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Chapter 3 Perfect and not so perfect numbers（第1页）

Chapter3Perfeotsoperfeumbers

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&ioninanumber

Itisofteofindpeculiarpropertiesofsmallcharacterizethemforiheoisthesumofallthepreviousnumbers，while2istheonlyevenprime（makiprimeofall）.Thenumber6hasatrulyuyinthatitisboththesumandproductofallofitssmallerfactors:6=1+2+3=1×2×3.

&hagoreansumberlike6perfegthatthehesumofitsproperfactors，asweshallcallthem，whicharethedivisorsstrictlysmallerthantheself.Thiskiionisindeedveryrare.Thefirstfiveperfeumbersare6，28，496，8128，and33，550，336.Alotisknownabouttheeveothisday，noooahebasioftheAowhetherthereareinfinitelymanyofthesespeumbers.Whatismore，noonehasfoundanoddone，herearenone.Abeextremelylargeandthereisalonglistofspecialpropertiesthatsuumbermustpossessisoddperfe.However，alltheserestrishavelegislatedsuumberoutofexistehesespecialpropertiesservetodirectoursearchfortheelusivefirstoddperfeumber，whichmayyetbeawaitiheeveswerekohaveatightecialsequenes，knowntousastheMersenneprimeserMarinMersenne（1588–1648），a17th-turyFrenk.

AMersennenumbermisoheform2p-1，wherepisitselfaprime.Ifyoutake，byle，thefirstfourprimes，2，3，5，afourMersennenumbersareseentobe:3，7，31，and127，whichthereaderquicklyverifyasprime.Ifpwerenotprime，supposep=absay，thenm=2p-1islyher，asitbeverifiedthatiahenumbermhas2a-1asafactor.However，ifpisprimethentheerseenaprime，orsoitseems.

AndEuclidexplained，ba300BceyouhaveaprimeMersehereisaperfeumberthatgoeswithit，thatnumberbeingP=2p-1（2p-1）.ThereaderverifythatthefirstfourMersenneprimesdoihefirstfourperfeumberslistedabove:forexample，usihirdprime5asourseedwegettheperfeumberP=24（25-1）=16×31=496，thethirdperfeumberinthepreviouslist.（ThefactorsofParethepowersof2upto2p-1，togetherwiththesamelistofipliedbytheprime2p-1.Itisnowanexersummingwhatarekricseries（explainediocheckthattheproperfactorsofPdoioP.）

Whatismore，ihturythegreatSwissmathematihardEuler（1707–83）（pronounced‘Oiler’）provedthereverseimplithateveryevehistype.Inthisway，EudEulertogetherestablishedaotheMerseheevenumbers.However，theuralquestioheMersennenumbersprime?Sadlynot，andfailureiscloseathahMersennenumberequals211-1=2，047=23×89.IevenknowifthesequenersenneprimesrunsoutorerapointalltheMerseurnouttobeposite.

TheMersennenumbersarenaturalprimedidatesallthesame，asitbeshoerdivisor，ifos，ofaMerseheveryspe2kp+1.Forexample，whenp=11，bydentofthisresult，weneedonlycheckfordivisioheform22k+1.Thetwoprimefactors，23and89，dtothevaluesk=1aively.ThisfactaboutdivisorsofMersennenumbersalsoprovidesabonusinthatitaffordsusasedwayofseeingthattheremustbeinfinitelymashowsthatthesmallestprimedivisorof2p-1exceedsp，andsopotbethelargestprime.

&hisappliestoeveryprimep，wecludethatthereisprimeandtheprimesequenforever.Sincewehavenorodugprimesatwill，thereis，ataime，alargestknownprimeandnowadaystheisalwaysaMersehaionalGIMPSveerMersennePrimeSearch）.Thisisacollaborativeprojectofvolunteers，whiin1996.TheprojectusesthousandsofpersonalputerswiestMersennenumbersforprimalityusingaspeciallydevisedcocktailorithms.Thepion，announAugust2008，is2p-1wherep=43，112，609，althoughanewMersenneprimewasfoundinApril2009withp=42，643，801.Thesenumbershaveabout13milliondigitsandwouldtakethousandsofpagestowritedowninordiation.

&hanumbers

Traditionalnumberloreoftenfoindividualhoughttohavespeotmagical，propertiessuchasthosethatareperfect.Hoairwithasimilartraitis220and284，thefirstamicablepair，meaningthattheproperfactorsofeachsumstotheother–akiiooacouple.TherekeurFreiPierredeFermat（1601–65）foundotheramicablepairs，suchas17，296and18，416，whileEulerdisly，theybothmissedthesmallpairof1184and1210，foundby16-year-oldNiiniin1866.Weofcobeyondpairsandlookforperfecttriples，quadruples，andsoon.Longercyclesarerarebutdocropup.

Wewithahesumofitsproperdivisors，aheprwhatisknownasthenumber’saliquotsequeisoftenalittledisappointinginthattypicallywegetathatheadsto1quiterapidly，atwhittheprocessstalls.Forexample，evenbeginningromising-lookingnumbersuchas12，theisshort:

12→（1+2+3+4+6）=16→（1+2+4+8）=15→（1+3+5）=9

9→（1+3）=4→（1+2）=3→1.

&roubleis，oaprime，youarefiheperfeumbersareofcourseexs，eagusalittleloop，airleadstoatwo-cycle:220→284→220→….leadtogerthantwoarecalledsociable.Theywerealluuryasnoonehadeverfouoday，leadstoathree-cyclehasbeehoughtherearenow120knowncyclesoflengthfour.ThefirstexampleswerefoundbyP.Pouletin1918.Thefirstisafive-cycle:

12，496→14，288→15，472→14，536→14，264→12，496.

&’ssepleisquitestunning，andtothisdaynoothercyclehasbeenfoundthatatgit:startingwith14，316weobtaih28.Allotherknowncycleshavelehan10.Tothepresentday，therearenotheoremsonamidsoumbersasbeautifulasthoseofEudEuleronumbers.However，modernputingpowerhasledtosomethialrehiskindoftopidthereismorethatbesaid.

Wedivideallhreetypes，defit，perfedabundantagtowhetherthesumoftheirproperdivisorsislessthao，orexceedstheself.Forexample，aswehavealreadyseen，12isanabundantnumber，asare18and24astherespectivesumsoftheirproperdivisorsare21and36.