Mathematics – Dynamical Systems
Scientific paper
2006-09-28
Mathematics
Dynamical Systems
Notes pour l'expos\'{e} No 944 (2004-2005) au S\'{e}minaire Bourbaki
Scientific paper
B. Green and T. Tao have recently proved that 'the set of primes contains arbitrary long arithmetic progressions', answering to an old question with a remarkably simple formulation. The proof does not use any "transcendental" method and any of the deep theorems of analytic number theory. It is written in a 'spirit' close to ergodic theory and in particular of Furstenberg's proof of Szemer\'{e}di's Theorem, but it does not use any result of this theory. Therefore the method can be considered as elementary, which does not mean easy. We entend here to present the mains ideas of this proof.
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