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1、THEPRIMESCONTAINARBITRARILYLONGARITHMETICPROGRESSIONSBENGREENANDTERENCETAOAbstract.Weprovethattherearearbitrarilylongarithmeticprogressionsofprimes.Therearethreemajoringredients.ThefirstisSzemer´edi’stheorem,whichassertsthatanysubsetoftheintegersofpositivedensitycontainsprogressionsofar
2、bitrarylength.Thesecond,whichisthemainnewingredientofthispaper,isacertaintrans-ferenceprinciple.ThisallowsustodeducefromSzemer´edi’stheoremthatanysubsetofasufficientlypseudorandomset(ormeasure)ofpositiverelativedensitycontainsprogressionsofarbitrarylength.Thethirdingredientisarecentresul
3、tofGoldstonandYıldırım,whichwereproducehere.Usingthis,onemayplace(alargefractionof)theprimesinsideapseudorandomsetof“almostprimes”(ormoreprecisely,apseudorandommeasureconcentratedonalmostprimes)withpositiverelativedensity.1.IntroductionItisawell-knownconjecturethattherearearbitrarilylo
4、ngarithmeticprogressionsofprimenumbers.Theconjectureisbestdescribedas“classical”,ormaybeeven“folklore”.InDickson’sHistoryitisstatedthataround1770LagrangeandWaringinvestigatedhowlargethecommondifferenceofanarithmeticprogressionofLprimesmustbe,anditishardtoimaginethattheydidnotatleastwond
5、erwhethertheirresultsweresharpforallL.Itisnotsurprisingthattheconjectureshouldhavebeenmade,sinceasimpleheuristicbasedontheprimenumbertheoremwouldsuggestthatthereare≫N2/logkNk-tuplesofprimesp1,...,pkinarithmeticprogression,eachpibeingatmostN.HardyandLittlewood[24],intheirfamouspaperof19
6、23,advancedaverygeneralconjecturewhich,asaspecialcase,containsthehypothesisthatthenumberofsuchk-termprogressionsisasymptoticallyCN2/logkNforacertainexplicitnumericalfactorC>0(wedokknotcomeclosetoestablishingthisconjecturehere,obtaininginsteadalowerbound(γ(k)+o(1))N2/logkNforsomeverysma
7、llγ(k)>0).arXiv:math/0404188v6[math.NT]23Sep2007ThefirsttheoreticalprogressontheseconjectureswasmadebyvanderCorput[42](seealso[8])who,in1939,usedVinogradov’smethodofprimenumbersumstoestablishthecasek=3,thatistosaythatthereareinfinitelymanytriplesofprimesinarithmeticprogression.However,
