1513位唯一循环周期大素数

yangchuanju · 发表于 2021-5-7 12:29

本帖最后由 yangchuanju 于 2021-5-7 12:33 编辑

99个唯一循环周期素数的循环周期和位数（A051627）按位数排序：
周期位数
1 1 1
2 2 2
3 3 2
4 4 3
5 10 4
6 12 4
7 9 6
8 14 6
9 24 8
10 36 12
11 48 16
12 38 18
13 19 19
14 23 23
15 39 24
16 62 30
17 120 33
18 150 41
19 106 52
20 93 60
21 134 66
22 294 84
23 196 84
24 320 128
25 654 217
26 738 241
27 385 241
28 586 292
29 317 317
30 597 396
31 1404 433
32 945 433
33 1452 441
34 1836 577
35 1752 577
36 1172 584
37 1812 601
38 1282 640
39 1426 661
40 2232 721
41 1862 757
42 1844 920
43 1521 936
44 2134 961
45 3750 1001
46 1031 1031
47 2264 1128
48 2667 1513
49 4354 1861
50 3927 1920
51 4274 2136
52 6522 2520
53 3903 2600
54 6022 3010
55 6682 3073
56 6135 3265
57 9550 3801
58 5877 3912
59 15456 4224
60 8696 4344
61 13092 4361
62 13370 4560
63 6437 6240
64 14714 6301
65 10887 6841
66 19240 6912
67 23664 7168
68 18910 7200
69 24174 7488
70 18668 8593
71 36630 8641
72 20644 9505
73 23912 10081
74 26570 10625
75 40120 14848
76 47184 15713
77 52062 17353
78 18827 18480
79 59772 18688
80 47498 20160
81 79710 21248
82 32481 21600
83 70842 23613
84 47468 23732
85 73094 29832
86 92712 30897
87 82360 31360
88 73808 31585
89 92092 31681
90 68102 32033
91 103248 34273
92 88676 37993
93 100480 39937
94 55601 43057
95 145878 47360
96 49081 49081
97 141230 54432
98 77077 55441
99 198552 66177

yangchuanju · 发表于 2021-5-7 12:36

106个唯一循环周期素数的循环周期和位数（A007498）按周期排序：
周期位数
1 1 1
2 2 2
3 3 2
4 4 3
5 9 6
6 10 4
7 12 4
8 14 6
9 19 19
10 23 23
11 24 8
12 36 12
13 38 18
14 39 24
15 48 16
16 62 30
17 93 60
18 106 52
19 120 33
20 134 66
21 150 41
22 196 84
23 294 84
24 317 317
25 320 128
26 385 241
27 586 292
28 597 396
29 654 217
30 738 241
31 945 433
32 1031 1031
33 1172 584
34 1282 640
35 1404 433
36 1426 661
37 1452 441
38 1521 936
39 1752 577
40 1812 601
41 1836 577
42 1844 920
43 1862 757
44 2134 961
45 2232 721
46 2264 1128
47 2667 1513
48 3750 1001
49 3903 2600
50 3927 1920
51 4274 2136
52 4354 1861
53 5877 3912
54 6022 3010
55 6135 3265
56 6437 6240
57 6522 2520
58 6682 3073
59 8696 4344
60 9550 3801
61 10887 6841
62 13092 4361
63 13370 4560
64 14714 6301
65 15456 4224
66 18668 8593
67 18827 18480
68 18910 7200
69 19240 6912
70 20644 9505
71 23664 7168
72 23912 10081
73 24174 7488
74 26570 10625
75 32481 21600
76 36630 8641
77 40120 14848
78 47184 15713
79 47468 23732
80 47498 20160
81 49081 49081
82 52062 17353
83 55601 43057
84 59772 18688
85 68102 32033
86 70842 23613
87 73094 29832
88 73808 31585
89 77077 55441
90 79710 21248
91 82360 31360
92 86453 86453
93 88676 37993
94 92092 31681
95 92712 30897
96 100480 39937
97 103248 34273
98 109297 109297
99 141230 54432
100 145878 47360
101 157825 122961
102 159495 70560
103 175364 75145
104 177231 118152
105 180683 179760
106 198552 66177

ysr · 发表于 2021-5-8 20:01

10000099999999989999899999000000000100001这是素数有41位，用时2.210938秒.

这么大的素数你们是怎么判定的呢？

ysr · 发表于 2021-5-8 20:25

"专业网站提供的！ "

谢谢！终于知道了，那就是专业高手弄出来的在线程序，运行平台可能是服务器等大型计算机，速度快，比不了。

yangchuanju · 发表于 2021-5-12 12:47

本帖最后由 yangchuanju 于 2021-5-12 12:53 编辑

各种类型清一色数的φ因子
不同类型的清一色数（10^n-1）/9有着不同类型的φ因子，以下简单分类如下：
（一）指数是素数时，该类清一色数的φ因子最简单，也最大，就等于清一色数本身；除少数几个是素数外（φ2，φ19，φ23等），其余都是合数：
Φ2(10) = 11 = 11(100.00%)
Φ3(10) = 111 = 3 × 37(100.00%)
Φ5(10) = 11111 = 41 × 271(100.00%)
Φ7(10) = 1111111 = 239 × 4649(100.00%)
Φ11(10) = 11111111111<11> = 21649 × 513239(100.00%)
Φ13(10) = 1111111111111<13> = 53 × 79 × 265371653(100.00%)
Φ17(10) = 11111111111111111<17> = 2071723 × 5363222357<10>(100.00%)
Φ19(10) = 1111111111111111111<19> = 1111111111111111111<19>(100.00%)
Φ23(10) = 11111111111111111111111<23> = 11111111111111111111111<23>(100.00%)
Φ29(10) = 11111111111111111111111111111<29> = 3191 × 16763 × 43037 × 62003 × 77843839397<11>(100.00%)
Φ31(10) = 1111111111111111111111111111111<31> = 2791 × 6943319 × 57336415063790604359<20>(100.00%)
Φ37(10) = 1111111111111111111111111111111111111<37> = 2028119 × 247629013 × 2212394296770203368013<22>(100.00%)
Φ41(10) = 11111111111111111111111111111111111111111<41> = 83 × 1231 × 538987 × 201763709900322803748657942361<30>(100.00%)
Φ43(10) = 1111111111111111111111111111111111111111111<43> = 173 × 1527791 × 1963506722254397<16> × 2140992015395526641<19>(100.00%)
Φ47(10) = 11111111111111111111111111111111111111111111111<47> = 35121409 × 316362908763458525001406154038726382279<39>(100.00%)
Φ53(10) = 11111111111111111111111111111111111111111111111111111<53> = 107 × 1659431 × 1325815267337711173<19> × 47198858799491425660200071<26>(100.00%)
Φ59(10) = 11111111111111111111111111111111111111111111111111111111111<59> = 2559647034361<13> × 4340876285657460212144534289928559826755¬746751<46>(100.00%)
Φ61(10) = 1111111111111111111111111111111111111111111111111111111111111<61> = 733 × 4637 × 329401 × 974293 × 1360682471<10> × 106007173861643<15> × 7061709990156159479<19>(100.00%)
Φ67(10) = 1111111111111111111111111111111111111111111111111111111111111111111<67> = 493121 × 79863595778924342083<20> × 28213380943176667001263153660999177245677<41>(100.00%)
Φ71(10) = 11111111111111111111111111111111111111111111111111111111111111111111111<71> = 241573142393627673576957439049<30> × 45994811347886846310221728895223034301839<41>(100.00%)
Φ73(10) = 1111111111111111111111111111111111111111111111111111111111111111111111111<73> = 12171337159<11> × 1855193842151350117<19> × 49207341634646326934001739482502131487446637<44>(100.00%)
Φ79(10) = 1111111111111111111111111111111111111111111111111111111111111111111111111111111<79> = 317 × 6163 × 10271 × 307627 × 49172195536083790769<20> × 3660574762725521461527140564875080461079917<43>(100.00%)
Φ83(10) = 11111111111111111111111111111111111111111111111111111111111111111111111111111111111<83> = 3367147378267<13> × 9512538508624154373682136329<28> × 346895716385857804544741137394505425384477<42>(100.00%)
Φ89(10) = 11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111<89> = 497867 × 103733951 × 104984505733<12> × 5078554966026315671444089<25> × 403513310222809053284932818475878953159<39>(100.00%)
Φ97(10) = 1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111<97> = 12004721 × 846035731396919233767211537899097169<36> × 109399846855370537540339266842070119107662296580348039<54>(100.00%)

yangchuanju · 发表于 2021-5-12 12:54

本帖最后由 yangchuanju 于 2021-5-12 12:58 编辑

（二）指数是2的幂时，清一色数的φ因子都是100…01型数字，除φ2=11、φ4=101外都是合数，位数等于n/2+1：
Φ2(10) = 11 = 11(100.00%)
Φ4(10) = 101 = 101(100.00%)
Φ8(10) = 10001 = 73 × 137(100.00%)
Φ16(10) = 100000001 = 17 × 5882353(100.00%)
Φ32(10) = 10000000000000001<17>= 353 × 449 × 641 × 1409 × 69857(100.00%)
Φ64(10) = 100000000000000000000000000000001<33> = 19841 × 976193 × 6187457 × 834427406578561<15>(100.00%)
Φ128(10) = 1000000000...<65> = 1265011073<10> × 15343168188889137818369<23> × 515217525265213267447869906815873<33>(100.00%)
Φ256(10) = 1000000000...<129> = 257 × 15361 × 453377 × 55871187633753621225794775009016131346430842253464047463157158784732544216230781165223702155223678309562822667655169<116>(100.00%)
Φ512(10) = 1000000000...<257> = 10753 × 8253953 × 9524994049<10> × 73171503617<11> × 1616596633...<225>(100.00%)
Φ1024(10) = 1000000000...<513> = 1514497 × 302078977 × 3611707318387778163302401<25> × [6051994595...<473>](7.66%)
Φ2048(10) = 1000000000...<1025> = 1856104284667693057<19> × 315827195278624446663038977<27> × 1567629031101501414376301131777<31> × [1088190187...<950>](7.32%)
Φ4096(10) = 1000000000...<2049> = 106907803649<12> × 3835256936681473<16> × 9027276203313319937<19> × 969220351349023606432543884535631873<36> × [2787512819...<1967>](3.98%)

（三）指数是3的幂时，清一色数的φ因子都是10…010…01型数字，数字首尾和中间为1，其余都是0，都是合数，位数等于2n/3+1：
Φ3(10) = 111 = 3 × 37(100.00%)
Φ9(10) = 1001001 = 3 × 333667(100.00%)
Φ27(10) = 1000000001000000001<19> = 3 × 757 × 440334654777631<15>(100.00%)
Φ81(10) = 1000000000000000000000000001000000000000000000000000001<55> = 3 × 163 × 9397 × 2462401 × 676421558270641<15> × 130654897808007778425046117<27>(100.00%)
Φ243(10) = 1000000000...<163> = 3 × 411361786890737698932559<24> × 810316718654935254370114242173495345974018609204623064367629310709846283851718998731856856949061264781563433202832207926987960406521712613<138>(100.00%)
Φ729(10) = 1000000000...<487> = 3 × 313471 × 2064529 × 386953775911<12> × 1331071238...<463>(100.00%)
Φ2401(10) = 1000000000...<2059> = 1051639 × [9508966479...<2052>](0.29%)

（四）指数是5的幂时，清一色数的φ因子都是10…010…010…010…01型数字，数字由4节100…（一个1加n/5个0），末尾再加一个1组成，都是合数，位数等于4n/5+1：
Φ5(10) = 11111 = 41 × 271(100.00%)
Φ25(10) = 100001000010000100001<21> = 21401 × 25601 × 182521213001<12>(100.00%)
Φ125(10) = 1000000000...<101> = 751 × 1797655751<10> × 176144543406001<15> × 42051775804956304559810859008305819975199677041099230574273451704628125001<74>(100.00%)
Φ625(10) = 1000000000...<501> = 2782057126261476352728985117193142348250001<43> × [3594462495...<458>](8.49%)
Φ3125(10) = 1000000000...<2501> = 532525001 × 3852075247993751<16> × [4874894663...<2476>](0.97%)
Φ15625(10) = 1000000000...<12501> = 27018407327495281843751<23> × [3701180413...<12478>](0.18%)

yangchuanju · 发表于 2021-5-12 12:59

本帖最后由 yangchuanju 于 2021-5-12 13:03 编辑

（五）指数是4+4k时，清一色数的φ因子是100…01型数字或100…01除以某个小一点的100…01型数字的商，亦即可将该类清一色数字的φ因子转换成100…01型数字：
Φ4(10) = 101 = 101(100.00%)
Φ8(10) = 10001 = 73 × 137(100.00%)
Φ12(10) = 1000001/101= 9901 = 9901(100.00%)
Φ16(10) = 100000001 = 17 × 5882353(100.00%)
Φ20(10) =10000000001/101=99009901
Φ24(10) = 1000000000001/10001= 99990001 = 99990001(100.00%)
Φ28(10) =100000000000001/101= 990099009901<12> = 29 × 281 × 121499449(100.00%)
Φ32(10) = 10000000000000001<17> = 353 × 449 × 641 × 1409 × 69857(100.00%)
Φ36(10) = 1000000000000000001/1000001 = 999999000001<12> = 999999000001<12>(100.00%)
Φ40(10) = 100000000000000000001/10001 = 9999000099990001<16> = 1676321 × 5964848081<10>(100.00%)
Φ44(10) = 10000000000000000000001/101 = 99009900990099009901<20> = 89 × 1052788969<10> × 1056689261<10>(100.00%)
Φ48(10) = 1000000000000000000000001/100000001 =9999999900000001<16> = 9999999900000001<16>(100.00%)
Φ52(10) = 100000000000000000000000001/ 101 =990099009900990099009901<24> = 521 × 1900381976777332243781<22>(100.00%)
Φ56(10) = 10000000000000000000000000001/10001 =999900009999000099990001<24> = 7841 × 127522001020150503761<21>(100.00%)
Φ60(10) = 1000000000000000000000000000001/99010000009901 = 10099989899000101<17> = 255522961 × 39526741 = 61 × 4188901 × 39526741 (100.00%) (99010000009901=101*3541*9901*27961)
Φ64(10) = 100000000000000000000000000000001<33> = 19841 × 976193 × 6187457 × 834427406578561<15>(100.00%)

（六）指数是6+4k时，清一色数的φ因子是90…91型数字或90…91除以某个小一点的90…91型数字的商，亦即可将该类清一色数字的φ因子转换成90…91型数字；指数每增加4，90…91型数字就增加一节90：
Φ6(10) = 91 = 7 × 13(100.00%)
Φ10(10) = 9091 = 9091(100.00%)
Φ14(10) = 909091 = 909091(100.00%)
Φ18(10) = 90909091/91 =999001 = 19 × 52579(100.00%)
Φ22(10) = 9090909091<10> = 11 × 23 × 4093 × 8779(100.00%)
Φ26(10) = 909090909091<12> = 859 × 1058313049<10>(100.00%)
Φ30(10) = 90909090909091<14>/827281 = 109889011 = 211 × 241 × 2161(100.00%)（827281=91*9091）
Φ34(10) = 9090909090909091<16> = 103 × 4013 × 21993833369<11>(100.00%)
Φ38(10) = 909090909090909091<18> = 909090909090909091<18>(100.00%)
Φ42(10) = 90909090909090909091<20>/82727281 = 1098900989011<13> = 7 × 127 × 2689 × 459691(100.00%) （82727281=91*909091）
Φ46(10) = 9090909090909090909091<22> = 47 × 139 × 2531 × 549797184491917<15>(100.00%)
Φ50(10) = 909090909090909090909091<24>/9091 = 99999000009999900001<20> = 251 × 5051 × 78875943472201<14>(100.00%)
Φ54(10) = 90909090909090909090909091<26>/90909091 =999999999000000001<18> = 70541929 × 14175966169<11>(100.00%)
Φ58(10) = 9090909090909090909090909091<28> = 59 × 154083204930662557781201849<27>(100.00%)
Φ62(10) = 909090909090909090909090909091<30> = 909090909090909090909090909091<30>(100.00%)

yangchuanju · 发表于 2021-5-12 13:04

本帖最后由 yangchuanju 于 2021-5-12 13:06 编辑

（七）指数是6k时，清一色数的φ因子是99…90…01型数字或99…90…01除以某个小一点的99…90…01型数字的商，亦即可将该类清一色数字的φ因子转换成99…90…01型数字；9的个数等于n/6，0的个数等于n/6-1，末尾一个1；指数每增加6，9和0各增加一个：
Φ6(10) = 91 = 7 × 13(100.00%)
Φ12(10) = 9901 = 9901(100.00%)
Φ18(10) = 999001 = 19 × 52579(100.00%)
Φ24(10) = 99990001 = 99990001(100.00%)
Φ30(10) = 9999900001<10>/91 = 109889011 = 211 × 241 × 2161(100.00%)
Φ36(10) = 999999000001<12> = 999999000001<12>(100.00%)
Φ42(10) = 99999990000001<14>/91 =1098900989011<13> = 7 × 127 × 2689 × 459691(100.00%)
Φ48(10) = 9999999900000001<16> = 9999999900000001<16>(100.00%)
Φ54(10) = 999999999000000001<18> = 70541929 × 14175966169<11>(100.00%)
Φ60(10) = 99999999990000000001<20>/9901 = 10099989899000101<17> = 255522961 × 39526741 = 61 × 4188901 × 39526741 (100.00%)
Φ66(10) = 9999999999900000000001<22>/91 =109890109889010989011<21> = 599144041 × 183411838171<12> (100.00%)
Φ72(10) = 999999999999000000000001<24> = 3169 × 98641 × 3199044596370769<16>(100.00%)
Φ78(10) = 99999999999990000000000001<26>/91=1098901098900989010989011<25> = 13 × 157 × 6397 × 216451 × 388847808493<12>(100.00%)
Φ84(10) = 9999999999999900000000000001<28>/9901 =1009998990000999899000101<25> = 226549 × 4458192223320340849<19>(100.00%)

（八）指数是12k时，清一色数的φ因子也是99…90…01型数字或99…90…01除以某个小一点的99…90…01型数字的商；9的个数等于n/6，0的个数等于n/6-1，末尾一个1；指数每增加12，9和0各增加二个（实际上它是七中的双号部分）：
Φ12(10) = 9901 = 9901(100.00%)
Φ24(10) = 99990001 = 99990001(100.00%)
Φ36(10) = 999999000001<12> = 999999000001<12>(100.00%)
Φ48(10) = 9999999900000001<16> = 9999999900000001<16>(100.00%)
Φ60(10) = 99999999990000000001<20>/9901 = 10099989899000101<17> = 255522961 × 39526741 = 61 × 4188901 × 39526741 (100.00%)
Φ72(10) = 999999999999000000000001<24> = 3169 × 98641 × 3199044596370769<16>(100.00%)
Φ84(10) = 9999999999999900000000000001<28>/9901 =1009998990000999899000101<25> = 226549 × 4458192223320340849<19>(100.00%)
Φ96(10) = 99999999999999990000000000000001<32> = 97 × 206209 × 66554101249<11> × 75118313082913<14>(100.00%)
Φ108(10) = 9999999999...<36> = 109 × 153469 × 59779577156334533866654838281<29>(100.00%)

yangchuanju · 发表于 2021-5-12 13:07

本帖最后由 yangchuanju 于 2021-5-12 13:09 编辑

（九）指数是18k时，清一色数的φ因子还是99…90…01型数字或99…90…01除以某个小一点的99…90…01型数字的商；9的个数等于n/6，0的个数等于n/6-1，末尾一个1；指数每增加18，9和0各增加三个：
Φ18(10) = 999001 = 19 × 52579(100.00%)
Φ36(10) = 999999000001<12> = 999999000001<12>(100.00%)
Φ54(10) = 999999999000000001<18> = 70541929 × 14175966169<11>(100.00%)
Φ72(10) = 999999999999000000000001<24> = 3169 × 98641 × 3199044596370769<16>(100.00%)
Φ90(10) = 999999999999999000000000000001<30>/999001 =1000999998998999000001001<25> = 29611 × 3762091 × 8985695684401<13>(100.00%)
Φ108(10) = 9999999999...<36> = 109 × 153469 × 59779577156334533866654838281<29>(100.00%)

（十）指数是14k时，清一色数的φ因子是990099009901型数字（未加省略号，由二组9900和一组9901组成）或990099009901除以某个小一点的990099009901型数字的商；每小节9的个数等于(n-2)/12，0的个数与9的个数相同，但第3节末尾改为1；指数每增加14，9和0各增加一个：
Φ14(10) = 909091 = 909091(100.00%)
Φ28(10) = 990099009901<12> = 29 × 281 × 121499449(100.00%)
Φ42(10) = 999000999000999001<18>/909091 =1098900989011<13>  = 7 × 127 × 2689 × 459691(100.00%)
Φ56(10) = 999900009999000099990001<24> = 7841 × 127522001020150503761<21>(100.00%)
Φ70(10) = 999990000099999000009999900001<30>/909091 = 1099988890111109888900011<25> = 4147571 × 265212793249617641<18>(100.00%)
Φ84(10) = 999999000000999999000000999999000001<36>/990099009901 =1009998990000999899000101<25> = 226549 × 4458192223320340849<19>(100.00%)
Φ98(10) = 9999999000000099999990000000999999900000<42> = 197 × 5076141624365532994918781726395939035533<40>(100.00%)

yangchuanju · 发表于 2021-5-12 13:09

本帖最后由 yangchuanju 于 2021-5-12 13:12 编辑

（十一）指数是10k时，清一色数的φ因子是99009901型数字（未加省略号，由一组9900和一组9901组成）或990099009901除以某个小一点的990099009901型数字的商；每小节9的个数等于n/10，0的个数与9的个数相同，但第2节末尾改为1；指数每增加10，9和0各增加一个：
Φ10(10) = 9091 = 9091(100.00%)
Φ20(10) = 99009901= 3541 ×27961(100.00%)
Φ30(10) = 999000999001<12>/9091 = 109889011 = 211 × 241 × 2161(100.00%)
Φ40(10) = 9999000099990001<16> = 1676321 × 5964848081<10>(100.00%)
Φ50(10) = 99999000009999900001<20> = 251 × 5051 × 78875943472201<14>(100.00%)
Φ60(10) = 999999000000999999000001<24>/ 99009901 = 10099989899000101<17> = 255522961 × 39526741 = 61 × 4188901 × 39526741 (100.00%)
Φ70(10) = 9999999000000099999990000001<28>/9091 =1099988890111109888900011<25> = 4147571 × 265212793249617641<18>(100.00%)
Φ80(10) = 99999999000000009999999900000001<32> = 5070721 × 19721061166646717498359681<26>(100.00%)

（十二）指数是15+18k时，清一色数的φ因子是900-90-991型数字（未加省略号，由数节900和数节990组成）或900-90-991除以某个小一点的900-90-991型数字的商；前半段都是900，后半段都是990，末节改为991，中间加一节90；指数每增加18，900和990各增加一节：
Φ15(10) = 90090991 = 31 × 2906161(100.00%)
Φ33(10) = 90090090090990990991<20> = 67 × 1344628210313298373<19>(100.00%)
Φ51(10) = 90090090090090090990990990990991<32> = 613 × 210631 × 52986961 × 13168164561429877<17> (100.00%)
Φ69(10) = 90090090090090090090090990990990990990990991<44> = 277 × 203864078068831<15> × 1595352086329224644348978893<28>(100.00%)
Φ87(10) = 90090090090090090090090090090990990990990990990990990991<56> = 4003 × 72559 × 310170251658029759045157793237339498342763245483<48>(100.00%)
Φ105(10) = 90090090090090090090090090090090090990990990990990990990990990990991<58>/81163063071261262081 = 1109988789...<49> = 30703738801<11> × 625437743071<12> × 57802050308786191965409441<26>(100.00%) (81163063071261262081=31*43*1933*2906161*10838689)
Φ123(10) = 9009009009...<80> = 1811791 × 626920594693<12> × 9425856976319889649<19> × 8414640003465161203119978906558054839526493<43>(100.00%)
Φ141(10) = 9009009009...<92> = 283 × 721030498171501831<18> × 441506346488360048482114135141919313523563714948107161215664533500695867<72>(100.00%)
Φ159(10) = 9009009009...<104> = 351391 × 3852774594841<13> × 185328247384490197483<21> × 3590634232366608194325908088526698673242¬58623298339641467853849067<66>(100.00%)
Φ177(10) = 9009009009...<116> = 3187 × 232199557797964905143390121842739619033812919<45> × 1217400762439061404401283287717503878956¬38651224514935329511899287347<69>(100.00%)

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1513位唯一循环周期大素数

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