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时间：2019-03-10
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1、March4,200913:54WSPC/203-IJNT00209InternationalJournalofNumberTheoryVol.5,No.2(2009)281–292cWorldScientificPublishingCompanyABINARYADDITIVEEQUATIONINVOLVINGFRACTIONALPOWERSANGELV.KUMCHEVDepartmentofMathematics,TowsonUniversity7800YorkRoad,Towson,MD21252-0001,US
2、Aakumchev@towson.eduReceived27September2007Accepted26October2007Letcbearealnumberwith13、eywords:Binaryadditiveproblem;fractionalpowers;primenumbers.MathematicsSubjectClassification2000:11P32,11L20,11P051.IntroductionItiswellknownthatthenumberofintegersn≤xthatcanbeexpressedassumsoftwosquaresisOx(logx)−1/2.Ontheotherhand,Deshouillers[2]showedthatwh4、en15、ssolutionsinintegersm1,m2for16、(inthesenseusuallyusedinanalyticnumbertheory)canberepresentedintheform(1.1)withprimesm1,m2.Recently,Balanzario,GaraevandZuazua[1]consideredtheequation[mc]+[pc]=n,(1.2)281March4,200913:54WSPC/203-IJNT00209282A.V.Kumchevwherepisaprimenumberandmisaninteger.Theysho7、wedthatwhen18、emistrivial.Themainpurposeofthepresentnoteistoaddressthisissue.Weestablishthefollowingtheorem.Theorem1.1.Supposethat1
3、eywords:Binaryadditiveproblem;fractionalpowers;primenumbers.MathematicsSubjectClassification2000:11P32,11L20,11P051.IntroductionItiswellknownthatthenumberofintegersn≤xthatcanbeexpressedassumsoftwosquaresisOx(logx)−1/2.Ontheotherhand,Deshouillers[2]showedthatwh
4、en15、ssolutionsinintegersm1,m2for16、(inthesenseusuallyusedinanalyticnumbertheory)canberepresentedintheform(1.1)withprimesm1,m2.Recently,Balanzario,GaraevandZuazua[1]consideredtheequation[mc]+[pc]=n,(1.2)281March4,200913:54WSPC/203-IJNT00209282A.V.Kumchevwherepisaprimenumberandmisaninteger.Theysho7、wedthatwhen18、emistrivial.Themainpurposeofthepresentnoteistoaddressthisissue.Weestablishthefollowingtheorem.Theorem1.1.Supposethat1
5、ssolutionsinintegersm1,m2for16、(inthesenseusuallyusedinanalyticnumbertheory)canberepresentedintheform(1.1)withprimesm1,m2.Recently,Balanzario,GaraevandZuazua[1]consideredtheequation[mc]+[pc]=n,(1.2)281March4,200913:54WSPC/203-IJNT00209282A.V.Kumchevwherepisaprimenumberandmisaninteger.Theysho7、wedthatwhen18、emistrivial.Themainpurposeofthepresentnoteistoaddressthisissue.Weestablishthefollowingtheorem.Theorem1.1.Supposethat1
6、(inthesenseusuallyusedinanalyticnumbertheory)canberepresentedintheform(1.1)withprimesm1,m2.Recently,Balanzario,GaraevandZuazua[1]consideredtheequation[mc]+[pc]=n,(1.2)281March4,200913:54WSPC/203-IJNT00209282A.V.Kumchevwherepisaprimenumberandmisaninteger.Theysho
7、wedthatwhen18、emistrivial.Themainpurposeofthepresentnoteistoaddressthisissue.Weestablishthefollowingtheorem.Theorem1.1.Supposethat1
8、emistrivial.Themainpurposeofthepresentnoteistoaddressthisissue.Weestablishthefollowingtheorem.Theorem1.1.Supposethat1
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