EXPLICIT WARNING AGAINST CONFLATING THE TWO: THIS SHOULD NOT BE CONFUSED WITH THE TERM PSEUDORANDOM, WHICH WILL BE RESERVED FOR MEASURES. Two neighbouring technical words, one used loosely and one reserved, and the boundary between them is stated rather than left to context	`mechanisms[3]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: A METHOD IS GIVEN A COINED NAME AND THEN DESCRIBED AS BASICALLY ONE RESTRICTION: we get around this using something WHICH WE REFER TO AS THE W -TRICK, which BASICALLY CONSISTS OF RESTRICTING THE PRIMES TO an arithmetic progression. The name is minted for a manoeuvre, and the word `basically` marks the description as a simplification of it	`mechanisms[4]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: AND THE AUTHORS CALL THEIR OWN MANOEUVRE A TRICK AND THEN INSIST IT IS NOT ONE: Quotation withheld The word they chose for it is the word they then have to argue against, which is what a name does when it is picked for how a step looks rather than for what it does	`mechanisms[5]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: A SECTION IS INTRODUCED AS DOING NOTHING BUT FIXING WHAT A TERM MEANS: in this section WE SPECIFY EXACTLY WHAT WE MEAN BY A PSEUDORANDOM MEASURE. The definitional work is given a section of its own and announced as its whole content - and the same word, `pseudorandom`, is the one reserved elsewhere in the paper against confusion with `Gowers uniform`	`mechanisms[6]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: A PROPERTY IS DEFINED BY NAMING A CONJUNCTION OF NUMBERED HYPOTHESES, SO THE TERM IS AN ABBREVIATION FOR A LIST: WE SAY THAT ν IS k-PSEUDORANDOM IF IT SATISFIES the linear forms condition AND ALSO the correlation condition. What the word denotes is exactly two earlier definitions holding at once, with parameters fixed - a term whose content is entirely inherited	`mechanisms[7]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. about this quote: The span contains “Szemer´edi’s” - the acute accent detached from its vowel by the extractor. Quoted verbatim, because the quote must be the staged file's characters; the paper spells it Szemerédi. also in: A CROSS-FIELD ANALOGY IS DRAWN AND HEDGED TWICE IN ONE SENTENCE: VERY ROUGHLY SPEAKING, Gowers uniform functions are ANALOGOUS TO the notion of “weakly mixing” functions that appear in ergodic theory proofs, whereas Gowers anti-uniform functions are SOMEWHAT ANALOGOUS TO the notion of almost periodic functions. Two fields' vocabularies mapped onto each other, with the strength of the mapping qualified separately for each half	`mechanisms[8]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: A NEW METHOD IS PLACED INSIDE AN OLD TRADITION RATHER THAN AGAINST IT: the idea of using a majorant to study the primes is BY NO MEANS NEW - indeed IN SOME SENSE SIEVE THEORY IS PRECISELY THE STUDY OF SUCH OBJECTS. The novelty claim is withdrawn in the same breath as the method is introduced, and an existing field is named as having been doing it all along	`mechanisms[9]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: A CONDITION IS RESTATED IN ANOTHER FIELD'S LANGUAGE, WITH THE BORROWED WORDS IN SCARE QUOTES: Quotation withheld. The quotation marks concede that the objects are not random variables, and the translation is offered as a point of view rather than as an identity	`mechanisms[10]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. about this quote: The span contains “ergodictheoretical” - the space between “ergodic” and “theoretical” lost by the extractor. Quoted verbatim for the same reason. The guards on this paper test for glyph codes, split capitals, running heads and mathematics, and NOT for a fused word, which has no detector (damage shape 8). also in: A HOPED-FOR THEORY IS NAMED IN SCARE QUOTES BEFORE IT EXISTS, AS A SUSPICION: particularly exciting is THE SUSPICION that the notion of a k-step nilsystem might be ANALOGOUS TO A KIND OF “higher order Fourier analysis” which COULD BE USED to deal with systems of linear equations that cannot be handled by conventional Fourier analysis. A name is coined for something that does not yet exist, and every hedge in the sentence marks that	`mechanisms[11]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: A STRUCTURE IS DEFINED AS A CONJUNCTION OF TWO CHECKABLE PROPERTIES: then we say that Φ is a UNIFORM COVER of B by A if Φ is SURJECTIVE and all the fibers HAVE THE SAME CARDINALITY. The term abbreviates exactly those two tests, either of which a reader can apply without knowing what the term is for	`mechanisms[12]`	0.85	the tagger's words
limitations	The proof yields only a weak lower bound on the count of k-term progressions, far from the conjectured Hardy-Littlewood asymptotic. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`limitations[0]`		the tagger's words
limitations	The methods give no progress on the Hardy-Littlewood prime tuples conjecture, which is why a pseudorandom majorant is used instead of the von Mangoldt function itself. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`limitations[1]`		the tagger's words
limitations	The argument neither requires nor delivers progress on the prime tuple conjecture. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`limitations[2]`		the tagger's words
limitations	A restriction-type property used in the k=3 case does not follow from pseudorandomness, and its generalisation to higher k is unclear. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`limitations[3]`		the tagger's words
limitations	The paper marks the size of its own assumption in a bracket, with an exclamation mark: 'Thus, in this paper, we must assume Szemerédi's theorem. However with this one (rather large!) caveat our paper is self-contained.' The result is a conditional one - long progressions in the primes GIVEN Szemerédi - and the conditionality is carried by a parenthesis rather than by the statement of the theorem. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`limitations[4]`	0.95	the tagger's words
transfers	adapting ergodic-theory and Szemeredi-style combinatorial methods, plus a transference principle, to prove the primes contain arbitrarily long arithmetic progressions Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`transfers[0]`		the tagger's words
concepts	transference principle Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: transference principle	`concepts[4]`	0.95	the paper's words
concepts	positive density Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: positive density	`concepts[5]`	0.95	the paper's words
concepts	positive relative density Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: positive relative density	`concepts[6]`	0.95	the paper's words
concepts	almost primes Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: almost primes	`concepts[7]`	0.95	the paper's words
concepts	k-pseudorandom Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: k-pseudorandom	`concepts[8]`	0.95	the paper's words
concepts	linear forms condition Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: linear forms condition	`concepts[9]`	0.9	the paper's words
concepts	correlation condition Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: correlation condition	`concepts[10]`	0.9	the paper's words
concepts	uniform cover Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: uniform cover	`concepts[11]`	0.95	the paper's words
unhoused_relations	the (linear forms condition AND correlation condition) conjunction → k-pseudorandom Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. IS-A-DEFINING-CONDITION-OF / CONSTITUTES, AND THE VOCABULARY HAS NO TYPE FOR A DEFINIENS. 'We say that ν is…' The two conditions do not sit INSIDE k-pseudorandomness the way a part sits in a whole - they ARE it, jointly and by stipulation. I tried 'part-of' and the gate refused it for want of a containment cue ('within / part of / comprises / consists of / includes / belongs to'), correctly: 'if it satisfies A and also B' is a biconditional definition, not a mereology. 'operationalised-as' fails the same way - its cue list wants 'measured / defined as / quantified / computed as' and this paper writes 'Definition 3.3' and 'we say that'. This is the FOURTH distinct missing relation type met in one evening's tagging, after 'is-an-aspect-of' (bouchaud2008), 'a claim about a usage' (marill1956) and 'prerequisite-for' (botker2018, already live and mistyped). the project, 2026-08-26.	`unhoused_relations[0]`	0.9	the tagger's words
senses	density - TWO SENSES IN ONE ABSTRACT, AND THE PAPER'S CONTRIBUTION IS THE MOVE BETWEEN THEM: density relative to THE INTEGERS, which is what Szemerédi's theorem requires and the primes do not have, versus RELATIVE density inside a pseudorandom set of almost primes, which they do Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`senses[0]`	0.95	the tagger's words
senses	`k-pseudorandom` - A NAME THAT PROMISES RANDOMNESS AND DELIVERS A CONJUNCTION OF TWO TESTS. `Quotation withheld` Nothing in the definition is about randomness; the word is a promise about what the two conditions are FOR. The paper's own remark says the parameter values were chosen for convenience, so even the constants are not load-bearing - a reader who imports `pseudorandom` from another paper imports a different object. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`senses[1]`	0.95	the tagger's words
senses	`uniform cover` - A DEFINITION ADOPTED FOR A PROOF STEP AND SAID SO OUTRIGHT. `To facilitate this we adopt the following definition ... Φ is a uniform cover of B by A if Φ is surjective and all the fibers ... have the same cardinality.` The term is local to this argument and the paper does not claim it for the field. A definition introduced to make an expectation swap legal is a different kind of object from one introduced to name a phenomenon, and both live in the same layer. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`senses[2]`	0.9	the tagger's words
claim_relations	builds-on → Szemeredi theorem Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_relations[0]`		the tagger's words
claim_relations	builds-on → ergodic theory (dynamics) Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_relations[1]`		the tagger's words
claim_relations	builds-on → Szemeredi's theorem [37], [38] Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_relations[2]`		the tagger's words
claim_relations	builds-on → Goldston and Yildirim [17] Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_relations[3]`		the tagger's words
claim_relations	extends → Green [21], the first author's k=3 case Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_relations[4]`		the tagger's words
claim_relations	extends → Gowers [19], the generalised von Neumann theorem for the constant measure Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_relations[5]`		the tagger's words
concept_edges	positive relative density —distinct-from→ positive density Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. differs in: WHAT THE DENSITY IS MEASURED AGAINST. Szemerédi's theorem is about 'any subset of THE INTEGERS of positive density'; the primes are not such a subset. The transference principle is about 'any subset of a sufficiently PSEUDORANDOM SET (or measure) of POSITIVE RELATIVE DENSITY'. Same word, two ambient sets, and the whole proof lives in the gap between them · strength of the relation: 0.9 · note: CHANGING THE DENOMINATOR IS THE PROOF TECHNIQUE, NOT AN ARTEFACT OF ONE. The primes are too sparse for Szemerédi; so '…(a large fraction of) THE PRIMES INSIDE A PSEUDORANDOM SET…', and the transference principle carries the theorem across. The object is not changed and the theorem is not reproved - THE AMBIENT SET AGAINST WHICH DENSITY IS COUNTED IS CHANGED. also in: positive relative density · positive density	`concept_edges[0]`	0.95	the tagger's words
concept_edges	prime numbers —part-of→ almost primes Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. strength of the relation: 0.85 · note: The containment is the load-bearing step: 'one may place (a large fraction of) the primes inside…'. Note A LARGE FRACTION, not all - the edge holds for the fraction the argument needs, which is what makes the relative density positive. Audit 2026-09-28: DOUBTFUL - read against its full span and kept, but the primes lie 'inside' a pseudorandom set such as the almost primes; part-of is loose. also in: prime numbers · almost primes	`concept_edges[1]`	0.85	the tagger's words
content_tags	transference principle Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. about this quote: QUOTE RECOVERED FROM THE PAPER'S OWN CHARACTERS. The recorded quote was LETTERS-ONLY IDENTICAL to this stretch of the staged text but not an exact substring of it - spaces, hyphens, ligature glyphs or mathematical operators differed, so it could never anchor. This is the file's own text between the same two letters: not produced by a rule about the damage, copied out of the file. Round-trip checked (the recovered span is letters-only equal to the original) and length-checked. The text as originally recorded is kept in evidence as extracted. also in: transference principle	`content_tags.definitions[1]`		the tagger's words
content_tags	pseudorandom measure Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: pseudorandom measure	`content_tags.definitions[2]`		the tagger's words

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