质数公式

yangchuanju · 发表于 2021-5-10 09:52

本帖最后由 yangchuanju 于 2021-5-10 10:01 编辑

另外根据素因子7和19出现的规律，可以推断：

当s=931,2989,5047，…，2058s+931时6666666666^s*18-1都能被2401*19整除；
当s=2695,8281,13867，…，5586s+2695时6666666666^s*18-1都能被343*361整除；
当s=19453,58555,97657，…，39102s+19453时6666666666^s*18-1都能被2401*361整除。

6666666666^s*18-1的素因子19出现在s=1处，19^2=361出现在s=35处，
以后在38t+35处都有361的身影，估计在s=38*19+35=757时会出现19的立方因子6859，
在s=38*19*19+757=13718+757=14475时可能会出现19的四次方因子130321，……

推断当s=49，343，637，…，294t+49时6666666666^7^k*18-1只能被6517=343*19整除，但都不能被2407*19或343*361整除！

对于（6666666666^7^k*18-1)/6517，其中可能存在少量素数，但为数不会很多，
太阳先生认定它是一个求算（寻找）大素数的公式，是没有任何道理的！

yangchuanju · 发表于 2021-5-10 11:57

本帖最后由 yangchuanju 于 2021-5-10 12:07 编辑

早已知晓，太阳先生非常想找到一些大素数，现谨提供已知的5万多个百位以上的前100个大素数，供参考！
如太阳先生想要更多的大素数，可通过邮箱发送！
已知的5万个100位以上的大素数之前100个：
rank    digits flabel wlabel          expression
序号位数结构形式             表达式
1 846521 95999 959w             96*10^846519-1
2 608990 79999 79w             8*10^608989-1
3 511057 49999 49w             5*10^511056-1
4 464844 49999 49w             5*10^464843-1
5 445774 49999 49w             5*10^445773-1
6 424861 33331 3w1             (10^424861-7)/3
7 414509 59999 59w             6*10^414508-1
8 383644 89999 89w             9*10^383643-1
9 380976 33331 3w1             (10^380976-7)/3
10 373264 —— 85789w          8579*10^373260-1
11 364522 89999 89w             9*10^364521-1
12 360410 —— 99999919w       9999992*10^360403-1
13 319890 59999 59w             6*10^319889-1
14 293135 59999 59w             6*10^293134-1
15 270659 59999 59w             6*10^270658-1
16 270343 11111 1w                (10^270343-1)/9
17 267599 59999 59w             6*10^267598-1
18 260254 89999 89w             9*10^260253-1
19 260198 77771 7w1             (7*10^260198-61)/9
20 255862 —— 451328423796179w 45132842379618*10^255848-1
21 255045 —— 99999989w       9999999*10^255038-1
22 249876 —— 99999979w       9999998*10^249869-1
23 237743 —— 51664093993218128449w 5166409399321812845*10^237724-1
24 236892 79999 79w             8*10^236891-1
25 234013 —— 99999929w       9999993*10^234006-1
26 232599 99997 9w7             10^232599-3
27 231618 60001 60w1             6*10^231617+1
28 229218 —— 451328423796179w 45132842379618*10^229204-1
29 225620 95999 959w             96*10^225618-1
30 225044 —— 451328423796179w 45132842379618*10^225030-1
31 225003 —— 9999989w          999999*10^224997-1
32 223872 89999 89w             9*10^223871-1
33 223474 —— 999989w          99999*10^223469-1
34 223167 —— 999929w          99993*10^223162-1
35 221631 99991 9w1             10^221631-9
36 218979 —— 99999949w       9999995*10^218972-1
37 218824 —— 451328423796179w 45132842379618*10^218810-1
38 214373 —— 9999919w          999992*10^214367-1
39 214340 —— 9949w             995*10^214337-1
40 212904 89999 89w             9*10^212903-1
41 211985 95999 959w             96*10^211983-1
42 211030 49999 49w             5*10^211029-1
43 205963 —— 99919w          9992*10^205959-1
44 200252 —— 51664093993218128449w 5166409399321812845*10^200233-1
45 198530 58889 58w9             (53*10^198529+1)/9
46 196685 48889 48w9             (44*10^196684+1)/9
47 195538 —— 99999929w       9999993*10^195531-1
48 191074 38889 38w9             (35*10^191073+1)/9
49 189587 —— 9989w             999*10^189584-1
50 183454 95999 959w             96*10^183452-1
51 181176 —— 99999979w       9999998*10^181169-1
52 180239 —— 9999999989w       999999999*10^180230-1
53 180215 —— 999999929w       99999993*10^180207-1
54 179973 49999 49w             5*10^179972-1
55 173135 —— 451328423796179w 45132842379618*10^173121-1
56 172394 —— 451328423796179w 45132842379618*10^172380-1
57 171167 —— 999929w          99993*10^171162-1
58 168758 —— 99999959w       9999996*10^168751-1
59 168728 —— 99949w          9995*10^168724-1
60 167503 92999 929w             93*10^167501-1
61 167449 —— 9929w             993*10^167446-1
62 161576 —— 9999949w          999995*10^161570-1
63 154742 —— 999919w          99992*10^154737-1
64 153262 —— 9999979w          999998*10^153256-1
65 149359 —— 51664093993218128449w 5166409399321812845*10^149340-1
66 148861 79997 79w7                   8*10^148860-3
67 148328 88889 8w9                   (8*10^148328+1)/9
68 146822 —— 51664093993218128449w 5166409399321812845*10^146803-1
69 144683 88889 8w9             (8*10^144683+1)/9
70 142365 —— 9999929w          999993*10^142359-1
71 141163 —— 9999959w          999996*10^141157-1
72 140948 —— 451328423796179w 45132842379618*10^140934-1
73 140328 95999 959w             96*10^140326-1
74 139672 98999 989w             99*10^139670-1
75 139290 28889 28w9             (26*10^139289+1)/9
76 138939 —— 9959w             996*10^138936-1
77 137251 78889 78w9             (71*10^137250+1)/9
78 134964 —— 999989w          99999*10^134959-1
79 134809 99899 9w89w             10^134809-10^67404-1
80 133581 98889 98w9             (89*10^133580+1)/9
81 132747 —— 999929w          99993*10^132742-1
82 131932 —— 9999989w          999999*10^131926-1
83 131396 97999 979w             98*10^131394-1
84 130622 94999 949w             95*10^130620-1
85 130567 95999 959w             96*10^130565-1
86 130134 —— 99999919w       9999992*10^130127-1
87 129024 —— 99949w          9995*10^129020-1
88 125877 99299 9w29w             10^125877-7*10^62938-1
89 125058 97999 979w             98*10^125056-1
90 123281 —— 9999989w          999999*10^123275-1
91 121622 18889 18w9             (17*10^121621+1)/9
92 121243 18889 18w9             (17*10^121242+1)/9
93 119931 79997 79w7             8*10^119930-3
94 119293 29999 29w             3*10^119292-1
95 118951 95999 959w             96*10^118949-1
96 118217 58889 58w9             (53*10^118216+1)/9
97 118142 —— 99919w          9992*10^118138-1
98 115811 78887 78w7             (71*10^115810-17)/9
99 115774 —— 999949w          99995*10^115769-1
100 113942 20009 20w9             2*10^113941+9

太阳 · 发表于 2021-5-10 12:00

本帖最后由太阳于 2021-5-10 12:01 编辑

找大素数确实很困难，找素数公式难度太大了，有可能素数公式不存在，假设有素数公式存在，也是个相当复杂式

yangchuanju · 发表于 2021-5-10 12:16

太阳发表于 2021-5-10 12:00
找大素数确实很困难，找素数公式难度太大了，有可能素数公式不存在，假设有素数公式存在，也是个相当复杂式

素数公式太简单了，2^n-1中就存在无穷多个素数，请太阳先生找一找第52个好吗？

太阳 · 发表于 2021-5-10 22:21

100 113942 20009 20w9 判断素数，2*10^113941+9 ，根据什么方法判断是素数，是试除法判断吗？

yangchuanju · 发表于 2021-5-11 08:09

本帖最后由 yangchuanju 于 2021-5-11 08:13 编辑

换个课题，求(67^k*18-1)/833中的素数

67^1=67，*18-1=1205=5*241
67^2=4489，*18-1=80801=7*7*17*97
67^3=300763，*18-1=5413733=13*416441
67^4=20151121，*18-1=362720177=17*337*63313
67^5=1350125107，*18-1=24302251925=5*5*7*7*59*336247
67^6=90458382169，*18-1=1628250879041=17*62071*1543063
67^7=6060711605323，*18-1=109092808895813=素数
67^8=406067677556641，*18-1=7309218196019537=7*7*17*79*383*290001377
67^9=27206534396294947，*18-1=489717619133309045=5*19*5154922306666411<P16>
67^10=1822837804551761449，*18-1=32811080481931706081=17*127*70705969*214937311

67^11=122130132904968017083，
*18-1=2198342392289424307493=7*7*36899*49627*24500031109<P11>
67^12=8182718904632857144561，
*18-1=147288940283391428602097=17*30809*281218322918229449<P18>
67^13=548242166610401428685587，
*18-1=9868358998987225716340565=5*333533*5917470834362552261<P19>
67^14=36732225162896895721934329，
*18-1=661180052932144122994817921=7*7*7*17*73*1553294631979157509567<P22>
67^15=2461059085914092013369600043，
*18-1=44299063546453656240652800773=13*53*83*1381*1563623*358733143128733<P15>
67^16=164890958756244164895763202881，
*18-1=2968037257612394968123737651857=17*23*7590888126885920634587564327<C28>

67^17=11047694236668359048016134593027，
*18-1=198858496260030462864290422674485
=5*7*7*6060809*133920625394001646641017<P24>
67^18=740195513856780056217081017732809，
*18-1=13323519249422041011907458319190561
=17*4703*42433*125083561*31397211577499447<P17>
67^19=49593099428404263766544428188098203，
*18-1=892675789711276747797799707385767653
=1697*1176936439993*446949961839521904493<P21>
67^20=3322737661703085672358476688602579601，
*18-1=59809277910655542102452580394846432817
=7*7*17*919*2729*33510555583*854324542033766753<P18>
67^21=222623423334106740048017938136372833267，
*18-1=4007221620013921320864322886454710998805
=5*79*1043467*9722267077727170596842546030077<C31>

67^25=4486111541039808345623155281556563464276142307，
*18-1=80750007738716550221216795068018142356970561525，该数以25结尾，肯定含因子25了；
67^27=20138154707727699663502344058907413391135602816123，
*18-1=362486784739098593943042193060333441040440850690213——内含一个素因子19，不含361。

在67^k*18-1的分解式中，素数7总是成对成双地出现，最早出现在k=2时，随后指数k每增加3又出现一次；
素数17最早与7同时出现在k=2中，随后指数k每增加2再出现一次；
3和2的最小公倍数是6，指数k每增加6，7*7*17总要同时出现一次，如k=2,8,14,20……。

素因子5最早出现在k=1时，指数k每增加4，素因子5再出现一次（周期等于4，即基为67时，素数5的循环周期是4，最大值），其中k=5时出现5的平方；预计下一个5的平方要在k=25时出现。（经验证正确）
素因子19最早出现在k=9时，在k=20以内没在出现，说明素因子19个出现周期不是2,3,6,9，据此它的周期应该是18，预计要在k=9+18=27时才会再出现。（经验证正确）

7*7的周期是3，起步点是2；19的周期是18，起步点是9；素因子7和19不会同时出现，即代数式67^k*18-1不能既被7整除，又被19整除。

若将分母定为7*7*17=833，则当k=2时代数式(67^k*18-1)/833是一个素数97，最小的。下一个素数出现在哪儿？即k等于多少时，(67^k*18-1)/833又是素数？不详。或许不再有这样的指数k！

另外，当k=7时67^k*18-1本身是素数，下一个本身是素数的出现在哪儿？亦不详。

yangchuanju · 发表于 2021-5-11 16:30

本帖最后由 yangchuanju 于 2021-5-11 17:24 编辑

几种特殊的形式的素数

（一）清一色数：已知的清一色素数有11、1111111111111111111<19>、11111111111111111111111<23>、111…1<317>、111…1<1031>等9个。
n=2,19,23,317,1031,49081,86453,109297,270343时都是素数。

（二）101型素数：已知的101型素数只有两个，它们的11和101，有没有第3个101型素数尚不知道。

（三）9901型素数：已知的9901型素数有四个，9901、99990001、999999000001、9999999900000001，有没有更多个9901型素数，尚不知道。

（四）9091型素数：已知的9091型素数有12个，它们是由(10^n+1)/11得到的，前6个是：
9091, 909091, 909090909090909091, 909090909090909090909090909091, 9090909090909090909090909090909090909090909090909091<52>, 909090909090909090909090909090909090909090909090909090909090909091<66>
12个素数对应的指数n等于：5, 7, 19, 31, 53, 67, 293, 641, 2137, 3011, 268207, 1600787；
分别含90和91的个数是：2, 3, 9, 15, 26, 33, 146, 320, 1068, 1505, 134103, 800393。
肯定还有更多个9091型素数。

（五）9001型素数：已知的900.型素数有23个，它们是：
Known (probable) prime numbers
9×10^3+1 = 9001 is prime.
9×10^4+1 = 90001 is prime.
9×10^5+1 = 900001 is prime.
9×10^9+1 = 9000000001<10> is prime.
9×10^22+1 = 9(0)21  1<23> is prime.
9×10^27+1 = 9(0)26  1<28> is prime.
9×10^36+1 = 9(0)35  1<37> is prime.
9×10^57+1 = 9(0)56  1<58> is prime.
9×10^62+1 = 9(0)61  1<63> is prime.
9×10^78+1 = 9(0)77  1<79> is prime.
9×10^201+1 = 9(0)200  1<202> is prime.
9×10^537+1 = 9(0)536  1<538> is prime.
9×10^696+1 = 9(0)695  1<697> is prime.
9×10^790+1 = 9(0)789  1<791> is prime.
9×10^905+1 = 9(0)904  1<906> is prime.
9×10^1038+1 = 9(0)1037  1<1039> is prime. (Harvey Dubner / Cruncher / December 31, 1984 )
9×10^66886+1 = 9(0)66885  1<66887> is prime. (Peter Benson / NewPGen, OpenPFGW, Proth.exe / December 31, 2004 )
9×10^70500+1 = 9(0)70499  1<70501> is prime. (Peter Benson / NewPGen, OpenPFGW, Proth.exe / March 10, 2005 )
9×10^91836+1 = 9(0)91835  1<91837> is prime. (Dmitry Domanov / Prime95 v25.11, pfgw / March 8, 2010 )
9×10^100613+1 = 9(0)100612  1<100614> is prime. (Predrag Kurtovic / September 23, 2013 )
9×10^127240+1 = 9(0)127239  1<127241> is prime. (Bob Price / PFGW / January 22, 2015 )
9×10^380734+1 = 9(0)380733  1<380735> is prime. (Predrag Kurtovic / llr64 / September 18, 2019 )
9×10^583696+1 = 9(0)583695  1<583697> is prime. (Predrag Kurtovic / Srsieve, Prime95, LLR / June 25, 2020 )

太阳 · 发表于 2021-5-11 18:13

已知的9901型素数有四个，9901、99990001、999999000001、9999999900000001，有没有更多个9901型素数，尚不知道，
这样型素数应该只有4个，找不到其它的素数

yangchuanju · 发表于 2021-5-12 14:34

(b^n-1/(b-1)型素数（广义梅森素数）
A000043
Numbers n such that 2^n - 1 is prime.梅森素数
2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583
A******
Numbers n such that (3^n - 1)/2 is prime.
3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843
A004061
Numbers n such that (5^n - 1)/4 is prime.
3, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, 13241, 13873, 16519, 201359, 396413
A004062
Numbers n such that (6^n - 1)/5 is prime.
2, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, 10613, 19889, 79987, 608099
A004063
Numbers n such that (7^n - 1)/6 is prime.
5, 13, 131, 149, 1699, 14221, 35201, 126037, 371669
A004023
Indices of prime repunits: numbers n such that 11...111 = (10^n - 1)/9 is prime.
2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343
A005808
Numbers n such that (11^n - 1)/10 is prime.
17, 19, 73, 139, 907, 1907, 2029, 4801, 5153, 10867, 20161, 293831
A004064
Numbers n such that (12^n - 1)/11 is prime.
2, 3, 5, 19, 97, 109, 317, 353, 701, 9739, 14951, 37573, 46889
A016054
Numbers n such that (13^n - 1)/12 is prime.
5, 7, 137, 283, 883, 991, 1021, 1193, 3671, 18743, 31751, 101089
A006032
Numbers n such that (14^n - 1)/13 is prime.
3, 7, 19, 31, 41, 2687, 19697, 59693, 67421
A006033
Numbers n such that (15^n - 1)/14 is prime.
3, 43, 73, 487, 2579, 8741, 37441, 89009
A006034
Numbers n such that (17^n-1)/16 is prime.
3, 5, 7, 11, 47, 71, 419, 4799, 35149, 54919, 74509
A006035
Numbers n such that (19^n-1)/18 is prime.
19, 31, 47, 59, 61, 107, 337, 1061, 9511, 22051, 209359
A127995
Numbers n such that (20^n - 1)/19 is prime.
3, 11, 17, 1487, 31013, 48859, 61403
A127996
Numbers n such that (21^n - 1)/20 is prime.
3, 11, 17, 43, 271, 156217
A127997
Numbers n such that (22^n - 1)/21 is prime.
2, 5, 79, 101, 359, 857, 4463, 9029, 27823
A127998
Numbers n such that (24^n - 1)/23 is prime.
3, 5, 19, 53, 71, 653, 661, 10343, 49307
A127999
Numbers n such that (26^n - 1)/25 is prime.
7, 43, 347, 12421, 12473, 26717
A128000
Numbers n such that (28^n - 1)/27 is prime.
2, 5, 17, 457, 1423
A181979
Numbers n such that (29^n - 1)/28 is prime.
5, 151, 3719, 49211, 77237
A098438
Numbers n such that (30^n-1)/29 is prime.
2, 5, 11, 163, 569, 1789, 8447, 72871, 78857, 82883
A128002
Numbers n such that (31^n - 1)/30 is prime.
7, 17, 31, 5581, 9973, 101111
A209120
Numbers n such that (33^n - 1)/32 is prime.
3, 197, 3581, 6871
A185073
Numbers n such that (34^n - 1)/33 is prime.
13, 1493, 5851, 6379
A128003
Numbers n such that (37^n - 1)/36 is prime.
13, 71, 181, 251, 463, 521, 7321, 36473, 48157, 87421
A128004
Numbers n such that (38^n - 1)/37 is prime.
3, 7, 401, 449
A181987
Numbers n such that (39^n - 1)/38 is prime.
349, 631, 4493, 16633, 36341
A128005
Numbers n such that (40^n - 1)/39 is prime.
2, 5, 7, 19, 23, 29, 541, 751, 1277

yangchuanju · 发表于 2021-5-12 14:35

本帖最后由 yangchuanju 于 2021-5-12 14:38 编辑

(b^n+1/(b+1)型素数（Wagstaff numbers）
A001562
Numbers n such that (10^n + 1)/11 is a prime.
2, 3, 9, 15, 26, 33, 146, 320, 1068, 1505
A000978
Wagstaff numbers: numbers n such that (2^n + 1)/3 is prime.
3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, 4031399
A007658
Numbers n such that (3^n + 1)/4 is prime.
3, 5, 7, 13, 23, 43, 281, 359, 487, 577, 1579, 1663, 1741, 3191, 9209, 11257, 12743, 13093, 17027, 26633, 104243, 134227, 152287, 700897, 1205459, 1896463, 2533963
A057171
Numbers n such that (5^n+1)/6 is a prime.
5, 67, 101, 103, 229, 347, 4013, 23297, 30133, 177337, 193939, 266863, 277183, 335429, 1856147
A057172
Numbers n such that (6^n + 1)/7 is a prime.
3, 11, 31, 43, 47, 59, 107, 811, 2819, 4817, 9601, 33581, 38447, 41341, 131891, 196337, 1313371
A057173
Numbers n such that (7^n + 1)/8 is a prime.
3, 17, 23, 29, 47, 61, 1619, 18251, 106187, 201653, 1178033
A057175
3, 59, 223, 547, 773, 1009, 1823, 3803, 49223, 193247, 703393, 860029
A057177
Numbers n such that (11^n + 1)/12 is a prime.
5, 7, 179, 229, 439, 557, 6113, 223999, 327001
A057178
Numbers k such that (12^k + 1)/13 is a prime.
5, 11, 109, 193, 1483, 11353, 21419, 21911, 24071, 106859, 139739, 495953
A057179
Numbers n such that (13^n + 1)/14 is a prime.
3, 11, 17, 19, 919, 1151, 2791, 9323, 56333, 1199467
A057180
Numbers n such that (14^n + 1)/15 is a prime.
7, 53, 503, 1229, 22637, 1091401
A057181
Numbers n such that (15^n + 1)/16 is a prime.
3, 7, 29, 1091, 2423, 54449, 67489, 551927
A057182
Numbers n such that (16^n + 1)/17 is a prime.
3, 5, 7, 23, 37, 89, 149, 173, 251, 307, 317, 30197, 1025393
A057183
Numbers n such that (17^n + 1)/18 is a prime.
7, 17, 23, 47, 967, 6653, 8297, 41221, 113621, 233689, 348259
A057184
Numbers n such that (18^n + 1)/19 is a prime.
3, 7, 23, 73, 733, 941, 1097, 1933, 4651, 481147
A057185
Numbers n such that (19^n + 1)/20 is a prime.
17, 37, 157, 163, 631, 7351, 26183, 30713, 41201, 77951, 476929
A057186
Numbers n such that (20^n+1)/21 is a prime.
5, 79, 89, 709, 797, 1163, 6971, 140053, 177967, 393257
A057187
Numbers n such that (21^n+1)/22 is a prime.
3, 5, 7, 13, 37, 347, 17597, 59183, 80761, 210599, 394579
A057188
Numbers n such that (22^n+1)/23 is a prime.
3, 5, 13, 43, 79, 101, 107, 227, 353, 7393, 50287
A057189
Numbers n such that (23^n+1)/24 is a prime.
11, 13, 67, 109, 331, 587, 24071, 29881, 44053
A057190
Numbers n such that (24^n+1)/25 is a prime.
7, 11, 19, 2207, 2477, 4951
A057191
Numbers n such that (25^n+1)/26 is a prime.
A001562
Numbers n such that (10^n + 1)/11 is a prime.
5, 7, 19, 31, 53, 67, 293, 641, 2137, 3011, 268207, 1600787

		自动登录	找回密码
密码			注册

质数公式

点评