欧拉计算素数公式的方法可以用来计算双生素数。

柳林 · 发表于 2020-4-11 15:48

世界数学历史上，最伟大的数学家是欧拉。

我们坚信欧拉计算素数的公式是正确的。

欧拉计算素数的公式可以用来计算双生素数。

我们用（P+2）的平方作为计算区间。双生素数连乘积作为密度。

得到的误差率结果，全部都是负值。

区间=19^2=361时，负误差率最大。为-14.73%，以后误差率绝对值逐渐下降；

区间=2659^2=7070281时，负误差率最小。为-0.004%，是拐点。

以后误差率绝对值逐渐上升。

请读者看一看计算结果：

柳林 · 发表于 2020-4-11 15:49

密度区间计算值双序号相邻2 欧拉误差率
10 25 3.49 4 19 -12.822
7.142857143 49 5.47 6 43 -8.754
4.945054945 169 11.27 12 151 -6.083
3.903990746 361 17.91 21 349 -14.726
3.104576091 961 34.34 35 883 -1.886
2.663574082 1849 54.30 56 1789 -3.041
2.29307766 3721 90.41 98 3673 -7.746
2.102730179 5329 117.31 131 5281 -10.451
1.840488869 10609 198.79 217 10531 -8.390
1.772947993 11881 214.52 233 11833 -7.929
1.639397266 19321 317.80 331 19213 -3.989
1.595969524 22801 364.15 386 22741 -5.660
1.486302072 32761 483.05 505 32719 -4.346
1.455497884 37249 536.97 561 37201 -4.283
1.426241645 39601 559.41 589 39511 -5.024
1.37559754 52441 710.37 732 52363 -2.955
1.34115432 58081 766.04 789 57901 -2.910
1.290738204 73441 928.10 945 73363 -1.788
1.263306869 80089 989.52 1007 79999 -1.736
1.230579941 97969 1175.13 1203 97861 -2.317
1.194366467 121801 1413.01 1440 121789 -1.874
1.122239463 177241 1919.78 1948 177211 -1.449
1.111872355 187489 2011.07 2057 187471 -2.233
1.082666715 214369 2234.49 2283 214213 -2.125
1.044083657 273529 2739.99 2785 273313 -1.616
1.021821835 326041 3189.63 3232 325891 -1.311
1.004656554 361201 3469.79 3513 361093 -1.230
0.991629659 383161 3630.19 3670 383101 -1.085
0.9823374 413449 3877.51 3910 413143 -0.831
0.970393392 436921 4044.81 4109 436741 -1.562
0.919137339 657721 5737.59 5763 657661 -0.441
0.914670086 677329 5878.46 5911 677323 -0.551
0.91025672 687241 5934.68 5984 687163 -0.824
0.901739036 737881 6307.70 6327 737719 -0.305
0.89353148 779689 6600.18 6616 779659 -0.239
0.85883216 1042441 8451.89 8486 1042333 -0.402
0.855506576 1067089 8616.40 8650 1066981 -0.388
0.850610083 1104601 8864.91 8919 1104559 -0.607
0.847409292 1129969 9032.45 9101 1129861 -0.753
0.841177889 1194649 9473.80 9558 1194343 -0.881
0.829261432 1329409 10380.54 10449 1328911 -0.655
0.814189279 1515361 11599.40 11709 1514659 -0.936
0.807747905 1635841 12413.91 12519 1635499 -0.839
0.803989947 1666681 12585.50 12703 1666309 -0.925
0.800285864 1697809 12757.84 12907 1697743 -1.156
0.796641685 1745041 13048.91 13229 1744993 -1.361
0.78524389 2042041 15030.22 15150 2041201 -0.791
0.779823054 2111209 15424.28 15575 2110951 -0.968
0.775597635 2199289 15973.86 16114 2199181 -0.870
0.773514096 2217121 16057.78 16214 2217001 -0.963
0.756790344 2588881 18309.96 18477 2588821 -0.904
0.753986831 2627641 18510.66 18709 2627563 -1.060
0.748530368 2785561 19469.03 19668 2785513 -1.012
0.7458859 2886601 20097.96 20256 2886511 -0.780
0.743283437 2968729 20591.98 20728 2968711 -0.656
0.735728933 3200521 21954.68 22140 3200359 -0.837
0.728572164 3508129 23809.34 23954 3507841 -0.604
0.727021186 3530641 23908.00 24088 3530539 -0.747
0.722470693 3736489 25129.50 25338 3736393 -0.823
0.720989461 3806401 25543.21 25732 3806191 -0.734
0.716648245 3996001 26640.07 26838 3995923 -0.737
0.713102863 4116841 27298.87 27555 4116781 -0.930
0.708968049 4338889 28589.53 28801 4338223 -0.734
0.707610523 4363921 28695.86 28931 4363633 -0.813
0.705598026 4464769 29268.68 29511 4464643 -0.821
0.704273582 4541161 29709.20 29933 4541041 -0.748
0.702301132 4592449 29954.41 30214 4591723 -0.859
0.696578612 5013121 32406.25 32534 5012929 -0.393
0.694113337 5148361 33152.72 33235 5148289 -0.248
0.68988634 5340721 34164.09 34292 5340661 -0.373
0.688117145 5480281 34959.05 35069 5480119 -0.314
0.684056843 5678689 35992.57 36157 5678341 -0.455
0.673051907 6507601 40520.70 40663 6507343 -0.350
0.670967285 6723649 41724.37 41788 6723643 -0.152
0.66740992 7070281 43621.10 43623 7070209 -0.004

柳林 · 发表于 2020-4-11 15:53

密度区间计算值双序号相邻2 欧拉误差率
0.6644252 7230721 44394.44 44475 7230649 -0.181
0.661972163 7360369 45009.53 45162 7360261 -0.338
0.660516385 7458361 45500.16 45687 7458169 -0.409
0.657181164 7789681 47259.12 47491 7788931 -0.488
0.655774091 7856809 47556.15 47860 7856803 -0.635
0.646781132 8826841 52624.95 52958 8826823 -0.629
0.645919044 9006001 53614.80 53886 9005881 -0.503
0.640042227 9740641 57410.15 57679 9740503 -0.466
0.638422865 10042561 59025.43 59211 10042183 -0.313
0.634461239 10582009 61773.32 61938 10581763 -0.266
0.63368252 10621081 61919.39 62138 10620949 -0.352
0.632527668 10896601 63398.56 63554 10896553 -0.245
0.630245077 11095561 64302.29 64510 11095531 -0.322
0.628742474 11296321 65295.43 65534 11296003 -0.364
0.627996856 11377129 65678.05 65917 11376751 -0.362
0.627256075 11498881 66295.76 66521 11498719 -0.339
0.624707955 11992369 68832.43 68964 11992069 -0.191
0.623987623 12033961 68985.16 69174 12033949 -0.273
0.621858749 12453841 71124.76 71271 12453829 -0.205
0.620804651 12538681 71477.48 71667 12538609 -0.264
0.61975727 12666481 72073.38 72276 12666133 -0.280
0.618718666 12837889 72915.07 73142 12837823 -0.310
0.614982445 13490929 76116.14 76311 13490419 -0.255
0.611030695 14205361 79581.43 79804 14205241 -0.279
0.60910498 14615329 81594.85 81802 14615269 -0.253
0.607838973 14845609 82691.94 82903 14845531 -0.255
0.605346813 15358561 85163.90 85412 15358243 -0.290
0.604422541 15452761 85543.79 85866 15452599 -0.375
0.602600095 16024009 88411.29 88629 16023769 -0.246
0.601400292 16168441 89014.25 89315 16168291 -0.337
0.600508074 16410601 90200.46 90450 16410469 -0.276
0.599037163 16752649 91832.89 92130 16751719 -0.322
0.597873833 17048641 93256.21 93533 17048461 -0.296
0.596433928 17297281 94366.63 94731 17296999 -0.385
0.595016866 17799961 96854.63 97140 17799841 -0.294
0.594454335 17901361 97306.05 97629 17901271 -0.331
0.593893926 18003049 97758.27 98121 18002923 -0.370
0.593057389 18156121 98437.63 98805 18155941 -0.372
0.592502222 18258529 98891.86 99284 18258283 -0.395
0.590855193 18826921 101657.75 101991 18826783 -0.327
0.588435226 19562929 105154.99 105423 19562749 -0.254
0.586590759 20097289 107654.43 107873 20097223 -0.203
0.585291204 20421361 109124.10 109387 20421253 -0.240
0.58451797 20693401 110417.20 110643 20693083 -0.204
0.582234569 21520321 114334.38 114514 21520243 -0.157
0.581483244 21631801 114764.51 115040 21631699 -0.239
0.579501887 22306729 117900.50 118076 22306213 -0.149
0.577800136 22934521 120825.92 120974 22934239 -0.122
0.577077836 23049601 121266.10 121544 23049493 -0.229
0.574245652 24334489 127330.90 127469 24334399 -0.108
0.572856061 24690961 128852.22 129094 24690793 -0.187
0.571252834 25110121 130635.93 131019 25109923 -0.292
0.570797924 25230529 131147.97 131523 25230511 -0.285
0.569000548 26020201 134781.99 135137 26020039 -0.263
0.565929085 27384289 141000.97 141217 27384211 -0.153
0.564855246 27888961 143298.63 143467 27888649 -0.117
0.561704876 29365561 149954.62 150075 29365543 -0.080
0.56087899 29626249 151040.96 151206 29626153 -0.109
0.560058988 30019441 152798.61 152972 30019219 -0.113
0.559447754 30283009 153954.90 154158 30282823 -0.132
0.558839401 30481441 154778.38 155048 30481261 -0.174
0.556640474 31820881 160875.08 161004 31820863 -0.080
0.556049595 31956409 161371.86 161609 31956103 -0.147
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0.551240695 34234201 171221.01 171612 34233673 -0.228
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0.544510032 38427601 189595.43 189970 38426809 -0.197
0.542766962 39325441 193337.34 193761 39324361 -0.219
0.542077075 39702601 194917.28 195467 39702409 -0.281
0.540539258 40462321 198023.00 198740 40462309 -0.361
0.53902225 41615401 203031.63 203631 41615293 -0.294
0.537534293 42941809 208860.13 209283 42941683 -0.202
0.537043369 43178041 209797.31 210305 43177789 -0.241
0.53558518 44368921 214933.49 215325 44368789 -0.182
0.534944336 44769481 216586.61 217022 44769079 -0.201
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0.533673528 45738169 220690.37 221144 45737893 -0.205
0.533358723 45981961 221722.10 222159 45981283 -0.197
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0.530413503 48288601 231418.49 231869 48288343 -0.194
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0.523771778 54037201 255365.94 255846 54036799 -0.188
0.522643967 55636681 262293.25 262478 55636501 -0.070
0.522085474 56085121 264093.79 264359 56085061 -0.100
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0.514558161 63218401 292908.86 293685 63218371 -0.264
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0.513024071 65431921 302156.81 302711 65431843 -0.183
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0.495842935 88755241 394567.87 396184 88755049 -0.408
0.495632676 88981489 395387.08 397101 88981033 -0.432
0.49542264 89094721 395703.85 397554 89094463 -0.465
0.495213225 89548369 397531.14 399342 89548213 -0.453
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0.489835038 98624761 432505.47 434935 98624221 -0.559
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0.488468969 101868649 445336.76 447509 101868523 -0.485
0.487890098 102839881 448986.37 451256 102839731 -0.503
0.486363925 105534529 459135.74 461869 105534439 -0.592
0.486080597 106151809 461520.27 464251 106151701 -0.588
0.485704064 106770889 463809.60 466711 106770739 -0.622
0.484955992 108764041 471651.18 474324 108763819 -0.564
0.484584858 109390681 473962.37 476755 109390579 -0.586
0.484122881 110271001 477266.41 480192 110270971 -0.609
0.483846932 110901961 479691.02 482652 110901733 -0.613
0.482211979 114725521 494347.11 497231 114725113 -0.580
0.480873425 117961321 506705.90 509639 117960991 -0.576
0.48052002 118613881 509089.59 512153 118613611 -0.598
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0.479208921 122301481 523308.40 526028 122300641 -0.517
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0.478174991 124567921 531717.07 534702 124567141 -0.558
0.478003801 124835929 532647.87 535694 124835863 -0.569
0.476479911 128890609 547977.82 551106 128890273 -0.568
0.475314619 132043081 559837.30 563106 132042739 -0.580
0.474819899 133425601 565037.45 568342 133425559 -0.581
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0.473598753 137334961 579911.23 583078 137334739 -0.543
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0.470739687 145033849 608266.03 611977 145033831 -0.606
0.470505611 145757329 610958.91 614646 145757239 -0.600
0.4701947 146627881 614151.64 617897 146627251 -0.606
0.469652892 147938569 618838.79 622799 147937579 -0.636
0.469191869 149842081 626108.45 629823 149841931 -0.590
0.469038701 150136009 627107.18 630887 150135991 -0.599
0.468049744 153239641 638551.37 642333 153238621 -0.589
0.466550514 157276681 653010.45 657396 157276519 -0.667
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0.464274789 164429329 678957.13 683612 164429263 -0.681
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0.444815446 247527289 973827.97 982778 247527199 -0.911
0.444702398 247716121 974291.63 983415 247716109 -0.928
0.443803541 252460321 990677.37 1000134 252460099 -0.946
0.443302434 255136729 999899.33 1009701 255136363 -0.971
0.442915424 258019969 1010198.19 1019684 258019501 -0.930
0.442805171 258212761 1010669.10 1020352 258212443 -0.949
0.44231059 260531881 1018456.25 1028473 260531701 -0.974
0.442146653 262083721 1024092.25 1034086 262083259 -0.966
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0.441278426 267747769 1043895.63 1053864 267747661 -0.946
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0.439784421 277255801 1076816.06 1087046 277255753 -0.941
0.439520746 278656249 1081519.94 1091964 278655661 -0.956
0.438892087 283282561 1097690.81 1108257 283282369 -0.953
0.438528234 285711409 1106061.25 1116724 285710611 -0.955
0.438114399 288354361 1115097.74 1126007 288353929 -0.969
0.437805438 289986841 1120514.09 1131747 289986331 -0.993
0.436987509 295530481 1139512.88 1151095 295530241 -1.006
0.436835146 296149681 1141449.55 1153247 296149663 -1.023
0.436582145 299047849 1151861.43 1163214 299047753 -0.976
0.435978562 302377321 1162858.32 1174731 302377111 -1.011
0.435778233 303421561 1166266.29 1178292 303421501 -1.021
0.435329356 305935081 1174552.77 1187040 305935033 -1.052
0.434882973 309091561 1185289.33 1198146 309091243 -1.073

雷明85639720 · 发表于 2020-4-11 18:16

本帖最后由雷明85639720 于 2020-4-11 10:21 编辑

一个自然数是不是素数是绝对的，是就是，不是就不是，不存在误差和误差率的问题！这是我的一点看法，不一定正确，请参考。

ysr · 发表于 2020-4-14 10:48

老是打不开网页终于打开一次，楼主的研究是有意义的，误差率叫法不好，与实际的差也可以叫理论误差，但可能是不合适。由于孪生素数对的个数的实际函数上不减函数，我已经严格证明了，实际函数图像的曲线是台阶式上升的，所以理论值函数不可在任何一点误差(暂时这么叫)都是0的，误差为0就是好公式是不对的，误差是增大的才是对的，反之就不对了，比如上界，误差越来越小反而会在某点以后变成了下限公式，这样的公式是错误的没有价值的，误差随着实际值增大而增大，始终是上限或始终是下限才是对的，才是好公式有价值的公式。界线必须是明确的，在规定范围就是公式适用范围内没有反例。

lusishun · 发表于 2020-4-14 11:31

小于156，而大于11的孪生素数对有，
（156-2）·1/2·1/3·3/5·5/7·9/11·=9
实际有17，19
29，31
41，43
59，61
71，73
101，103
107，109
137，139
149，151

ysr · 发表于 2020-4-14 11:39

咋少了一对呢？11，13

lusishun · 发表于 2020-4-14 11:45

11，13被筛掉了，
11平方121，小于156，根据倍数含量筛法，最大素数，要筛到11，这样一来，11，13一对被筛掉了

lusishun · 发表于 2020-4-14 11:47

实际还有3，5
5，7两对也被筛掉了。

ysr · 发表于 2020-4-14 14:26

嗯，哈哈哈哈！连乘积公式写法不规范。有限个范围内是准确的。
文章短信的网页才能打开？哈

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