### 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.