The primes contain arbitrarily long arithmetic progressions

• Autores: Benjamin Green, Terence Tao
• Localización: Annals of mathematics, ISSN 0003-486X, Vol. 167, Nº 2, 2008, págs. 481-547
• Idioma: inglés
• DOI: 10.4007/annals.2008.167.481
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• Resumen

  • We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemerédi�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 Szemerédi�s theorem that any subset of a suficiently 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.