坎宁安第一类素数链

yangchuanju · 发表于 2021-8-5 13:06

本帖最后由 yangchuanju 于 2021-8-5 13:07 编辑

蔡家雄发表于 2021-8-5 10:22
素数链：Q(n+1)=3*Q(n) +2，求证：这条素数链一定是9字尾素数。

素数差(首项差2p+2)等比3 的8生素数是存 ...

存在大量的各种类型的累加或连乘式素数链，如p，a1*p±b1，a2*p±b2，a3*p±b3，……。
如a1=2,a2=4,a3=8,……；b1=1,b2=3,b3=7,……；各项都取加号，就是上面所谈坎宁安第一类素数链；各项都取减号，就是坎宁安第二类素数链。坎宁安素数链已被研究的很透彻。
如a1=3,a2=9,a3=27,……；b1=2,b2=8,b3=26,……；各项都取加号，就是蔡家雄老师给出的第二个素数链，没有专用名称。
网上可搜索到的累加或连乘式素数链大多是2-3级的，间或有个别4-5级的。包括蔡老师构成的3p+2型的。

yangchuanju · 发表于 2021-8-5 13:33

本帖最后由 yangchuanju 于 2021-8-5 13:34 编辑

网页A083388给出了1-14级3p+2型连乘式素数链的最小首素数：
2, 3, 5, 29, 1129, 10009, 575119, 32694619, 586130089, 1030544269, 427744892959, 18736892249459, 221530676461709, 221530676461709
蔡老师给出的问题是8级，最小首素数是32694619；各个素数连乘7次都是素数。
1 98083859 is prime 50 30395815379 is prime
2 147977549 is prime 51 47476537289 is prime
3 184785929 is prime 52 54701092529 is prime
4 500068799 is prime 53 59366822129 is prime
5 595314089 is prime 54 63775570319 is prime
6 665633429 is prime 55 67276250999 is prime
7 688238459 is prime 56 72974515949 is prime
8 1125770939 is prime 57 7944792659 is prime
9 1758390269 is prime 58 11986181549 is prime
10 2025966389 is prime 59 14967660329 is prime
11 2198771189 is prime 60 40505572799 is prime
12 2362058159 is prime 61 48220441289 is prime
13 2491712999 is prime 62 53916307829 is prime
14 2702759849 is prime 63 55747315259 is prime
15 294251579 is prime 64 91187446139 is prime
16 443932649 is prime 65 142429611869 is prime
17 554357789 is prime 66 164103277589 is prime
18 1500206399 is prime 67 178100466389 is prime
19 1785942269 is prime 68 191326710959 is prime
20 1996900289 is prime 69 201828752999 is prime
21 2064715379 is prime 70 218923547849 is prime
22 3377312819 is prime 71 23834377979 is prime
23 5275170809 is prime 72 35958544649 is prime
24 6077899169 is prime 73 44902980989 is prime
25 6596313569 is prime 74 121516718399 is prime
26 7086174479 is prime 75 144661323869 is prime
27 7475138999 is prime 76 161748923489 is prime
28 8108279549 is prime 77 167241945779 is prime
29 882754739 is prime 78 273562338419 is prime
30 1331797949 is prime 79 427288835609 is prime
31 1663073369 is prime 80 492309832769 is prime
32 4500619199 is prime 81 534301399169 is prime
33 5357826809 is prime 82 573980132879 is prime
34 5990700869 is prime 83 605486258999 is prime
35 6194146139 is prime 84 656770643549 is prime
36 10131938459 is prime 85 71503133939 is prime
37 15825512429 is prime 86 107875633949 is prime
38 18233697509 is prime 87 134708942969 is prime
39 19788940709 is prime 88 364550155199 is prime
40 21258523439 is prime 89 433983971609 is prime
41 22425416999 is prime 90 485246770469 is prime
42 24324838649 is prime 91 501725837339 is prime
43 2648264219 is prime 92 820687015259 is prime
44 3995393849 is prime 93 1281866506829 is prime
45 4989220109 is prime 94 1476929498309 is prime
46 13501857599 is prime 95 1602904197509 is prime
47 16073480429 is prime 96 1721940398639 is prime
48 17972102609 is prime 97 1816458776999 is prime
49 18582438419 is prime 98 1970311930649 is prime

yangchuanju · 发表于 2021-8-5 13:33

网页A023208给出p,3p+2都是素数的首素数10000个；
3, 5, 7, 13, 17, 19, 23, 29, 37, 43, 59, 79, 83, 89, 97, 103, 127, 139, 149, 163, 167, 173, 197, 199, 227, 233, 239, 257, 269, 293, 313, 317, 337, 349, 353, 367, 383, 397, 409, 419, 433, 439, 457, 479, 499, 503, 523, 569, 577, 607, 643, 659, 709, 757, 769, 797, 859, 863
网页A023246给出p,3p+2都是素数的3级素数链的首素数10000个；
5, 7, 19, 29, 79, 89, 97, 127, 139, 167, 317, 337, 397, 419, 607, 659, 709, 877, 929, 1069, 1129, 1409, 1699, 1777, 2029, 2099, 2267, 2339, 2557, 2617, 2707, 2837, 2917, 2939, 3019, 3067, 3389, 3407, 3529, 3617, 3659, 3719, 4229, 4549, 4919, 5209, 5227, 5417
网页A023277给出p,3p+2都是素数的4级素数链的首素数1000个；
29, 139, 1129, 1409, 2339, 2939, 3389, 4549, 7309, 10009, 10039, 11489, 12539, 13859, 15259, 17159, 17569, 20939, 22079, 24229, 24509, 27799, 28099, 29339, 30029, 32059, 33599, 34469, 37379, 37619, 39419, 41579, 42359, 44549, 52919, 55259, 57679, 58109
网页A023307给出p,3p+2都是素数的5级素数链的首素数10000个；
1129, 10009, 11489, 12539, 13859, 30029, 63079, 77359, 99119, 121039, 124669, 169409, 194749, 205589, 246329, 330329, 349519, 351829, 354839, 361279, 369539, 384589, 395719, 399769, 416989, 429109, 446819, 527599, 532489, 544259, 575119

在各个网页中没有找到6级及6级以上素数链的首素数，但从网页A023307给出的10000个5级首素数中可以找到6级及6级以上的素数链。

yangchuanju · 发表于 2021-8-5 15:01

3p+2型6级素数链（首尾素数）909个（仅发前50个）：
序号 p1 p6
2 10009 2432429
31 575119 139754159
35 724499 176053499
47 1373819 333838259
52 1725359 419262479
75 3188569 774822509
84 3715669 902907809
98 4980499 1210261499
107 5939779 1443366539
146 9202199 2236134599
150 9565709 2324467529
187 13707469 3330915209
196 14483839 3519573119
202 14886829 3617499689
210 15412739 3745295819
219 16158589 3926537369
222 16740709 4067992529
236 18207349 4424386049
244 19429409 4721346629
270 22428209 5450055029
273 22602719 5492460959
276 22722419 5521548059
277 22749019 5528011859
279 22877119 5559140159
291 24171209 5873604029
292 24348869 5916775409
301 24914119 6054131159
322 27344729 6644769389
327 27606599 6708403799
333 28170169 6845351309
338 28392139 6899290019
357 30400789 7387391969
373 32047399 7787518199
375 32694619 7944792659
383 33003389 8019823769
386 33802649 8214043949
410 37434109 9096488729
423 40004509 9721095929
425 40258399 9782791199
433 41407939 10062129419
434 41580559 10104076079
446 43145549 10484368649
458 44113999 10719701999
474 45140969 10969255709
481 45904249 11154732749
487 46902239 11397244319
501 48581539 11805314219
507 49325849 11986181549
516 50222129 12203977589
530 52332209 12716727029

yangchuanju · 发表于 2021-8-5 15:02

3p+2型7级素数链（首尾素数）155个（仅发前50个）：
序号 p1 p7
31 575119 419262479
75 3188569 2324467529
146 9202199 6708403799
222 16740709 12203977589
273 22602719 16477382879
279 22877119 16677420479
375 32694619 23834377979
383 33003389 24059471309
433 41407939 30186388259
507 49325849 35958544649
578 57982469 42269220629
607 61595309 44902980989
673 71059169 51802134929
873 98083859 71503133939
927 105328789 76784687909
945 108796099 79312356899
949 109824469 80062038629
1000 117674899 85785002099
1048 124313209 90624330089
1208 147977549 107875633949
1255 155618539 113445915659
1259 156158449 113839510049
1279 158801719 115766453879
1330 166689599 121516718399
1336 167460299 122078558699
1423 183881039 134049278159
1434 184785929 134708942969
1503 196931909 143563362389
1512 198438029 144661323869
1518 199048219 145106152379
1519 199086509 145134065789
1629 221877809 161748923489
1674 229412819 167241945779
1724 238850149 174121759349
1734 239755039 174781424159
1874 265783909 193756470389
1879 266264459 194106791339
1926 272852299 198909326699
1963 278594189 203095164509
2096 305430509 222658841789
2311 348852769 254313669329
2448 374486209 273000447089
2450 375256979 273562338419
2551 397466999 289753442999
2565 400868089 292232837609
2718 431899649 314854844849
2739 435910649 317778863849
2926 481269179 350845232219
2959 487844419 355638582179
3013 500068799 364550155199

yangchuanju · 发表于 2021-8-5 15:02

3p+2型8级素数链（首尾素数）28个：
序号 p1 p8
375 32694619 71503133939
507 49325849 107875633949
607 61595309 134708942969
1330 166689599 364550155199
1512 198438029 433983971609
1629 221877809 485246770469
1674 229412819 501725837339
2450 375256979 820687015259
3413 586130089 1281866506829
3849 675322129 1476929498309
4093 732923729 1602904197509
4322 787352719 1721940398639
4509 830570999 1816458776999
4807 900919949 1970311930649
5343 1030544269 2253800318489
5542 1084419419 2371625271539
5912 1170965879 2560902379559
5973 1189331149 2601067225049
6106 1222799759 2674263075119
6550 1337215879 2924491129559
6645 1363722149 2982460342049
7084 1475653339 3227253854579
7684 1632041669 3569275132289
8152 1758390269 3845599520489
8364 1818192179 3976386297659
8545 1870211909 4090153447169
9026 2011330859 4398780590819
9666 2198771189 4808712592529

前14个素数计算蔡家雄给出的14个素数。

yangchuanju · 发表于 2021-8-5 15:02

3p+2型9级素数链（首尾素数）5个：
序号 p1 p9
3413 586130089 3845599520489
4093 732923729 4808712592529
5343 1030544269 6761400955469
5542 1084419419 7114875814619
7084 1475653339 9681761563739
9026 2011330859 13196341772459

yangchuanju · 发表于 2021-8-5 15:03

3p+2型10级素数链（全链）1个：
序号 5343
p1 1030544269
p2 3091632809
p3 9274898429
p4 27824695289
p5 83474085869
p6 250422257609
p7 751266772829
p8 2253800318489
p9 6761400955469
p10 20284202866409

wlc1 · 发表于 2021-8-5 19:03

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wlc1 · 发表于 2021-8-5 19:04

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