相邻素数间隔规律

yangchuanju · 发表于 2023-11-7 11:16

p#-1前，p#+1后的素数间隙不小于p；
素数p的下一个素数是q时，p#-1前，p#+1后的素数间隙不小于q-1。

yangchuanju · 发表于 2023-11-7 11:31

本帖最后由 yangchuanju 于 2023-11-7 04:29 编辑

The Top-20 Prime Gaps
前20个素数间隙
http://primerecords.dk/primegaps/gaps20.htm

Top-20 merits  按M参数排序的前20个最大素数间隙
Rank Size Gap start Merit Discoverer Year
序号       间隙  小素数    M参数    发现者          年份
1 18306 P209 = 650094367*491#/2310 - 8936 38.07 Dana Jacobsen 2017
2 10716 P127 = 7910896513*283#/30 - 6480 36.86 Dana Jacobsen 2016
3 13692 P163 = 1037600971*383#/210 - 8776 36.59 Dana Jacobsen 2016
4 26892 P321 = 59740589*757#/210 - 14302 36.42 Dana Jacobsen 2016
5 11924 P147 = 4588394369*347#/30 - 7200 35.45 Dana Jacobsen 2016
6 66520 P816 = 1931*1933#/7230 - 30244 35.42 Michiel Jansen 2012
7 1476 P19 = 1425172824437699411 35.3103 Tomás Oliveira e Silva 2009
8 19474 P241 = 1485582109*571#/210 - 7576 35.20 Dana Jacobsen 2016
9 1442 P18 = 804212830686677669 34.9757 Siegfried Herzog & Tomás Oliveira e Silva 2005
10 24008 P299 = 171346649*701#/210 - 6918 34.96 Dana Jacobsen 2016
11 1550 P20 = 18361375334787046697 34.9439 Bertil Nyman 2014
12 10942 P137 = 6436615289*313#/30 - 1942 34.91 Dana Jacobsen 2016
13 18840 P236 = 962682899*563#/30 - 3918 34.76 Dana Jacobsen 2016
14 11350 P142 = 693236519*337#/210 - 2778 34.75 Dana Jacobsen 2016
15 19170 P241 = 820696571*571#/210 - 9428 34.69 Dana Jacobsen 2016
16 13704 P172 = 558020653*409#/2310 - 4196 34.68 Dana Jacobsen 2016
17 27666 P347 = 46349070720025037040361858900129... 34.66 Gapcoin 2017
18 21126 P266 = 844298401*619#/210 - 10510 34.62 Dana Jacobsen 2016
19 12176 P153 = 577924573*359#/30 - 7508 34.61 Dana Jacobsen 2016
20 22040 P278 = 1371215149*647#/30 - 9708 34.52 Dana Jacobsen 2017

Top-20 gaps with merit above 10  M参数大于10的前20个最大素数间隙
Rank Size Gap start Merit Discoverer Year
序号       间隙  小素数    M参数    发现者          年份
1 5103138 PRP216849 = 281*499979#/46410 - 2702372 10.22 Robert Smith 2016
2 4680156 PRP99750 = 230077#/2229464046810 - 3131794 20.38 Martin Raab 2016
3 3311852 PRP97953 = 226007#/2310 - 2305218 14.68 Michiel Jansen & Jens K. Andersen 2012
4 2945060 PRP99750 = 230077#/2227174919970 - 1072622 12.82 Martin Raab 2016
5 2765878 PRP100006 = 230567#/2310 - 939244 12.01 Michiel Jansen, Pierre Cami, Jens K. Andersen 2013
6 2724214 PRP100000 = 230563#/2310 - 44352 11.83 Michiel Jansen & Jens K. Andersen 2013
7 2586246 PRP99750 = 230077#/2226554139810 - 1616044 11.26 Martin Raab 2016
8 2559528 PRP99750 = 230077#/2228280684630 - 961364 11.14 Martin Raab 2016
9 2493532 PRP99750 = 230077#/2227349514390 - 1441218 10.86 Martin Raab 2016
10 2435476 PRP99750 = 230077#/2226961526790 - 994222 10.60 Martin Raab 2016
11 2254930 PRP86853 = 11224835119429687764118930339136... 11.28 Hans Rosenthal & Jens K. Andersen 2004
12 2055816 PRP56962 = 6887*(131591#)/2730 - 1381994 15.67 Pierre Cami 2010
13 1575828 PRP45334 = 104729#/2310 - 1282742 15.10 Michiel Jansen 2012
14 1462522 PRP39448 = 91229#/46056680670 - 853776 16.10 Martin Raab 2015
15 1452592 PRP43291 = 451*99991#/46410 - 1136946 14.57 Robert Smith 2016
16 1286500 PRP21608 = 1111111111111111111*49999#/(510510*499) - 525318 25.86 Hans Rosenthal 2016
17 1224222 PRP43293 = 8263*99991#/46410 - 600164 12.28 Robert Smith 2016
18 1217460 PRP39448 = 91229#/46093437390 - 495038 13.40 Martin Raab 2015
19 1176666 PRP39443 = 91199#/46473256830 - 547454 12.96 Martin Raab 2015
20 1117820 PRP43291 = 149*99991#/46410 - 813426 11.21 Robert Smith 2016

Top-20 gaps with merit above 20  M参数大于20的前20个最大素数间隙
Rank Size Gap start Merit Discoverer Year
序号       间隙  小素数    M参数    发现者          年份
1 4680156 PRP99750 = 230077#/2229464046810 - 3131794 20.38 Martin Raab 2016
2 1286500 PRP21608 = 1111111111111111111*49999#/(510510*499) - 525318 25.86 Hans Rosenthal 2016
3 1113106 P18662 = 587*43103#/2310 - 455704 25.90 Michiel Jansen, Pierre Cami, Jens K. Andersen 2013
4 984108 PRP16901 = 39161#/2310 - 510478 25.29 Michiel Jansen 2012
5 973764 PRP18648 = 431*43063#/2310 - 278398 22.68 Michiel Jansen, Pierre Cami, Jens K. Andersen 2013
6 865056 PRP18636 = 44633*(43037#)/2310 - 394442 20.16 Pierre Cami 2013
7 834114 PRP16497 = 38231#/2310 - 393706 21.96 Michiel Jansen 2012
8 818886 PRP17071 = 39541#/2310 - 380216 20.83 Michiel Jansen 2012
9 784246 PRP16979 = 39323#/2310 - 490362 20.06 Michiel Jansen 2012
10 571948 PRP12411 = 19*28751#/30 - 295468 20.01 Dana Jacobsen 2016
11 566040 PRP10449 = 24137#/2310 - 311774 23.53 Michiel Jansen 2012
12 538328 PRP10699 = 24821#/2310 - 362006 21.85 Michiel Jansen 2012
13 535836 PRP10569 = 24469#/30 - 374362 22.02 Dana Jacobsen 2015
14 440020 PRP9527 = 19*22091#/30 - 297762 20.06 Dana Jacobsen 2015
15 418250 PRP7713 = 11*17923#/46410 - 156344 23.55 Robert Smith 2015
16 384902 PRP7473 = 17383#/30 - 77702 22.37 Michiel Jansen 2012
17 383796 P7183 = 9527*(16673#)/2310 - 175622 23.21 Pierre Cami 2010
18 379304 PRP6261 = 79*14593#/46410 - 129600 26.31 Robert Smith 2015
19 360072 PRP7790 = 29*18089#/46410 - 190054 20.08 Robert Smith 2015
20 358374 PRP7581 = 17599#/2310 - 116826 20.53 Michiel Jansen 2012

Gaps which are the largest with at least that merit  间隙最大，M参数最小数表
Size Gap start Merit Discoverer Year
间隙  小素数    M参数    发现者          年份
5103138 PRP216849 = 281*499979#/46410 - 2702372 10.22 Robert Smith 2016
4680156 PRP99750 = 230077#/2229464046810 - 3131794 20.38 Martin Raab 2016
1286500 PRP21608 = 1111111111111111111*49999#/(510510*499) - 525318 25.86 Hans Rosenthal 2016
1113106 P18662 = 587*43103#/2310 - 455704 25.90 Michiel Jansen, Pierre Cami, Jens K. Andersen 2013
379304 PRP6261 = 79*14593#/46410 - 129600 26.31 Robert Smith 2015
309030 PRP4223 = 1111111111111111111*9787#/(7#*641) - 130308 31.78 Hans Rosenthal 2015
157178 PRP2145 = 422569*5009#/30 - 96332 31.83 Dana Jacobsen 2016
120664 PRP1622 = 4999*3803#/510510 - 71716 32.32 Michiel Jansen, Pierre Cami, Jens K. Andersen 2013
71046 P948 = 961*2267#/39669630 - 53540 32.56 Dana Jacobsen 2015
66520 P816 = 1931*1933#/7230 - 30244 35.42 Michiel Jansen 2012
26892 P321 = 59740589*757#/210 - 14302 36.42 Dana Jacobsen 2016
18306 P209 = 650094367*491#/2310 - 8936 38.07 Dana Jacobsen 2017

yangchuanju · 发表于 2023-11-7 11:41

本帖最后由 yangchuanju 于 2023-11-7 04:30 编辑

The merit of the prime gap is size / ln p1, where ln is the natural logarithm. Some people use p2 or a number between p1 and p2, but the difference is microscopic for large primes.

素数间隙中的merit（M参数）是——间隙/ln(p1)。

yangchuanju · 发表于 2023-11-7 12:05

本帖最后由 yangchuanju 于 2023-11-7 04:09 编辑

Let P(g) be the least prime such that P(g)+g is the smallest prime larger than P(g). The values of P(g) are bounded, for our empirical data, by the functions
                     0.5                                  0.5
               0.5 g                               0.5 g
P_min(g) = 0.12  g    e       and P_max(g) = 30.83  g    e    .

令P(g)是一个小素数，P(g)+g是一个比P(g)大的最小素数，则
具有间隙g的最小素数P-min(g)=  0.12*g^0.5*e^(g^0.5)，
具有间隙g的最大素数P-max(g)=30.83*g^0.5*e^(g^0.5)。

yangchuanju · 发表于 2023-11-7 12:36

Top 50 前50个素数间隙
The following table presents the 50 largest g (prime gaps) found so far, with P(g)<4·10^18, either by contributors to this project or by other discoverers (those have an * before the discoverer name). Repeated values of g were excluded from this list.

  Rank       g       P(g)       Discoverer
序号间隙       最小素数    发现者
  1       1476       1425 17282 44376 99411       Tomás Oliveira e Silva
  2       1454       3219 10718 24928 71783       Silvio Pardi
  3       1442       804 21283 06866 77669       Siegfried "Zig" Herzog
  4       1418       3725 23553 35041 01511       Tomás Oliveira e Silva
  5       1416       3750 99252 93399 78877       Tomás Oliveira e Silva
  6       1410       2635 28193 24815 39903       Siegfried "Zig" Herzog
  7       1400       3431 65779 58583 78003       Silvio Pardi
  8       1398       2424 70872 97267 67749       Tomás Oliveira e Silva
  9       1392       1480 03203 79396 34731       Tomás Oliveira e Silva
  10       1390       3492 65766 10051 61107       Silvio Pardi
  11       1388       2975 20552 41233 01149       Tomás Oliveira e Silva
  12       1386       3449 34008 02746 51527       Silvio Pardi
  13       1380       1031 50183 31302 43273       Siegfried "Zig" Herzog
  14       1374       2812 81423 52816 09869       Silvio Pardi
  15       1370       418 03264 59367 12127       * D. Knuth
  16       1364       1051 14088 80512 30423       Siegfried "Zig" Herzog
  17       1362       3937 45795 06462 69397       Tomás Oliveira e Silva
  18       1360       1153 27764 73035 40597       Siegfried "Zig" Herzog
  19       1358       523 25522 06146 45319       Siegfried "Zig" Herzog
  20       1356       401 42992 59991 53707       * D. Knuth
  21       1354       2805 56288 34484 62853       Silvio Pardi
  22       1350       1180 35175 22041 37089       Tomás Oliveira e Silva
  23       1348       3754 93042 77306 28273       Tomás Oliveira e Silva
  24       1344       753 91763 53808 95597       Siegfried "Zig" Herzog
  25       1342       1578 16910 61411 87727       Tomás Oliveira e Silva
  26       1340       1954 31746 71273 10787       Tomás Oliveira e Silva
  27       1338       2927 64805 72684 73633       Tomás Oliveira e Silva
  28       1336       2221 89928 82387 68093       Siegfried "Zig" Herzog
  29       1334       2722 01256 97476 81653       Silvio Pardi
  30       1332       3103 25146 45367 97599       Silvio Pardi
  31       1330       2918 47807 68491 34103       Tomás Oliveira e Silva
  32       1328       352 52122 34513 64323       Tomás Oliveira e Silva
  33       1326       2075 06944 41432 79153       Tomás Oliveira e Silva
  34       1324       2937 02084 97138 56977       Tomás Oliveira e Silva
  35       1322       1106 02843 61874 67937       Siegfried "Zig" Herzog
  36       1320       605 04633 00290 26447       Siegfried "Zig" Herzog
  37       1318       2216 27085 98031 42601       Siegfried "Zig" Herzog
  38       1316       2187 39466 98657 44117       Siegfried "Zig" Herzog
  39       1314       1214 11964 65476 13277       Tomás Oliveira e Silva
  40       1312       1396 48771 92534 03577       Siegfried "Zig" Herzog
  41       1310       1918 58679 96576 17591       Tomás Oliveira e Silva
  42       1308       749 56545 75543 71299       Siegfried "Zig" Herzog
  43       1306       3278 01806 91024 80227       Silvio Pardi
  44       1304       1188 13437 93823 95323       Tomás Oliveira e Silva
  45       1302       1731 08087 63942 28577       Tomás Oliveira e Silva
  46       1300       1141 04186 63848 33123       Siegfried "Zig" Herzog
  47       1298       1739 01802 68170 67239       Tomás Oliveira e Silva
  48       1296       1799 23519 83799 03447       Christian Kern
  49       1294       1868 32618 47640 55803       Siegfried "Zig" Herzog
  50       1292       494 65339 43054 48051       Tomás Oliveira e Silva

重生888@ · 发表于 2023-11-7 15:06

不用验证，怎样证明偶数994哥猜必定成立？

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