below is a list that includes record ("maximal") prime gaps, gaps that tie the record up until that point, and gaps that achieve or tie for second place up until that point.
Pari/GP source code:
g1=0; g2=0; oldp=2; forprime(p=3, +oo, newgap=p-oldp; if(newgap>=g2, print(oldp" "p" "newgap); if(newgap>g1, g2=g1; g1=newgap, if(newgap<g1, g2=newgap))); oldp=p)
also given in the fourth column is the "merit" newgap/log(oldp), and in the fifth column the Cramer-Shanks-Granville ratio newgap/(log(oldp))^2. these values were calculated afterwards, not by the Pari/GP script.
the merits seem to generally increase on this list, as do the CSG ratios. is this true? are there an infinite number of maximal prime gaps whose merit is less than (say) 10, or whose CSG ratio is less than (say) 0.5? if so, that would stymie efforts to parallelize the search, recording only interesting prime gaps and collating them after all parallel threads have completed. on the other hand if the merit and CSG ratio steadily increase, we would like to know their growth rates, in order to decrease noise during (hypothetical) parallel search.
there are probably good ways to avoid calling floating-point log for each new prime gap.
previously, factoring the composites in a large prime gap.
columns are:
- first prime
- last prime
- gap length
- merit
- merit/ln(first prime)
2 3 1 1.44 2.08
3 5 2 1.82 1.66
5 7 2 1.24 0.77
7 11 4 2.06 1.06
11 13 2 0.83 0.35
13 17 4 1.56 0.61
17 19 2 0.71 0.25
19 23 4 1.36 0.46
23 29 6 1.91 0.61
31 37 6 1.75 0.51
37 41 4 1.11 0.31
43 47 4 1.06 0.28
47 53 6 1.56 0.40
53 59 6 1.51 0.38
61 67 6 1.46 0.36
67 71 4 0.95 0.23
73 79 6 1.40 0.33
79 83 4 0.92 0.21
83 89 6 1.36 0.31
89 97 8 1.78 0.40
113 127 14 2.96 0.63
139 149 10 2.03 0.41
181 191 10 1.92 0.37
199 211 12 2.27 0.43
211 223 12 2.24 0.42
293 307 14 2.46 0.43
317 331 14 2.43 0.42
467 479 12 1.95 0.32
509 521 12 1.93 0.31
523 541 18 2.88 0.46
773 787 14 2.11 0.32
839 853 14 2.08 0.31
863 877 14 2.07 0.31
887 907 20 2.95 0.43
1069 1087 18 2.58 0.37
1129 1151 22 3.13 0.45
1327 1361 34 4.73 0.66
1669 1693 24 3.23 0.44
2179 2203 24 3.12 0.41
2477 2503 26 3.33 0.43
2971 2999 28 3.50 0.44
3271 3299 28 3.46 0.43
4297 4327 30 3.59 0.43
4831 4861 30 3.54 0.42
5351 5381 30 3.49 0.41
5591 5623 32 3.71 0.43
8467 8501 34 3.76 0.42
9551 9587 36 3.93 0.43
9973 10007 34 3.69 0.40
11743 11777 34 3.63 0.39
12163 12197 34 3.61 0.38
12853 12889 36 3.80 0.40
14107 14143 36 3.77 0.39
15683 15727 44 4.55 0.47
15823 15859 36 3.72 0.39
16141 16183 42 4.33 0.45
19609 19661 52 5.26 0.53
25471 25523 52 5.13 0.51
28229 28277 48 4.68 0.46
31397 31469 72 6.95 0.67
34061 34123 62 5.94 0.57
89689 89753 64 5.61 0.49
107377 107441 64 5.52 0.48
134513 134581 68 5.76 0.49
155921 156007 86 7.19 0.60
188029 188107 78 6.42 0.53
265621 265703 82 6.57 0.53
338033 338119 86 6.76 0.53
360653 360749 96 7.50 0.59
370261 370373 112 8.74 0.68
396733 396833 100 7.76 0.60
492113 492227 114 8.70 0.66
1349533 1349651 118 8.36 0.59
1357201 1357333 132 9.35 0.66
1561919 1562051 132 9.26 0.65
1671781 1671907 126 8.79 0.61
1889831 1889957 126 8.72 0.60
2010733 2010881 148 10.20 0.70
3826019 3826157 138 9.10 0.60
4652353 4652507 154 10.03 0.65
7230331 7230479 148 9.37 0.59
8421251 8421403 152 9.53 0.60
11113933 11114087 154 9.49 0.59
15203977 15204131 154 9.31 0.56
17051707 17051887 180 10.81 0.65
17983717 17983873 156 9.34 0.56
20285099 20285263 164 9.75 0.58
20831323 20831533 210 12.46 0.74
36271601 36271783 182 10.46 0.60
46006769 46006967 198 11.22 0.64
47326693 47326913 220 12.45 0.70
90438133 90438343 210 11.46 0.63
122164747 122164969 222 11.92 0.64
189695659 189695893 234 12.28 0.64
191912783 191913031 248 13.00 0.68
216668603 216668839 236 12.30 0.64
367876529 367876771 242 12.27 0.62
387096133 387096383 250 12.64 0.64
428045491 428045741 250 12.58 0.63
436273009 436273291 282 14.18 0.71
516540163 516540413 250 12.46 0.62
630045137 630045389 252 12.44 0.61
649580171 649580447 276 13.60 0.67
1294268491 1294268779 288 13.73 0.65
1453168141 1453168433 292 13.84 0.66
1692326849 1692327137 288 13.55 0.64
1948819133 1948819423 290 13.56 0.63
2114305601 2114305891 290 13.51 0.63
2300942549 2300942869 320 14.84 0.69
2433630109 2433630413 304 14.07 0.65
3842610773 3842611109 336 15.22 0.69
4275912661 4275912997 336 15.15 0.68
4302407359 4302407713 354 15.96 0.72
8605261447 8605261787 340 14.86 0.65
10420857673 10420858013 340 14.74 0.64
10726904659 10726905041 382 16.54 0.72
11705477863 11705478217 354 15.27 0.66
16161669787 16161670163 376 16.00 0.68
20678048297 20678048681 384 16.17 0.68
22367084959 22367085353 394 16.53 0.69
25056082087 25056082543 456 19.04 0.80
36172730063 36172730507 444 18.26 0.75
42652618343 42652618807 464 18.96 0.77
127976334671 127976335139 468 18.30 0.72
182226896239 182226896713 474 18.28 0.71
241160624143 241160624629 486 18.54 0.71
297501075799 297501076289 490 18.55 0.70
303371455241 303371455741 500 18.91 0.72
304599508537 304599509051 514 19.44 0.74
416608695821 416608696337 516 19.29 0.72
update: we search for prime gaps whose CSG ratio exceeds 0.7.
the code below illustrates how to avoid computing a logarithm for every prime gap. the "reset" lines are instances of gaps which don't quite exceed the threshold but induced a call to log to the compute a new threshold. we parallelized the search in chunks of 10^12, so resets that occurred between chunks are not listed in the output below.
oldp=2; thresh=0; forprime(p=3, +oo, newgap=p-oldp; if(newgap>thresh, l=log(oldp)^2; newthresh=0.7*l; if(newgap>newthresh, print(oldp" "p" "newgap" "newgap/l), print("reset "oldp" "thresh" "newgap" "newthresh)); thresh=floor(newthresh)); oldp=p)
gaps not listed above start appearing at 63816175447:
42652618343 42652618807 464 0.77450550917510106283898165530310145891
63816175447 63816175897 450 0.72700460026485189234254814344900404585
reset 101328529441 433 438 449.53888403776908677348047135064719134
127976334671 127976335139 468 0.71550190064280205206726129619634528501
131956235563 131956236023 460 0.70158985315822980713847897278610067860
reset 148473908887 458 460 463.19522180780304931246179264144094677
182226896239 182226896713 474 0.70505490514034954724945024999568966404
241160624143 241160624629 486 0.70752916656704371883122661164782236260
297501075799 297501076289 490 0.70205919248542594891246853930689529100
303371455241 303371455741 500 0.71532837838938398432416433408115084798
304599508537 304599509051 514 0.73513289437876846212218346846413021856
416608695821 416608696337 516 0.72081924871050504611664927563569497575
461690510011 461690510543 532 0.73749503203690276859525575553476210934
614487453523 614487454057 534 0.72475600873340796557679856426917046343
738832927927 738832928467 540 0.72304833425828650820311194969265061448
1346294310749 1346294311331 582 0.74615935559780324306159196779729964364
1408695493609 1408695494197 588 0.75141171068636691753662939458953311422
1480064231153 1480064231717 564 0.71820189417775311082224708369207221396
1769203875643 1769203876207 564 0.70914186179536510230568455972583107995
1968188556461 1968188557063 602 0.75123183484167506862091672869393379970
reset 2081209441279 561 562 563.16036793714184663245614702674442335
reset 2370659939419 563 564 568.34318419526495714283905742440471558
2614941710599 2614941711251 652 0.79753644510442432635173641072734875057
3410069454097 3410069454689 592 0.71088059543911277903218020098717671524
4165633395149 4165633395767 618 0.73191448230710788948328506668960705578
4872634110067 4872634110667 600 0.70299089923994787281153407459362276726
5120731250207 5120731250857 650 0.75899083362902819095810369731993698612
5439564948583 5439564949187 604 0.70237522555254898565725472105856344232
7177162611713 7177162612387 674 0.76916582817817330898294465591279714196
7753325440403 7753325441053 650 0.73792231761442722186985063699533690456
9483480841753 9483480842389 636 0.71232680321689097648138536205342607944
9787731507761 9787731508409 648 0.72423535466862031219658369631941001719
might be a subset of OEIS A211075 (found by google search of 1769203875643).
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