文档介绍：该【数学专业英语 】是由【1485173816】上传分享，文档一共【14】页，该文档可以免费在线阅读，需要了解更多关于【数学专业英语 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。:§,,,+1isdenotedby2,thenumber2+1by3,,2,3,…,obtainedinthiswaybyrepeatedadditionof1areallpositive,,pletebecausewehavenotexplainedindetailswhatwemeanbytheexpressions“andsoon”,or“repeatedadditionof1”.Althoughtheintuitivemeaningofexpressionsmayseemclear,inacarefultreatmentofthereal-:,thenumberx+,+.(a)itcontains1,and(b)itcontainsx+,,,togetherwiththenegativeintegersand0(zero),-numbersystem,,thesum,difference,orproductoftwointegersisaninteger,,(whereb10),denotedbyQ,,,wheredivisionalgebra,alsocalleda"divisionring"or"skewfield,"meansaringinwhicheverynonzeroelementhasamultiplicativeinverse,(or"totallyorderedset,"or"linearlyorderedset")isasetplusarelationontheset(calledatotalorder)￡isatotalorderonasetS("￡totallyordersS"):a￡aforalla?:a￡bandb￡aimpliesa=:a￡bandb￡cimpliesa￡(trichotomylaw):Foranya,b?S,eithera￡borb￡,,totherightof0,torepresent1,asillustratedinFigure2-4-,theneachrealnumbercorrespondstoexactlyonepointonthislineand,conversely,,<y,thepointxliestotheleftofthepointyasshowninFigure2-4-<b,apointxsatisfiestheinequalitiesa<x<,,thegeometryoftensuggeststhemethodofproofofaparticulartheorem,andsometimesageometricargumentismoreilluminatingthanapurelyanalyticproof(onedependingentirelyontheaxiomsfortherealnumbers).Inthisbook,,(orprimeinteger,oftensimplycalleda"prime"forshort)isapositiveintegerp>,theonlydivisorsof13are1and13,making13aprimenumber,whilethenumber24hasdivisors1,2,3,4,6,8,12,and24(correspondingtothefactorization24=23×3),,moreprecisely,,thesmallestprimeistherefore2andsince2istheonlyevenprime,,,sop1=2,p2=3,andsoon,putedinMathematicaasPrime[n].ThesetofprimesissometimesdenotedP,"Mathematicianshavetriedinvaintothisdaytodiscoversomeorderinthesequenceofprimenumbers,rate".Ina1975lecture,",despitetheirsimpledefinitionandroleasthebuildingblocksofthenaturalnumbers,theprimenumbersgrowlikeweedsamongthenaturalnumbers,seemingtoobeynootherlawthanthatofchance,,foritstatesjusttheopposite:thattheprimenumbersexhibitstunningregularity,thattherearelawsgoverningtheirbehavior,andthattheyobeytheselawswithalmostmilitaryprecision".LargeprimesincludethelargeMersenneprimes,Ferrier'-1(Weisstein2005).Primenumberscanbegeneratedbysievingprocesses(suchasthesieveofEratosthenes),andluckynumbers,whicharealsogeneratedbysieving,(n)(n)',itisnotknownifthereareaninfinitenumberofprimesoftheformn2+1,whetherthereareaninfinitenumberoftwinprimes(thetwinprimeconjecture),orifaprimecanalwaysbefoundbetweenn2and(n+1)'(ingafter9)include2,3,5,7,23,67,89,4567,78901,....Primesconsistingofdigitsthatarethemselvesprimesinclude23,37,53,73,223,227,233,257,277,337,353,373,523,557,...,,identallyreferredtoatrivialoperationwhenhestated"Becauseboththesystem'sprivacyandthesecurityofdigitalmoneydependonencryption,[emphasisadded]"(Gates1995,).NotationsMersenneprimeAMersenneprimeisaMersennenumber,.,anumberoftheformMn=2n-,,n=,2n-1canbewrittenas2rs-1,whichisabinomialnumberthatalwayshasafactor2r-,7,31,127,8191,131071,524287,2147483647,...correspondingtoindicesn=1,3,5,7,13,17,19,31,61,89,....,,the1963discoverythatM11213isprimewasheraldedbyaspecialpostalmeterdesign,illustratedabove,issuedinUrbana,(MersennePrimeSearch),2003,aGIMPSvolunteerreporteddiscoveryofthe40thMersenneprime,,,,,GIMPSparticipantshavetestedanddouble-checkedallexponentsbelow9889900andtestedallexponentsbelow15130000atleastonce(GIMPS).ThetablebelowgivestheindexpofknownMersenneprimesMp,togetherwiththenumberofdigits,discoveryyears,,soidentificationof"the"40thMersenneprimeistentative(GIMPS).#pdigitsyeardiscoverer(reference)121antiquity?231antiquity?352antiquity?473antiquity?51341461Reguis(1536),Cataldi(1603)61761588Cataldi(1603)71961588Cataldi(1603)831101750Euler(1772)961191883Pervouchine(1883),Seelhoff(1886)1089271911Powers(1911)**********Powers(1914)**********Lucas(1876),,,,,,,,1961Hurwitz2196892917May11,1963Gillies(1964)2299412993May16,1963Gillies(1964),1963Gillies(1964),1971Tuckerman(1971),1978NollandNickel(1980),1979Noll(NollandNickel1980),1979NelsonandSlowinski(Slowinski1978-79),1982Slowinski29**********,1988ColquittandWelsh(1991)30**********,,,,,,1996JoelArmengaud/,1997GordonSpence/GIMPS(Devlin1997),1998RolandClarkson/,1999NayanHajratwala/,2001MichaelCameron/GIMPS(Whitehouse2001,Weisstein2001),2003MichaelShafer/GIMPS(Weisstein2003)41May15,2004JoshFindley/GIMPS(Weisstein2004),2005MartinNowak/GIMPS(Weisstein2005)Ferrier'ordingtoHardyandWright(1979),the44-digitFerrier'sprimeis(2148+1)/17=24593863921determinedtobeprimeusingonlyamechanicalcalculator,(small)fractionofasecond,:1,3,5,7,9,11,13,15,17,19,....Thefirstoddnumbergreaterthan1is3,sostrikeouteverythirdnumberfromthelist:1,3,7,9,13,15,19,....Thefirstoddnumbergreaterthan3inthelistis7,sostrikeouteveryseventhnumber:1,3,7,9,13,15,21,25,31,....,3,7,9,13,15,21,25,31,33,37,....,justastheprimenumbertheorem,(n),?(1808)suggestedthatforlargen,withB=-(whereBissometimescalledLegendre'sconstant).In1792,whenonly15yearsold,Gaussproposedthat