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Re: [PrimeNumbers] Is there another prime gap...

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  • Andrey Kulsha
    ... gap of all, not ... most unusual ... up to ... digits at ... http://www.trnicely.net/#TPG Best, Andrey
    Message 1 of 6 , Feb 5, 2008
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      > The fact that the gap between 1327 and 1361 is the most persistent primal
      gap of all, not
      > being equalled until between 8467 and 8501 makes me think of it as the
      most unusual
      > prime gap that probably exists among numbers of any size. A gap persisting
      up to
      > numbers six times its size is unknown among numbers with fewer than twenty
      digits at
      > any level - I would in fact doubt it occurs at any larger size of number.

      http://www.trnicely.net/#TPG

      Best,

      Andrey
    • Jens Kruse Andersen
      ... Ahem. http://hjem.get2net.dk/jka/math/primegaps/maximal.htm lists the known maximal gaps. The largest ratio between the starting prime of successive
      Message 2 of 6 , Feb 5, 2008
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        julienbenney wrote:
        > The fact that the gap between 1327 and 1361 is the most persistent primal
        > gap of all, not being equalled until between 8467 and 8501 makes me think
        > of it as the most unusual prime gap that probably exists among numbers
        > of any size. A gap persisting up to numbers six times its size is unknown
        > among numbers with fewer than twenty digits at any level - I would in
        > fact doubt it occurs at any larger size of number.

        Ahem. http://hjem.get2net.dk/jka/math/primegaps/maximal.htm lists the
        known maximal gaps.
        The largest ratio between the starting prime of successive maximal gaps is
        43841547845541059 / 1693182318746371 = 25.9
        The special gap of 1132 after 1693182318746371 is the closest known
        counter example to Cramer's conjecture: http://wvwright.net/

        --
        Jens Kruse Andersen
      • Andrey Kulsha
        ... The unusuality could be measured dividing the gap length by log(p)*log(p)*sqrt(log(log(p))), where p is the prime after the gap. Thus, the most unusual
        Message 3 of 6 , Feb 5, 2008
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          > The fact that the gap between 1327 and 1361 is the most persistent primal gap of all, not
          > being equalled until between 8467 and 8501 makes me think of it as the most unusual
          > prime gap that probably exists among numbers of any size.

          The "unusuality" could be measured dividing the gap length by log(p)*log(p)*sqrt(log(log(p))), where p is the prime after the gap. Thus, the most "unusual" gaps (with unusuality larger than 0.45) are:

          6 between 23 and 29
          14 between 113 and 127
          34 between 1327 and 1361
          1132 between 1693182318746371 and 1693182318747503

          I don't count the gap of 4 between 7 and 11, the gap of 2 between 3 and 5 and the gap of 1 between 2 and 3, because log(log(p)) becomes inadmissibly small here.

          Note that the gap of 1132 is more unusual than the gap of 34, and, who knows, it may appear the most unusual gap at all!

          P.S. The factorization of the product of all 1131 composites in this gap looks as
          2^1127*3^568*5^285*7^189*11^112*13^94*17^70*19^63*23^51*29^40*31^37*37^31*41^27*43^27*47^24*53^23*59^20*61^19*67^18*71^16*73^15*79^15*83^15*89^14*97^12*101^11*103^11*107^11*109^10*113^10*127^9*131^9*137^9*139^8*149^7*151^9*157^7*163^6*167^7*173^6*179^7*181^6*191^5*193^6*197^6*199^5*211^5*223^5*227^5*229^5*233^5*239^4*241^4*251^4*257^5*263^5*269^4*271^5*277^5*281^4*283^4*293^4*307^3*311^3*313^4*317^3*331^4*337^4*347^3*349^3*353^3*359^3*367^3*373^3*379^3*383^3*389^3*397^3*401^3*409^3*419^3*421^2*431^2*433^4*439^2*443^2*449^2*457^2*461^3*463^2*467^3*479^2*487^3*491^3*499^2*503^2*509^2*521^2*523^2*541^2*547^2*557^2*563^2*569^2*571^2*577^2*587^2*593^2*599^2*601^2*607^2*613^2*617^2*619^2*631*641^2*643^2*647^2*653^2*659^2*661*673^2*677*683^2*691^2*701*709^2*719*727^2*733^2*739^2*743^2*751*757^2*761^2*769*773^2*787^2*797^2*809*811^2*821^2*823^2*827^2*829^2*839*853*857^2*859^2*863*877^2*881*883*887^2*907*911*919*929*937*941*947*953*967*971*977*983*991*997^2*1009*1013*1019*1021*1031*1033*1039*1049*1051*1061*1063*1069*1087*1091*1093*1097*1103*1109*1117*1123*1129*1151*1153*1163*1171*1181*1187*1193*1201*1217*1223*1229*1231*1237*1249*1259*1277*1279*1283*1289*1291*1297*1301*1303*1307*1319*1321*1327*1367*1373*1381*1399*1423*1427*1433*1439*1447*1451*1453*1471*1481*1483*1487*1489*1493*1499*1511*1543*1549*1553*1567*1571*1583*1597*1601*1607*1609*1613*1619*1627*1637*1657*1667*1669*1693*1699*1709*1733*1741*1753*1783*1787*1801*1811*1823*1861*1867*1871*1877*1879*1889*1901*1907*1931*1933*1949*1973*1979*1987*1993*1997*1999*2017*2027*2039*2069*2083*2111*2129*2131*2143*2153*2161*2213*2239*2243*2251*2269*2273*2281*2287*2311*2333*2339*2341*2351*2357*2371*2383*2389*2411*2417*2423*2447*2503*2521*2543*2551*2591*2593*2617*2621*2633*2647*2663*2671*2683*2693*2729*2741*2797*2801*2833*2851*2917*2971*2999*3011*3041*3049*3067*3119*3181*3187*3191*3221*3253*3299*3319*3331*3343*3371*3391*3407*3433*3449*3457*3461*3467*3499*3533*3583*3607*3613*3617*3659*3671*3697*3719*3733*3739*3769*3833*3847*3851*3889*3911*3919*4051*4091*4099*4127*4139*4177*4253*4271*4327*4391*4421*4423*4441*4447*4457*4483*4567*4591*4721*4729*4787*4793*4799*4877*4937*4967*4987*5021*5101*5153*5167*5189*5351*5417*5441*5479*5483*5501*5563*5569*5573*5623*5647*5717*5737*5743*5801*5861*5881*5939*5981*6089*6091*6199*6247*6277*6547*6571*6637*6679*6703*6709*6793*6841*6871*6899*6907*6949*6983*7043*7109*7129*7151*7177*7229*7321*7369*7457*7517*7577*7603*7757*8087*8117*8171*8387*8431*8467*8543*8669*8741*8837*8933*8969*8999*9103*9323*9413*9463*9521*9613*9619*9697*9733*9829*9851*10139*10177*10181*10337*10357*10501*10513*10529*10531*10567*10601*10657*10709*10781*10889*10949*10973*10987*10993*11087*11113*11177*11273*11443*11813*11831*11867*11959*11971*12269*12289*12409*12517*12539*12757*12823*12973*13009*13457*13499*13523*13709*13721*14029*14153*14461*14717*14779*14783*14813*14891*15139*15349*15401*15511*15649*15787*15809*16217*16267*16301*16607*16649*16661*16747*16943*16963*17027*17029*17209*17393*17477*17807*17929*17977*17987*18119*18257*18311*18379*19013*19139*19207*19531*20359*20719*20731*20749*20771*20807*20921*21139*21157*21277*21649*21673*21787*22031*22073*22193*22229*22571*22691*22769*22901*22961*23053*23291*23581*23599*23869*23957*23981*24061*24229*24691*24749*24763*24793*25439*25463*25633*25639*25703*26267*26699*26921*27011*27127*27281*27361*27527*27541*28607*28879*29201*29387*29429*29527*29671*30941*31039*31159*31391*31393*31481*31567*31573*31957*32359*32363*32843*33181*33329*33647*33829*34171*34429*34607*34667*34703*34841*35053*35227*35449*35597*36241*36263*37273*37423*37537*37657*37691*38069*38699*38917*39499*39901*40039*40127*40163*40993*41333*41941*41947*42019*42187*42443*42989*43633*43997*44777*44893*45949*46187*46273*47189*47563*47741*47791*48049*49603*50069*50849*51479*51787*52021*52121*52391*53239*53401*54139*54293*54583*55259*56543*56747*57107*57367*57373*57601*58789*59447*59699*61297*61657*62969*63029*63737*63803*63977*64399*64601*66047*67021*67073*67139*68633*69767*71129*71429*71537*73277*73571*76667*76943*77167*78059*78607*78643*79103*79579*80273*80557*82031*83207*84053*84631*87281*87509*88589*89003*91807*92899*93307*93581*95813*96797*97301*98947*99907*100559*101323*102217*102481*105251*105517*106163*110947*112799*113453*114761*115421*116491*118369*121013*121349*127681*128509*129263*129629*130003*130651*131101*132437*132851*133709*134081*136403*136987*138727*139709*139987*141601*141629*146221*146347*148429*149249*151817*151841*152111*154073*154409*154681*158261*163393*165559*165817*168353*172981*174257*175709*176159*176419*177623*177841*181607*181813*183809*185483*188189*194101*198337*199211*200689*203341*207029*211339*215249*219937*222461*227191*228251*230611*233069*233419*238759*238943*242591*246707*246811*247697*251897*252877*254377*263647*265957*267593*269237*271501*276517*277021*277789*280037*283873*284819*292793*293473*295247*296969*299191*299903*305867*308809*308851*311153*311447*316499*317857*319567*323413*323789*324953*329419*332567*336103*348709*353201*354791*361469*362221*362419*364643*367949*371087*373463*374083*380929*386437*387977*388231*389629*389723*392599*404429*404693*405577*408137*425101*435649*442789*452531*454039*454507*455269*474709*476023*478901*481681*486181*493657*495589*501503*504767*507077*509801*511831*514001*518239*521021*526739*530063*531989*558539*559067*564491*566701*572687*580711*581947*596419*604729*632669*635519*640049*647963*687031*696523*698039*699191*701479*702817*705161*709901*727471*746797*749773*756853*761347*763027*764893*765287*779231*780223*780347*794557*796921*808523*811501*813811*814199*823457*833737*843607*844867*844891*850567*869863*880283*909901*912047*931981*939613*945341*951151*955613*967229*974657*976951*979177*984913*...

          [Non-text portions of this message have been removed]
        • julienbenney
          I have had a look at your factorisation of all the 1132 composites after 1693182318746371 and it still does not seem as efficient in using all the primes
          Message 4 of 6 , Feb 6, 2008
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            I have had a look at your factorisation of all the 1132 composites after
            1693182318746371 and it still does not seem as "efficient" in using all the primes below
            the square root of that sixteen-digit number. Have you thought of, as I did in my
            demonstration, of excluding all the numbers it that gap that are divisible by 2, 3 or 5 and
            then comparing the factors?

            As it was I found 975 primes less than 975,000 occurring as factors, and that is as little as
            1.27 percent of all the primes over the duration. This is a far cry from a situation where
            every possible prime less than the square root of a number occurs! Even with the relativel
            persistent gap from 31397 to 31469 14 primes of 36 below the square root of 31469 are
            required to divide all those numbers not divisible by 2, 3 or 5.

            I am really curious as to how much less probable it becomes that every potential prime
            factor could occur over an unusually large prime gap. I do imagine that there does exist a
            difficulty combining factors in a precise manner to create a large prime gap, and that this
            difficulty increases immensely as one moves from four digits to sixteen. Nonetheless, it is
            hard to believe this difficulty increases so much as to make even 1 percent of possible
            factors being used impossible with sixteen digits when all can be used in a large four-digit
            gap is quite hard to believe.

            Is there any way of working how hard precisely combining factors to create an unusually
            large gap is??

            --- In primenumbers@yahoogroups.com, "Andrey Kulsha" <Andrey_601@...> wrote:
            >
            > > The fact that the gap between 1327 and 1361 is the most persistent primal gap of all,
            not
            > > being equalled until between 8467 and 8501 makes me think of it as the most
            unusual
            > > prime gap that probably exists among numbers of any size.
            >
            > The "unusuality" could be measured dividing the gap length by log(p)*log(p)*sqrt(log
            (log(p))), where p is the prime after the gap. Thus, the most "unusual" gaps (with
            unusuality larger than 0.45) are:
            >
            > 6 between 23 and 29
            > 14 between 113 and 127
            > 34 between 1327 and 1361
            > 1132 between 1693182318746371 and 1693182318747503
            >
            > I don't count the gap of 4 between 7 and 11, the gap of 2 between 3 and 5 and the
            gap of 1 between 2 and 3, because log(log(p)) becomes inadmissibly small here.
            >
            > Note that the gap of 1132 is more unusual than the gap of 34, and, who knows, it
            may appear the most unusual gap at all!
            >
            > P.S. The factorization of the product of all 1131 composites in this gap looks as
            >
            2^1127*3^568*5^285*7^189*11^112*13^94*17^70*19^63*23^51*29^40*31^37*37^3
            1*41^27*43^27*47^24*53^23*59^20*61^19*67^18*71^16*73^15*79^15*83^15*89^1
            4*97^12*101^11*103^11*107^11*109^10*113^10*127^9*131^9*137^9*139^8*149^7
            *151^9*157^7*163^6*167^7*173^6*179^7*181^6*191^5*193^6*197^6*199^5*211^5*
            223^5*227^5*229^5*233^5*239^4*241^4*251^4*257^5*263^5*269^4*271^5*277^5*
            281^4*283^4*293^4*307^3*311^3*313^4*317^3*331^4*337^4*347^3*349^3*353^3*
            359^3*367^3*373^3*379^3*383^3*389^3*397^3*401^3*409^3*419^3*421^2*431^2*
            433^4*439^2*443^2*449^2*457^2*461^3*463^2*467^3*479^2*487^3*491^3*499^2*
            503^2*509^2*521^2*523^2*541^2*547^2*557^2*563^2*569^2*571^2*577^2*587^2*
            593^2*599^2*601^2*607^2*613^2*617^2*619^2*631*641^2*643^2*647^2*653^2*65
            9^2*661*673^2*677*683^2*691^2*701*709^2*719*727^2*733^2*739^2*743^2*751*7
            57^2*761^2*769*773^2*787^2*797^2*809*811^2*821^2*823^2*827^2*829^2*839*85
            3*857^2*859^2*863*877^2*881*883*887^2*907*911*919*929*937*941*947*953*967*
            971*977*983*991*997^2*1009*1013*1019*1021*1031*1033*1039*1049*1051*1061*1
            063*1069*1087*1091*1093*1097*1103*1109*1117*1123*1129*1151*1153*1163*1171
            *1181*1187*1193*1201*1217*1223*1229*1231*1237*1249*1259*1277*1279*1283*12
            89*1291*1297*1301*1303*1307*1319*1321*1327*1367*1373*1381*1399*1423*1427*
            1433*1439*1447*1451*1453*1471*1481*1483*1487*1489*1493*1499*1511*1543*154
            9*1553*1567*1571*1583*1597*1601*1607*1609*1613*1619*1627*1637*1657*1667*1
            669*1693*1699*1709*1733*1741*1753*1783*1787*1801*1811*1823*1861*1867*1871
            *1877*1879*1889*1901*1907*1931*1933*1949*1973*1979*1987*1993*1997*1999*20
            17*2027*2039*2069*2083*2111*2129*2131*2143*2153*2161*2213*2239*2243*2251*
            2269*2273*2281*2287*2311*2333*2339*2341*2351*2357*2371*2383*2389*2411*241
            7*2423*2447*2503*2521*2543*2551*2591*2593*2617*2621*2633*2647*2663*2671*2
            683*2693*2729*2741*2797*2801*2833*2851*2917*2971*2999*3011*3041*3049*3067
            *3119*3181*3187*3191*3221*3253*3299*3319*3331*3343*3371*3391*3407*3433*34
            49*3457*3461*3467*3499*3533*3583*3607*3613*3617*3659*3671*3697*3719*3733*
            3739*3769*3833*3847*3851*3889*3911*3919*4051*4091*4099*4127*4139*4177*425
            3*4271*4327*4391*4421*4423*4441*4447*4457*4483*4567*4591*4721*4729*4787*4
            793*4799*4877*4937*4967*4987*5021*5101*5153*5167*5189*5351*5417*5441*5479
            *5483*5501*5563*5569*5573*5623*5647*5717*5737*5743*5801*5861*5881*5939*59
            81*6089*6091*6199*6247*6277*6547*6571*6637*6679*6703*6709*6793*6841*6871*
            6899*6907*6949*6983*7043*7109*7129*7151*7177*7229*7321*7369*7457*7517*757
            7*7603*7757*8087*8117*8171*8387*8431*8467*8543*8669*8741*8837*8933*8969*8
            999*9103*9323*9413*9463*9521*9613*9619*9697*9733*9829*9851*10139*10177*10
            181*10337*10357*10501*10513*10529*10531*10567*10601*10657*10709*10781*10
            889*10949*10973*10987*10993*11087*11113*11177*11273*11443*11813*11831*11
            867*11959*11971*12269*12289*12409*12517*12539*12757*12823*12973*13009*13
            457*13499*13523*13709*13721*14029*14153*14461*14717*14779*14783*14813*14
            891*15139*15349*15401*15511*15649*15787*15809*16217*16267*16301*16607*16
            649*16661*16747*16943*16963*17027*17029*17209*17393*17477*17807*17929*17
            977*17987*18119*18257*18311*18379*19013*19139*19207*19531*20359*20719*20
            731*20749*20771*20807*20921*21139*21157*21277*21649*21673*21787*22031*22
            073*22193*22229*22571*22691*22769*22901*22961*23053*23291*23581*23599*23
            869*23957*23981*24061*24229*24691*24749*24763*24793*25439*25463*25633*25
            639*25703*26267*26699*26921*27011*27127*27281*27361*27527*27541*28607*28
            879*29201*29387*29429*29527*29671*30941*31039*31159*31391*31393*31481*31
            567*31573*31957*32359*32363*32843*33181*33329*33647*33829*34171*34429*34
            607*34667*34703*34841*35053*35227*35449*35597*36241*36263*37273*37423*37
            537*37657*37691*38069*38699*38917*39499*39901*40039*40127*40163*40993*41
            333*41941*41947*42019*42187*42443*42989*43633*43997*44777*44893*45949*46
            187*46273*47189*47563*47741*47791*48049*49603*50069*50849*51479*51787*52
            021*52121*52391*53239*53401*54139*54293*54583*55259*56543*56747*57107*57
            367*57373*57601*58789*59447*59699*61297*61657*62969*63029*63737*63803*63
            977*64399*64601*66047*67021*67073*67139*68633*69767*71129*71429*71537*73
            277*73571*76667*76943*77167*78059*78607*78643*79103*79579*80273*80557*82
            031*83207*84053*84631*87281*87509*88589*89003*91807*92899*93307*93581*95
            813*96797*97301*98947*99907*100559*101323*102217*102481*105251*105517*10
            6163*110947*112799*113453*114761*115421*116491*118369*121013*121349*1276
            81*128509*129263*129629*130003*130651*131101*132437*132851*133709*134081
            *136403*136987*138727*139709*139987*141601*141629*146221*146347*148429*1
            49249*151817*151841*152111*154073*154409*154681*158261*163393*165559*165
            817*168353*172981*174257*175709*176159*176419*177623*177841*181607*18181
            3*183809*185483*188189*194101*198337*199211*200689*203341*207029*211339*
            215249*219937*222461*227191*228251*230611*233069*233419*238759*238943*24
            2591*246707*246811*247697*251897*252877*254377*263647*265957*267593*2692
            37*271501*276517*277021*277789*280037*283873*284819*292793*293473*295247
            *296969*299191*299903*305867*308809*308851*311153*311447*316499*317857*3
            19567*323413*323789*324953*329419*332567*336103*348709*353201*354791*361
            469*362221*362419*364643*367949*371087*373463*374083*380929*386437*38797
            7*388231*389629*389723*392599*404429*404693*405577*408137*425101*435649*
            442789*452531*454039*454507*455269*474709*476023*478901*481681*486181*49
            3657*495589*501503*504767*507077*509801*511831*514001*518239*521021*5267
            39*530063*531989*558539*559067*564491*566701*572687*580711*581947*596419
            *604729*632669*635519*640049*647963*687031*696523*698039*699191*701479*7
            02817*705161*709901*727471*746797*749773*756853*761347*763027*764893*765
            287*779231*780223*780347*794557*796921*808523*811501*813811*814199*82345
            7*833737*843607*844867*844891*850567*869863*880283*909901*912047*931981*
            939613*945341*951151*955613*967229*974657*976951*979177*984913*...
            >
            > [Non-text portions of this message have been removed]
            >
          • Mark Underwood
            ... after ... all the primes below ... as I did in my ... divisible by 2, 3 or 5 and ... and that is as little as ... from a situation where ... Even with the
            Message 5 of 6 , Feb 7, 2008
            • 0 Attachment
              --- In primenumbers@yahoogroups.com, "julienbenney" <jpbenney@...> wrote:
              >
              > I have had a look at your factorisation of all the 1132 composites
              after
              > 1693182318746371 and it still does not seem as "efficient" in using
              all the primes below
              > the square root of that sixteen-digit number. Have you thought of,
              as I did in my
              > demonstration, of excluding all the numbers it that gap that are
              divisible by 2, 3 or 5 and
              > then comparing the factors?
              >
              > As it was I found 975 primes less than 975,000 occurring as factors,
              and that is as little as
              > 1.27 percent of all the primes over the duration. This is a far cry
              from a situation where
              > every possible prime less than the square root of a number occurs!
              Even with the relativel
              > persistent gap from 31397 to 31469 14 primes of 36 below the square
              root of 31469 are
              > required to divide all those numbers not divisible by 2, 3 or 5.
              >
              > I am really curious as to how much less probable it becomes that
              every potential prime
              > factor could occur over an unusually large prime gap. I do imagine
              that there does exist a
              > difficulty combining factors in a precise manner to create a large
              prime gap, and that this
              > difficulty increases immensely as one moves from four digits to
              sixteen. Nonetheless, it is
              > hard to believe this difficulty increases so much as to make even 1
              percent of possible
              > factors being used impossible with sixteen digits when all can be
              used in a large four-digit
              > gap is quite hard to believe.
              >
              > Is there any way of working how hard precisely combining factors to
              create an unusually
              > large gap is??


              I'm supposing that there isn't such a way (now), or else we would have
              a much better grip on how large gaps can be and where to find them.

              Regarding the "efficiency" of prime factors in the gap, I noticed that
              the gap of 13 composites between the primes 113 and 127 contains all
              the primes up to and including 31.

              The gap of 33 composites between the primes 1327 and 1361 contains all
              the primes up to and including 43. This gap (I'm supposing) is the
              last gap which contains all the primes up to the square root of where
              the gap starts. So it *is* special. :)


              Personally, I think the gap of 1131 composites between the primes
              1693182318746371 and 1693182318747503 (which Jens mentioned) to be
              the most outstanding, because it has the highest gap/(ln(p1))^2 ratio
              (namely .9206).

              The conventional "merit" scale of gap/ln(p1) seems to me to be more a
              measure of the merit of those who are capable of searching high enough
              to find such gaps, rather than of the gap itself! :)

              Mark
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