Progressions arithmétiques dans les nombres premiers, d'après B. Green et T. Tao

Abstract

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 Szemerédi'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.