整數(shù)矩陣中的一些算術(shù)問(wèn)題.pdf

1、Y1212148四川大學(xué)碩士學(xué)位論文題目整數(shù)矩陣中的一些算術(shù)問(wèn)題作者鄭麗娟完成時(shí)間2007年5月7日培養(yǎng)單位指導(dǎo)教師專業(yè)研究方向授予學(xué)位時(shí)間四川大學(xué)年月日SomearithmeticalproblemsinintegermatricesMajoring：MathematicsGraduatestudent：LijuanZhengSupervisor：ProfessorShaofemgHongAbstract：Letnbeapositiv

2、eintegerands=￡1，，￡”)asetof幾distinctpositiveintegersFor茁∈S，defineGs(z)：=d∈Sldo，dIo強(qiáng)d(d[vlx，y∈s)辛y∈d，z))The幾，lmatrixwhose0，j)eficryi毫thegreatesteoialnondivisor(z，碼)of托andz，iscalledtheGCDmatrixonS，denotedby(S)Similarlywecan

3、definetheLCMmatrix㈣In1992，BourqueandLighshowedthat(s)舊holdsintheringofnnmatricesovertheintegersifSisfactorclosed(iewehaved∈Sforanydlxandforanyz∈s)ThesetSissaidtobegcdclosedif(zI，z，)∈Sforalll≤i，J≤nClearlyafactorclosedseti

4、sgcdclosedbutnotconverselyIn2002，HongshowedthatforanygcdclosedsetswithIsl≤3，wehave(s)l【s】MeanwhileHongfoundthatthereisagcdclosedsetSwithmsxz∈sIas(。)f)=2suchthat(s)十[司Inthispaper，weobtainasufficientconditiononthegcdclosed

5、setSwithmaxz∈slGs(z)J)=2snchthattheLCMmatrixf卅iSdivisiblebytheGCDmatrix(S)ItsolvespartiallyanopenproblemraisedbyHongin2002WeshowalsothatifSisconsistingoftwelativelyprimedivisorchain8，thentheGCDmatrix(S)dividestheLCMmatri