Geodesic
The primes contain arbitrarily long arithmetic progressions
Ben Green1 ; Terence Tao2
1 Center for Mathematical Sciences<br/>University of Cambridge<br/>Cambridge CB3 0WB<br/>United Kingdom
2 Department of Mathematics <br/>University of California at Los Angeles<br/>Los Angeles, CA 90095<br/>United States
Annals of mathematics, Tome 167 (2008) no. 2, pp. 481-547.

Voir la notice de l'article provenant de la source Annals of Mathematics website

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.
DOI : 10.4007/annals.2008.167.481

Ben Green 1 ; Terence Tao 2

1 Center for Mathematical Sciences<br/>University of Cambridge<br/>Cambridge CB3 0WB<br/>United Kingdom
2 Department of Mathematics <br/>University of California at Los Angeles<br/>Los Angeles, CA 90095<br/>United States 
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Ben Green; Terence Tao. The primes contain arbitrarily long arithmetic progressions. Annals of mathematics, Tome 167 (2008) no. 2, pp. 481-547. doi : 10.4007/annals.2008.167.481. http://geodesic.mathdoc.fr/articles/10.4007/annals.2008.167.481/

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