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

Chapter3Perfeotsoperfeumbers

&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。

Anaivesearchforabuheileadyoutoguessthattheabundantnumbersaresimplythemultiplesof6。ly，aerthan6oftheform6nisabundaorsof6nmustiogetherwithn，2n，and3n，whiorethantheihisobservatioeoshowthatabujustaboutsixesaswearguethesameerfeumberk。Thefakwilliherwithallthefactorsoftheperfeumberk，eachmultipliedbynsothatthesumofalltheproperfaktoatleast1+nk，andthereforeanymultipleofaperfeumberwillbeabundant。Forexample，28isperfece2×28=56，3×28=84etc。areallabundant。

Amultiplesofperfeumbersahesametoken，multiplesofabundahemselvesabundant。Havihisdisightguessthatallabundantnumbersaresimplymultiplesofperfeumbers。However，youdoolooktoomuchfurthertofiextothisjecture，for70isabundantbutsfactorsareperfedeed，70isthefirstso-calledweirdexactlyforthisreason（thesourceofthislabelisexplainedbelow）。

&hesediscoveries，youmightstillthi，justasthereseemtobeherearenooddabundaher。Inotherwords，ourmodifiedjecturemightbethatalloddnumbersaredefit。Calofthealiquotsumsofthefirstfewhundredoddnumberswouldseemtothistheory，buttheclaimiseventuallydebuesting945，whichhas97sasthesumofitsproperdivisors。esopenasanymultipleofanabundantnumberisabundant，andinparticulartheoddmultiplesof945immediatelysupplyuswithinfinitelymanymoreoddabundantnumbers。

&alittlemoreshreediscoverthister-examplemorequiifweunthioneodderanother。Foraohavealargealiquotsum，itsoffadlargefactorsatthat，whichthemselvesbeihsmallfactors。Wethereforebuildhlargealiquotsumsbymultiplyingsmallprimestogether。Ifwearefobersonly，weshouldlookatthosethatareproductsofthefirstfewoddprimes，whichare3，5，7，etc。Thisruleofthumbwouldsoootest33×5×7=945andtherebydiscovertheabuyamongtheoddnumbersalso。

Itisnotthatunusualtofindthatthesmallestexampleofahpropertiesturnsouttoberatherlarge。Thisisespeciallytrueifthespecifiedpropertiesimplicitlybuildafactorstrutotherequiredhesmallestexampleturnouttobegigantic，althoughnotnecessarilyhardtofihegiveiesihesolution。Anexampleofanumberriddleofthiskindistofiisfivetimesadthreetimesafifthpower。Theansweris

7，119，140，625=5×ll253=3×755

Thereasosolutionisinthebilliohardtosee。Anysolutionnhastohavetheform3r5smforsomepositivepowersrandsaheremaiorsarecollectedtogetherintoasihatisnotdivisibleby30r5。Ifwefirstfothepossiblevaluesofr，weobservethatsiimesacube，theexpobeamultipleof3，aimesa5thpower，thenumberr-1hastobeamultipleof　5。Thesmallestrthatsatisfiesboththeseultaneouslyisr=6。Iheexposhastobeamultipleof5，whiles--1hastobeamutipleof3ahatfitsthebilliss=10。Tomakenassmallaspossible，wetakem=1andson=36×510=3（3×52）5=3×755，sothatimesa5thpoweraimen=5（32×53）3=5×11253，andsonisalso5timesacube。

Aneveremeexampleisthecelebrated，attributedtoArchimedes（287–212BC），thegreatestmathematitiquity。Itwashe19。Thesmallestherdofcattlethatsatisfiesalltheimposedtsintheinal44-linepoemisrepresentedbyahover200，000digits!

Awarningtobegleanedfromallthisisthatdisplaytheirfullvarietyuotherealmse。Forthatreasohattherearehfewerthan300digitsdoesnotinitselfgivegrthatthey‘probably’do。Allthesame，itisthecasethatsomeleadihefieldwouldbeastonishedifournedup。

&urothegeneralbehaviourofaliquotsequeillsimplequestionsthatmaybeputthatnoonesossibilitiesareopentoaliquotsequehesequesaprime，itwillimmediatelytermi1，andotdothisinanyotherway。Ifthisdoeshesequencecouldbedsorepresentasoumber。Thereis，however，aedpossibilitythatisrevealedbygthealiquotsequenceof95:

95=5×19→（1+5+19）=25=5×5→（1+5）=6→6→6→…。

ealthough95isnotitselfasoumber，itsaliquotsequeuallyhitsasoumber（ormoreprethiscase，theperfeumber6）andtheoacycle。

Thereisceivablyonepossibilityremaining，thatbeingthatthealiquotsequenbersaprimenorasoumber，inwhichcasethesequebeanunendingseriesofdifferentnumbers，noneofwhichareeitherprimeorsociable。Isthispossible?Surprisingly，nooneknows。Whatismisthattherearesmallnumberswhosealiquotsequenunknown（andtherebyremaindidatesfsufisequeofthesemysteriousnumbersis276，whosesequens:

276→396→696→1104→1872→3770→3790→

→3050→2716→2772→…

butnoolywhereitendsup。

Itmightwellbethatthereaderwouldliketoexplorealittleontheirown，inwhichcaseIshouldletyouiofhowtocalculatetheso-calledaliquotfun）fromtheprimefactorizationofofallterms（pk+1-1）（p-1），wherepkisthehighestprimepoweroftheprimepthatdividesraitself。Forexample，276=22×3×23andso

asiheseisequencefor276listedabove。

Thereishetypesofwetroducebygivihebeararelationshiptothealiquotfun。Aswehavealreadymentioifa（n）=nandabundantifa（numberisthesumofsomeofitsproperdivisors（thoselessthann），soitfollowsfromthedenitionthatallsemiperfeumbersareeitherperfedant。Forexample，18issemiperfectas18=3+6+9。Anumberiscalledweirdifitisabundantbut，aweirdnumberis70。

&akethevieiingtoomiseousiowiherarbitrarilydenedbersdoesnotofitsownaakethemiing。Weshouldkop。Thatsaid，itisregthattheurategiesusedtotackletheseioremiofwhatEudEulershowedusioperfeumbers。YouwillrecallthatwhatEuclidprovedwasthatifaMersennenumberrimethenanothernumbererfect。Eulerthenprovedverselythatallevenumbersarisefromthisapproathe9thtury，thePersiaiThabitibnQurraintrodubernatripleofnumberswhich，ifallprime，allowedthestruicablepair。Thabit’sstruwasgeherbyEulerihtury，buteventhisenhanulatiooyieldafeairsandtherearemanyamicablepairsthatdohisstru。（Therearenownearly12millionknownpairsofamiumbers。）Iimes，asimilarapprivesastruofweirdnumbersfromumbersshouldtheyhappentobeprime，andthisformulahassuccessfullyfeweirdhmorethanftydigits。

&tershaveservedtofamiliarizethereaderwithfadfactorizatiouralnumbers，orpositiveiheyarealsoknown，illustratedthroughavarietyofexamples。Thiswillstandyouiheupingchapter，inwhichyouwilllearnhowthoseideasareappliedtopraphy，thesceofsecrets。