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Arithmetic progressions and the primes. (English) Zbl 1109.11043
The author surveys the methods of proof of the theorem: Let \(A\subset \mathbb P\) be a subset of primes with positive relative upper density: \[ \lim\sup_{N\rightarrow\infty}\frac{A\cap[1,N]} {\mathbb P\cap[1,N]}>0, \] and let \(k\geq 3\). Then \(A\) contains infinitely many arithmetic progressions of length \(k\). In particular, the primes contain arbitrarily long arithmetic progressions.

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