大素数，试除法

太阳

太阳

太阳

yangchuanju

太阳 发表于 2021-3-4 19:01

90…01型正整数中有素数，但大多数是合数；在这些合数中，只有当0的个数是0,6,12,18……时含有素因子7和13，
当0的个数是36，78时含2个7；0的个数是60时含2个13。0的个数是多少时含多个7，多个13，暂时不好估计。
谨提供含0-79个0的90…01的分解式，供太阳老师求解参考！

yangchuanju

数字        分解式
91        91=P1 * P2
        P1 = 7
        P2 = 13
901        901=P2 * P2
        P2 = 17
        P2 = 53
9001        9001 is prime
90001        90001 is prime
900001        900001 is prime
9000001        9000001=P2 * P6
        P2 = 61
        P6 = 147541
90000001        90000001=P1 * P2 * P6
        P1 = 7
        P2 = 13
        P6 = 989011
900000001        900000001=P3 * P7
        P3 = 409
        P7 = 2200489
9000000001        9000000001 is prime
90000000001        90000000001=P3 * P9
        P3 = 113
        P9 = 796460177
9E+11        900000000001=P6 * P7
        P6 = 634939
        P7 = 1417459
9E+12        9000000000001=P7 * P7
        P7 = 1072157
        P7 = 8394293
9E+13        90000000000001=P1 * P2 * P3 * P10
        P1 = 7
        P2 = 13
        P3 = 103
        P10 = 9602048437
9E+14        900000000000001=P4 * P12
        P4 = 1097
        P12 = 820419325433
9E+15        9000000000000001=P2 * P2 * P13
        P2 = 23
        P2 = 53
        P13 = 7383100902379
9E+16        90000000000000001=P2 * P5 * P6 * P6
        P2 = 29
        P5 = 12553
        P6 = 286469
        P6 = 863017
9E+17        900000000000000001=P2 * P17
        P2 = 19
        P17 = 47368421052631579
9E+18        9000000000000000001=P2 * P3 * P6 * P11
        P2 = 17
        P3 = 173
        P6 = 264601
        P11 = 11565276061
9E+19        90000000000000000001=P1 * P2 * P7 * P12
        P1 = 7
        P2 = 13
        P7 = 9642641
        P12 = 102566401571
9E+20        900000000000000000001=P10 * P12
        P10 = 1669850621
        P12 = 538970365781
9E+21        9000000000000000000001=P6 * P17
        P6 = 101477
        P17 = 88690047991170413
9E+22        90000000000000000000001 is prime
9E+23        900000000000000000000001=P2 * P2 * P3 * P18
        P2 = 59
        P2 = 89
        P3 = 419
        P18 = 409059485884947929
9E+24        9000000000000000000000001=P4 * P5 * P18
        P4 = 1277
        P5 = 34781
        P18 = 202632707706349273
9E+25        90000000000000000000000001=P1 * P2 * P24
        P1 = 7
        P2 = 13
        P24 = 989010989010989010989011
9E+26        900000000000000000000000001=P6 * P10 * P12
        P6 = 876653
        P10 = 2462224309
        P12 = 416953065713
9E+27        9000000000000000000000000001 is prime
9E+28        90000000000000000000000000001=P2 * C28
        P2 = 53
        C28 = 1698113207547169811320754717
9E+29        900000000000000000000000000001 is not prime
9E+30        9000000000000000000000000000001=P3 * P7 * P22
        P3 = 661
        P7 = 9338569
        P22 = 1458010722709494472789
9E+31        90000000000000000000000000000001=P1 * P2 * P3 * P4 * P8 * P17
        P1 = 7
        P2 = 13
        P3 = 487
        P4 = 1493
        P8 = 28195807
        P17 = 48242279334679103
9E+32        900000000000000000000000000000001=P5 * P5 * P25
        P5 = 16901
        P5 = 48817
        P25 = 1090834891658648232287453
9E+33        9000000000000000000000000000000001=P2 * P9 * P10 * P15
        P2 = 47
        P9 = 909983219
        P10 = 1065587651
        P15 = 197479531885607
9E+34        90000000000000000000000000000000001=P2 * P8 * P27
        P2 = 17
        P8 = 41043601
        P27 = 128987650159127692753171553
9E+35        900000000000000000000000000000000001=P2 * P4 * P14 * P19
        P2 = 19
        P4 = 1289
        P14 = 36477404545339
        P19 = 1007423463151225049
9E+36        9000000000000000000000000000000000001 is prime
9E+37        90000000000000000000000000000000000001=P1 * P1 * P2 * P2 * P5 * P6 * P24
        P1 = 7
        P1 = 7
        P2 = 13
        P2 = 23
        P5 = 87407
        P6 = 197339
        P24 = 356136186044107373212487
9E+38        900000000000000000000000000000000000001 is not prime
9E+39        9000000000000000000000000000000000000001=P5 * P36
        P5 = 38201
        P36 = 235595926808198738252925316091201801
9E+40        90000000000000000000000000000000000000001=P4 * C38
        P4 = 4129
        C38 = 21797045289416323565027851780092031969
9E+41        900000000000000000000000000000000000000001=P2 * P5 * P36
        P2 = 53
        P5 = 31081
        P36 = 546350892039242563405538662520875157
9E+42        9000000000000000000000000000000000000000001=P7 * C37
        P7 = 1032841
        C37 = 8713829137301869309990598746564088761
9E+43        90000000000000000000000000000000000000000001=P1 * P2 * C42
        P1 = 7
        P2 = 13
        C42 = 989010989010989010989010989010989010989011
9E+44        900000000000000000000000000000000000000000001=P2 * P2 * P3 * P4 * P5 * P8 * P24
        P2 = 29
        P2 = 97
        P3 = 313
        P4 = 8641
        P5 = 11813
        P8 = 16493429
        P24 = 607146191816885557369397
9E+45        9000000000000000000000000000000000000000000001=P4 * P10 * P11 * P24
        P4 = 1303
        P10 = 5532177317
        P11 = 10909587209
        P24 = 114444172630926939804739
9E+46        90000000000000000000000000000000000000000000001=P5 * P43
        P5 = 40697
        P43 = 2211465218566479101653684546772489372681033
9E+47        900000000000000000000000000000000000000000000001=P3 * P46
        P3 = 103
        P46 = 8737864077669902912621359223300970873786407767
9E+48        9000000000000000000000000000000000000000000000001 is not prime
9E+49        90000000000000000000000000000000000000000000000001=P1 * P2 * C48
        P1 = 7
        P2 = 13
        C48 = 989010989010989010989010989010989010989010989011
9E+50        900000000000000000000000000000000000000000000000001=P2 * P6 * P45
        P2 = 17
        P6 = 185177
        P45 = 285894989499712357874453344955494091662380889
9E+51        9000000000000000000000000000000000000000000000000001=P4 * P49
        P4 = 5303
        P49 = 1697152555157458042617386385065057514614369224967
9E+52        90000000000000000000000000000000000000000000000000001=P6 * C48
        P6 = 535637
        C48 = 168024240297066856845214202902338710731334840573
9E+53        900000000000000000000000000000000000000000000000000001=P2 * P3 * P3 * P9 * P39
        P2 = 19
        P3 = 317
        P3 = 373
        P9 = 621654619
        P39 = 644423915856720982631763433468126261601
9E+54        9000000000000000000000000000000000000000000000000000001=P2 * C54
        P2 = 53
        C54 = 169811320754716981132075471698113207547169811320754717
9E+55        90000000000000000000000000000000000000000000000000000001=P1 * P2 * C54
        P1 = 7
        P2 = 13
        C54 = 989010989010989010989010989010989010989010989010989011
9E+56        900000000000000000000000000000000000000000000000000000001=P5 * P53
        P5 = 22349
        P53 = 40270258177099646516622667680880576312139245603830149
9E+57        9000000000000000000000000000000000000000000000000000000001 is prime
9E+58        90000000000000000000000000000000000000000000000000000000001=P7 * C53
        P7 = 2836549
        C53 = 31728695679150968306910968222301113077898530926135949
9E+59        900000000000000000000000000000000000000000000000000000000001=P2 * P2 * P3 * P6 * P50
        P2 = 23
        P2 = 23
        P3 = 223
        P6 = 433241
        P50 = 17609718952061731403087007685427467197151800957383
9E+60        9000000000000000000000000000000000000000000000000000000000001=P8 * P12 * P13 * P30
        P8 = 52069729
        P12 = 165220125649
        P13 = 2903138521729
        P30 = 360351625548496622538364964689
9E+61        90000000000000000000000000000000000000000000000000000000000001=P1 * P2 * P2 * P3 * C57
        P1 = 7
        P2 = 13
        P2 = 13
        P3 = 173
        C57 = 439755886621160076028906620280564255664300128506442423739
9E+62        900000000000000000000000000000000000000000000000000000000000001 is prime
9E+63        9000000000000000000000000000000000000000000000000000000000000001=P7 * P7 * P51
        P7 = 1846367
        P7 = 6443011
        P51 = 756546466489705346851565812353711496009046424433973
9E+64        90000000000000000000000000000000000000000000000000000000000000001=P3 * C63
        P3 = 509
        C63 = 176817288801571709233791748526522593320235756385068762278978389
9E+65        900000000000000000000000000000000000000000000000000000000000000001=P4 * P4 * P59
        P4 = 4721
        P4 = 7459
        P59 = 25558060971253457331200579406921787420600688835179728118459
9E+66        9000000000000000000000000000000000000000000000000000000000000000001=P2 * P2 * C64
        P2 = 17
        P2 = 61
        C64 = 8678881388621022179363548698167791706846673095467695274831243973
9E+67        90000000000000000000000000000000000000000000000000000000000000000001=P1 * P2 * P2 * P2 * P3 * P60
        P1 = 7
        P2 = 13
        P2 = 53
        P2 = 89
        P3 = 967
        P60 = 216824706300274768218062939196360764018857398893392710120649
9E+68        900000000000000000000000000000000000000000000000000000000000000000001=P10 * P60
        P10 = 1937018929
        P60 = 464631494574310378357691309892201884620818798410410371369169
9E+69        9000000000000000000000000000000000000000000000000000000000000000000001=P6 * P14 * C51
        P6 = 223849
        P14 = 56519161534037
        C51 = 711363602530245908762799315716791946722922712239477
9E+70        90000000000000000000000000000000000000000000000000000000000000000000001=P6 * P66
        P6 = 177481
        P66 = 507096534276908514150810509293952592108451045464021500893053340921
9E+71        900000000000000000000000000000000000000000000000000000000000000000000001=P2 * P10 * C62
        P2 = 19
        P10 = 1021624687
        C62 = 46365775666334870926399629037773613381142209078813674896413717
9E+72        9000000000000000000000000000000000000000000000000000000000000000000000001=P2 * P7 * P15 * C51
        P2 = 29
        P7 = 1193737
        P15 = 369568350763637
        C51 = 703462716197286561893566473716153635964886210257401
9E+73        90000000000000000000000000000000000000000000000000000000000000000000000001=P1 * P2 * P3 * P3 * P10 * P20 * P39
        P1 = 7
        P2 = 13
        P3 = 131
        P3 = 157
        P10 = 3434714219
        P20 = 40505281728310780921
        P39 = 345643109794933143493746424616869980967
9E+74        900000000000000000000000000000000000000000000000000000000000000000000000001=P5 * P71
        P5 = 26513
        P71 = 33945611586768754950401689737110096933579753328555802813714027081054577
9E+75        9000000000000000000000000000000000000000000000000000000000000000000000000001=P3 * P5 * C69
        P3 = 761
        P5 = 27259
        C69 = 433858322793388134138773633889811266326872042020239105106469073445899
9E+76        90000000000000000000000000000000000000000000000000000000000000000000000000001=P8 * P70
        P8 = 39191309
        P70 = 2296427506414751290904827904574455525330883946744417238015703940891589
9E+77        900000000000000000000000000000000000000000000000000000000000000000000000000001=P3 * P3 * P10 * P63
        P3 = 797
        P3 = 997
        P10 = 1347545921
        P63 = 840514975997107693570604695977849746413871622498980856073069009
9E+78        9000000000000000000000000000000000000000000000000000000000000000000000000000001 is prime
9E+79        90000000000000000000000000000000000000000000000000000000000000000000000000000001=P1 * P1 * P2 * P2 * P4 * P6 * P9 * P58
        P1 = 7
        P1 = 7
        P2 = 13
        P2 = 47
        P4 = 1361
        P6 = 939179
        P9 = 362132527
        P58 = 6494282636470574357798239086613389616047164558265238476143
9E+80        900000000000000000000000000000000000000000000000000000000000000000000000000000001=P2 * P7 * C73
        P2 = 53
        P7 = 7228517
        C73 = 2349186157474859381697178988416478892519306675501416362182340796482848601

太阳

yangchuanju：网友，9000...01，能不能提供含有120个了0，数据值，参考一下

yangchuanju

当0的个数是0,6,12,18,24,30时都含1个素因子7；
当0的个数是36时含2个7（7的平方），从36开始每增加6*7=42个0，都出现一个7的平方；
7的立方何时出现？在9加140个0，再加1之中没有找到，
根据规律7的平方最多6个，之后便有一个7的立方因子，36+42*6=288，在288个0或之前的246,204,162个0之中一定有一个7的立方因子，但现在不能确定。（已查明78、120个0中各有一个7的平方因子）

当0的个数是0,6,12,18,24,30……54时都含1个素因子13；
当0的个数是60时含2个13（13的平方），从60开始每增加6*13=78个0，都出现一个13的平方；
13的立方何时出现？在9加140个0，再加1之中没有找到，
根据规律13的平方最多12个，之后便有一个13的立方因子，60+78*13=1074，在1074个0或之前的996,918,840,……个0之中一定有一个13的立方因子，但现在不能确定。（已查明138个0中有一个13的平方因子）
单个素因子7和单个素因子13总是成对出现，但7的平方和13的平方一般不同时出现，据推测它只有在9加372个0，再加1的分解式中同时出现；以后每增加42*13=78*7=546个0时将循环出现一次。

因7的立方和13的立方出现的起点都未找到，两个立方同时出现更无从谈起！
如果找到了含两个立方因子的数字之和，下一个双立方因子便迎刃而解，它一定也会循环出现，周期是：546*91=49686。
肯定还有含更高次素因子7、素因子13的900…001型的整数，它们或单独出现，或同时出现；他俩的次数可以相等，但更多时候是不相等的。

yangchuanju

91由111111除以11*111得到，91的两个素因子7和13被保留；
901=17*53，素数17的周期是16，53的周期是13，16和13的最小公倍数是208，它是从208个1的分解式中得到的，208个1共有21个素因子，但没有7和13：

90001、900001、9000001都是素数，肯定不含素因子7和13了。

9000001=61*147541，素数61的周期是60，147541的周期是147540，60和147540的最小公倍数是147540，它是从147540个1的分解式中得到的，147540是6的倍数，但不是7和13的倍数，147540个1分解式中应含有素因子7和13各一个，只是在多次除法中除掉了。

90000001=7*13*989011，素数7和13的周期都是6，989011的周期不知，暂按989010计算，6和989010的最小公倍数是989010，它可能是从989010个1的分解式中得到的，989010是6的倍数，但不是7和13 的倍数，989010个1分解式中应含有一个素因子7和13各一个，只是在多次除法中保留下来了。
其余的不再分析！

红树

数据中可以发现，7和13同时出现，17个19不能同时出现，13和17不能同时出现，7和17不能同时出现
7和19不能同时出现，13和19不能同时出现

yangchuanju

红树 发表于 2021-3-5 14:05
数据中可以发现，7和13同时出现，17个19不能同时出现，13和17不能同时出现，7和17不能同时出现
7和19不能 ...

17和19可同时出现在16*18=288个1的分解式中；
7,13和17可同时出现在6*16=96个1的分解式中；
7,13和19可同时出现在6*18=108个1的分解式中。