Challenging Perfect Number.docx

Challenging Perfect Number.docx

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ChallengingPerfectNumber

ChallengingPerfectNumber

　　PassageandpicturesbyHuangHongtaowhoisthepost-doctoratefromUniversitàdiRomaLaSapienzaanddoesresearchesoncomputationalmathematics

　　OnJanuary7thisyear，amathematiciannamedCooperfromtheUniversityofCentralMissouriCooper，throughtheprojectcalledInternetMersennePrimeSearch（GIMPS），foundthelargestknownperfectnumber2^74，207，280（2^74207281-1）whichmeansthatafter74，207，281thpowerof2minus1andthenmultipliedby2tothepowerof74，207，280（NOTE：

^isthecomputerlanguagepowersymbol）.Thisnumberisthe49thperfectnumberhumandiscoveredduringthepast2500yearsandithas44，677，235digits；ifitwereprintedintheordinarysize，thelengthwillbe200km！

Readerscanimaginemoreaboutit......Eventhewell-knownAustralianmathematicianParkeragreesthatthisisatremendousscientificachievement.Well，willitbeanexplorativestorysimilartotheclassicmovieGoodWillHunting？

Whatkindofcharmonearthdoestheperfectnumberboast，attractingsomanymathematiciansonafteranothertodevotethemselves？

　　WhatisPerfectNumber？

　　PerfectnumberhassomeothernamesinChinesewhosedefinitionisthatthesumofallfactorsexceptitselfequalsthenumberitself.Thisdefinitionseemstobeconfusing，solet’scitetwoexamples，thetwosmallestperfectnumbersare6whoseallfactorsare1，2，3，6and28whoseallfactorsare1，2，4，7，14，28），bothofwhicharethesumoftheirallfactorsinadditiontoitself：

6=1+2+3and28+1=2+4+7+14.Duetosomecommonmagicalproperties，scientistsentitlethemwithawonderfulnamecalledtheperfectnumber.

　　Forthousandyears，theintriguingpropertiesandincomparablecharmoftheperfectnumberhasattractedmanymathematiciansandcountlessmathematicamateurstoexploreit.In17thcentury，FrenchmathematicianandphilosopherDescartespubliclypredicted，“Thenumberoftheperfectnumberwillnotbelarge，justlikeit’snoteasytofindaperfectperson."Afteralongperiodoftime，peopleonlyfind49perfectnumbers.Thisnumberofrareyetbeautifulwhichisknownasthe"diamondsintreasurehouseofnumbertheory."

　　DiscoveryofMersennePrime

　　Later，thestructureof2P-1usingforfindingtheperfectnumberwasnamedasMersennePrimeinthemathematicalfield.ItwasnamedafterMersennewhoisaFrenchmathematicianbecausehissystematicanddeepresearchintothisspecialprime.Interestingly，the"superprimes"foundinthepastcenturyarealmostMersenneprimes.　　Mersenneprimeseemstobesimplebutactuallyit’shardtoexplore.Itrequiresnotonlyprofoundtheoriesandskillfultechniques，butalsocomplicatedcalculationsandgreatamountofcomputation.In1772whenEulerwasblind，hestillproved231-1（ie2147483647）tobetheeighthMersenneprimebymentalarithmetic.Thisprimewith10digitswasthelargestknownprimenumberatthattime.Theeighthperfectnumber--230（231-1）alsoemergedwhichwaslikelytobethebiggestperfectnumberpeoplefoundatthatmoment.Euler'sperseveranceandproblem-solvingskillswereimpressive.WhatFrenchmathematicianLaplacesaysmaybecanrepresentourvoice：

"ReadEuler’swritings.Heistheteacherofeveryoneofus."

　　Againstthebackdropwhereallthecalculationsandrecordmustbewrittendownbyhands，nomatterhowhardpeopletried，theyfoundonly12Mersenneprimes，thatistosay，only12perfectnumberswerediscovered.However，thebirthofcomputersgreatlyacceleratedtheprocessofresearchintoMersenneprimes.Forexample，in1952，theAmericanmathematicianRobinsoncompiled"Lucas-Lehmertest"intoacomputerprogram.ByusingtheSWACcomputerinafewmonths，theyfoundfiveMersenneprimes：

2521-1，2607-1，21279-1，22203-1and22281-1.

　　ExplorationintoMersenneprimeisnotonlychallenging.Fortheexplorers，theycanobtainasenseofpride.Perhapsthisiswhytherearecountlessmathematicianswillingtodevotethemselvesintoit.At20：

00onJune2，1963，whenthefirst23rdMersenneprime211213-1wasfoundbylargecomputer，theAmericanBroadcastingCorporation（ABC）interrupteditsregularbroadcastandtimelypublishedthisimportantmessage.AllthestudentsandfacultiesfromtheDepartmentofMathematics，UniversityofIllinoisstaffwhofoundtheprimewereveryproudandatthesametime，inordertolettheworldtosharethisgreatachievement，alltheenvelopessentfromthedepartmentarecoveredwitha"211213-1isaprimenumber"postmark，whichwasverycrazyandfantastic.

　　WiththeincreaseoftheindexP，thediscoveryofeveryMersenneprimeencounterstremendoushardships.However，theprofessionalmathematiciansandamateurmathematicsloverswerenottiredofthisandeventriedtotakethelead.OnFebruary23，1979，whenVinskiandNelsonwhoarethecomputerexp