证明哥德巴赫猜想的原文英文并附译后汉语.docx
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证明哥德巴赫猜想的原文英文并附译后汉语.docx
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证明哥德巴赫猜想的原文英文并附译后汉语
此文发表在:
AdvancesinTheoreticalandAppliedMathematics(ATAM),ISSN0793-4554,Vol.7,№4,2012,pp.417-424
ProvingGoldbach’sConjecturebyTwoNumberAxes’PositiveHalfLineswhichReversefromEachOther’sDirections
ZhangTianshu
Nanhaiwestoilcorporation,ChinaoffshorePetroleum,
Zhanjiangcity,Guangdongprovince,P.R.China
Email:
tianshu_zhang507@;
Abstract
Weknowthateverypositiveevennumber2n(n≥3)canexpressinasumwhich3plusanoddnumber2k+1(k≥1)makes.Andthen,foranyoddpoint2k+1(k≥1)atthenumberaxis,if2k+1isanoddprimepoint,ofcourseevennumber3+(2k+1)isequaltothesumwhichoddprimenumber2k+1plusoddprimenumber3makes;If2k+1isanoddcompositepoint,thenlet3
Keywords
Numbertheory,Goldbach’sConjecture,Evennumber,Oddprimenumber,Mathematicalinduction,Twonumberaxes’positivehalflineswhichreversefromeachother’sdirection,OD,PL,CL,andRPL.
BasicConcepts
Goldbach’sconjecturestatesthateveryevennumber2Nisasumoftwoprimenumbers,andeveryoddnumber2N+3isasumofthreeprimenumbers,whereN≥2.
WeshallprovetheGoldbach’sconjecturethereinafterbyoddpointsattwonumberaxes’positivehalflineswhichreversefromeachother’sdirectionsandwhichbeginwithoddpoint3.
Firstwemustunderstandlistedbelowbasicconceptsbeforetheproofoftheconjecture,inordertoapplythemintheproof.
Axiom.Eachandeveryevennumber2n(n≥3)canexpressinasumwhich3pluseachoddnumber2k+1(k≥1)makes.
Definition1.Alinesegmentwhichtakestwooddpointsastwoendsatthenumberaxis’spositivehalflinewhichbeginswithoddpoint3iscalledanodddistance.“OD”isabbreviatedfrom“odddistance”.
TheODbetweenoddpointNandoddpointN+2tiswrittenasODN(N+2t),whereN≥3,andt≥1.
AintegerwhichthelengthofODbetweentwoconsecutiveoddpointsexpressesis2.
AlengthofODbetweenoddpoint3andeachoddpointisunique.
Definition2.AnODbetweenoddpoint3andeachoddprimepointatthenumberaxis’spositivehalflinewhichbeginswithoddpoint3,otherwisecalledaprimelength.“PL”isabbreviatedfrom“primelength”,and“PLS”denotesthepluralofPL.
AnintegerwhicheachlengthoffromsmalltolargePLexpressesbesuccessively2,4,8,10,14,16,20,26...
Definition3.AnODbetweenoddpoint3andeachoddcompositepointatthenumberaxis’spositivehalflinewhichbeginswithoddpoint3,otherwisecalledacompositelength.“CL”isabbreviatedfrom“compositelength”.
AnintegerwhicheachlengthoffromsmalltolargeCLexpressesbesuccessively6,12,18,22,24,30,32...
Weknowthatpositiveintegersandpositiveintegers’pointsatthenumberaxis’spositivehalflineareone-to-onecorrespondence,namelyeachinteger’spointatthenumberaxis’spositivehalflinerepresentsonlyapositiveinteger.Thevalueofapositiveintegerexpressesthelengthofthelinesegmentbetweenpoint0andthepositiveinteger’spointhere.Whenthelinesegmentislonger,itcanexpressinasumofsomeshorterlinesegments;correspondinglythepositiveintegercanalsoexpressinasumofsomesmallerintegers.
Sinceeachandeverylinesegmentbetweentwoconsecutiveinteger’spointsandthelinesegmentbetweenpoint0andpoint1haveanidenticallength,hencewhenusethelengthasaunittomeasurealinesegmentbetweentwointeger’spointsorbetweenpoint0andanyinteger’spoint,thelinesegmenthassomesuchunitlength,thentheintegerwhichthelinesegmentexpressesisexactlysome.
Sincetheprooffortheconjecturerelatemerelytopositiveintegerswhicharenotlessthan3,hencewetakeonlythenumberaxis’spositivehalflinewhichbeginswithoddpoint3.Howeverwestipulatethatanintegerwhicheachinteger’spointrepresentsexpressesyetthelengthofthelinesegmentbetweentheinteger’spointandpoint0.Forexample,anoddprimevaluewhichtherightend’spointofanyPLrepresentsexpressesyetthelengthofthelinesegmentbetweentheoddprimepointandpoint0really.
Wecanprovenextthreetheoremseasieraccordingtoabove-mentionedsomerelationsamonglinesegments,integers’pointsandintegers.
Theorem1.IftheODwhichtakesoddpointFandoddprimepointPSastwoendsisequaltoaPL,thenevennumber3+Fcanexpressinasumoftwooddprimenumbers,whereF>PS.
Proof.OddprimepointPSrepresentsoddprimenumberPS,itexpressesthelengthofthelinesegmentfromoddprimepointPStopoint0.
Thoughlackthelinesegmentfromoddpoint3topoint0atthenumberaxis’spositivehalflinewhichbeginswithoddpoint3,butoddprimepointPSrepresentsyetoddprimePSaccordingtoabove-mentionedstipulation;
LetODPSF=PL3Pb,oddprimepointPbrepresentsoddprimenumberPb,itexpressesthelengthofthelinesegmentfromoddprimepointPbtopoint0.SincePL3Pblackthesegmentfromoddpoint3topoint0,thereforetheintegerwhichthelengthofPL3PbexpressesisevennumberPb-3,namelytheintegerwhichthelengthofODPSFexpressesisevennumberPb-3.
ConsequentlythereisF=PS+(Pb-3),i.e.3+F=oddprimePS+oddprimePb.
Theorem2.Ifevennumber3+Fcanexpressinasunoftwooddprimenumbers,thentheODwhichtakesoddpoint3andoddpointFasendscanexpressinasunoftwoPLS,whereFisanoddnumberwhichismorethan3.
Proof.SupposethetwooddprimenumbersarePbandPd,thentherebe3+F=Pb+Pd.
ItisobviousthattherebeOD3F=PL3Pb+ODPbFatthenumberaxis’spositivehalflinewhichbeginswithoddpoint3.
OddprimepointPbrepresentsoddprimenumberPbaccordingtoabove-mentionedstipulation,thenthelengthoflinesegmentPb(3+F)ispreciselyPd,neverthelessPdexpressesalsothelengthofthelinesegmentfromoddprimepointPdtopoint0.Thereuponcutdown3unitlengthsoflinesegmentPb(3+F),weobtainODPbF;againcutdown3unitlengthsofthelinesegmentfromoddprimepointPdtopoint0,weobtainPL3Pd,thentherebeODPbF=PL3Pd.
ConsequentlytherebeOD3F=PL3Pb+PL3Pd.
Theorem3.IftheODbetweenoddpointFandoddpoint3canexpressinasumoftwoPLS,thenevennumber3+Fcanexpressinasumoftwooddprimenumbers,whereFisanoddnumberwhichismorethan3.
Proof.SupposeoneofthetwoPLSisPL3PS,thentherebeF>PS,andtheODbetweenoddpointFandoddprimepointPSisanotherPL.Consequentlyevennumber3+Fcanexpressinasumoftwooddprimenumbersaccordingtotheorem1.
TheProof
FirstletusgiveordinalnumberKtofromsmalltolargeeachandeveryoddnumber2k+1,wherek≥1,thenfromsmalltolargeeachandeveryevennumberwhichisnotlessthan6isequalto3+(2k+1).
Weshallprovethisconjecturebythemathematicalinductionthereinafter.
1.Whenk=1,2,3and4,wegettingevennumberbeorderly3+(2*1+1)=6=3+3,3+(2*2+1)=8=3+5,3+(2*3+1)=10=3+7and3+(2*4+1)=12=5+7.Thisshowsthateachofthemcanexpressinasumoftwooddprimenumbers.
2.Supposek=m,theevennumberwhich3plus№moddnumbermakes,i.e.3+(2m+1)canexpressinasumoftwooddprimenumbers,wherem≥4.
3.Provethatwhenk=m+1,theevennumberwhich3plus№(m+1)oddnumbermakes,i.e.3+(2m+3)canalsoexpressinasumoftwooddprimenumbers.
Proof.Incase2m+3isanoddprimenumber,naturallyevennumber3+(2m+3)isthesumofoddprimenumber3plusoddprimenumber2m+3makes.
When2m+3isanoddcompositenumber,supposethatthegreatestoddprimenumberwhichislessthan2m+3isPm,thentheODbetweenoddprimepointPmandoddcompositepoint2m+3iseitheraPLoraCL.
WhentheODbetweenoddprimepointPmandoddcompositepoint2m+3isaPL,theevennumber3+(2m+3)canexpressinasumoftwooddprimenumbersaccordingtotheorem1.
IftheODbetweenoddprimepointPmandoddcompositepoint2m+3isaCL,thenweneedtoprovethatOD3(2m+3)canexpressinasumoftwoPLS,onpurposetousethetheorem3.
WhenODPm(2m+3)isaCL,fromsmalltolargeoddcompositenumber2m+3besuccessively95,119,125,145...
Firstletusadopttwonumberaxes’positivehalflineswhichreversefromeachother’sdirectionsandwhichbeginwithoddpoint3.
Atfirst,enableendpoint3ofeitherhalflinetocoincidewithoddpoint2m+1ofanotherhalfline.Please,seefirstillustration:
3572m-32m+1
2m+12m-3753
FirstIllustration
Suchacoincidentlinesegmentcanshortenorelongate,namelyendpoint3ofeitherhalflinecancoincidewithanyoddpointofanotherhalfline.
Thisproofwillperformatsomesuchcoincidentlinesegments.Andforcertainofoddpointsatsuchacoincidentlinesegment,weuseusuallynameswhichmarkattherightwarddirection’shalfline.
WecallPLSwhichbelongbothintheleftwarddirection’shalflineandinacoincidentlinesegment“reversePLS”.“RPLS”isabbreviatedfrom“reversePLS”,and“RPL”denotesthesingularofRPLS.
TheRPLSwherebyoddpoint2k+1attherightwarddirection’shalflineactsasthecommonrightendmostpointarewrittenasRPLS2k+1,andRPL2k+1denotesthesingular,wherek>1.
ThisisknownthateachandeveryODatalinesegmentwhichtakesoddpoint2m+1andoddpoint3astwoendscanexpressinasumofaPLandaRPLaccordingtoprecedi
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