The primes contain arbitrarily long arithmetic progressions. (English) Zbl 1191.11025

This paper needs little introduction: in 2004, the authors proved [arxiv:math/0404188] that the primes contain arbitrarily long arithmetic progressions, a startling result considering that the previous state of the art had been an infinitude of four term arithmetic progressions in which three elements were prime and the fourth a product of at most two primes. As frequently happens when an old problem falls, the solution also precipitated a vast new theory of linear forms in the primes which looks like it will lead to a resolution of the Hardy-Littlewood conjecture for essentially all systems except those describing structures such as twin primes or the Goldbach conjecture. This will be a major achievement, and although the theory has moved on somewhat from this opening of the door, it is still very much worth reading.
The main idea of the paper is a transference principle allowing the authors to transfer results from vanilla structures to pseudo-random versions. They then use some estimates of Goldston and Yıldırım to show that in a suitable sense subsets of the primes behave pseudo-randomly, which allows them to transfer Szemerédi’s theorem to subsets of the primes. A good introduction to this sphere of ideas may be found in the earlier paper of B. Green [Ann. Math. (2) 161, No. 3, 1609–1636 (2005; Zbl 1160.11307)] where this result is proved for three term progressions.


11B25 Arithmetic progressions
11N13 Primes in congruence classes
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
11A41 Primes


Zbl 1160.11307
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