用孪生素数个数和素数个数求偶数N以内素数对个数的方法

大傻8888888 · 发表于 2022-10-24 22:51

本帖最后由大傻8888888 于 2023-3-25 21:01 编辑

   用孪生素数个数的值求出偶数素数对个数的值，设偶数N以内孪生素数个数为Z，则偶数N以内素数对个数的值为Π[(p-1)/(p-2)]Z，单计法则为（1/2）Π[(p-1)/(p-2)]Z，这样得出的结果与偶数的实际值的精确度应该比哈李公式要好得多。
   由此我想既然如此也可以用N以内素数个数的值求出偶数素数对个数的值，设偶数N以内素数个数为M，则偶数N以内素数对个数的值为
2CΠ[(p-1)/(p-2)]M^2/N，以2^30为例，2^30以内素数个数为54400028
   54400028^2/2^30=2756121  单计法为1378060
   2C1378060=1819481
   2^30以内素数对个数为1817111
   1817111/1819481 =0.9986743
   可以看出这个方法得出的值也比哈李公式要好得多。

大傻8888888 · 发表于 2023-3-26 21:30

愚工688先生给出下列偶数的素数对的个数如下:
G(219980) = 1629
G(219982) = 1451
G(219984) = 2275
G(219986) = 1249
G(219988) = 1184
G(219990) = 3056
G(219992) = 1140
G(219994) = 1181
G(219996) = 2795
G(219998) = 1157
G(220000) = 1722
设Z(N)为N以内孪生素数的个数，则有：
Z（219980）一直到Z(220000)这11个偶数以内孪生素数个数都是2329
根据1楼的公式用孪生素数个数的值求出偶数素数对个数的值（四舍五入）如下：
219980=4×5×17×647
G（219980）=1656
219982=2×7×19×827
G（219982）=1473
219984=16×3×4583
G（219984）=2329
219986=2×13×8461
G（219986）=1270
219988=4×43×1279
G（219988）=1193
219990=2×3×5×7333
G（219990）=3105
219992=8×107×257
G（219992）=1180
219994=2×29×3793
G（219994）=1208
219996=4×81×7×97
G（219996）=2824
219998=2×317×347
G（219998）=1172
220000=32×625×11
G（220000）=1725
从上面的计算可知（小数点后面保留4位）：
219992的实际值和计算值之比0.9661，精确度最低。
220000的实际值和计算值之比0.9983，精确度最高。
用同样的计算可知：
100亿的实际值和计算值之比0.9959
1000亿的实际值和计算值之比0.9965
1万亿的实际值和计算值之比0.9973
10万亿的实际值和计算值之比0.9978
100万亿的实际值和计算值之比0.9981
1000万亿的实际值和计算值之比0.9980
从上面的数据可知由孪生素数个数计算素数对个数，随着数值增大，虽然有一些波动，但是实际值和计算值之比越来越接近1。
希望网友用更多的数据验证这个根据哈李公式得出的这个方法。

yangchuanju · 发表于 2023-3-27 12:24

2^n次方内孪生素数对数与哥德巴赫猜想素数对数的比值
n 双计素数对孪生素数对数孪/哥
2 2 1 0.5
3 2 2 1
4 4 3 0.75
5 4 5 1.25
6 10 7 0.7
7 6 10 1.666666667
8 16 17 1.0625
9 22 24 1.090909091
10 44 36 0.818181818
11 50 62 1.24
12 106 107 1.009433962
13 152 177 1.164473684
14 302 290 0.960264901
15 488 505 1.034836066
16 870 860 0.988505747
17 1498 1526 1.018691589
18 2628 2679 1.019406393
19 4734 4750 1.003379806
20 8478 8535 1.006723284
21 14942 15500 1.037344398
22 27410 27995 1.021342576
23 49856 50638 1.015685173
24 91492 92246 1.008241158
25 166934 168617 1.010081829
26 307700 309561 1.006048099
27 567492 571313 1.006733135
28 1050472 1056281 1.005529895
29 1951370 1961080 1.004975991
30 3634222 3650557 1.004494772
31 6780076 6810670 1.004512339
32 12682848 12739574 1.004472655
33 23783308 23878645 1.004008568
34 44672120 44849427 1.003969075
35 84068194 84384508 1.003762588
36 158575328 159082253 1.003196746
37 299422268 300424743 1.003348031
38 566554450 568237005 1.002969803
39 1073420200 1076431099 1.002804958
40 2036739786 2042054332 1.00260934

大傻8888888 · 发表于 2023-3-27 21:08

yangchuanju 发表于 2023-3-27 12:24
2^n次方内孪生素数对数与哥德巴赫猜想素数对数的比值
n 双计素数对孪生素数对数孪/哥
2 2 1 0.5

n 双计素数对孪生素数对数孪/哥
30 3634222 3650557 1.004494772
根据1楼2^30以内素数个数为54400028
2C54400028^2/2^30=3638972
3638972/3634222=1.0013069
看起来2^30用素数个数求偶数N以内素数对个数比用孪生素数个数求偶数N以内素数对个数的精确度要好一些。

lusishun · 发表于 2023-3-28 06:38

白先生的孪生素数合成法，在哪里啊，没查到

yangchuanju · 发表于 2023-3-29 12:17

偶数       素数对       波动因子k       哈李式       素数个数       素数平方式
100000000       291400       1.333333       259404.6675       5761455       292182.019
200000000       538290       1.333333       481863.2322       11078937       540200.9298
300000000       1547388       2.666667       1386156.354       16252325       1549989.925
400000000       999700       1.333333       897455.2807       21336326       1001770.925
500000000       1219610       1.333333       1096963.299       26355867       1222851.576
600000000       2874881       2.666667       2585430.614       31324703       2879000.94
700000000       1979689       1.600000       1782509.052       36252931       1983163.828
800000000       1859646       1.333333       1675584.063       41146179       1862765.443
900000000       4132595       2.666667       3727113.005       46009215       4140628.534
1000000000       2274205       1.333333       2049617.126       50847534       2275772.935
1100000000       2748595       1.481481       2482202.862       55662470       2754729.063
1200000000       5352052       2.666667       4833654.781       60454705       5361645.545
1300000000       3137228       1.454545       2834503.669       65228333       3142728.8
1400000000       3688114       1.600000       3334206.458       69985473       3695371.901
1500000000       6543613       2.666667       5915120.095       74726528       6553563.057
1600000000       3467407       1.333333       3135546.17       79451833       3472777.803
1700000000       3904453       1.422222       3533375.215       84163019       3912114.443
1800000000       7709245       2.666667       6977210.734       88862422       7722944.611
1900000000       4284979       1.411765       3879320.492       93547928       4292669.629
2000000000       4238417       1.333333       3838182.896       98222287       4245992.53
2100000000       10629971       3.200000       9628301.172       102886526       10648736.76
2200000000       5134372       1.481481       4649635.484       107540122       5141205.895
2300000000       5039234       1.396825       4564329.197       112184940       5045843.861
2400000000       9992839       2.666667       9056778.782       116818447       10009923.26
2500000000       5185533       1.333333       4699292.004       121443371       5192742.132
2600000000       5859790       1.454545       5312286.376       126062167       5869132.134
2700000000       11116031       2.666667       10078653.82       130670192       11132903.06
2800000000       6894453       1.600000       6250210.972       135270258       6902668.054
2900000000       6148656       1.382716       5576318.912       139864011       6157399.955
3000000000       12224533       2.666667       11090627.94       144449537       12244193.57
3100000000       6511564       1.379310       5909975.267       149028641       6523642.94
3200000000       6478775       1.333333       5880168.764       153600805       6489722.738
3300000000       14804731       2.962963       13437574.52       158165829       14828185.41
3400000000       7299554       1.422222       6627424.937       162725196       7312208.236
3500000000       8434100       1.600000       7654908.37       167279333       8444752.3
3600000000       14415400       2.666667       13089120.9       171827136       14437770.53
3700000000       7599208       1.371429       6901337.951       176369517       7611476.677
3800000000       8012125       1.411765       7278695.363       180906194       8026680.904
3900000000       16906288       2.909091       15357032.05       185436625       16933008.09
4000000000       7930427       1.333333       7202593.776       189961812       7940754.205
4100000000       8314407       1.367521       7555072.677       194481069       8328280.253
4200000000       19883421       3.200000       18070693.49       198996103       19917747.93
4300000000       8667672       1.365854       7880001.317       203507248       8684521.175
4400000000       9603218       1.481481       8727760.495       208013454       9617823.487
4500000000       17642968       2.666667       16034541.1       212514323       17667822.65
4600000000       9425212       1.396825       8568736.484       217011319       9440597.136
4700000000       9378444       1.362963       8526279.638       221504167       9392915.143
4800000000       18704238       2.666667       17004619.13       225991743       18731089.51
4900000000       11434201       1.600000       10396088.22       230475545       11450488.23
5000000000       9703556       1.333333       8824224.163       234954223       9718195.775
5100000000       21074753       2.844444       19167504.68       239429993       21107402.23
5200000000       10965645       1.454545       9976411.48       243902342       10985157.7
5300000000       10432025       1.359477       9487510.191       248370960       10445953
5400000000       20806971       2.666667       18929632.05       252834065       20839944
5500000000       11751032       1.481481       10693692.29       257294520       11771857.91
5600000000       12902169       1.600000       11740301.18       261751864       12922957.7
5700000000       23138994       2.823529       21054927.66       266206294       23174186.62
5800000000       11513223       1.382716       10475514.58       270655552       11528937.68
5900000000       11470516       1.356725       10439940.86       275101551       11488844.24
6000000000       22899781       2.666667       20836535.23       279545368       22928333.17
6100000000       11818360       1.355932       10755631.04       283984956       11834480.8
6200000000       12198951       1.379310       11104401.42       288422869       12217436.48
6300000000       28716471       3.200000       26140617.95       292856421       28758705.33
6400000000       12138392       1.333333       11049382.67       297285198       12155021.61
6500000000       13427034       1.454545       12225418.9       301711468       13447698.93
6600000000       27736743       2.962963       25252625.99       306137611       27775767.34
6700000000       12850259       1.353846       11697768.16       310558733       12865669.95
6800000000       13679782       1.422222       12455658.68       314977166       13698313.16
6900000000       27222854       2.793651       24794287.56       319391721       27265956.5
7000000000       15799407       1.600000       14387884.92       323804352       15821137.55
7100000000       13531786       1.352657       12322012.38       328215206       13548684.62
7200000000       27011820       2.666667       24603757.64       332620900       27051148.3
7300000000       13870610       1.352113       12633051.58       337024801       13888781.54
7400000000       14241533       1.371429       12973502.64       341426904       14262227.32
7500000000       28031543       2.666667       25536973.27       345826612       28072086.26
7600000000       15018224       1.411765       13683888.41       350221825       15041305.86
7700000000       19135150       1.777778       17438249.83       354615116       19166889.41
7800000000       31686056       2.909091       28873165.3       359006517       31733484.03
7900000000       14883005       1.350649       13562087.9       363395981       14904800.37
8000000000       14862150       1.333333       13542732.38       367783654       14882777.77
8100000000       30063256       2.666667       27394177.19       372168766       30103289.78
8200000000       15586610       1.367521       14206447.04       376549859       15610458.1
8300000000       15553036       1.349794       14178231.63       380930729       15578724.4
8400000000       37285242       3.200000       33982109.33       385307831       37336771.73
8500000000       16751667       1.422222       15267141.86       389682427       16773379.7
8600000000       16258121       1.365854       14819370.37       394055910       16280652.65
8700000000       33262802       2.765432       30322884.87       398425675       33311085.25
8800000000       18008203       1.481481       16414722.01       402793457       18031371.88
8900000000       16557548       1.348659       15097960.19       407159590       16584099.21
9000000000       33076258       2.666667       30158784.3       411523195       33125715.01
9100000000       21870485       1.745455       19940400.85       415885628       21901072.21
9200000000       17675475       1.396825       16117587.32       420243162       17701334.43
9300000000       35257514       2.758621       32146657.3       424603409       35304201.62
9400000000       17590961       1.362963       16038662.86       428958595       17613162.83
9500000000       18395699       1.411765       16774207.48       433311792       18420021.46
9600000000       35075543       2.666667       31988969.99       437663672       35126016.39
9700000000       17886783       1.347368       16316495.12       442014876       17915907.13
9800000000       21437787       1.600000       19558138.16       446362736       21474357.05
9900000000       40070955       2.962963       36556082.25       450708777       40135941.97
10000000000       18200488       1.333333       16601898.72       455052511       18226873.01
1E+11       149091160       1.333333       137205774.5       4118054813       149270292.3
1E+12       1243722370       1.333333       1152909633       37607912018       1244937594

yangchuanju · 发表于 2023-3-29 12:18

（接上楼）
偶数       孪生素数对数       孪生素数式       哈李比       素数比       孪素比
100000000       440312       293541.3333       0.89020133       1.002683662       1.007348433
200000000       813371       542247.3333       0.895174037       1.003550001       1.007351675
300000000       1166480       1555306.667       0.895803996       1.001681495       1.005117441
400000000       1507733       1005155.333       0.897724598       1.002071547       1.00545697
500000000       1840170       1226780       0.899437771       1.002657879       1.005878929
600000000       2166301       2888401.333       0.899317437       1.001433082       1.004702919
700000000       2486868       1989494.4       0.900398523       1.001755239       1.004953
800000000       2802751       1868500.667       0.901023132       1.001677439       1.00476148
900000000       3115262       4153682.667       0.901881991       1.001943944       1.005102766
1000000000       3424506       2283004       0.901245545       1.000689443       1.003869044
1100000000       3730451       2763297.037       0.903080615       1.002231709       1.005348928
1200000000       4033838       5378450.667       0.903140474       1.001792498       1.004932438
1300000000       4334388       3152282.182       0.903505792       1.001753395       1.004798562
1400000000       4632699       3706159.2       0.904041051       1.001967917       1.004892799
1500000000       4929508       6572677.333       0.903953228       1.001520575       1.004441634
1600000000       5223982       3482654.667       0.904291354       1.001548939       1.004397426
1700000000       5516430       3922794.667       0.904960366       1.001962232       1.004697628
1800000000       5808585       7744780       0.905044623       1.001777037       1.004609401
1900000000       6099613       4305609.176       0.905330106       1.001794788       1.004814534
2000000000       6388041       4258694       0.905569909       1.001787349       1.004784097
2100000000       6674812       10679699.2       0.905769279       1.001765363       1.004678112
2200000000       6960677       5156057.037       0.905589911       1.001331009       1.004223503
2300000000       7245806       5060562.921       0.905758533       1.00131168       1.004232572
2400000000       7528643       10038190.67       0.906326899       1.001709651       1.004538417
2500000000       7810997       5207331.333       0.906231241       1.001390239       1.004203682
2600000000       8092871       5885724.364       0.906565999       1.001594278       1.004425818
2700000000       8373302       11164402.67       0.906677376       1.001517813       1.004351523
2800000000       8652758       6922206.4       0.90655647       1.001191545       1.004025468
2900000000       8932233       6175370.963       0.906916717       1.001422092       1.004344846
3000000000       9210144       12280192       0.907243486       1.001608288       1.004553057
3100000000       9487353       6543002.069       0.907612252       1.001854998       1.004828037
3200000000       9762902       6508601.333       0.907605028       1.001689785       1.0046037
3300000000       10038577       14871965.93       0.907654082       1.001584251       1.004541449
3400000000       10313157       7333800.533       0.9079219       1.001733563       1.004691593
3500000000       10586929       8469543.2       0.907614134       1.001263004       1.004202369
3600000000       10859464       14479285.33       0.907995678       1.00155185       1.004431742
3700000000       11131895       7633299.429       0.908165423       1.001614468       1.004486182
3800000000       11403183       8049305.647       0.908460036       1.001816735       1.004640548
3900000000       11674153       16980586.18       0.908362146       1.001580482       1.004394707
4000000000       11944438       7962958.667       0.908222694       1.001302226       1.004102133
4100000000       12214163       8351564.444       0.908672462       1.00166858       1.004469043
4200000000       12484112       19974579.2       0.908832212       1.00172641       1.004584634
4300000000       12752991       8709359.707       0.909125463       1.00194391       1.004809562
4400000000       13020820       9645051.852       0.908837068       1.001520895       1.004356233
4500000000       13288127       17717502.67       0.908834676       1.001408757       1.00422461
4600000000       13555113       9467063.048       0.90912931       1.001632338       1.00444033
4700000000       13821362       9419002.252       0.909135848       1.001543022       1.004324625
4800000000       14087306       18783074.67       0.909131884       1.001435584       1.004214909
4900000000       14353023       11482418.4       0.909209854       1.001424431       1.004216945
5000000000       14618166       9745444       0.909380454       1.001508702       1.004316768
5100000000       14883703       21167933.16       0.909500798       1.001549211       1.004421412
5200000000       15147862       11016626.91       0.909787931       1.001779439       1.004649239
5300000000       15411824       10476011.08       0.90946007       1.001335119       1.004216447
5400000000       15674423       20899230.67       0.909773559       1.001584709       1.004434075
5500000000       15937153       11805298.52       0.910021545       1.001772262       1.004618022
5600000000       16199441       12959552.8       0.909947868       1.001611256       1.004447609
5700000000       16462030       23240512.94       0.909932716       1.001520923       1.004387353
5800000000       16723078       11561634.17       0.909868121       1.001364925       1.004204832
5900000000       16983784       11521163.42       0.910154422       1.001597857       1.004415443
6000000000       17244409       22992545.33       0.90990107       1.001246831       1.004050883
6100000000       17504488       11867449.49       0.910078136       1.001364047       1.004153664
6200000000       17764537       12251404.83       0.910275107       1.001515334       1.004299864
6300000000       18024376       28839001.6       0.910300502       1.001470735       1.00426691
6400000000       18283435       12188956.67       0.910283889       1.001370001       1.004165681
6500000000       18542454       13485421.09       0.910507779       1.001539054       1.004348473
6600000000       18800026       27851890.37       0.910439484       1.001406955       1.004151438
6700000000       19058240       12900962.46       0.910313805       1.001199272       1.003945715
6800000000       19315571       13735517.16       0.910515875       1.001354639       1.004074272
6900000000       19572967       27340017.4       0.910789426       1.001583321       1.004303862
7000000000       19830161       15864128.8       0.910659806       1.001375403       1.00409647
7100000000       20087571       13585796.81       0.910597639       1.001248809       1.003991403
7200000000       20343905       27125206.67       0.910851532       1.001455966       1.004197669
7300000000       20600009       13926766.65       0.910778371       1.001310075       1.004048607
7400000000       20855304       14300779.89       0.910962509       1.001453096       1.004160148
7500000000       21110363       28147150.67       0.911008476       1.001446344       1.004124199
7600000000       21365320       15081402.35       0.911152238       1.001536923       1.004206779
7700000000       21620694       19218394.67       0.911320258       1.001658697       1.004350353
7800000000       21875356       31818699.64       0.911226228       1.001496811       1.004186183
7900000000       22129653       14944700.73       0.911246613       1.001464447       1.004145381
8000000000       22384176       14922784       0.911222964       1.00138794       1.00407976
8100000000       22638090       30184120       0.911217906       1.001331651       1.004020323
8200000000       22891711       15652451.97       0.911452012       1.001530038       1.004224265
8300000000       23145559       15620871.09       0.911605401       1.001651664       1.004361534
8400000000       23398612       37437779.2       0.911409113       1.001382041       1.004091088
8500000000       23651769       16819035.73       0.911380453       1.001296152       1.004021614
8600000000       23904154       16324788.1       0.911505725       1.001385871       1.004100541
8700000000       24156114       33401046.52       0.91161547       1.001451569       1.00415613
8800000000       24407781       18079837.78       0.911513603       1.001286574       1.003977897
8900000000       24659928       16628916.97       0.911847587       1.001603571       1.004310358
9000000000       24911210       33214946.67       0.911795533       1.001495242       1.004192997
9100000000       25162461       21959965.96       0.911749367       1.001398561       1.004091403
9200000000       25413080       17748817.78       0.911861623       1.001463012       1.004149409
9300000000       25664044       35398681.38       0.911767554       1.001324189       1.004003895
9400000000       25914706       17660392.24       0.911755921       1.001262116       1.003946984
9500000000       26165152       18469519.06       0.911854857       1.001322182       1.004012898
9600000000       26415111       35220148       0.912002132       1.001438991       1.004122673
9700000000       26665038       17963815.07       0.912209597       1.001628249       1.004306648
9800000000       26913838       21531070.4       0.912320762       1.001705869       1.004351354
9900000000       27163538       40242278.52       0.912283779       1.001621797       1.004275504
10000000000       27412679       18275119.33       0.912167779       1.001449687       1.004100513
1E+11       224376048       149584032       0.920281085       1.001201495       1.003305843
1E+12       1870585220       1247056813       0.926983112       1.000977086       1.002681019
————       ————       最小       0.89020133       1.000689443       1.002681019
————       ————       最大       0.926983112       1.003550001       1.007351675

用孪生素数对求哥猜素数对的精度不如用素数个数的平方求哥猜素数对的精度！

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用孪生素数个数和素数个数求偶数N以内素数对个数的方法

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