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此文发表在：GeneralMathematics,ISSN1221 5023,Vol.20,Issue2 3,2012,pp.86 100Legendre-Zhang’sConjecture&Gilbreath’sConjectureandProofsThereofZhangTianshuNanhaiwestoilcorporation,ChinaoffshorePetroleum,Zhanjiangcity,Guangdongprovince,P.R.ChinaEmail:tianshu_zhang507@aliyun.com;AbstractIfreducelimitswhichcontainoddprimesbyahalfforLegendre’sconjecture,thenthereisatleastanoddprimewithintheeitherhalflikewise,thisisexactlytheLegendre Zhang’sconjecture.WeshallfirstprovetheLegendre Zhang’sconjecturebymathematicalinductionwiththeaidoftwonumberaxes’positivehalflineswhosedirectionsreversefromeachother.Successively,provetheGilbreath’sconjecturebymathematicalinductionwiththeaidofthegotresult.KeywordsLegendre Zhang’sconjecture,Mathematicalinduction,Numberaxis’spositivehalfline,Oddprimepoints,PLS,RPLSP,SCRPAB,Gilbreath’sconjecture.BasicConceptsTheGilbreath’sconjecturewasfirstsuggestedin1958bytheAmericanmathematicianandamateurmagicianNormanL.Gilbreathfollowingsomedoodlingonanapkin.Hestartedbywritingdownthefirstfewprimes.2,3,5,7,11,13,17,19,23,29,31,…1Undertheseheputtheirdifferences:1,2,2,4,2,4,2,4,6,2,…Undertheseheputtheunsigneddifferenceofthedifferences.1,0,2,2,2,2,2,2,4,…Andhecontinuedthisprocessoffindingiterateddifferences:2,3,5,7,11,13,17,19,23,29,31,…1,2,2,4,2,4,2,4,6,2,…1,0,2,2,2,2,2,2,4,…1,2,0,0,0,0,0,2,…1,2,0,0,0,0,2,…1,2,0,0,0,2,…1,2,0,0,2,…1,2,0,2,…1,2,2,…1,0,…1,…TheGilbreath’sConjectureisthatthenumbersinthefirstcolumnexceptforfirstnumberoffirstrankareallone.TheLegendre’sconjectureisknownasanage oldproblem.Istherealwaysaprimebetweenn2and(n+1)2foreverypositiveintegern?Ifoddnumbersbetweenn2and(n+1)2aredividedintotwopartsbyn(n+1)exceptforn=1,thenLegendre Zhang’sconjectureassertsthatthereisalwaysanoddprimeineithersuchpart.ManifestlytheLegendre Zhang’sconjectureisbetterthantheLegendre’sconjecture.2Please,seethesequenceofintegersinrelationtotheLegendre Zhang’sconjectureaslistedbelow.1(
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