kummer判别法的证明

kummer判别法的证明

Kummer'stheoremisanimportantresultinnumbertheoryandalgebrawhichprovidesacriterionfordeterminingwhetheragivenprimepdividesthebinomialcoefficientC(n,m)forfixedintegersnandm.ThetheoremwasfirstintroducedbyErnstEduardKummerinthemid-19thcentury.Inthisproof,wewillnotuseanylinkstoexternalsourcesbutwillusethenotationcommonlyusedinnumbertheory.

Tobegin,let'sstateKummer'stheoremforbinomialcoefficients:

Kummer'sTheorem:Letpbeaprimeandn,mbenon-negativeintegers.Then,thelargestpowerofpthatdividesC(n,m)isgivenbythenumberofcarriesthatoccurwhenaddingnand(n-m)inbasep.

Now,let'sprovideanoutlineoftheproof:

1.First,weneedtounderstandthenotionofacarryinbasepaddition.Whenaddingtwonumbersinbasep,acarryoccurswheneverthesumoftwodigitsatthesameplacevaluepositionexceedsp-1.Forexample,whenadding12and17inbase10,thereisacarryinthetensplace,resultinginthesum2carry1.Wewillusethisconcepttocountthenumberofcarriesintheadditionofnand(n-m)inbasep.

2.AsthebinomialcoefficientC(n,m)representsthenumberofwaystochooseasubsetofsizemfromasetofsizen,itcanalsobeexpressedcombinatoriallyasC(n,m)=n!/(m!*(n-m)!).Notethattheprimepdividesafactork!forany0<k<pifandonlyifkisamultipleofp.Thisisbecausewhencalculatingthefactorial,theprimepwillappearasafactorineverymultipleofp.ThisobservationwillhelpusdeterminethepowerofpdividingthefactorialsinC(n,m).

3.Werewriten!/(m!*(n-m)!)as(n*(n-1)*...*(n-m+1))/(m*(m-1)*...*1)andanalyzeeachfactorindividually.Let'sintroducethenotationv_p(x)todenotethelargestpowerofpthatdividesx.Wewanttofindv_p(C(n,m)).

4.Weobservethateachfactorinthenumerator,(n*(n-1)*...*(n-m+1)),isdivisiblebypifandonlyifitisamultipleofp,i.e.,ifv_p(n*(n-1)*...*(n-m+1))>0.Similarly,eachfactorinthedenominator,(m*(m-1)*...*1),isdivisiblebypifandonlyifitisamultipleofp,whichoccurswhenv_p(m*(m-1)*...*1)>0.

5.Usingtheconceptofcarries,wecanrewrite(n*(n-1)*...*(n-m+1))asthesumofmterms,whereeachtermrepresentstheproductofmintegerschosenfromn,(n-1),...,(n-m+1).Ifwecalculatethesumoftheseterms(withoutcarryinganydigits)inbasep,weobtainthenumeratorofC(n,m).Similarly,wecanrewrite(m*(m-1)*...*1)asthesumofmtermsinbasep,representingthedenominatorofC(n,m).

6.ThenumberofcarriesthatoccurwhenaddingthenumeratoranddenominatorinbasepisdirectlyrelatedtothepowerofpthatdividesC(n,m).Specifically,thelargestpowerofpthatdividesC(n,m)isequaltothenumberofcarriesthatoccurwhenaddingthenumeratoranddenominatorinbasep.

7.Bycountingthenumberofcarries,wecandeterminewhetherpdividesC(n,m)ornot.Ifthenumberofcarriesisgreaterthanzero,thenpdividesC(n,m).Otherwise,pdoesnotdivideC(n,m).

Inconclusion,wehaveoutlinedtheproofofKummer'stheoremforbinomialcoefficients.Byanalyzingthenumberofcarriesthatoccurwhenaddingthenumeratoranddenomin