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Chapter3Perfeotsoperfeumbers
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&ioninanumber
Itisofteofindpeculiarpropertiesofsmallcharacterizethem–foriheoisthesumofallthepreviousnumbers，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，whichmayyetbeawaitingdiscovery.
Theeveswerekohaveatightecialsequenes，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）.ThereaderverifythatthefirstfourMersenneprimesdoifourperfeumberslistedabove: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=1ahisfactaboutdivisorsofMersennenumbersalsoprovidesabonusinthatitaffordsusasedwayofseeingthattheremustbeinfinitelymashowsthatthesmallestprimedivisorof2p-1exdsopotbethelargestprime.Sihisappliestoeveryprimep，wecludethatthereisprimeandtheprimesequenforever.
Sincewehavenorodugprimesatwill，thereis，ataime，alargestknownprimeandnowadaystheisalwaysaMersehaionalGIMPSveerMersennePrimeSearch）.Thisisacollaborativeprojectofvolunteers，whiin1996.TheprojectusesthousandsofpersonalputerswiestMersennenumbersforprimalityusingaspeciallydevisedcocktailorithms.Thepion，announAugust2008，is2p-1wherep=43，112，609，althoughanewMersenneprimewasfoundinApril2009withp=42，643，801.Thesenumbershaveabout13milliondigitsandwouldtakethousandsofpagestowritedowninordiation.
&hanumbers
Traditionalnumberloreoftenfoindividualhoughttohavespeotmagical，propertiessuchasthosethatareperfect.Hoairwithasimilartraitis220aamicablepair，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:
&roubleis，oaprime，youarefiheperfeumbersareofcourseexs，eagusalittleloop，airleadstoatwo-cycle:220→284→220→···.leadtogerthantwoarecalledsociable.Theywerealluuryasnoonehadeverfouoday，leadstoathree-cyclehasbeehoughtherearenow120knowncyclesoflengthfour.ThefirstexampleswerefoundbyP.Pouletin1918.Thefirstisafive-cycle:
&’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，whichhas975asthesumofitsproperdivisors.esopenasanymultipleofanabundantnumberisabundant，andinparticulartheoddmultiplesof945immediatelysupplyuswithinfinitelymanymoreoddabundantnumbers.
&alittlemoreshreediscoverthister-examplemorequiifweunthioneodderanother.Foraohavealargealiquotsum，itsoffadlargefactorsatthat，whichthemselvesbeihsmallfactors.Wethereforebuildhlargealiquotsumsbymultiplyingsmallprimestogether.Ifwearefobersonly，weshouldlookatthosethatareproductsofthefirstfewoddprimes，whichare3，5，7，etc.Thisruleofthumbwouldsoootest33×5×7=945andtherebydiscovertheabuyamongtheoddnumbersalso.
Itisnotthatunusualtofindthatthesmallestexampleofahpropertiesturnsouttoberatherlarge.Thisisespeciallytrueifthespecifiedpropertiesimplicitlybuildafactorstrutotherequiredhesmallestexampleturnouttobegigantic，althoughnotnecessarilyhardtofihegiveiesihesolution.Anexampleofanumberriddleofthiskindistofiisfivetimesadthreetimesafifthpower.Theansweris
7，119，140，125=5×11253=3×755.
Thereasosolutionisinthebilliohardtosee.Anysolutionnhastohavetheform3r5smforsomepositivepowersrandsaheremaiorsarecollectedtogetherintoasiisnotdivisibleby3or5.Ifwefirstfothepossiblevaluesofr，weobservethatsiimesacube，theexpobeamultipleof3，aimesa5thpower，thenumberr-1hastobeamultipleof5.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.Itwashe19thtury.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:
butnoolywhereitendsup.
Itmightwellbethatthereaderwouldliketoexplorealittleontheirown，inwhichcaseIshouldletyouiofhowtocalculatetheso-calledaliquotfun）fromtheprimefactorizationofofallterms（pk+1-1）（p-1），wherepkisthehighestprimepoweroftheprimepthatdividesraitself.Forexample，276=22×3×23andsoasiheseisequencefor276listedabove.
Thereishetypesofwetroducebygivihebeararelationshiptothealiquotfun.Aswehavealreadymentioned，ifa（n）=nandabundantifa（numberisthesumofsomeofitsproperdivisors（thoselessthann），soitfollowsfromthedefinitionthatallsemiperfeumbersareeitherperfedant.Forexample，18issemiperfectas18=3+6+9.Anumberiscalledweirdifitisabundantbut，aweirdnumberis70.
&akethevieiingtoomiseousiowiherarbitrarilydefinedbersdoesnotofitsownaakethemiing.Weshouldkop.Thatsaid，itisregthattheurategiesusedtotackletheseioremiofwhatEudEulershowedusioperfeumbers.YouwillrecallthatwhatEuclidprovedwasthatifaMersennenumberrimethenanothernumbererfect.Eulerthenprovedverselythatallevenumbersarisefromthisapproathe9thtury，thePersiaiThabitibnQurraintrodubernatripleofnumberswhich，ifallprime，allowedthestruicablepair.Thabit’sstruwasgeherbyEulerihtury，buteventhisenhanulatiooyieldafeairsandtherearemanyamicablepairsthatdohisstru.（Therearenownearly12millionknownpairsofamiumbers.）Iimes，asimilarapprivesastruofweirdnumbersfromumbersshouldtheyhappentobeprime，andthisformulahassuccessfullyfeweirdhmorethanfiftydigits.
&tershaveservedtofamiliarizethereaderwithfadfactorizatiouralnumbers，orpositiveiheyarealsoknown，illustratedthroughavarietyofexamples.Thiswillstandyouiheupingchapter，inwhichyouwilllearnhowthoseideasareappliedtopraphy，thesceofsecrets.
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