GlobalJournalofPureandAppliedMathematics,ISSN07931768,OnlineISSN07939750,Vol.9,№2,2013,pp.143149LetusContinuetoCompletetheProofforthePolignac’sConjectureHereofZhangTianshuNanhaiwestoilcorporation,ChinaoffshorePetroleum,Zhanjiangcity,Guangdongprovince,P.R.ChinaEmail:tianshu_zhang507@aliyun.comAbstractLetusconsideroddnumberswhichshareaprimefactor>>>1asakind,thenthenumberaxis’spositivehalflinewhichbeginswithoddpoint3consistsofinfinitemanyequivalentlinesegmentsonsamepermutationofχχχkinds’oddpointsplusoddpointsamongsttheχχχkinds’oddpoints,whereχχχ≥1.Inthisarticle,weshallprovetheunprovedhalfofthePolignac’sconjecturebymathematicalinductionwiththeaidofsuchequivalentlinesegmentsandkindsofoddpointsthereon.KeywordsNumberaxis’spositivehalfline,Kindsofoddpoints,Mathematicalinduction,TheCoexistingtheorem,BasicConceptsIn1849,Polignacconjecturedthatforeveryevennumber2narethereinfinitelymanypairsofconsecutiveprimeswhichdifferby2n,wheren≥1.Yetthisisanunprovedconjectureuptonow.Happily,Ihadprovenahalfoftheconjecturebytheendoflastyear,andthehalfstatesthatthereareinfinitelymanypairsofconsecutiveoddprimenumbers.Thepaperwithrelationtothehalfispublishedatpp.1726,Number1(2013)oflearnedjournal“AdvancesinTheoreticalandAppliedMathematics”ofResearchIndiaPublications.Whatweneedistosuccessivelyproveanotherhalfoftheconjectureby1now,namelyprovethateverypositiveevennumber2nisadifferenceoftwoconsecutiveoddprimenumbers.Neverthelessapplymoreorlessalikemethodascomparedwiththeprovenhalf,thereforefollowingbasicconceptsexpoundedhavemorerepeats.Ofcourse,thisisinevitableandindispensableforthenewproof.Everyoneknows,eachandeveryoddpointatpositivehalflineofthenumberaxisexpressesapositiveoddnumber.Alsoinfinitemanyadistancebetweentwoconsecutiveoddpointsatthepositivehalflineequaloneanother.Afterwards,thenumberaxis’spositivehalflinewhichbeginswithoddpoint3iscalledthehalflineforshort.Letususesymbol“•”todenoteanoddpointatthebeginning’shalfline