The DOI matches its Crossref record; 10 of the 10 references it cites were found in a registry or an index.
The primes contain arbitrarily long arithmetic progressions
Open accessAbstract
We prove that there are arbitrarily long arithmetic progressions of primes.There are three major ingredients.The first is Szemerédi's theorem, which asserts that any subset of the integers of positive density contains progressions of arbitrary length.The second, which is the main new ingredient of this paper, is a certain transference principle.This allows us to deduce from Szemerédi's theorem that any subset of a sufficiently pseudorandom set (or measure) of positive relative density contains progressions of arbitrary length.The third ingredient is a recent result of Goldston and Yıldırım, which we reproduce here.Using this, one may place (a large fraction of) the primes inside a pseudorandom set of "almost primes" (or more precisely, a pseudorandom measure concentrated on almost primes) with positive relative density.
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Cites
Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions
Furstenberg, H. (1977) · Journal d Analyse Mathématique · doi:10.1007/bf02813304 · Cited by 804
Roth, K. F. (1953) · Journal of the London Mathematical Society · doi:10.1112/jlms/s1-28.1.104 · Cited by 586
A New Proof of Szemer�di's Theorem for Arithmetic Progressions of Length Four
Gowers, W. T. (1998) · Geometric And Functional Analysis · doi:10.1007/s000390050065 · Cited by 417
Polynomial extensions of van der Waerden’s and Szemerédi’s theorems
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On Triples in Arithmetic Progression
Bourgain, J. (1999) · Geometric And Functional Analysis · doi:10.1007/s000390050105 · Cited by 246
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Self-citationOn sets of integers containing no four elements in arithmetic progression
Szemerédi, E. (1969) · Acta Mathematica Academiae Scientiarum Hungaricae · doi:10.1007/bf01894569 · Cited by 159
A Szemeredi-type regularity lemma in abelian groups, with applications
Green, B. (2003) · Bristol Research (University of Bristol) · doi:10.48550/arxiv.math/0310476 · Cited by 111
Self-citationÜber Summen von Primzahlen und Primzahlquadraten
van der Corput, J. G. (1939) · Mathematische Annalen · doi:10.1007/bf01597346 · Cited by 78
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Boosting and Differential Privacy
Dwork, C., Rothblum, G. N., & Vadhan, S. (2010) · 2010 IEEE 51st Annual Symposium on Foundations of Computer Science · doi:10.1109/focs.2010.12 · Cited by 784
New applications of the polynomial method: The cap set conjecture and beyond
Grochow, J. (2018) · Bulletin of the American Mathematical Society · doi:10.1090/bull/1648 · Cited by 16
“What makes Tao’s version more symmetric is that each of the three variables is put on an equal footing.”
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Metadata
- Authors
- Green, B., & Tao, T.
- Venue
- Annals of Mathematics
- Publisher
- Annals of Mathematics
- Published
- 2008-03-01
- Volume
- 167
- Issue
- 2
- Pages
- 481-547
- ISSN
- 0003-486X (ISSN-L)
- OpenAlex
- W2097673562
- Cited by
- 803
- References
- 10
- FWCI
- 96.00
- Type
- journal article
- Language
- en
- Open access
- bronze
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