相邻素数间隔规律

yangchuanju · 发表于 2025-8-31 21:13

在整数序列在线百科全书OEIS中网页A005250和A002386分别给出83条
相邻二素数的间隙记录（第一次出现的大间隙）和对应的间隙记录前的小素数；
查阅和分析互联网上有关素数间隙的论文，比较可靠的间隙计算式有：
1、间隙D<ln(p)^2，p——间隙D前的小素数；
2、当素数p相当大时间隙D趋近于ln(p)^2*0.61*ln ln ln p；
3、当素数p相当大时间隙D趋近于ln(p)^2*(1+1/1ln ln ln p)。
2和3式中的两个校正系数哪个好用，条件不详；
2的系数0.61*ln ln ln p由零点几逐渐增大到10^66时的0.78，10^76时的1.0，再最大到10^300时的1.14；
3的系数1+1/ln ln ln p由3点几逐渐减小到10^76时的178，10^76时的1.60，再减小到10^300时的1.53。
p 0.61*ln ln ln p 1+1/1ln ln ln p
10 -0.110704614 -4.510158759
100 0.258287818 3.361706428
1000 0.40192253 2.517705415
10000 0.486569181 2.253675785
100000 0.544985777 2.119295266
1000000 0.588883345 2.035858809
10000000 0.62368247 1.978061802
100000000 0.652301169 1.935150861
1000000000 0.676476209 1.901731637
10000000000 0.697318451 1.874779663
1E+11 0.715578053 1.85245767
1E+12 0.731783561 1.833579807
1E+13 0.746320309 1.817343428
1E+14 0.75947722 1.803184064
1E+15 0.771476072 1.790692054
1E+16 0.782490535 1.779562145
1E+17 0.792658958 1.769561732
1E+18 0.802093205 1.76051012
1E+19 0.810884921 1.752264574
1E+20 0.819110056 1.744710672
1E+21 0.826832205 1.737755492
1E+22 0.834105114 1.731322695
1E+23 0.840974584 1.725348913
1E+24 0.847479938 1.71978105
1E+25 0.853655176 1.714574242
1E+26 0.859529875 1.709690283
1E+27 0.865129915 1.705096413
1E+28 0.870478064 1.700764356
1E+29 0.875594446 1.696669563
1E+30 0.880496933 1.692790602
1E+31 0.885201462 1.689108668
1E+32 0.889722301 1.685607183
1E+33 0.894072271 1.682271467
1E+34 0.898262932 1.67908847
1E+35 0.902304743 1.676046541
1E+36 0.90620719 1.673135246
1E+37 0.909978908 1.670345208
1E+38 0.913627773 1.66766797
1E+39 0.917160991 1.665095884
1E+40 0.920585167 1.662622017
1E+41 0.923906374 1.66024006
1E+42 0.927130203 1.657944265
1E+43 0.930261813 1.655729378
1E+44 0.933305978 1.653590585
1E+45 0.936267117 1.651523469
1E+46 0.939149332 1.649523967
1E+47 0.941956439 1.647588333
1E+48 0.944691988 1.645713108
1E+49 0.947359292 1.643895094
1E+50 0.949961443 1.642131325
1E+51 0.952501336 1.640419049
1E+52 0.95498168 1.638755709
1E+53 0.957405016 1.637138922
1E+54 0.959773732 1.635566467
1E+55 0.96209007 1.634036271
1E+56 0.964356142 1.632546394
1E+57 0.966573939 1.63109502
1E+58 0.968745338 1.62968045
1E+59 0.97087211 1.628301085
1E+60 0.972955933 1.626955424
1E+61 0.974998391 1.625642058
1E+62 0.977000986 1.624359656
1E+63 0.978965142 1.623106967
1E+64 0.980892208 1.621882807
1E+65 0.982783468 1.620686061
1E+66 0.98464014 1.619515674
1E+67 0.986463383 1.618370647
1E+68 0.988254301 1.617250033
1E+69 0.990013944 1.616152938
1E+70 0.991743316 1.615078509
1E+71 0.99344337 1.614025941
1E+72 0.995115021 1.612994465
1E+73 0.996759139 1.611983353
1E+74 0.998376558 1.61099191
1E+75 0.999968074 1.610019475
1E+76 1.001534451 1.609065419
……
1E+300 1.14533783 1.532593951
1E+301 1.145648245 1.532449644
1E+302 1.145957473 1.532305966
1E+303 1.146265523 1.532162913
1E+304 1.146572404 1.53202048
1E+305 1.146878123 1.531878661
1E+306 1.147182689 1.531737452
1E+307 1.14748611 1.531596849
1E+308 1.147788394 1.531456846

yangchuanju · 发表于 2025-8-31 21:14

根据1式计算的相关小素数p、间隙D及ln(p)^2、ln(p)^2/D数值见下表——
序号小素数p 间隙D ln(p)^2 ln(p)^2/D
1 2 1 0.480453014 0.480453014
2 3 2 1.206948961 0.60347448
3 7 4 3.786566308 0.946641577
4 23 6 9.831323978 1.638553996
5 89 8 20.14785646 2.518482057
6 113 14 22.34819559 1.596299685
7 523 18 39.18236011 2.176797784
8 887 20 46.0748395 2.303741975
9 1129 22 49.40807198 2.245821454
10 1327 34 51.70582183 1.520759466
11 9551 36 83.98624826 2.33295134
12 15683 44 93.32202598 2.120955136
13 19609 52 97.68839395 1.87862296
14 31397 72 107.2149998 1.48909722
15 155921 86 142.9723539 1.662469232
16 360653 96 163.7292106 1.70551261
17 370261 112 164.4027465 1.467881665
18 492113 114 171.7793892 1.506836748
19 1349533 118 199.2408236 1.688481556
20 1357201 132 199.4008066 1.510612172
21 2010733 148 210.6564831 1.423354615
22 4652353 154 235.711037 1.530591149
23 17051707 180 277.2811402 1.540450779
24 20831323 210 283.9888364 1.352327793
25 47326693 220 312.3202616 1.419637553
26 122164747 222 346.7372121 1.561879334
27 189695659 234 363.3191116 1.552645776
28 191912783 248 363.7622235 1.466783159
29 387096133 250 391.0183381 1.564073352
30 436273009 282 395.7624339 1.403412886
31 1294268491 288 440.211236 1.528511236
32 1453168141 292 445.0839126 1.524259975
33 2300942549 320 464.6863431 1.452144822
34 3842610773 336 487.0592047 1.449580966
35 4302407359 354 492.0606689 1.39000189
36 10726904659 382 533.4261803 1.396403613
37 20678048297 384 564.1735845 1.469202043
38 22367084959 394 567.9097117 1.441395207
39 25056082087 456 573.3334501 1.257310198
40 42652618343 464 599.0919296 1.2911464
41 1.27976E+11 468 654.0863128 1.397620327
42 1.82227E+11 474 672.288068 1.418329257
43 2.41161E+11 486 686.8974778 1.413369296
44 2.97501E+11 490 697.9468473 1.424381321
45 3.03371E+11 500 698.9796786 1.397959357
46 3.046E+11 514 699.1933077 1.360298264
47 4.16609E+11 516 715.8521376 1.387310344
48 4.61691E+11 532 721.3607914 1.355941337
49 6.14487E+11 534 736.7996865 1.379774694
50 7.38833E+11 540 746.8380389 1.383033405
51 1.34629E+12 582 779.9942407 1.34019629
52 1.4087E+12 588 782.5270642 1.330828341
53 1.96819E+12 602 801.3504914 1.331146996
54 2.61494E+12 652 817.517499 1.253861195
55 7.17716E+12 674 876.273978 1.30010976
56 1.3829E+13 716 915.5340048 1.278678778
57 1.95813E+13 766 936.7026223 1.222849376
58 4.28423E+13 778 985.2408622 1.26637643
59 9.08743E+13 804 1033.011655 1.284840367
60 1.71231E+14 806 1074.137478 1.332676771
61 2.18209E+14 906 1090.087707 1.203187314
62 1.18946E+15 916 1204.942091 1.315438964
63 1.68699E+15 924 1229.324574 1.330437851
64 1.69318E+15 1132 1229.581308 1.086202569
65 4.38415E+16 1184 1468.373224 1.240180088
66 5.53508E+16 1198 1486.292718 1.240645007
67 8.08736E+16 1220 1515.674473 1.242356125
68 2.03986E+17 1224 1588.566906 1.297848779
69 2.18035E+17 1248 1593.88032 1.277147692
70 3.05406E+17 1272 1620.901247 1.274293433
71 3.52521E+17 1328 1632.474103 1.229272668
72 4.0143E+17 1356 1642.989703 1.211644324
73 4.18033E+17 1370 1646.276736 1.201661851
74 8.04213E+17 1442 1699.800785 1.178780017
75 1.42517E+18 1476 1747.308951 1.183813652
76 5.73324E+18 1488 1865.619075 1.25377626
77 6.78799E+18 1510 1880.235848 1.245189303
78 1.55706E+19 1526 1952.92562 1.279767772
79 1.76787E+19 1530 1964.163976 1.283767305
80 1.83614E+19 1550 1967.524018 1.269370334
81 1.84701E+19 1552 1968.047608 1.268071912
82 1.85717E+19 1572 1968.534435 1.252248368
83 2.07337E+19 1676 1978.318665 1.180381065

yangchuanju · 发表于 2025-8-31 21:15

应用
给定一个大素数p，其后的最大素数间隙D可用ln(p)^2估算；
给定一个大素数间隙D，其前的素数p的大小可用e^(D^0.5)估算。

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