InternationalJournalofMathematicsandComputerScience,(2012),no.2,101 - PDF document

InternationalJournalofMathematicsandComputerScience,(2012),no.2,101…112 SubsetsofPrimeNumbersBadihGhusayniDepartmentofMathematicsFacultyofScience-1LebaneseUniversityHadath,Lebanonemail:badih@future-in-tech.net(ReceivedFebruary10,2012,AcceptedOctober14,2012)TheFundamentalTheoremofArithmeticshowstheimportanceofprimenumbers.Awell-knownresultisthatthesetofprimenumbersisin“nite(thesubsetofevenprimenumbersisobviously“nitewhilethatofoddprimenumbersisthereforein“nite).ThesubsetofRamanujanprimesisin“nite.Thesetoftripletprimenumbersis“nitewhileitisnotknownwhetherornotthesubsetoftwinprimenumbersisin“niteeventhoughitissoconjectured.Wegivemanyresultsinvolvingthedierenttypesofprimenumbers.1Introductionprimetripletisatripletoftheform(p,p+4)consistingofthreeprimes.Thesetofprimetripletsis“nite.Indeed,itisasingletonsetcon-tainingonly(3Toseethis,noticethat(37)isaprimetriplet.Toproveitistheonlyone,weusethecontradictionmethod.Supposetherewasaprimetriplet(P,P+4)withcanbewrittenas+1or3forsomeinteger,...However,isnotprime.+2isnotprimebeing3(+1) Keywordsandphrases:Primenumbers,RamanujanPrimes,BertrandPostulate,TwinPrimeConjecture,GoldbachConjecture,Dierentkindsofprimes,RiemannHypothesis.AMS(MOS)SubjectClassi“cations:11M06,11M38,11M50. SubsetsofPrimeNumbersNotethatBertrandPostulateisRecallthatifareprimenumbers,thenthispairissaidtobeaTwinPrimePair.ThesmallestsuchpairisTheTwinPrimeConjecturestatesthattherearein“nitelymanyTwinPrimePairs.Similarly,wede“neaTwinRamanujanPrimePair.Thesmallestsuchpairisaswecanseefromthebeginningofthissection.AswemightexpecttherearefewerTwinRamanujanPrimePairsthanTwinPrimepairs.Asamatteroffact,amongthe“rstprimenumbers,thereareTwinRamanujanPrimePairsandTwinPrimePairs.Notethat 1861 Thisobservationisimportantforthenextsection.3ResultsThefollowingconjecturewasstatedin[12]:thenthenumberofpairsofTwinRamanujanPrimesnotexceed-isgreaterthanonequarterofthenumberofpairsofTwinPrimesnotInparticular,iftherearein“nitelymanyTwinPrimes,thentherearealsoin“nitelymanyTwinRamanujanPrimes.Ontheotherhand,ifthereare“nitelymanyTwinRamanujanPrimes,thenthereare“nitelymanytwinprimes.Therefore,ifSondowConjectureisprovenandifitisprovenalsothatthereareonly“nitelymanyTwinRa-manujanPrimes,thenthatwouldsettlethefamousTwinPrimeConjecture.InalettertoLeonhardEulerin1742ChristianGoldbachstatedthathebelievesthat:G:Everyintegergreaterthan5isthesumofthreeprimes.Eulerreplied:E:ŽThateveryevennumber4isasumoftwoprimes,IconsideranentirelycertaintheoreminspiteofthatIamnotabletodemonstrateitŽ.Asamatteroffact,if(G)issatis“edand2then2+2=withprimesthatcannotallbeodd,forotherwisetheirsumwouldbeodd.Soatleastoneofthem,sayis2Then(E)obtains.Conversely,if(E)issatis“edand5,then26andweconsider SubsetsofPrimeNumbersThereforeforallwehaveven+k]+[(+2)Since2+2=wecannowchoosesuchthat+2)Inaddition,thefollowingtheoremprovidesanequivalentstatementtotheTwinPrimeconjecture:Theorem3.2.Foreachthereisatleastonesuchthat+2)+areprimenumbersifandonlyiftherearein“nitelymanypairsp,p+2)suchthatarebothprimenumbers.Proof.():Thisfollowssinceisin“niteandtherepresentation[(2++n+k]=2holdswithand(2+primenumbers(Noticethatdistinctvaluesyielddistinctvaluesof,forifthereissuchthatk,n+2)=(k,n+2)):LetUsingthehypothesis,wecanchoosesuchthatand(+2)+areprimenumbers.Remark3.3.Lookingatthestatementsontheleft-handsidesintheprevioustwotheorems,weobservetheunusualfeaturethatprovingoneoftheGoldbachortheTwinPrimeconjectureprovestheother.4Connectionbetweenvariouskindsofprimes)denotingthenumberofprimesdenotingthethprimenumber,letusstateandprovethefollowingresultsLemma4.1.)log xn Proof.():LetThenlim pn/n(pn Thenthereissuchthat 2pn n.n0pn2nn. SubsetsofPrimeNumbersRemark4.2.ThePrimeNumberTheoremstatesthat)log BytheabovelemmaitfollowsthatthePrimeNumberTheoremisequivalentn,nDe“nition4.3.(a)Asetinavectorspaceissaidtobeconvexifwhen-x,y(b)TheconvexhullconvexenvelopeofasetofpointssionsistheintersectionofallconvexsetscontainingTheorem4.4.[10]Let...beasequenceofnumberswith Thentherearein“nitelymanyforwhichforallpositivei.Proof.Clearlytheboundaryoftheconvexhullofthesetn,a,...ispolygonal.Theadditionalassumptionlim =0impliesthatthenonverticalportionofthispolygonalboundaryisconvexandhasin“nitelymanyvertices.Sinceeachoftheseverticesisoftheform(n,a)fortheresultfollows.Corollary4.5.Therearein“nitelymanyforwhichallpositivei.Proof.Clearly0...Inaddition,usingtheabovelemma,itfollowsthatlim Theorem4.6.Let...beasequenceofnumberswith Thentherearein“nitelymanyforwhichforallpositivei.Proof.Clearlytheboundaryoftheconvexhullofthesetn,a,...ispolygonal.Theadditionalassumptionlim =0impliesthatthenonhorizontalportionofthispolygonalboundaryisconcaveandhasin“nitelymanyvertices.Sinceeachoftheseverticesisoftheform(n,a)fortheresultfollows.Remark4.7.Since isnotnecessarilywecannothopetogetthattherearein“nitelymanyforwhichforallpositivefromtheabovetheorem.However,wecanprovethefollowing B.Ghusayni5LandauOpenProblemswithRemarksAtthe1912InternationalCongressofMathematicians,EdmundLandau[7]raisedthefollowingfourmajoropenquestions(conjectures)aboutprimenumberswhichwerecharacterizedinhisspeechatthattimeasŽunattackableatthepresentstateofscienceŽ:1.GoldbachConjecture.2.TwinPrimeConjecture.3.LegendreConjecture:Foreachnaturalnumberthereisalwaysaprimenumbersuchthat+1)4.Therearein“nitelymanyprimenumbersoftheformaninteger.Almostacenturylater,theseproblemshaveremainedde“ant.Indeed,aswementionedearlierthe“rstonehadbeenopensinceGoldbachraiseditin1742inalettertoEulerandthesecondoneisa2300-year-oldmysterynow.ThethirdoneresistedallattemptseventhoughitisinthesamespiritasBertrandPostulate(Theorem)andthefollowingtheoremduetoIngham[6]:Foreverylargeenoughnaturalnumberthereisaprimenumbersuch+1)Moreover,in[4]M.ElBachraouiraisedaquestionwhichwerephraseas:Isittruethatforallintegers1anda“xedintegerthereexistsaprimenumberintheinterval(kn,+1)Thecase=1isBertrandPostulate.ElBachraoui[4]gaveapositiveanswerforthecaseAndyLoo[5]gaveapositiveanswerforthecaseApositiveanswertotheproblemwouldimplyapositiveanswertoLegendreConjecture(bytakingFinally,concerningthefourthproblem,itcanbeobviouslybephrasedas:Therearein“nitelymanyprimenumbersoftheformisanaturalnumber.itisknownthattherearein“nitelymanyprimenumbersoftheformsisarealnumber.Ontheotherhand,in1978HendrikIwaniecshowedthattherearein“nitelymanyvaluesof+1thatareeitherprimesoraproductoftwoprimes.Finally,Ishowamethodwhichcouldleadtotheproofofproblem4Suppose,togetacontradiction,thattherewereonly“nitelymanyprimesofthatform: B.Ghusayni[9]PaulErd¨BeweiseinesSatzesvonTschebyschef,ActaLitt.Sci.Sze-(1932),194-198.[10]CarlPomerance,ThePrimeNumberGraph,Math.Computation,(1979),399-408.[11]SrinivasaRamanujan,AproofofBertrandPostulate,J.IndianMath.Soc.,(1919),181-182.[12]JonathanSondow,RamanujanPrimesandBertrandsPostulate,Math.Monthly,(2009),630-635.