EXPECTED GAPS BETWEEN PRIME NUMBERS

EXPECTEDGAPSBETWEENPRIMENUMBERSFREDB.HOLTAbstract.Westudythegapsbetweenconsecutiveprimenumbersdi-rectlythroughEratosthenessieve.Usingelementarymethods,weiden-tifyarecursiverelationforthesegapsandforspecicsequencesofcon-secutivegaps,knownasconstellations.Usingthisrecursionwecanestimatethenumbersofagaporofaconstellationthatoccurbetweenaprimeanditssquare.Thisrecursionalsohasexplicitimplicationsforopenquestionsaboutgapsbetweenprimenumbers,includingthreequestionsposedbyErdosandTuran.Keywords:primes,twinprimes,gaps,distributionofprimes,Eratosthenessieve.1.IntroductionWeworkwiththeprimenumbersinascendingorder,denotingthekthprimebypk.Accompanyingthesequenceofprimesisthesequenceofgapsbetweenconsecutiveprimes.Wedenotethegapbetweenpkandpk+1bygk=pk+1pk:Thesesequencesbeginp1=2;p2=3;p3=5;p4=7;p5=11;p6=13;:::g1=1;g2=2;g3=2;g4=4;g5=2;g6=4;:::Anumberdisthedierencebetweenprimenumbersiftherearetwoprimenumbers,pandq,suchthatqp=d.Therearealreadymanyinter-estingresultsandopenquestionsaboutdierencesbetweenprimenumbers;aseminalandinspirationalworkaboutdierencesbetweenprimesisHardyandLittlewood's1923paper[10].Anumbergisagapbetweenprimenumbersifitisthedierencebetweenconsecutiveprimes;thatis,p=piandq=pi+1andqp=g.Dierencesoflength2or4arealsogaps;soopenquestionsliketheTwinPrimeConjecture,thatthereareaninnitenumberofgapsgk=2,canbeformulatedasquestionsaboutdierencesaswell.Date:2Feb06.1991MathematicsSubjectClassication.11N05,11A41,11A07.Keywordsandphrases.primes,twinprimes,gaps,primeconstellations,Eratothenessieve.1arXiv:0706.0889v1[math.NT]6Jun20072FREDB.HOLTAconstellationamongprimes[22]isasequenceofconsecutivegapsbe-tweenprimenumbers.Lets=a1a2


akbeasequenceofknumbers.Thensisaconstellationamongprimesifthereexistsasequenceofk+1consec-utiveprimenumberspipi+1


pi+ksuchthatforeachj=1;:::;k,wehavethegappi+jpi+j1=aj.Equivalently,sisaconstellationifforsomeiandallj=1;:::;k,aj=gi+j.Wewillwritetheconstellationswithoutmarkingaseparationbetweensingle-digitgaps.Forexample,aconstellationof24denotesagapofgk=2followedimmediatelybyagapgk+1=4.Thenumberofgapsafterkiterationsofthesieveisk=Qki=1(pi1):Forthesmallprimeswewillconsiderexplicitly,mostofthesegapsaresingledigits,andtheseparatorsintroducealotofvisualclutter.Weusecommasonlytoseparatedouble-digitgapsinthecycle.Forexample,aconstellationof2;10;2denotesagapof2followedbyagapof10,followedbyanothergapof2.WeuseelementarymethodstostudythegapsgeneratedbyEratosthenessievedirectly.Bystudyingthissieve,wecanestimatetheoccurrenceofcertaingapsandconstellationsbetweenpkandp2k.Fromthemethodsdevelopedbelow,wecancalculateexactlyhowmanytimesasequencesofgapsoccursafterkstagesofEratosthenes'sieve.Wedon'tknowhowmanyoftheseoccurrenceswillsurvivesubsequentstagesofthesievetobecomeconstellationsamongprimenumbers.However,foraprimepwecanmakeestimatesforthenumberthatoccurbeforep2,allofwhichwillsurviveasconstellationsamongprimes.Thusourestimatesandcountsareonlycoincidentallycommensuratewithtabulationsagainstpowersoften.Theproductoftherstkprimeswillbedenotedbyk=Qki=1pi:Bythepk-sieve,wemeanthosepositiveintegersremainingafterremovingallthemultiplesoftherstkprimenumbers.Thepk-sievehasafundamen-talcycleofkelementsmodulok.MostoftenwepicturethisfundamentalcycleasthegeneratorsforZmodk,althoughitisalsoattractivetovisu-alizetheseastheprimitivethkrootsofunityinC.1.1.Organizationofthematerial.Weproceedasfollows.WeidentifyarecursivealgorithmforproducingeachcycleofgapsG(pk+1)fromtheprecedingcycleG(pk).Thisrecursionenablesustoenumeratevariousgapsandconstellationsinthepk-sieve.InthecycleofgapsG(pk)ofcourse,allthegapsfrompk+1andp2k+1areactuallygapsbetweenprimenumbers.Wemakeaconjectureabouttheuniformityofthedistributionofthesegapsandconstellations.Fromthisconjecturewecanmakestatisticalesti-matesabouttheexpectednumberofoccurrencesofthesegapsandconstel-lationsbelowp2k+1,andwecomparetheseestimateswithactualcounts.EXPECTEDGAPSBETWEENPRIMENUMBERS3WemakeaweakerconjecturethateveryconstellationinG(pk)occursinnitelyoftenasaconstellationamongprimes,providedthesumofthegapsintheconstellationislessthan2pk+1.Fromthisweakerconjectureweaddressseveralquestionsaboutgapsanddierencesbetweenprimenumbers.WeshowthatHardyandLittlewood'sk-tupleconjectureonthedierencesbetweenprimenumbers[10]isequivalenttoaconjectureongaps.WearealsoabletogiveexactanswerstothreequestionsposedbyErdosandTuran[6].1.2.Newresults.ThispaperapplieselementarymethodstoEratosthenessieve.Inageneralsense,thisiswell-troddenground.However,thespecicinsightofidentifyingtherecursionongapsappearstobenew.WecastEratosthenessieveasarecursiveoperationdirectlyonthecycleofgaps.Bystudyingthisrecursionwecanenumerateparticulargapsateverystage.Moreoverweobservehowmuchstructureofthecycleofgapsatonestageofthesieveispreservedinsubsequentstages.Wecantherebyeasilyenumeratetheoccurrencesofspecicconstellationsofprimesthathavenotpreviouslybeenapproachable(e.g.2;10;2).Theconclusionsontherecursionofgapsareprecise.Togofurther,weneedtosupplementourrigorousworkwithanappropriateconjecture.Werstmakea