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1、南京师范大学博士学位论文关于2k+p和k2n+1形式的整数问题研究姓名：孙学戈申请学位级别：博士专业：基础数学指导教师：陈永高20092k+pk2n+11.1849dePolignac321934Romanoff2k+pkp1950vanderCorput2k+pkpErdos2k+pkp2004ChenSun2k+p0.0868kp0.0868Lu0.09322HabsiegerRoblot0.0933Pintz0.09368.1.2k+p2.2k+p02k+p12.ABA2k+pBAPolignac-Romanoff-Corput-ErdosB=PRCE(A).PRCE(A)APRCEPR
2、CEm.m=2rm2me(m)2(modm)e(m)2l1(modm)l.mu2um02|m.(a)l1le(m)(u2lm)=1u+mkk=1PRCE0.0851(e(m)(m)(b)l1le(m)(u2lm)=1u+mkk=1PRCE0.1.PRCE02k+p2.u+mkk=1PRCE0u+mkk=1PRCE0.0851(e(m)(m).2.Yong-GaoChenOnintegersofthesk2nandk2n1J.NumberTheory125(2007)14-25.(1)2npnp(2)p2nnp1.2np0.0283np2.p2n0.0283np3.xx2np9.63104pn1
3、.4427logxn1.4437logx.II4.xxp2n9.63104pn1.4427logxn1.4437logx.3.1960Sierpinskikk2n+1n1979ErdosOdlyzkonk2n+1kErdosOdlyzko(p1)2nkk2n+11.(p1)2n2.(p1)2n0k2n+1m.m=2rm2me(m)2(modm)e(m)2l1(modm)l.ms2s2|m.(a)n01n0e(m)(2n0s+1m)=1s+mkk=1(p1)2n(b)n01n0e(m)(2n0s+1m)=1s+mkk=1(p1)2n0.k2n+1III(p1)2n0.ErdosChen2k+pk
4、2n+1IVAbstractInthisdissertationweinvestigateintegersofthesof2k+pk2n+1andsomerelatedproblems.Themainresultsaresummarizedasfollows.1.In1849dePolignacconjecturedthateveryoddnumberlargerthan3canbewrittenasthesumofanoddprimeandapowerof2.In1934Romanoffprovedthatthereareapositiveproportionnaturalnumberswh
5、ichcanbeexpressedinthe2k+pwherekisapositiveintegerandpisanoddprime.Ontheotherhandin1950vanderCorputprovedthatthecounterexamplesofdePolignacsconjectureasetofpositivelowerdensity.ByemployingacoveringsystemErdosprovedthatthereisaninfinitearithmeticprogressionofpositiveoddnumberseachofwhichhasnorepresen
6、tationofthe2k+pwherekisapositiveintegerandpisanoddprime.In2004ChenandSunshowedthattheproportionofnaturalnumberswhichcanbeexpressedasthe2k+pismorethan0.0868wherekisapositiveintegerandpisanoddprime.Recently0.0868hasbeenimprovedbyLuto0.09322HabsiegerandRoblotto0.0933andPintzto0.09368.Inthispaperweconsi
7、derthefollowingproblems.Problem1.Characterizeallarithmeticprogressionsinwhichthereareapositiveproportionnaturalnumbersthatcanbeexpressedinthe2k+p.Problem2.Cananyarithmeticprogressionofoddnumbersbeobtainedby2k+pfromacongruentcoveringsystemiftheasymptoticdensityofintegersofthe2k+pinthearithmeticprogre
8、ssioniszeroInthispaperweanswerProblems1and2.ForanysetAofintegersletBbethesetofallintegersofAwhichcanberepresentedasthe2k+p.WecallBthePolignac-Romanoff-Corput-ErdossubsetofAandwriteB=PRCE(A).The(lower)(upper)asymptoticdensityofPRCE(A)iscalled(lower)(upper)PRCEdensityofA.Forapositiveintegermletm=2rm2m
9、ande(m)betheorderof2(modm)inotherwordse(m)isthesmallestpositiveintegerlsuchthat2l1(modm).Theorem.Letmubetwointegerswith2uandm02|m.(a)Ifthereexistsanintegerlwith1le(m)and(u2lm)=1thenthelowerPRCEdensityofthearithmeticprogressionu+mkk=1isatleast0.0851(e(m)(m)(b)Ifthereisnoanyintegerlwith1le(m)and(u2lm)
10、=1thenthePRCEdensityofthearithmeticprogressionu+mkk=1iszero.Corollary1.Anarithmeticprogressionofpositiveoddnumbershasthe(lower)PRCEdensityzeroifandonlyifitcanbeobtainedby2k+pfromacongruentcoveringsystem.Corollary2.IftheupperPRCEdensityofanarithmeticprogressionofpositiveoddnumbersisnotzerothenthelowe
11、rPRCEdensityofthearithmeticprogressionisatleast0.0851(e(m)(m).2.RecentlyChenOnintegersofthesk2nandk2n1J.NumberTheory125(2007)14-25.posedthetwofollowingconjectures:(1)thesetofpositiveoddnumberswhichcanbeexpressedinthe2nphasapositivelowerasymptoticdensityinthesetofallpositiveoddnumberswherekisapositiv
12、eintegerandpisanoddprime.(2)thesetofpositiveoddnumberswhichcanbeexpressedinthep2nhasapositivelowerasymptoticdensityinthesetofallpositiveoddnumberswherekisapositiveintegerandpisanoddprime.InthispaperweprovethattheanswerstotheaboveChensconjecturesareaffirmative.Wehavethefollowingmainresults.VITheorem1
13、.Theproportionofnaturalnumberswhichcanbeexpressedinthe2npismorethan0.0283wherenisapositiveintegerandpisapositiveprime.Theorem2.Theproportionofnaturalnumberswhichcanbeexpressedinthep2nismorethan0.0283wherenisapositiveintegerandpisapositiveprime.Theorem3.Forsufficientlylargextheproportionofnaturalnumb
14、erslessthanxwhichhasonlyonerepresentationofthe2npismorethan9.63104wherenisapositiveintegersatisfying1.4427logxn1.4437logxandpisapositiveprime.Theorem4.Forsufficientlylargextheproportionofnaturalnumberslessthanxwhichhasonlyonerepresentationofthep2nismorethan9.63104wherenisapositiveintegersatisfying1.
15、4427logxn1.4437logxandpisapositiveprime.3.Letknbeintegersandpbeaprime.In1960Sierpinskiprovedthatthereareinfinitelymanypositiveoddintegersksuchthatk2n+1iscompositeforallpositiveintegersn.In1979ErdosandOdlyzkoprovedthatthelowerasymptoticdensityofoddintegersksuchthatk2n+1isprimeforsomepositiveintegerni
16、spositive.ErdosandOdlyzkoalsoproposedthequestion:whetheralloddintegerskwhicharenotrepresentableas(p1)2nactuallyfailtobeofthisbecauseofacoveringsystemwherenisapositiveintegerandpisaprime.Inthispaperwecompletelyanswerthequestioninthearithmeticprogression.Weconsiderthefollowingproblems.Problem1.Characterizeallarithmeticprogressionsinwhichthereareapositiveproportionn
