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1、THEPRIMESCONTAINARBITRARILYLONGARITHMETICPROGRESSIONSBENGREENANDTERENCETAOAbstract.Weprovethattherearearbitrarilylongarithmeticprogressionsofprimes.Therearethreemajoringredients.ThefirstisSzemer´edi’stheorem,whichassertsthatanysubsetoftheintegersofpositivedensitycontainsprogres
2、sionsofarbitrarylength.Thesecond,whichisthemainnewingredientofthispaper,isacertaintrans-ferenceprinciple.ThisallowsustodeducefromSzemer´edi’stheoremthatanysubsetofasufficientlypseudorandomset(ormeasure)ofpositiverelativedensitycontainsprogressionsofarbitrarylength.Thethirdingred
3、ientisarecentresultofGoldstonandYıldırım,whichwereproducehere.Usingthis,onemayplace(alargefractionof)theprimesinsideapseudorandomsetof“almostprimes”(ormoreprecisely,apseudorandommeasureconcentratedonalmostprimes)withpositiverelativedensity.1.IntroductionItisawell-knownconjectu
4、rethattherearearbitrarilylongarithmeticprogressionsofprimenumbers.Theconjectureisbestdescribedas“classical”,ormaybeeven“folklore”.InDickson’sHistoryitisstatedthataround1770LagrangeandWaringinvestigatedhowlargethecommondifferenceofanarithmeticprogressionofLprimesmustbe,anditisha
5、rdtoimaginethattheydidnotatleastwonderwhethertheirresultsweresharpforallL.Itisnotsurprisingthattheconjectureshouldhavebeenmade,sinceasimpleheuristicbasedontheprimenumbertheoremwouldsuggestthatthereare≫N2/logkNk-tuplesofprimesp1,...,pkinarithmeticprogression,eachpibeingatmostN.
6、HardyandLittlewood[24],intheirfamouspaperof1923,advancedaverygeneralconjecturewhich,asaspecialcase,containsthehypothesisthatthenumberofsuchk-termprogressionsisasymptoticallyCN2/logkNforacertainexplicitnumericalfactorC>0(wedokknotcomeclosetoestablishingthisconjecturehere,obtain
7、inginsteadalowerbound(γ(k)+o(1))N2/logkNforsomeverysmallγ(k)>0).arXiv:math/0404188v6[math.NT]23Sep2007ThefirsttheoreticalprogressontheseconjectureswasmadebyvanderCorput[42](seealso[8])who,in1939,usedVinogradov’smethodofprimenumbersumstoestablishthecasek=3,thatistosaythattherear
8、einfinitelymanytriplesofprimesinarithmeticprogression.However,