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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.
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