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1、IntegersRepresentedasaSumofPrimesandPowersofTwoD.R.Heath-BrownandJ.-C.PuchtaMathematicalInstitute,Oxford1IntroductionItwasshownbyLinnik[10]thatthereisanabsoluteconstantKsuchthateverysufficientlylargeevenintegercanbewrittenasasumoftwoprimesandatmostKpowersoftwo.Thisisaremarkablystrongapproximationtothe
2、GoldbachConjecture.ItgivesusaveryexplicitsetK(x)ofintegersn≤xofcardinalityonlyO((logx)K),suchthateverysufficientlylargeevenintegerN≤xcanbewrittenasN=p+p0+n,withp,p0primeandn∈K(x).Incontrast,ifonetriestoarrangesucharepresentationusinganintervalinplaceofthesetK(x),allknownresultswouldrequireK(x)tohaveca
3、rdinalityatleastapositivepowerofx.LinnikdidnotestablishanexplicitvalueforthenumberKofpowersof2thatwouldbenecessaryinhisresult.However,suchavaluehasbeencomputedbyLiu,LiuandWang[12],whofoundthatK=54000isacceptable.Thisresultwassubsequentlyimproved,firstlybyLi[8]whoobtainedK=25000,thenbyWang[18],whofoun
4、dthatK=2250isacceptable,andfinallybyLi[9]whogavethevalueK=1906.OnecandobetterifoneassumestheGeneralizedRiemannHypothesis,andLiu,LiuandWang[13]showedthatK=200isthenadmissible.Theobjectofthispaperistogivearatherdifferentapproachtothisproblem,whichleadstodramaticallyimprovedboundsonthenumberofpowersof2th
5、atarerequiredforLinnik’stheorem.Theorem1Everysufficientlylargeevenintegerisasumoftwoprimesandexactly13powersof2.Theorem2AssumingtheGeneralizedRiemannHypothesis,everysufficientlylargeevenintegerisasumoftwoprimesandexactly7powersof2.Quiteindependentlyofthiswork,andataboutthesametime,PintzandRuzsahavealsoi
6、nvestigatedLinnik’stheorem.InapaperinpreparationtheyestablishTheorem2withthesamevalueK=7.MoreovertheygiveaversionofTheorem1requiringonly8powersof2.Weonlylearntofthisworkafterthepresentpaperwasessentiallycompleted.Indeed,althoughwewereunawareofit,Pintzhadalreadyin2000announcedthevaluesK=12uncondition
7、ally,andK=10ontheGeneralizedRiemannHypothesis.WehavenotseenPintzandRuzsa’spaper,butweunderstandthatthatourapproachisdifferentinanumberofrespects.1WeshouldalsoreportthatElsholtz,inunpublishedwork,hassho
