素数和孪生素数数量和几率

yangchuanju · 发表于 2021-4-16 15:40

白新岭发表于 2021-4-16 09:40
yangchuanju先生理解能力，渗透能力，搜集资料能力，提供数据材料能力都是超强的。
祝你在哥德巴赫猜想 ...

第二间距继续增大到128,512,2048，……怎么样？
一不做，二不休，再算一算第二间距是2,8,32,128……的情况，
（第二间距只能是2的奇次幂，因为没有间距4,4,4；4,16,4；……这样的四生素数）
前已算出，间距4,2,4的四生素数126万以内共387个；
间距4,8,4的四生素数126万以内共188个；
间距4,32,4的四生素数126万以内共352个；加在一起进行分析。
第2间距 2的指数个数
2 1 387
8 3 188
32 5 352
128 7 275
512 9 450
2048 11 194
8192 13 384
32768 15 204
131072 17 439
524288 19 174
2097152 21 345
8388608 23 150
33554432 25 250
134217728 27 122
536870912 29 180
2147483648 31 77
8589934592 33 141
34359738368 35 59
137438953472 37 136
549755813888 39 74
2199023255552 41 57
8796093022208 43 41
35184372088832 45 82
140737488355328 47 46
562949953421312 49 53
当第2间距是2,8,32……时，总体上随着加数的增大，四生素数个数波动式地越来越小；且波动曲线成明显锯齿状，一大一小；
其中当间距是4,2,4时，由双表兄弟素数构成的非最密四生素数数量最多。

ysr老师计算的间距由双孪生素数构成的四生素数个数有什么规律？

yangchuanju · 发表于 2021-4-16 17:07

本帖最后由 yangchuanju 于 2021-4-16 17:16 编辑

四生素数个数为什么呈标准式的锯齿状？
序号素数1 素数2 素数3 素数4
210以内间距4,2,4的有5组
1 7 11 13 17
2 13 17 19 23
3 37 41 43 47
4 97 101 103 107
5 103 107 109 113
210以内间距4,8,4的有3组
1 7 11 19 23
2 67 71 79 83
3 97 101 109 113
210以内间距4,32,4的有5组
1 7 11 43 47
2 43 47 79 83
3 67 71 103 107
4 127 131 163 167
5 193 197 229 233
210以内间距4,128,4的只1组
1 97 101 229 233
210以内间距4,512,4的也是1组
1 97 101 613 617
各组四生素数的素数1、素数3模6皆余1；素数2、素数4模6皆余5，完全相同！
它不是形成锯齿状的原因。

210以内的表兄弟素数共14对，它们的间距都是6的倍数（第1,2组间距等于4除外）
序号素数1 素数2 间距
1 3 7
2 7 11 4
3 13 17 6
4 19 23 6
5 37 41 18
6 43 47 6
7 67 71 24
8 79 83 12
9 97 101 18
10 103 107 6
11 109 113 6
12 127 131 18
13 163 167 36
14 193 197 30
这也不是形成锯齿状的原因。

另观察分析得知，间距等于2,32,512……的素数1的尾数包括3和7，间距等于8,128,2048……的素数 1的尾数值仅包括7，这可能是形成锯齿状的真正原因。

大傻8888888 · 发表于 2021-4-16 23:06

yangchuanju 发表于 2021-4-16 04:57
间距分别是2,6,30,210,2310,30030，…41#的（二生）素数对的数量和规律二
昨日对231000前20050个奇素数 ...

扩大样本范围后，根据素数对2，加数是510510时比加数是30030时大5160/4910≈1.051倍小于并接近16/15倍。如果再扩大样本范围，可以期待加数是510510时比加数是30030时更接近16/15倍。

yangchuanju · 发表于 2021-4-17 06:43

由两对孪生素数构成的四生素数个数变化规律与由两对表兄弟素数构成的四生素数个数变化规律完全相同，
本样本基数为第1-10000#和90001-100000#孪生素数，各一万对孪生素数；对应整数区间为1-126万和1631-1841万。
当素数2和素数3之间的邻距由4,16,64,256……逐步增大时，四生素数个数呈锯齿状波动式变化，先是稍微增大，随后逐渐变小，
其中起步点（最密4生素数）的个数不是很多，分别为孪生素数对数的2%和1%；
至两孪生素数对之间的邻距变到256时增大到4.9%和2.6%，达到最大值。
两样本相比，当两孪生素数对之间的邻距较小时，样本1的公式多于样本2的公式；
当邻距增大到2的24次方以后，样本2的个数变的与样本1的公式相当或更大些。
相关数表如下：
邻距 2的指数个数1 个数2 总个数个数1/个数2
4 2 212 101 313 2.0990
16 4 403 228 631 1.7675
64 6 291 149 440 1.9530
256 8 490 258 748 1.8992
1024 10 225 115 340 1.9565
4096 12 425 237 662 1.7932
16384 14 186 129 315 1.4419
65536 16 507 316 823 1.6044
262144 18 185 100 285 1.8500
1048576 20 373 255 628 1.4627
4194304 22 146 129 275 1.1318
16777216 24 201 230 431 0.8739
67108864 26 120 136 256 0.8824
268435456 28 172 194 366 0.8866
1073741824 30 82 74 156 1.1081
4294967296 32 159 165 324 0.9636
17179869184 34 55 58 113 0.9483
68719476736 36 126 171 297 0.7368
274877906944 38 59 44 103 1.3409

yangchuanju · 发表于 2021-4-17 07:05

下面是第一样本区的第1组四生素数
邻距孪生原序号素数1 素数2 素数3 素数4
4 2 5 7 11 13
16 3 11 13 29 31
64 2 5 7 71 73
256 3 11 13 269 271
1024 2 5 7 1031 1033
4096 5 29 31 4127 4129
16384 29 641 643 17027 17029
65536 6 41 43 65579 65581
262144 2 5 7 262151 262153
1048576 20 311 313 1048889 1048891
4194304 32 821 823 4195127 4195129
16777216 8 71 73 16777289 16777291
67108864 125 4931 4933 67113797 67113799
268435456 32 821 823 268436279 268436281
1073741824 2 5 7 1073741831 1073741833
4294967296 37 1031 1033 4294968329 4294968331
17179869184 358 21011 21013 17179890197 17179890199
68719476736 106 4049 4051 68719480787 68719480789
274877906944 128 5021 5023 274877911967 274877911969
以上是不同邻距时的最小的四生素数，请ysr老师核对！

yangchuanju · 发表于 2021-4-17 07:06

下面是第二样本区的第1组四生素数
邻距孪生原序号素数1 素数2 素数3 素数4
4 90029 16322231 16322233 16322237 16322239
16 90065 16330031 16330033 16330049 16330051
64 90035 16322921 16322923 16322987 16322989
256 90200 16360109 16360111 16360367 16360369
1024 90028 16321871 16321873 16322897 16322899
4096 90003 16315421 16315423 16319519 16319521
16384 90049 16326461 16326463 16342847 16342849
65536 90077 16332929 16332931 16398467 16398469
262144 90081 16334321 16334323 16596467 16596469
1048576 90021 16319621 16319623 17368199 17368201
4194304 90049 16326461 16326463 20520767 20520769
16777216 90023 16319969 16319971 33097187 33097189
67108864 90270 16374431 16374433 83483297 83483299
268435456 90013 16317641 16317643 284753099 284753101
1073741824 90043 16323971 16323973 1090065797 1090065799
4294967296 90040 16323611 16323613 4311290909 4311290911
17179869184 90028 16321871 16321873 17196191057 17196191059
68719476736 90156 16351061 16351063 68735827799 68735827801
274877906944 90192 16358891 16358893 274894265837 274894265839

yangchuanju · 发表于 2021-4-17 09:21

1万个最小的15位素数，用分解软件算出，在1*10^14开始的35万中间共有10768个，最小的一个是10^14+31；
在这10768个连续素数中共有421对孪生素数，最小的一对是10^14+97和+99；共有449对表兄弟素数，最小的一对是10^14+1173和+1177。
在这10768个连续素数中共有50对最密三生素数，其中间距2+4的30组，4+2的20组；
最小的间距2+4的最密三生首素数是10^14+20671；最小的间距4+2的最密三生首素数是10^14+15243。
在这10768个连续素数中没有间距2+4+2的最密四生素数，也没有间距4+2+4的跨度等于10的四生素数。

yangchuanju · 发表于 2021-4-17 09:37

孪生素数421对（仅给出首素数）
100000000000097 100000000085471 100000000179419 100000000263197
100000000000841 100000000087991 100000000179851 100000000263257
100000000001087 100000000088279 100000000180721 100000000264397
100000000003037 100000000088489 100000000181867 100000000264961
100000000003997 100000000089809 100000000182209 100000000265519
100000000005287 100000000089881 100000000183451 100000000266449
100000000006259 100000000090181 100000000184171 100000000267229
100000000007267 100000000090589 100000000185227 100000000267577
100000000007519 100000000092401 100000000185791 100000000267829
100000000008791 100000000093349 100000000185977 100000000268297
100000000008911 100000000094669 100000000186211 100000000271651
100000000009907 100000000097447 100000000188629 100000000273079
100000000010891 100000000097747 100000000189361 100000000273799
100000000011179 100000000098107 100000000189541 100000000274297
100000000012457 100000000098809 100000000189751 100000000275479
100000000014569 100000000098827 100000000190999 100000000276247
100000000015247 100000000098989 100000000191137 100000000276817
100000000015337 100000000100087 100000000193417 100000000277321
100000000016561 100000000102397 100000000193747 100000000278509
100000000018109 100000000103891 100000000193867 100000000279847
100000000020671 100000000104281 100000000194671 100000000280891
100000000020761 100000000104467 100000000194971 100000000281497
100000000021541 100000000107149 100000000195211 100000000281719
100000000021979 100000000107569 100000000195481 100000000284449
100000000022171 100000000107641 100000000197341 100000000284551
100000000022369 100000000110287 100000000197707 100000000285037
100000000024049 100000000110971 100000000197761 100000000285751
100000000024499 100000000111589 100000000198079 100000000287569
100000000025129 100000000112069 100000000198169 100000000288187
100000000025411 100000000113989 100000000199507 100000000288289
100000000025489 100000000114031 100000000199597 100000000289261
100000000026401 100000000115177 100000000202099 100000000289921
100000000027451 100000000115579 100000000202231 100000000290107
100000000027727 100000000115759 100000000203539 100000000291361
100000000028081 100000000116479 100000000205249 100000000291379
100000000029119 100000000116761 100000000206791 100000000293617
100000000029737 100000000117091 100000000206941 100000000294151
100000000030241 100000000117247 100000000207337 100000000296179
100000000030961 100000000118741 100000000208177 100000000297289
100000000031549 100000000119491 100000000208399 100000000298477
100000000031717 100000000120589 100000000208417 100000000298879
100000000031819 100000000120961 100000000208831 100000000300691
100000000032089 100000000121171 100000000208957 100000000301567
100000000032971 100000000121249 100000000209017 100000000303589
100000000036199 100000000121447 100000000209167 100000000303727
100000000036241 100000000121471 100000000209407 100000000304879
100000000036619 100000000123271 100000000209971 100000000305209
100000000037297 100000000124459 100000000210349 100000000309787
100000000037657 100000000126307 100000000211117 100000000311059
100000000037891 100000000126811 100000000211621 100000000311611
100000000038401 100000000126991 100000000212137 100000000311647
100000000038479 100000000128227 100000000212479 100000000313069
100000000039547 100000000129871 100000000213361 100000000313711
100000000039757 100000000130447 100000000214381 100000000313891
100000000039781 100000000130747 100000000215149 100000000314131
100000000040999 100000000132139 100000000216559 100000000314497
100000000041101 100000000132871 100000000217321 100000000314629
100000000041647 100000000133081 100000000218449 100000000315361
100000000043771 100000000133309 100000000220801 100000000317827
100000000045787 100000000134299 100000000222481 100000000318037
100000000047347 100000000134719 100000000224497 100000000319699
100000000047551 100000000135421 100000000225577 100000000321451
100000000047587 100000000137341 100000000226339 100000000321577
100000000048181 100000000139399 100000000226417 100000000322039
100000000050857 100000000139717 100000000226681 100000000323029
100000000050947 100000000141841 100000000231097 100000000324241
100000000051121 100000000142687 100000000231457 100000000325411
100000000053269 100000000142699 100000000231721 100000000325879
100000000053377 100000000142759 100000000233191 100000000327457
100000000054259 100000000144787 100000000234997 100000000328069
100000000055609 100000000144841 100000000235141 100000000329239
100000000055729 100000000145039 100000000236599 100000000330511
100000000057301 100000000145327 100000000237127 100000000330817
100000000057379 100000000145609 100000000238219 100000000331921
100000000057679 100000000146329 100000000238867 100000000332299
100000000058381 100000000147547 100000000240739 100000000332887
100000000058507 100000000148567 100000000243151 100000000333799
100000000059227 100000000148831 100000000244789 100000000334627
100000000059299 100000000149041 100000000245641 100000000334831
100000000059539 100000000149659 100000000246259 100000000335089
100000000059761 100000000151879 100000000249187 100000000335329
100000000060589 100000000152467 100000000249337 100000000335899
100000000061147 100000000152749 100000000249649 100000000336139
100000000061387 100000000154309 100000000251059 100000000336211
100000000061807 100000000154837 100000000251089 100000000336889
100000000061891 100000000155569 100000000251311 100000000337087
100000000063421 100000000156361 100000000251509 100000000337129
100000000063751 100000000159049 100000000252037 100000000337429
100000000064177 100000000161221 100000000252679 100000000338059
100000000066451 100000000162079 100000000252949 100000000341719
100000000069637 100000000162211 100000000252979 100000000342601
100000000069859 100000000163969 100000000253969 100000000342817
100000000074461 100000000164191 100000000254059 100000000343531
100000000074701 100000000165619 100000000254827 100000000343687
100000000074881 100000000166021 100000000255091 100000000343699
100000000076297 100000000166147 100000000257329 100000000344239
100000000078541 100000000166219 100000000257431 100000000344371
100000000081181 100000000168091 100000000258661 100000000344899
100000000081229 100000000169009 100000000258799 100000000345337
100000000081649 100000000170977 100000000259207 100000000346009
100000000081871 100000000171031 100000000259477 100000000346381
100000000081997 100000000172459 100000000260749 100000000347041
100000000082249 100000000173947 100000000261277 100000000349399
100000000083671 100000000174379 100000000262549
100000000084349 100000000177601 100000000262597
100000000085171 100000000178891 100000000262969

yangchuanju · 发表于 2021-4-17 09:38

表兄弟素数449对（仅给出首素数）
100000000001173 100000000098073 100000000171789 100000000263469
100000000003513 100000000098109 100000000173877 100000000265419
100000000003789 100000000098223 100000000174453 100000000267987
100000000003807 100000000098679 100000000174999 100000000268173
100000000004083 100000000098793 100000000175419 100000000269769
100000000004503 100000000098997 100000000176337 100000000270549
100000000006993 100000000099579 100000000176913 100000000271527
100000000007623 100000000099687 100000000177393 100000000271599
100000000008049 100000000099927 100000000178167 100000000271653
100000000008589 100000000101853 100000000178857 100000000272943
100000000008763 100000000102129 100000000179577 100000000274197
100000000008859 100000000103749 100000000181047 100000000274299
100000000011559 100000000104019 100000000181383 100000000275169
100000000011763 100000000106029 100000000181863 100000000275859
100000000012537 100000000106077 100000000181989 100000000276177
100000000013467 100000000107253 100000000182553 100000000276819
100000000015117 100000000107583 100000000182847 100000000277227
100000000015243 100000000107643 100000000183063 100000000277743
100000000016173 100000000108447 100000000183579 100000000279177
100000000016197 100000000109419 100000000183969 100000000279447
100000000016569 100000000109497 100000000184233 100000000279633
100000000017013 100000000110397 100000000185793 100000000279843
100000000018027 100000000113367 100000000186213 100000000280359
100000000018207 100000000113589 100000000188379 100000000280569
100000000020013 100000000114123 100000000189153 100000000281217
100000000020223 100000000114429 100000000189363 100000000281553
100000000020559 100000000114669 100000000190917 100000000281889
100000000020589 100000000116277 100000000191907 100000000284799
100000000020673 100000000116613 100000000192237 100000000284937
100000000022977 100000000117687 100000000192537 100000000287613
100000000023949 100000000118269 100000000192723 100000000288369
100000000024567 100000000118587 100000000192867 100000000288579
100000000025209 100000000119259 100000000193413 100000000289659
100000000028083 100000000119559 100000000194967 100000000290103
100000000028779 100000000119667 100000000196263 100000000290427
100000000030519 100000000119793 100000000196623 100000000290829
100000000030603 100000000119919 100000000196713 100000000290997
100000000030867 100000000119997 100000000197187 100000000291357
100000000031713 100000000120789 100000000197859 100000000291489
100000000031803 100000000122457 100000000197967 100000000292653
100000000032847 100000000122859 100000000198069 100000000293379
100000000032913 100000000123519 100000000198099 100000000293619
100000000034653 100000000123729 100000000198177 100000000294219
100000000034899 100000000124863 100000000201249 100000000294387
100000000035409 100000000126129 100000000202983 100000000296769
100000000036243 100000000126717 100000000203133 100000000297009
100000000036549 100000000127323 100000000203919 100000000299187
100000000037383 100000000127689 100000000204459 100000000300957
100000000037887 100000000129873 100000000204777 100000000301563
100000000039129 100000000131523 100000000204873 100000000306303
100000000039549 100000000131883 100000000204939 100000000306513
100000000039609 100000000131943 100000000205383 100000000307083
100000000039987 100000000132003 100000000205593 100000000307167
100000000040029 100000000132387 100000000206853 100000000308277
100000000040977 100000000133023 100000000207183 100000000309207
100000000041037 100000000134103 100000000207417 100000000309423
100000000041367 100000000134889 100000000208149 100000000309789
100000000041559 100000000135687 100000000208179 100000000313467
100000000043773 100000000135939 100000000209943 100000000313683
100000000044799 100000000136737 100000000210597 100000000315363
100000000045423 100000000137343 100000000210753 100000000316383
100000000046407 100000000139467 100000000211347 100000000316497
100000000048579 100000000140727 100000000211389 100000000316899
100000000048759 100000000140919 100000000212139 100000000316959
100000000051093 100000000141417 100000000213177 100000000318033
100000000051729 100000000142593 100000000213873 100000000318129
100000000052353 100000000144207 100000000214029 100000000320889
100000000055467 100000000144699 100000000215343 100000000321489
100000000057717 100000000145557 100000000216327 100000000321579
100000000057957 100000000147609 100000000218847 100000000323199
100000000058503 100000000147699 100000000221307 100000000325629
100000000059517 100000000147753 100000000222669 100000000326379
100000000060063 100000000148053 100000000224277 100000000327519
100000000061077 100000000148833 100000000224493 100000000327693
100000000061257 100000000149103 100000000224853 100000000327819
100000000062253 100000000149403 100000000225393 100000000328467
100000000062373 100000000150003 100000000225579 100000000329913
100000000063153 100000000150927 100000000229803 100000000329949
100000000063237 100000000151143 100000000229989 100000000330507
100000000063879 100000000151629 100000000230187 100000000330687
100000000066597 100000000152289 100000000230337 100000000330819
100000000068259 100000000152847 100000000230943 100000000331893
100000000068649 100000000153033 100000000231639 100000000332643
100000000069813 100000000155973 100000000233223 100000000333027
100000000070137 100000000156207 100000000234969 100000000334173
100000000072057 100000000157047 100000000235077 100000000336207
100000000074253 100000000157149 100000000235167 100000000336723
100000000075537 100000000158973 100000000236133 100000000336993
100000000076293 100000000159687 100000000237327 100000000337089
100000000077619 100000000160239 100000000239829 100000000338529
100000000078357 100000000160257 100000000240369 100000000338937
100000000079323 100000000160449 100000000241377 100000000339117
100000000079389 100000000161073 100000000242517 100000000339507
100000000082233 100000000161133 100000000243357 100000000341019
100000000083913 100000000161949 100000000243519 100000000341529
100000000084147 100000000162213 100000000243537 100000000341949
100000000084573 100000000162999 100000000243639 100000000342009
100000000085857 100000000163197 100000000244509 100000000342519
100000000086673 100000000163233 100000000244857 100000000343527
100000000088827 100000000164187 100000000244953 100000000343689
100000000088893 100000000166323 100000000245373 100000000344109
100000000089313 100000000166497 100000000246039 100000000344337
100000000089889 100000000166533 100000000246099 100000000344883
100000000090099 100000000166719 100000000246477 100000000344997
100000000090183 100000000167163 100000000249927 100000000345543
100000000091149 100000000168093 100000000253323 100000000345699
100000000091287 100000000169437 100000000253869 100000000345939
100000000092403 100000000169959 100000000254379 100000000346647
100000000093303 100000000170019 100000000254463 100000000346947
100000000094539 100000000170397 100000000255087 100000000348513
100000000095367 100000000171087 100000000255297
100000000096963 100000000171273 100000000257403
100000000097509 100000000171447 100000000262593

yangchuanju · 发表于 2021-4-17 09:38

最密三生素数共50组，其中间距2+4的30组，间距4+2的20组（仅给出首素数）
间距2+4的三生素数间距4+2的三生素数
100000000020671 100000000015243
100000000028081 100000000031713
100000000036241 100000000037887
100000000039547 100000000058503
100000000043771 100000000076293
100000000090181 100000000164187
100000000092401 100000000181863
100000000098107 100000000193413
100000000107641 100000000194967
100000000129871 100000000224493
100000000137341 100000000255087
100000000148831 100000000262593
100000000162211 100000000279843
100000000168091 100000000290103
100000000185791 100000000291357
100000000186211 100000000301563
100000000189361 100000000318033
100000000208177 100000000330507
100000000212137 100000000336207
100000000225577 100000000343527
100000000271651
100000000274297
100000000276817
100000000293617
100000000309787
100000000315361
100000000321577
100000000330817
100000000337087
100000000343687

		自动登录	找回密码
密码			注册

素数和孪生素数数量和几率

点评

点评

浏览过的版块