zbMATH — the first resource for mathematics

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)
Full Text: