TAILIEUCHUNG - Đề tài " The primes contain arbitrarily long arithmetic progressions "

We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemer´di’s theorem, which ase serts that any subset of the integers of positive density contains progressions of arbitrary length. The second, which is the main new ingredient of this paper, is a certain transference principle. This allows us to deduce from Szemer´di’s e theorem that any subset of a sufficiently pseudorandom set (or measure) of positive relative density contains progressions of arbitrary length. . | Annals of Mathematics The primes contain arbitrarily long arithmetic progressions By Ben Green and Terence Tao Annals of Mathematics 167 2008 481 547 The primes contain arbitrarily long arithmetic progressions By Ben Green and Terence Tao Abstract We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemeredi s theorem which asserts that any subset of the integers of positive density contains progressions of arbitrary length. The second which is the main new ingredient of this paper is a certain transference principle. This allows us to deduce from Szemeredi s theorem that any subset of a sufficiently pseudorandom set or measure of positive relative density contains progressions of arbitrary length. The third ingredient is a recent result of Goldston and Yildirim which we reproduce here. Using this one may place a large fraction of the primes inside a pseudorandom set of almost primes or more precisely a pseudorandom measure concentrated on almost primes with positive relative density. 1. Introduction It is a well-known conjecture that there are arbitrarily long arithmetic progressions of prime numbers. The conjecture is best described as classical or maybe even folklore . In Dickson s History it is stated that around 1770 Lagrange and Waring investigated how large the common difference of an arithmetic progression of L primes must be and it is hard to imagine that they did not at least wonder whether their results were sharp for all L. It is not surprising that the conjecture should have been made since a simple heuristic based on the prime number theorem would suggest that there are N2 logk N fc-tuples of primes p1 . pk in arithmetic progression each pi being at most N. Hardy and Littlewood 24 in their famous paper of 1923 advanced a very general conjecture which as a special case contains the hypothesis that the number of such fc-term progressions is asymptotically While this work was carried

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