AnIntroductiontoTheTwinPrimeConjectureAllisonBerkeDecember12,2006AbstractTwinprimesareprimesoftheform(p,p+2).
Therearemanyproofsfortheinnitudeofprimenumbers,butitisverydiculttoprovewhetherthereareaninnitenumberofpairsoftwinprimes.
Mostmathematiciansagreethattheevidencepointstowardthisconclusion,butnumerousattemptsataproofhavebeenfalsiedbysubsequentreview.
Theproblemitself,oneofthemostfamousopenproblemsinmathematics,hasyieldedanumberofrelatedresults,includingBrun'sconjecture,Mertens'theorems,andtheHardy-LittlewoodConjecture.
Alongwiththeseconjectures,thereareanumberofresultswhichareeasiertoarriveat,butneverthelesshelpmathematiciansthinkabouttheinnitudeofprimes,andthespecialpropertiesoftwinprimes.
Thispaperwillintroducetheaforementionedconjecturesassociatedwiththetwinprimeconjecture,andworkthroughsomeexercisesthatilluminatethedicultiesandintricaciesofthetwinprimeconjecture.
1Introduction:TheOriginalConjectureandFailedProofsThetermtwinprimewascoinedbyPaulStackelinthelatenineteenthcentury.
Sincethattime,mathematicianshavebeeninterestedinthepropertiesofrelatedprimes,bothinrelationtonumbertheoryasawhole,andasspecic,well-denedproblems.
Oneoftherstresultsoflookingattwinprimeswasthediscoverythat,asidefrom(3,5),alltwinprimesareoftheform6n±1.
Thiscomesfromnoticingthatanyprimegreaterthan3mustbeoftheform6n±1.
Toshowthis,notethatanyintegercanbewrittenas6x+y,wherexisanyinteger,andyis0,1,2,3,4or5.
Nowconsidereachyvalueindividually.
Wheny=0,6x+y=6xandisdivisibleby6.
Wheny=1therearenoimmediatelyrecognizablefactors,sothisisacandidateforprimacy.
Wheny=2,6x+2=2(3x+1),andsoisnotprime.
Forthecasewhen·y=3:6x+3=3(2x+1)andisnotprime.
Wheny=4:6x+4=2(3x+2)··andisnotprime.
Wheny=5,6x+5hasnoimmediatelyrecognizablefactors,andisthesecondcandidateforprimacy.
Thenallprimescanberepresentedaseither6n+1or6n1,andtwinprimes,sincetheyareseparatedbytwo,willhavetobe6n1and6n+1.
1TwinPrimeConjecture2Furtherresearchintotheconjecturehasbeenconcernedwithndingexpressionsforaformoftheprimecountingfunctionπ(x)thatdependonthetwinprimeconstant.
Theprimecountingfunctionisdenedasπ(x)={N(p)|px}whereN(p)denotesthenumberofprimes,p.
Onemotivationfordeningtheprimecountingfunctionisthatitcanbeusedtodetermineaformulaforthesizeoftheintervalsbetweenprimes,aswellasgivingusanindicationoftherateofdecaybywhichprimesthinoutinhighernumbers.
Ithasbeenshownalgebraicallythattheprimecountingfunctionincreasesasymptoticallywiththelogarithmicintegral[12].
Inthefollowingexpression,π2(x)referstothenumberofprimesoftheformpandp+2greaterthanx,andisthetwin2primeconstant,whichisdenedbytheexpression(19p11)2)overprimesp2.
ThetermO(x),meaning"ontheorderofx,"isdenedasfollows:iff(x)andg(x)aretwofunctionsdenedonthesameset,thenf(x)isO(g(x))asxgoestoinnityifandonlyifthereexistssomex0andsomeMsuchthat|f(x)|M|g(x)|forxgreaterthanx0.
Thisexpressionforthetwinprimecountingfunctionisπ2(x)cΠ2x[1+O(ln(ln(x)))](1)(ln(x))2ln(x)whichisthebestthathasbeenproventhusfar.
Theconstantcin(1)hasbeenreducedto6.
8325,downfrompreviousvaluesashighas9[12].
TheformationofthisinequalityinvolvestwoofMerten'stheoremswhichwillbediscussedinthefollowingsection.
HardyandLittlewood[3]haveconjecturedthatc=2,andusingthisassumptionhaveformulatedwhatisnowcalledtheStrongTwinPrimeConjecture.
Inthefollowingexpression,abmeansthataapproaches1batthelimitsoftheexpressionsaandb.
Inthiscase,thelimitisasxapproachesinnity.
xdxπ2(x)2Π2(ln(x))2.
(2)2Anecessaryconditionforthestrongconjecture(2)isthattheprimegapsconstant,Δ≡limsupn→∞pn+1pnbeequaltozero.
ThemostrecentattemptedpnproofofthetwinprimeconjecturewasthatofArenstorf,in2004[1],butanerrorwasfoundshortlyafteritspublication,anditwaswithdrawn,leavingtheconjectureopentothisday.
2Mertens'TheoremsAnumberofimportantresultsaboutthespacingofprimenumberswerederivedbyFranzMertens,aGermanmathematicianofthelatenineteenthandearlytwentiethcentury.
ThefollowingproofsofMertens'conjecturesleaduptotheresultthatthesumofthereciprocalsofprimesdiverges,whichwillcontrast3TwinPrimeConjecturewithBrun'sconjecture,thatthesumofthereciprocalsoftwinprimesconverges.
First,weshouldbrieyshowthattheprimesareinnite,forotherwisetheimplicationsofMertens'theoremsarenotobvious.
Euclid'sproofofthispostulate,hissecondtheorem,isasfollows.
Let2,3,5,.
.
.
,pbeanenumerationofallprimenumbersuptop,andletq=(235·.
.
.
p)+1.
Thenqisnotdivisiblebyanyoftheprimesup···toandincludingp.
Therefore,itiseitherprimeordivisiblebyaprimebetweenpandq.
Intherstcase,qisaprimegreaterthanp.
Inthesecondcase,thedivisorofqbetweenpandqisaprimegreaterthanp.
Thenforanyprimep,thisconstructiongivesusaprimegreaterthanp.
Thus,thenumberofprimesmustbeinnite[4].
NowwecanresumewithMertens'theorems.
MertensTheorem1:Foranyrealnumberx≥1,x0≤ln(n)0suchthat11p=ln(ln(x))+b1+O(ln(x)),x≥2.
(6)p≤x6TwinPrimeConjectureProof:Wecanwrite1=ln(p)1=u(n)f(n)ppln(p)p≤xp≤xn≤xwhereu(n)=ln(pp)ifn=p,and0otherwise,andf(t)=ln(1t).
WedenenewfunctionsU(t)andg(t)asfollowsln(p)U(t)=u(n)==ln(t)+g(t)pn≤tp≤tThenU(t)=0fort3TheformulationoftheHardy-LittlewoodconjecturebuildsuponsomeofthetechniquesusedtoproveBrun'sconjecture,namelytheBrunsievetechniques.
TheBrunsievecanbeconstructedasfollows:LetXbeanonempty,nitesetofNobjects,andletP1,PrberdierentpropertiesthattheelementsofthesetXmighthave.
LetN0denotethenumberofelementsofXthathavenoneoftheseproperties.
ForanysubsetI={i1,ik}of{1,2,r},letN(1)=N(i1,ik)denotethenumberofelementsofXthathaveeachofthepropertiesPi1,Pi2Pik.
LetN()=|X|=N.
Ifmisanonnegativeeveninteger,thenmN0≤(1)kN(I).
(9)k=0|I|=kIfmisanonnegativeoddinteger,thenmN0≥(1)kN(I).
[8](10)k=0|I|=kTheproofgiveninNathanson[8]isasfollows.
LetxbeanelementofthesetX,andsupposethatxhasexactlylpropertiesPi.
Ifl=0,thenxiscountedonceinN0andonceinN(),butisnotcountedinN(I)ifIisnonempty.
Ifl≥1,thenxisnotcountedinN0.
Byrenumberingtheproperties,wecanassumethatxhasthepropertiesP1,P2,Pl.
LetI{1,2,l,r}.
Ifi∈Iforsomei>l,thenxisnotcountedinN(I).
IfI{1,2,l}thenxcontributes1toN(I).
Foreachk=0,1,l,thereareexactlyklsuchsubsetswith|I|=k.
Ifm≥l,thentheelementxcontributesll(1)k=0kk=0TwinPrimeConjecture9totherightsidesoftheinequalities.
Ifm2cln(ln(x)),then·rrcln(ln(x)))k1xy(·m≤x2k2cln(ln(x)).
Ifweletc=max{2c,(ln(2)1)},andlet·ln(y)1x=e(3c·ln(ln(y)))=y3c·ln(ln(y))m=2[cln(ln(y))]·Thensinceln(y)ln(x)=3c·ln(ln(y))yy(ln(ln(y)))22c·ln(ln(y))2,y4y4y4y2m<22c·ln(ln(y))=(ln(y))2c·ln(2)≤(ln(y))2Thenm2cln(ln(y))2c·ln(ln(y)ln(y))32x≤x·=exp(ln(ln(y)))=y3c·Finally,x(ln(ln(x)))2π2(x)<<.
(ln(x))2TwinPrimeConjecture126ConclusionThetwinprimeconjecturemayneverbeproven,butstudyingthepropertiesoftwinprimesiscertainlyarewardingexercise.
RecentworkonthetwinprimeconjecturebyDanGoldstonandCemYilidrimhasfocusedoncreatingexpressionsforthegapsizebetweenprimes,andinparticularfocusingontheexpressionΔ=liminfpn+1pn=1n→∞ln(pn)ResearchintobetterexpressionsfortheintervalbetweenconsecutiveprimesiscurrentlybeingconductedatStanford,sponsoredbytheAmericanInstituteofMathematics[12].
Thoughnumbertheoryhasbeenthefoundationofmanydierentbranchesofhighermathematics,itsfundamentalproblemsremaininterestingandfruitfulforresearchersinterestedinthepropertiesofprimenumbers.
References[1]Arenstorf,R.
F.
"ThereAreInnitelyManyPrimeTwins.
"26May2004.
http://arxiv.
org/abs/math.
NT/0405509.
[2]Guy,R.
K.
"GapsbetweenPrimes.
TwinPrimes.
"A8inUnsolvedProblemsinNumberTheory,2nded.
NewYork:Springer-Verlag,pp.
19-23,1994.
[3]Hardy,G.
H.
andLittlewood,J.
E.
"SomeProblemsof'PartitioNumerorum.
'III.
OntheExpressionofaNumberasaSumofPrimes.
"ActaMath.
44,1-70,1923.
[4]Hardy,G.
H.
andWright,E.
M.
AnIntroductiontotheTheoryofNumbers,5thed.
Oxford,England:ClarendonPress,1979.
[5]Havil,J.
Gamma:ExploringEuler'sConstant.
Princeton,NJ:PrincetonUniversityPress,pp.
30-31,2003.
[6]Miller,S.
J.
andTakloo-Bighash,R.
AnInvitationtoNumberTheory.
Princeton,NJ:PrincetonUniversityPress,pp.
326-328,2006.
[7]Narkiewicz,W.
TheDevelopmentofPrimeNumberTheory.
Berlin,Germany:SpringerPress,2000.
[8]Nathanson,M.
B.
AdditiveNumberTheory.
NewYork,NewYork:SpringerPress,1996.
[9]Ribenboim,P.
TheNewBookofPrimeNumberRecords.
NewYork:Springer-Verlag,pp.
261-265,1996.
[10]Shanks,D.
SolvedandUnsolvedProblemsinNumberTheory,4thed.
NewYork:Chelsea,p.
30,1993.
13TwinPrimeConjecture[11]Tenenbaum,G.
"ReArenstorf'spaperontheTwinPrimeConjecture.
"8Jun2004.
[12]Weisstein,EricW.
"TwinPrimeConjecture"http://mathworld.
wolfram.
com/TwinPrimeConjecture.
html,2006.
[13]Young,R.
M.
ExcursionsinCalculus.
TheMathematicalAssociationofAmerica,1992.
7月份已经过去了一半,炎热的夏季已经来临了,主机圈也开始了大量的夏季促销攻势,近期收到一些商家投稿信息,提供欧美或者亚洲地区主机产品,价格优惠,这里做一个汇总,方便大家参考,排名不分先后,以邮件顺序,少部分因为促销具有一定的时效性,价格已经恢复故暂未列出。HostMem部落曾经分享过一次Hostmem的信息,这是一家提供动态云和经典云的国人VPS商家,其中动态云硬件按小时计费,流量按需使用;而经典...
水墨云怎么样?本站黑名单idc,有被删除账号风险,建议转出及数据备份!水墨云ink cloud Service是成立于2017年的商家,自2020起开始从事香港、日本、韩国、美国等地区CN2 GIA线路的虚拟服务器租赁,同时还有台湾、国内nat vps相关业务,也有iplc专线产品,相对来说主打的是大带宽服务器产品。注意:本站黑名单IDC,有被删除账号风险,请尽量避免,如果已经购买建议转出及数据备...
A400互联怎么样?A400互联是一家成立于2020年的商家,A400互联是云服务器网(yuntue.com)首次发布的云主机商家。本次A400互联给大家带来的是,全新上线的香港节点,cmi+cn2线路,全场香港产品7折优惠,优惠码0711,A400互联,只为给你提供更快,更稳,更实惠的套餐,香港节点上线cn2+cmi线路云服务器,37.8元/季/1H/1G/10M/300G,云上日子,你我共享。...
let美人双胞胎姐妹为你推荐
域名价格请问域名有什么价值吗?vps虚拟主机请问VPS和虚拟主机有什么不一样,为什么VPS贵那么多。广告的别来!手机网站空间手机网页空间需要多大?长沙虚拟主机在长沙,哪个兼职网站最最可靠??shopex虚拟主机我有一个PHP1G的虚拟主机,请问做什么站比较合适?根域名服务器什么是根域名服务器的任播技术?ip查域名如何用IP地址反查域名qq域名邮箱QQ域名邮箱怎么开通和设置?域名投资现在投资域名还有前途吗泛域名泛域名可以解析多少次?
山东vps 美国加州vps 主机屋 vir 香港服务器99idc 百度云100as linode themeforest 轻博客 ssh帐号 线路工具 帽子云 卡巴斯基试用版 中国联通宽带测试 创速 好看的空间 贵州电信 香港博客 shuangcheng 湖南铁通 更多