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Is there another prime gap...

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  • julienbenney
    in which so many of the possible prime factors are used as in the gap between 1327 and 1361. If you factorise all the numbers not divisibel by 2, 3, or 5
    Message 1 of 6 , Feb 5, 2008
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      in which so many of the possible prime factors are used as in the gap between 1327 and
      1361. If you factorise all the numbers not divisibel by 2, 3, or 5 between 1330 and 1370
      (just after the gap) every prime up to 37 occurs:

      - 11 divides 1331
      - 31 divides 1333
      - 7 divides 1337 and 1351
      - 13 divides 1339
      - 17 divides 1343
      - 19 divides 1349
      - 23 divides 1357
      - 29 divides 1363
      - 37 (squared) divides 1369

      What has always interested me is how large a prime gap would be possible were all the
      primes below the square root of a number to occur in near-successive fashion as occurs in
      the numbers between 1330 and 1360. I have walsys imagined that it would be possible to
      achieve (theoretically, of course) much larger gaps than actually occur between larger
      numbers.

      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.
    • Andrey Kulsha
      ... gap of all, not ... most unusual ... up to ... digits at ... http://www.trnicely.net/#TPG Best, Andrey
      Message 2 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 3 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 4 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 5 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 6 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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