Maximal gaps between prime triplets

© 2011-2013 by Alexei Kourbatov, JavaScripter.net/math
Main article: Maximal gaps between prime k-tuples

Prime triplets (3-tuples) are the densest permissible clusters of 3 consecutive primes. There are two types of prime triplets:

  • {p, p+2, p+6} (OEIS A022004, A201598, A201599, A233434)
  • {p, p+4, p+6} (OEIS A022005, A201596, A201597, A233435).

    The observed maximal gaps between prime triplets near p are at most log p times the average gap.
    The approximate size of a maximal gap that ends at p is given by the following empirical formula:

    E(max g3(p))  =  a(log(p/a) − 2/3)  =  O(log4p)  
    where a = 0.34986 log3p is the average gap, as predicted by the Hardy-Littlewood k-tuple conjecture.

    Maximal gaps between prime triplets of each type are listed below.

    Maximal gaps between prime triplets {p, p+2, p+6}

       1st triplet:    2nd triplet:   Gap g3(p): 
                  5              11           6
                 17              41          24
                 41             101          60
                107             191          84
                347             461         114
                461             641         180
                881            1091         210
               1607            1871         264
               2267            2657         390
               2687            3251         564
               6197            6827         630
               6827            7877        1050
              39227           40427        1200
              46181           47711        1530
              56891           58907        2016
              83267           86111        2844
             167621          171047        3426
             375251          379007        3756
             381527          385391        3864
             549161          553097        3936
             741677          745751        4074
             805031          809141        4110
             931571          937661        6090
            2095361         2103611        8250
            2428451         2437691        9240
            4769111         4778381        9270
            4938287         4948631       10344
           12300641        12311147       10506
           12652457        12663191       10734
           13430171        13441091       10920
           14094797        14107727       12930
           18074027        18089231       15204
           29480651        29500841       20190
          107379731       107400017       20286
          138778301       138799517       21216
          156377861       156403607       25746
          242419361       242454281       34920
          913183487       913222307       38820
         1139296721      1139336111       39390
         2146630637      2146672391       41754
         2188525331      2188568351       43020
         3207540881      3207585191       44310
         3577586921      3577639421       52500
         7274246711      7274318057       71346
        33115389407     33115467521       78114
        97128744521     97128825371       80850
        99216417017     99216500057       83040
       103205810327    103205893751       83424
       133645751381    133645853711      102330
       373845384527    373845494147      109620
       412647825677    412647937127      111450 
       413307596957    413307728921      131964
      1368748574441   1368748707197      132756
      1862944563707   1862944700711      137004
      2368150202501   2368150349687      147186
      2370801522107   2370801671081      148974
      3710432509181   3710432675231      166050
      5235737405807   5235737580317      174510
      8615518909601   8615519100521      190920
     10423696470287  10423696665227      194940
     10660256412977  10660256613551      200574
     11602981439237  11602981647011      207774
     21824373608561  21824373830087      221526
     36385356561077  36385356802337      241260
     81232357111331  81232357386611      275280
    186584419495421 186584419772321      276900
    297164678680151 297164678975621      295470
    428204300934581 428204301233081      298500
    450907041535541 450907041850547      315006
    464151342563471 464151342898121      334650
    484860391301771 484860391645037      343266
    666901733009921 666901733361947      352026
    
    

    Maximal gaps between prime triplets {p, p+4, p+6}

       1st triplet:    2nd triplet:   Gap g3(p): 
                  7              13           6
                 13              37          24
                 37              67          30
                103             193          90
                307             457         150
                457             613         156
                613             823         210
               2137            2377         240
               2377            2683         306
               2797            3163         366
               3463            3847         384
               4783            5227         444
               5737            6547         810
               9433           10267         834
              14557           15643        1086
              24103           25303        1200
              45817           47143        1326
              52177           54493        2316
             126487          130363        3876
             317587          321817        4230
             580687          585037        4350
             715873          724117        8244
            2719663         2728543        8880
            6227563         6237013        9450
            8114857         8125543       10686
           10085623        10096573       10950
           10137493        10149277       11784
           18773137        18785953       12816
           21297553        21311107       13554
           25291363        25306867       15504
           43472497        43488073       15576
           52645423        52661677       16254
           69718147        69734653       16506
           80002627        80019223       16596
           89776327        89795773       19446
           90338953        90358897       19944
          109060027       109081543       21516
          148770907       148809247       38340
         1060162843      1060202833       39990
         1327914037      1327955593       41556
         2562574867      2562620653       45786
         2985876133      2985923323       47190
         4760009587      4760057833       48246
         5557217797      5557277653       59856
        10481744677     10481806897       62220
        19587414277     19587476563       62286
        25302582667     25302648457       65790
        30944120407     30944191387       70980
        37638900283     37638972667       72384
        49356265723     49356340387       74664
        49428907933     49428989167       81234
        70192637737     70192720303       82566
        74734558567     74734648657       90090
       111228311647    111228407113       95466
       134100150127    134100250717      100590
       195126585733    195126688957      103224
       239527477753    239527584553      106800
       415890988417    415891106857      118440
       688823669533    688823797237      127704
       906056631937    906056767327      135390
       926175746857    926175884923      138066
      1157745737047   1157745878893      141846
      1208782895053   1208783041927      146874
      2124064384483   2124064533817      149334
      2543551885573   2543552039053      153480
      4321372168453   4321372359523      191070
      6136808604343   6136808803753      199410
     18292411110217  18292411310077      199860
     19057076066317  19057076286553      220236
     21794613251773  21794613477097      225324
     35806145634613  35806145873077      238464
     75359307977293  75359308223467      246174
     89903831167897  89903831419687      251790
    125428917151957 125428917432697      280740
    194629563521143 194629563808363      287220
    367947033766573 367947034079923      313350
    376957618687747 376957619020813      333066
    483633763994653 483633764339287      344634
    539785800105313 539785800491887      386574
    
    The ratio g3(p)/log4p is never greater than 0.35, i.e. maximal gap sizes are less than log p times the average gap, where p is the prime at the end of the gap.

    Copyright © 2011-2013, Alexei Kourbatov, JavaScripter.net.