Challenging Perfect NumberWord文档下载推荐.docx
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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
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