[原创]k生素数群的数量公式

白新岭 · 发表于 2019-2-14 14:29

10^n 素数个数 1.760431509 1.760431509 差值
13 3.46066E+11 21083166428 19647217418 1435949010
14 3.20494E+12 1.80825E+11 1.69407E+11 11418261931
15 2.98446E+13 1.56801E+12 1.47572E+12 92289169100
16 2.79238E+14 1.37268E+13 1.29702E+13 7.56564E+11
17 2.62356E+15 1.21171E+14 1.14892E+14 6.27939E+12
18 2.474E+16 1.0775E+15 1.02481E+15 5.26905E+13
19 2.34058E+17 9.64417E+15 9.19773E+15 4.46443E+14
20 2.22082E+18 8.68252E+16 8.30095E+16 3.81569E+15
21 2.11273E+19 7.85789E+17 7.52921E+17 3.28683E+16
22 2.01467E+20 7.14543E+18 6.86029E+18 2.85139E+17
23 1.92532E+21 6.52567E+19 6.27671E+19 2.4896E+18
24 1.84356E+22 5.9832E+20 5.76455E+20 2.18654E+19
25 1.76846E+23 5.50568E+21 5.31261E+21 1.93075E+20
26 1.69925E+24 5.08314E+22 4.9118E+22 1.71335E+21
27 1.63525E+25 4.70745E+23 4.5547E+23 1.52741E+22
这是从网上查到10^n内素数个数，第一列是n值，第二列是素数个数，第三列是用调节系数*素数个数^2/范围值得到的素数对，第四列是用调节系数*范围值/（LN(范围值））^2得到的素数对（也就是用哈代公式得到的值），最后一列是它们的差，数量级仅差一位，说明两种计算方式的相对误差在10%左右，非常大，不知道素数对的真值更接近那个值，我虽然没有真实数据，但是我倾向用素数个数计算出来的值更接近真实值。

白新岭 · 发表于 2019-2-16 17:56

10^n 素数个数 1.760431509 1.760431509 差值占比
1 3 1.584388358 3.320379744 -1.735991386 -1.095685522
2 24 10.14008549 8.300949359 1.839136132 0.181372843
3 167 49.09667435 36.89310826 12.20356609 0.24856197
4 1228 265.4702549 207.523734 57.94652087 0.218278771
5 9591 1619.373079 1328.151897 291.2211814 0.179835756
6 78497 10847.38992 9223.277066 1624.112853 0.149723838
7 664578 77751.90778 67762.85191 9989.055863 0.12847345
8 5761454 584363.8352 518809.335 65554.50025 0.112180967
9 50847533 4551545.692 4099234.251 452311.44 0.099375349
10 455052510 36453745.86 33203797.44 3249948.425 0.08915266
11 4118054812 298540584.6 274411549.1 24129035.49 0.080823301
12 37607912017 2489875188 2305819266 184055921.8 0.073921746
13 3.46066E+11 21083166428 19647217418 1435949009 0.068108793
14 3.20494E+12 1.80825E+11 1.69407E+11 11418261930 0.063145235
15 2.98446E+13 1.56801E+12 1.47572E+12 92289169100 0.058857382
16 2.79238E+14 1.37268E+13 1.29702E+13 7.56564E+11 0.055115861
从这里可以看出，除了10的时候，是负误差，其它10的次幂时，皆是正误差，随着次幂的增大相对误差在减小。

白新岭 · 发表于 2019-2-16 18:41

10^n →→ G（10^n）
1 →→ 3.00000000000000E+00
2 →→ 1.40000000000000E+01
3 →→ 5.60000000000000E+01
4 →→ 2.82000000000000E+02
5 →→ 1.66100000000000E+03
6 →→ 1.09930000000000E+04
7 →→ 7.83350000000000E+04
8 →→ 5.87153000000000E+05
9 →→ 4.56707300000000E+06
10 →→ 3.65485520000000E+07
11 →→ 2.99158483000000E+08
12 →→ 2.49407981800000E+09
13 →→ 2.11127977360000E+10
14 →→ 1.81040353189000E+11
15 →→ 1.56961132247700E+12
16 →→ 1.37389234059900E+13
17 →→ 1.21265111014650E+14
18 →→ 1.07823453532480E+15
19 →→ 9.65002408367160E+15
20 →→ 8.68723539049495E+16
21 →→ 7.86173278229553E+17
22 →→ 7.14859128401663E+18
23 →→ 6.52829861091019E+19
24 →→ 5.98540295787833E+20
25 →→ 5.50753926555204E+21
26 →→ 5.08471751563008E+22
27 →→ 4.70879543316114E+23
28 →→ 4.37308292519866E+24
29 →→ 4.07204193011379E+25
30 →→ 3.80105624750587E+26
31 →→ 3.55625329945931E+27
32 →→ 3.33436484838541E+28
33 →→ 3.13261723372195E+29
34 →→ 2.94864418675847E+30
35 →→ 2.78041706665415E+31
36 →→ 2.62618864920244E+32
37 →→ 2.48444754029461E+33
38 →→ 2.35388097907588E+34
39 →→ 2.23334431121631E+35
40 →→ 2.12183579935978E+36
41 →→ 2.01847573020612E+37
42 →→ 1.92248900049638E+38
43 →→ 1.83319053520075E+39
44 →→ 1.74997302339157E+40
45 →→ 1.67229656011145E+41
46 →→ 1.59967986303503E+42
47 →→ 1.53169279609694E+43
48 →→ 1.46794998243866E+44
49 →→ 1.40810532897322E+45
50 →→ 1.35184731682872E+46
51 →→ 1.29889493763103E+47
52 →→ 1.24899417634670E+48
53 →→ 1.20191495825314E+49
54 →→ 1.15744849133167E+50
55 →→ 1.11540494661338E+51
56 →→ 1.07561142823703E+52
57 →→ 1.03791019258972E+53
58 →→ 1.00215708220083E+54
59 →→ 9.68220145292820E+54
60 →→ 9.35978416253581E+55
61 →→ 9.05320835942033E+56
62 →→ 8.76145293797591E+57
63 →→ 8.48357776297965E+58
64 →→ 8.21871608481628E+59
65 →→ 7.96606777089306E+60
66 →→ 7.72489325438495E+61
这是用积分法得到的素数对，比其用素数个数计算出的素数对都多。

白新岭 · 发表于 2019-2-17 16:27

今天查了一下两个素数和的分布表：得到G（10）=3，G（100）=12，G（1000）=56，G（10000）=254；

当是10与1000时，素数对与积分值完全一致，100与10000时，实际素数对小于积分值，所以随位数增多，积分值应该大于实际值（也大于用素数个数求出来的素数对）。

白新岭 · 发表于 2019-2-18 10:10

10^n 积分值实际统计统计/积分值
1 6.00000000000000E+00 3.00000000000000E+00 0.500000000000000
2 2.90000000000000E+01 2.40000000000000E+01 0.827586206896552
3 1.77000000000000E+02 1.67000000000000E+02 0.943502824858757
4 1.24600000000000E+03 1.22800000000000E+03 0.985553772070626
5 9.62900000000000E+03 9.59100000000000E+03 0.996053588119223
6 7.86270000000000E+04 7.84970000000000E+04 0.998346623933254
7 6.64918000000000E+05 6.64578000000000E+05 0.999488658751906
8 5.76220900000000E+06 5.76145400000000E+06 0.999868973860546
9 5.08492340000000E+07 5.08475330000000E+07 0.999966548168651
10 4.55055614000000E+08 4.55052510000000E+08 0.999993178855717
11 4.11806640000000E+09 4.11805481200000E+09 0.999997186058000
12 3.76079502800000E+10 3.76079120170000E+10 0.999998982582148
13 3.46065645809000E+11 3.46065536838000E+11 0.999999685114656
14 3.20494206569200E+12 3.20494175080100E+12 0.999999901748302
15 2.98445714752870E+13 2.98445704226680E+13 0.999999964729968
16 2.79238344248557E+14 2.79238341033924E+14 0.999999988487853
17 2.62355716561082E+15 2.62355715765423E+15 0.999999996967251
18 2.47399543096904E+16 2.47399542877408E+16 0.999999999112787
19 2.34057667376222E+17 2.34057667276344E+17 0.999999999573276
20 2.22081960278366E+18 2.22081960256091E+18 0.999999999899699
21 2.11272694866161E+19 2.11272694860187E+19 0.999999999971724
22 2.01467286691248E+20 2.01467286689315E+20 0.999999999990405
23 1.92532039161405E+21 1.92532039160680E+21 0.999999999996234
24 1.84355997673663E+22 1.84355997673492E+22 0.999999999999072
25 1.76846309399199E+23 1.76846309399143E+23 0.999999999999683
26 1.69924675087259E+24 1.69924675087243E+24 0.999999999999906
27 1.63524604268422E+25 1.63524604268416E+25 0.999999999999963
28 1.57589269275975E+26
29 1.52069810971428E+27
30 1.46923988977204E+28
31 1.42115097348081E+29
32 1.37611086699377E+30
33 1.33383848331044E+31
34 1.29408626505159E+32
35 1.25663532881832E+33
36 1.22129142976194E+34
37 1.18788158912168E+35
38 1.15625126102652E+36
39 1.12626194055592E+37
40 1.09778913489828E+38
41 1.07072063488004E+39
42 1.04495503622646E+40
43 1.02040046944366E+41
44 9.96973504768770E+41
45 9.74598204664929E+42
46 9.53205301174764E+43
47 9.32731479347381E+44
48 9.13118751116142E+45
49 8.94313906580259E+46
50 8.76268031750784E+47
51 8.58936083553667E+48
52 8.42276514312127E+49
53 8.26250939115126E+50
54 8.10823840464122E+51
55 7.95962305413021E+52
56 7.81635791105613E+53
57 7.67815915194389E+54
58 7.54476268113578E+55
59 7.41592244592971E+56
60 7.29140892150319E+57
61 7.17100774599038E+58
62 7.05451848863218E+59
63 6.94175353610429E+60
64 6.83253708400362E+61
65 6.72670422208763E+62
66 6.62410010325301E+63
随着n的增大，很快积分值与实际统计值就有高度吻合，到n=27时，小数点后已经有13个9，可见接近度，精度是多高。

白新岭 · 发表于 2019-2-23 18:01

Pi10（n)→→1731.79315527582这是最密5家村的系数，素数式为0,2,4,2,10,2,10,2,4,2（数字表示相邻两个素数的距离）。
10^n 积分值
11 2.00000000000000E+00
12 1.00000000000000E+01
13 4.60000000000000E+01
14 2.12000000000000E+02
15 1.02800000000000E+03
16 5.24200000000000E+03
17 2.79050000000000E+04
18 1.54307000000000E+05
19 8.82439000000000E+05
20 5.19981300000000E+06
21 3.14739650000000E+07
22 1.95179436000000E+08
23 1.23724682000000E+09
24 8.00150300600000E+09
25 5.27038336390000E+10
26 3.53038179850000E+11
27 2.40181651006300E+12
28 1.65764381666330E+13
29 1.15937946231775E+14
30 8.20994235145442E+14
31 5.88129235487700E+15
32 4.25889702171590E+16
33 3.11542833857058E+17
34 2.30073375954359E+18
35 1.71434638090752E+19
36 1.28822339698477E+20
37 9.75752656922001E+20
38 7.44657421639768E+21
39 5.72359273307466E+22
40 4.42911477317930E+23
41 3.44948115921183E+24
42 2.70297932618277E+25
43 2.13037989362070E+26
44 1.68841397024750E+27
45 1.34523640875228E+28
46 1.07724298867865E+29
47 8.66819011459531E+29
48 7.00734182503486E+30
49 5.68989637496804E+31
50 4.63983783529086E+32
51 3.79904637193229E+33
52 3.12284235619149E+34
53 2.57669823412196E+35
54 2.13379736792040E+36
55 1.77321109816512E+37
56 1.47852982026902E+38
57 1.23682944706308E+39
58 1.03788702945369E+40
59 8.73582486657053E+40
60 7.37440082942806E+41
61 6.24275369986034E+42
62 5.29922112978888E+43
63 4.51020162176988E+44
64 3.84849975343357E+45
65 3.29203007003361E+46
66 2.82279791155501E+47
不知以前是否发表过，这是间距38的自对称5家村。

白新岭 · 发表于 2019-2-24 10:15

到素数19时，相邻素数式差为8的有128810组，2素数式相差为8的有378675组，它们之间的关系是什么？相邻素数差为8的是2素数差为8的一部分，2素数差为8的还包括3生素数为8的，4生素数为8的，所以相邻素数式差为8的=2素数差为8-3素数为8的-4素数为8的，而3素数为8的同样包括4素数为8的（且3素数为8的有两种形式），所以相邻素数为8的=2素数为8的-2倍3素数为8+4素数为8的（在减3素数为8的当中，有去4素数为8的情况，减减为加，所以等于加2倍的4素数为8的，再去一个4素数为8的后，等于加上一个4素数为8的）。

所以378675-2*143360（3素数为8的,在未去4素数为8的之前）+36855（4素数为8的）=128810
由此得到求相邻素数为8的公式=孪生素数对个数-2*最密3生素数个数+最密4生素数个数。

白新岭 · 发表于 2019-2-24 12:57

10^n 2生素数积分值 3生素数积分值 4生素数积分值相邻差为8的素数对
1 2.00000000000000E+00 0.00000000000000E+00 0.00000000000000E+00 2.00000000000000E+00
2 1.00000000000000E+01 4.00000000000000E+00 0.00000000000000E+00 2.00000000000000E+00
3 4.20000000000000E+01 1.60000000000000E+01 3.00000000000000E+00 1.30000000000000E+01
4 2.11000000000000E+02 6.00000000000000E+01 1.10000000000000E+01 1.02000000000000E+02
5 1.24600000000000E+03 2.70000000000000E+02 4.00000000000000E+01 7.46000000000000E+02
6 8.24500000000000E+03 1.43700000000000E+03 1.70000000000000E+02 5.54100000000000E+03
7 5.87510000000000E+04 8.58200000000000E+03 8.50000000000000E+02 4.24370000000000E+04
8 4.40365000000000E+05 5.54820000000000E+04 4.72200000000000E+03 3.34123000000000E+05
9 3.42530500000000E+06 3.79793000000000E+05 2.83840000000000E+04 2.69410300000000E+06
10 2.74114160000000E+07 2.71528400000000E+06 1.81062000000000E+05 2.21619100000000E+07
11 2.24368877000000E+08 2.00896490000000E+07 1.20994400000000E+06 1.85399523000000E+08
12 1.87055999000000E+09 1.52830589000000E+08 8.39456800000000E+06 1.57329338000000E+09
13 1.58345993750000E+10 1.18976333800000E+09 6.00754500000000E+07 1.35151481490000E+10
14 1.35780274094000E+11 9.44389232400000E+09 4.41290899000000E+08 1.17333780345000E+11
15 1.17720857164500E+12 7.62177954860000E+10 3.31455162500000E+09 1.02808753229800E+12
16 1.03041932528750E+13 6.24025314376000E+11 2.53794516430000E+10 9.08152207576600E+12
17 9.09488394251850E+13 5.17368778200500E+12 1.97622493144000E+11 8.07990863543190E+13
18 8.08675956302867E+14 4.33713953109410E+13 1.56177243700000E+12 7.23494938117985E+14
19 7.23751855328771E+15 3.67175721215464E+14 1.25056009324990E+13 6.51567271178928E+15
20 6.51542698446435E+16 3.13588380228085E+15 1.01318973598611E+14 5.89838212136804E+16
21 5.89629998635250E+17 2.69946618183503E+16 8.29588089136792E+14 5.36470263087686E+17
22 5.36144382639262E+18 2.34045046069285E+17 6.85772585781442E+15 4.90021146011186E+18
23 4.89622429003181E+19 2.04240297303743E+18 5.71826853616852E+16 4.49346196396049E+19
24 4.48905252266123E+20 1.79290708961908E+19 4.80603132430811E+17 4.13527713606172E+20
25 4.13065472912555E+21 1.58246230697159E+20 4.06872698422031E+18 3.81823099471545E+21
26 3.81353839519102E+22 1.40371521890257E+21 3.46758761873661E+19 3.53626293902924E+22
27 3.53159681423029E+23 1.25091388312333E+22 2.97351193606394E+20 3.28438754954169E+23
28 3.27981241619335E+24 1.11951512460133E+23 2.56440527376364E+21 3.05847379654685E+24
29 3.05403165457706E+25 1.00589817651362E+24 2.22331571235695E+22 2.85507533498669E+25
30 2.85079237884626E+26 9.07153321314230E+24 1.93711005712835E+23 2.67129882464054E+26
31 2.66719015536742E+27 8.20924131522523E+25 1.69552135401899E+24 2.50470085041693E+27
32 2.50077380578287E+28 7.45288486897691E+26 1.49045559838141E+25 2.35320656400171E+28
33 2.34946308452995E+29 6.78667964292004E+27 1.31548156560022E+26 2.21504497323715E+29
34 2.21148328995555E+30 6.19758208427067E+28 1.16545155188245E+27 2.08869709982202E+30
35 2.08531294132592E+31 5.67474971509747E+29 1.03621673520777E+28 1.97285416375918E+31
36 1.96964162039733E+32 5.20911963209356E+30 9.24411634219414E+28 1.86638363938968E+32
37 1.86333578151142E+33 4.79307680201278E+31 8.27289516674936E+29 1.76830153498784E+33
38 1.76541085396036E+34 4.42019118867108E+32 7.42595718236183E+30 1.67774962590517E+34
39 1.67500834693851E+35 4.08500804841166E+33 6.68469411571155E+31 1.59397665538185E+35
40 1.59137695737787E+36 3.78287959375271E+34 6.03366899229309E+32 1.51632273240205E+36
41 1.51385690025858E+37 3.50982906501237E+35 5.46001314650209E+33 1.44420632027298E+37
42 1.44186684809704E+38 3.26244035551446E+36 4.95294922061421E+34 1.37711333590881E+38
43 1.37489299458606E+39 3.03776790987013E+37 4.50341156725430E+35 1.31458797754538E+39
44 1.31247985649903E+40 2.83326280009472E+38 4.10374244595071E+36 1.25622497474173E+40
45 1.25422250509046E+41 2.64671178334713E+39 3.74744756347486E+37 1.20166301417986E+41
46 1.19975997859186E+42 2.47618683166226E+40 3.42899835163514E+38 1.15057914179378E+42
47 1.14876967493235E+43 2.32000315180180E+41 3.14367126065979E+39 1.10268397902238E+43
48 1.10096256144843E+44 2.17668412153300E+42 2.88741652544276E+40 1.05771762067031E+44
49 1.05607906830731E+45 2.04493188621645E+43 2.65675051885220E+41 1.01544610563487E+45
50 1.01388555633920E+46 1.92360260805324E+44 2.44866707393182E+42 9.75658370885528E+45
51 9.74171269249236E+46 1.81168555579433E+45 2.26056413234535E+43 9.38163614546584E+46
52 9.36745695749414E+47 1.70828537725722E+46 2.09018283137315E+44 9.02789006487407E+47
53 9.01436279786093E+48 1.61260701979356E+47 1.93555672919195E+45 8.69377695063141E+48
54 8.68086427334653E+49 1.52394286189039E+48 1.79496932759942E+46 8.37787067029605E+49
55 8.36553766658759E+50 1.44166169771357E+49 1.66691841245810E+47 8.07887224545734E+50
56 8.06708625853697E+51 1.36519927973583E+50 1.55008601732178E+48 7.79559648860713E+51
57 7.78432697201769E+52 1.29405017582088E+51 1.44331304196684E+49 7.52696024989548E+52
58 7.51617862592685E+53 1.22776073874045E+52 1.34557773785042E+50 7.27197205591661E+53
59 7.26165158186581E+54 1.16592302002200E+53 1.25597741680093E+51 7.02972295527821E+54
60 7.01983859768228E+55 1.10816948778414E+54 1.17371285518155E+52 6.79937841298063E+55
61 6.78990672976167E+56 1.05416843101521E+55 1.09807495928997E+53 6.58017111851792E+56
62 6.57109014884773E+57 1.00361995154042E+56 1.02843333348780E+54 6.37139459187313E+57
63 6.36268375347547E+58 9.56252460460582E+56 9.64226454099809E+54 6.17239748783745E+58
64 6.16403748138939E+59 9.11819608735616E+57 9.04953202321241E+55 5.98257851284459E+59
65 5.97455123310425E+60 8.70097592313475E+58 8.50165550451812E+56 5.80138188019201E+60
66 5.79367033346368E+61 8.30882781159487E+59 7.99462229506832E+57 5.62829323946129E+61
实际个数
0.00000000000000E+00
1.00000000000000E+00
1.50000000000000E+01
1.01000000000000E+02
7.73000000000000E+02
5.56900000000000E+03
4.23520000000000E+04
以上是用公式计算的邻差为8的素数对数量，后边给出了部分实际个数。

白新岭 · 发表于 2019-3-9 20:32

本帖最后由白新岭于 2019-3-9 12:33 编辑

4生素数必定产生在3生素数之中，不是三生素数的一定不是四生素数，而每一组四生素数都是两种三生素数；5生素数产生在对应的三生素数之中，有了3生素数表可以直接产生5生素数；6生素数产生在5生素数之中，也可以有三生素数直接产生；这种方法与在素数表中找k生素数是一样的；为了减少运算量，我们要一步一步的提炼，有了基础数列，我们就可以筛选更高一阶的素数，k生素数看似线性，实际上是以ln（N）的速度在下降，虽然比它慢点，但是有限。

白新岭 · 发表于 2019-3-21 13:11

808675888577436这是网上查到的10^18以下孪生素数对的数量，与用积分得到808675956302870 ，前5位数字是一致的

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