### Maximal gaps between prime quintuplets

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

Prime quintuplets (5-tuples) are the densest permissible clusters of 5 consecutive primes. There are two types of prime quintuplets:

• {p, p+2, p+6, p+8, p+12} (OEIS A022006, A201073, A201074, A233432)
• {p, p+4, p+6, p+10, p+12} (OEIS A022007, A201062, A201063, A233433).

The observed maximal gaps between prime quintuplets 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 g5(p))  =  a(log(p/a) − 2/5)  =  O(log6p)
where a = 0.098699 log5p is the average gap, as predicted by the Hardy-Littlewood k-tuple conjecture.

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

#### Maximal gaps between prime quintuplets {p, p+2, p+6, p+8, p+12}

```1st quintuplet:   2nd quintuplet:   Gap g5(p):
5               11             6
11              101            90
101             1481          1380
1481            16061         14580
22271            43781         21510
55331           144161         88830
536441           633461         97020
661091           768191        107100
1461401          1573541        112140
1615841          1917731        301890
5527001          5928821        401820
11086841         11664551        577710
35240321         35930171        689850
53266391         54112601        846210
72610121         73467131        857010
92202821         93188981        986160
117458981        119114111       1655130
196091171        198126911       2035740
636118781        638385101       2266320
975348161        977815451       2467290
1156096301       1158711011       2614710
1277816921       1281122231       3305310
1347962381       1351492601       3530220
2195593481       2199473531       3880050
3128295551       3132180971       3885420
4015046591       4020337031       5290440
8280668651       8286382451       5713800
9027127091       9033176981       6049890
15686967971      15693096311       6128340
18901038971      18908988011       7949040
21785624291      21793595561       7971270
30310287431      30321057581      10770150
107604759671     107616100511      11340840
140760439991     140772689501      12249510
162661360481     162673773671      12413190
187735329491     187749510491      14181000
327978626531     327994719461      16092930
508259311991     508275672341      16360350
620537349191     620554105931      16756740
667672901711     667689883031      16981320
1079628551621    1079646141851      17590230
1104604933841    1104624218981      19285140
1182148717481    1182168243071      19525590
1197151034531    1197173264711      22230180
2286697462781    2286720012251      22549470
2435950632251    2435980618781      29986530
3276773115431    3276805283951      32168520
5229301162991    5229337555061      36392070
9196865051651    9196903746881      38695230
14660925945221   14660966101421      40156200
21006417451961   21006458070461      40618500
22175175736991   22175216733491      40996500
22726966063091   22727007515411      41452320
22931291089451   22931338667591      47578140
31060723328351   31060771959221      48630870
85489258071311   85489313115881      55044570
90913430825291   90913489290971      58465680
96730325054171   96730390102391      65048220
199672700175071  199672765913051      65737980
275444947505591  275445018294491      70788900
331992774272981  331992848243801      73970820
465968834865971  465968914851101      79985130
686535413263871  686535495684161      82420290
761914822198961  761914910291531      88092570

```

#### Maximal gaps between prime quintuplets {p, p+4, p+6, p+10, p+12}

```1st quintuplet:   2nd quintuplet:   Gap g5(p):
7               97            90
97             1867          1770
3457             5647          2190
5647            15727         10080
19417            43777         24360
43777            79687         35910
101107           257857        156750
1621717          1830337        208620
3690517          3995437        304920
5425747          5732137        306390
8799607          9127627        328020
9511417          9933607        422190
16388917         16915267        526350
22678417         23317747        639330
31875577         32582437        706860
37162117         38028577        866460
64210117         65240887       1030770
119732017        120843637       1111620
200271517        201418957       1147440
203169007        204320107       1151100
241307107        242754637       1447530
342235627        344005297       1769670
367358347        369151417       1793070
378200227        380224837       2024610
438140947        440461117       2320170
446609407        448944487       2335080
711616897        714020467       2403570
966813007        970371037       3558030
2044014607       2048210107       4195500
3510456787       3514919917       4463130
4700738167       4705340527       4602360
5798359657       5803569847       5210190
7896734467       7902065527       5331060
12654304207      12659672737       5368530
13890542377      13896088897       5546520
14662830817      14668797037       5966220
15434185927      15440743597       6557670
17375054227      17381644867       6590640
17537596327      17544955777       7359450
25988605537      25997279377       8673840
66407160637      66416495137       9334500
74862035617      74871605947       9570330
77710388047      77723371717      12983670
144124106167     144138703987      14597820
210222262087     210238658797      16396710
585234882097     585252521167      17639070
926017532047     926036335117      18803070
986089952917     986113345747      23392830
2819808136417    2819832258697      24122280
3013422626107    3013449379477      26753370
3538026326827    3538053196957      26870130
4674635167747    4674662545867      27378120
5757142722757    5757171559957      28837200
7464931087717    7464961813867      30726150
8402871269197    8402904566467      33297270
9292699799017    9292733288557      33489540
10985205390997   10985239010737      33619740
12992848206847   12992884792957      36586110
15589051692667   15589094176627      42483960
24096376903597   24096421071127      44167530
37371241083097   37371285854467      44771370
38728669335607   38728728308527      58972920
91572717670537   91572784840627      67170090
109950817237357  109950886775827      69538470
325554440818297  325554513360487      72542190
481567288596127  481567361629087      73032960
501796510663237  501796584764467      74101230
535243109721577  535243185965557      76243980
657351798174427  657351876771637      78597210
818872754682547  818872840949077      86266530
991851356676277  991851464273767     107597490
```
The ratio g5(p)/log6p is never greater than 0.0987, 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.