大素数，试除法

yangchuanju · 发表于 2021-3-5 15:22

1111111111...<96> = 3 × 7 × 11 × 13 × 17 × 37 × 73 × 97 × 101 × 137 × 353 × 449 × 641 × 1409 × 9901 × 69857 × 206209 × 5882353 × 99990001 × 66554101249<11> × 75118313082913<14> × 9999999900000001<16>
1111111111...<108> = 33 × 7 × 11 × 13 × 19 × 37 × 101 × 109 × 757 × 9901 × 52579 × 153469 × 333667 × 70541929 × 14175966169<11> × 999999000001<12> × 440334654777631<15> × 59779577156334533866654838281<29>
1111111111...<288> = 32 × 7 × 11 × 13 × 17 × 19 × 37 × 73 × 97 × 101 × 137 × 353 × 449 × 641 × 1409 × 3169 × 8929 × 9901 × 13249 × 52579 × 69857 × 98641 × 206209 × 333667 × 1067329 × 5882353 × 99990001 × 66554101249<11> × 999999000001<12> × 75118313082913<14> × 3199044596370769<16> × 9999999900000001<16> × 1119946242580356142905139433307201254339-79169<45> × 7071614705525771137412357579842860859021-9877617672752248112085912824160137075376-116481<86>

yangchuanju · 发表于 2021-3-5 22:51

本帖最后由 yangchuanju 于 2021-3-6 07:25 编辑

接7楼
已经查明，第1的素因子7^3出现在“9+204个0+1”的分解式中，之后每增加297个0出现一个7的立方因子；
第1的素因子13^3出现在“9+216个0+1”的分解式中，之后每增加1014个0出现一个13的立方因子；
两个立方因子在“9+40776个0+1”的分解式中同时出现，之后需增加49686个0时在同时出现一次。

还查明，第1的素因子7^4出现在“9+1086个0+1”的分解式中，之后每增加297*7=2058个0出现一个7的四次方因子；
第1的素因子13^4出现在“9+5286个0+1”的分解式中，之后每增加1014*13=13182个0出现一个13的四次方因子；

经延伸分析，当0的个数达到7940850和12462276时，将同时出现7^4和13^4；
两数相差4521426=13*13*13*2058个0；另在3419424个0处出现了7^4和13^5一组数字。

yangchuanju · 发表于 2021-3-5 23:04

本帖最后由 yangchuanju 于 2021-3-7 07:07 编辑

900...01分解式（含80-139个0）：
9E+80 900000000000000000000000000000000000000000000000000000000000000000000000000000001=P2 * P7 * C73
P2 = 53
P7 = 7228517
C73 = 2349186157474859381697178988416478892519306675501416362182340796482848601
9E+81 9000000000000000000000000000000000000000000000000000000000000000000000000000000001=P2 * P2 * P3 * P4 * P5 * P69
P2 = 23
P2 = 59
P3 = 103
P4 = 9539
P5 = 15569
P69 = 433572640799482999983753283405797953951283348120523679061716819187841
9E+82 90000000000000000000000000000000000000000000000000000000000000000000000000000000001=P2 * P3 * P7 * P15 * C59
P2 = 17
P3 = 197
P7 = 4373989
P15 = 444342046542949
C59 = 13827136953803135915227779562819305309247744855725029151109
9E+83 900000000000000000000000000000000000000000000000000000000000000000000000000000000001=P8 * P8 * P11 * P58
P8 = 73235077
P8 = 82599773
P11 = 43082588561
P58 = 3453366913153286556653523424034883035479542624501723609121
9E+84 9000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P4 * P6 * C77
P4 = 1229
P6 = 166349
C77 = 44022067166610283366084365425399101982112650405733539680740523170278906355481
9E+85 90000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P1 * P2 * C84
P1 = 7
P2 = 13
C84 = 989010989010989010989010989010989010989010989010989010989010989010989010989010989011
9E+86 900000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P7 * P8 * C74
P7 = 2684897
P8 = 16342553
C74 = 20511384588721237948664065474561600385631983983167661345986948771736735561
9E+87 9000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P3 * P9 * C77
P3 = 887
P9 = 386371357
C77 = 26261163668678774102910222964726950487592747255637280662376555551200627554739
9E+88 90000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P3 * C87
P3 = 769
C87 = 117035110533159947984395318595578673602080624187256176853055916775032509752925877763329
9E+89 900000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P2 * P4 * P4 * P83
P2 = 19
P4 = 1013
P4 = 1637
P83 = 28564773432627871239776865954944655910167469236294439209861550034422138918173953059
9E+90 9000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P4 * P4 * C84
P4 = 1913
P4 = 6353
C84 = 740540276792562079285697888036728164696815816689622043876353141935487586940457023609
9E+91 90000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P1 * P2 * P3 * P6 * C82
P1 = 7
P2 = 13
P3 = 823
P6 = 288931
C82 = 4159174494750921523372414780632662165069892884998547063789420925456629331456173047
9E+92 900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P10 * C84
P10 = 1645448113
C84 = 546963464170930074171229731143761687245619029738435757044135976349683899148268068177
9E+93 9000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P2 * P4 * C89
P2 = 53
P4 = 3253
C89 = 52201451200343369545673369719678207054156105539734004605328028119181713251628395269388489
9E+94 90000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P4 * P11 * C82
P4 = 6113
P11 = 13693948969
C82 = 1075126108696142426199572830688768662376780750405651587919247117002743359677355833
9E+95 900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 is not prime
9E+96 9000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P3 * C95
P3 = 821
C95 = 10962241169305724725943970767356881851400730816077953714981729598051157125456760048721071863581
9E+97 90000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P1 * P2 * P5 * C92
P1 = 7
P2 = 13
P5 = 11251
C92 = 87904274198825794239535240335169230378545106124876811926851923296683762420141408675761177761
9E+98 900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P2 * P6 * P10 * P83
P2 = 17
P6 = 665117
P10 = 1302994037
P83 = 61087606833217334930936140025570390625015013586542691774287192644976828132114177457
9E+99 9000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P3 * P7 * P92
P3 = 647
P7 = 1159303
P92 = 11998895445679379799918978391750473540741698203219135396897417073903734149720758838785348561
9E+100 90000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P2 * P5 * P6 * P7 * P83
P2 = 29
P5 = 44497
P6 = 852673
P7 = 6267889
P83 = 13049983777688172742896635745449048831681908684773511663795549758829314812199285941
9E+101 900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P3 * P4 * C96
P3 = 929
P4 = 1087
C96 = 891245297443215296145958252089722654366161198546676001635930257084657410259025591613579805569887
9E+102 9000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P3 * P4 * P8 * P9 * P10 * P13 * P59
P3 = 857
P4 = 4729
P8 = 27724157
P9 = 146688169
P10 = 7824928201
P13 = 1082959301461
P59 = 64438650881125819456903404728709964010133394499896709654209
9E+103 90000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P1 * P2 * P2 * C101
P1 = 7
P2 = 13
P2 = 23
C101 = 43000477783086478738652651696129956999522216913521261347348303870043000477783086478738652651696129957
9E+104 900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P3 * P6 * P98
P3 = 173
P6 = 401017
P98 = 12972797010421811792073566252600451214059928931270279995193146281071704670643674584536446143483661
9E+105 9000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P6 * C101
P6 = 146173
C101 = 61570878342785603360401715775143152292146976527812933989177207829079241720427165071524837008202609237
9E+106 90000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P2 * P3 * P3 * C101
P2 = 53
P3 = 109
P3 = 601
C101 = 25921830703371594915519313636006229303938361342831476130170217141415395320879141768508259127292605913
9E+107 900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P2 * P12 * C96
P2 = 19
P12 = 140725352693
C96 = 336601899701529703434021871004838007731653651820999518518992617044366642164251527562746976439503
9E+108 9000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P6 * P7 * C97
P6 = 373777
P7 = 4126417
C97 = 5835214450292052431336126474646284560227499398302155339750431004300338522096120621448455426681889
9E+109 90000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P1 * P2 * P8 * P8 * P94
P1 = 7
P2 = 13
P8 = 18480611
P8 = 36737531
P94 = 1456715783890529448958356929556268261049304839622936526296628689514369386951968743985378548771
9E+110 900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P6 * C106
P6 = 535741
C106 = 1679916228177421552578578081572998893121862989765577023225775141346284865261385632236472474572601312947861
9E+111 9000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P2 * C111
P2 = 89
C111 = 101123595505617977528089887640449438202247191011235955056179775280898876404494382022471910112359550561797752809
9E+112 90000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 is not prime
9E+113 900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P5 * P7 * C104
P5 = 22699
P7 = 1184459
C104 = 33474627453340210450558941161791675296296994782280919086160013356355525225661891419848286880377772846761
9E+114 9000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P2 * C114
P2 = 17
C114 = 529411764705882352941176470588235294117647058823529411764705882352941176470588235294117647058823529411764705882353
9E+115 90000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P1 * P2 * P3 * P7 * C106
P1 = 7
P2 = 13
P3 = 103
P7 = 3391343
C106 = 2831340987036667571098801337230607550125181781769048544470896338148836442399891898973267857546594570354459
9E+116 900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P4 * P6 * C108
P4 = 3917
P6 = 719189
C108 = 319481637436666301862417530975811618189214620608706212631721564718698248156838639004348142600655848305387377
9E+117 9000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P5 * C114
P5 = 12527
C114 = 718448151991697932465873712780394348207870998642931268460126127564460764748144008940688113674463159575317314600463
9E+118 90000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P6 * C114
P6 = 196073
C114 = 459012714652195865825483365889235131813151224288912802884639904525355352343259908299459895039092582864545347906137
9E+119 900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P2 * P6 * C114
P2 = 53
P6 = 141413
C114 = 120081831765620544880651334529437327223925531118606292901736103096390086631036163444181761731227774020402436914009
9E+120 9000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 is not prime
9E+121 90000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P1 * P1 * P2 * P4 * P5 * P5 * P6 * C101
P1 = 7
P1 = 7
P2 = 13
P4 = 8329
P5 = 13099
P5 = 45779
P6 = 398459
C101 = 70994079165901357774302508596374222973187651738164763773225091228534809576347428381458718947064150183
9E+122 900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P3 * P3 * P7 * C113
P3 = 113
P3 = 229
P7 = 2618281
C113 = 13283494167595707594701695021323613648294145264418189068744715504356629299697769452353777609855703091779894048373
9E+123 9000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P4 * P4 * P5 * P8 * P105
P4 = 2383
P4 = 5569
P5 = 35081
P8 = 28502281
P105 = 678249788730965343155876794263068283826182488644313608841077587266701477227270380314651873177483730318783
9E+124 90000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P12 * P12 * P103
P12 = 168745086793
P12 = 360847422557
P103 = 1478045369332241890902656097128029833077561869578361464994129936119444146965960462980286762223022100301
9E+125 900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P2 * P2 * P2 * P4 * P7 * P113
P2 = 19
P2 = 23
P2 = 47
P4 = 1091
P7 = 2340179
P113 = 17162849181227637954686586305558908249546272559153601351238220969425254370779811594520448516673293863019660069131
9E+126 9000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P2 * P4 * P7 * C116
P2 = 61
P4 = 1609
P7 = 4480517
C116 = 20465789404222357840404381875745801274688927352651621232555433527637138892346348967984191087403221105854908526202697
9E+127 90000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P1 * P2 * P3 * P3 * P4 * P9 * P111
P1 = 7
P2 = 13
P3 = 167
P3 = 401
P4 = 1289
P9 = 112974047
P111 = 101416511916384986434377207218778382110001626648805153438789813840049798877618436058332277078236972412647025251
9E+128 900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P2 * P4 * P4 * P8 * P113
P2 = 29
P4 = 2833
P4 = 6121
P8 = 19705421
P113 = 90821744521678073858476364997021996717514800118315880931365217585390573943501211754500252286470470063005559638673
9E+129 9000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P5 * P7 * C120
P5 = 32801
P7 = 2379277
C120 = 115321536164667735888492935425573063879167720081542955638813507247273800701381449317112867250303185722672231194163915813
9E+130 90000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P2 * P5 * C126
P2 = 17
P5 = 24593
C126 = 215269289922287786338054109131962466603361549556186480610216680499711778339604048019402938664995539141936610369760883656516321
9E+131 900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 is not prime
9E+132 9000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P2 * P3 * C129
P2 = 53
P3 = 317
C129 = 535682399857151360038092970656508541158264389024462829593476578775072912326647223379560740432117135884768763764061662996250223201
9E+133 90000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P1 * P2 * C132
P1 = 7
P2 = 13
C132 = 989010989010989010989010989010989010989010989010989010989010989010989010989010989010989010989010989010989010989010989010989010989011
9E+134 900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P3 * P4 * P11 * P120
P3 = 181
P4 = 2753
P11 = 16531089961
P120 = 109258751591334591151303372851647591688655864712545289987012012810113495421933518921235388716934090390154756359979711637
9E+135 9000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P4 * P7 * P10 * C117
P4 = 5927
P7 = 3426823
P10 = 4329778969
C117 = 102341124002116886890802769572431620180185546879491021271962594087713193270517160649114163216150099920249933560241049
9E+136 90000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P5 * P5 * C129
P5 = 10037
P5 = 17333
C129 = 517326646039550392331618841935447509765129621565614254317238874101553784258498560231085444249745048495665558577899169944223163081
9E+137 900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P5 * C134
P5 = 12487
C134 = 72074957956274525506526787859373748698646592456154400576599663650196204052214302874989989589172739649235204612797309201569632417714423
9E+138 9000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P4 * P5 * P5 * C127
P4 = 2477
P5 = 16193
P5 = 95401
C127 = 2351994273131163932011470571366592962213409754392142945080247744180276601507505786882653964394865333813159089372234186818386141
9E+139 90000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P1 * P2 * P2 * P2 * P10 * C126
P1 = 7
P2 = 13
P2 = 13
P2 = 59
P10 = 4424608127
C126 = 291427774943488532047152968430812186544474227918461927420354897872451236915447049045959903935550375259390389351446759644503379
9E+140 900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001=P2 * P3 * P3 * P4 * C131
P2 = 97
P3 = 149
P3 = 293
P4 = 4733
C131 = 44903519496482035685920893374806943061833411079228781276315961742360447241922023072917417784572160800690565957656776306254923604493

太阳 · 发表于 2021-3-6 07:00

900...01，有没有可能同时出现7^2和13^2

yangchuanju · 发表于 2021-3-6 07:24

太阳发表于 2021-3-6 07:00
900...01，有没有可能同时出现7^2和13^2

请细看7楼的分析：
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。
12楼给出：
两个立方因子在“9+40776个0+1”的分解式中同时出现，之后需增加49686个0时在同时出现一次。
经延伸分析，当0的个数达到7940850和12462276时，将同时出现7^4和13^4；
两数相差4521426=13*13*13*2058个0；另在3419424个0处出现了7^4和13^5一组数字。

太阳 · 发表于 2021-3-6 07:34

900...01，有没有可能出17^2

太阳 · 发表于 2021-3-6 08:08

yangchuanju 发表于 2021-3-5 22:51
接7楼
已经查明，第1的素因子7^3出现在“9+204个0+1”的分解式中，之后每增加297个0出现一个7的立方因子； ...

7^2和13^2同时出现，900...01数据有多少位数？有多少个0在里面

太阳 · 发表于 2021-3-6 18:15

本帖最后由太阳于 2021-3-6 18:28 编辑

(9*10^373+1)*10^546*10^918，同时出现7^2，13^2

太阳 · 发表于 2021-3-6 21:30

太阳发表于 2021-3-6 18:15
(9*10^373+1)*10^546*10^918，同时出现7^2，13^2

900...01，有没有可能出17^2

太阳 · 发表于 2021-3-6 21:30

900...01，有没有可能出17^2

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