Table of contents 目次

  1. About 77...779 77...779 について
    1. Classification 分類
    2. Sequence 数列
    3. General term 一般項
  2. Prime numbers of the form 77...779 77...779 の形の素数
    1. Last updated 最終更新日
    2. Known (probable) prime numbers 既知の (おそらく) 素数
    3. Range of search 捜索範囲
    4. Prime factors that appear periodically 周期的に現れる素因数
    5. Difficulty of search 捜索難易度
  3. Factor table of 77...779 77...779 の素因数分解表
    1. Last updated 最終更新日
    2. Range of factorization 分解範囲
    3. Terms that have not been factored yet まだ分解されていない項
    4. Factor table 素因数分解表
  4. Related links 関連リンク

1. About 77...779 77...779 について

1.1. Classification 分類

Near-repdigit of the form AA...AAB AA...AAB の形のニアレプディジット (Near-repdigit)

1.2. Sequence 数列

7w9 = { 9, 79, 779, 7779, 77779, 777779, 7777779, 77777779, 777777779, 7777777779, … }

1.3. General term 一般項

7×10n+119 (1≤n)

2. Prime numbers of the form 77...779 77...779 の形の素数

2.1. Last updated 最終更新日

January 27, 2023 2023 年 1 月 27 日

2.2. Known (probable) prime numbers 既知の (おそらく) 素数

  1. 7×102+119 = 79 is prime. は素数です。
  2. 7×1066+119 = (7)659<66> is prime. は素数です。
  3. 7×1086+119 = (7)859<86> is prime. は素数です。
  4. 7×1090+119 = (7)899<90> is prime. は素数です。
  5. 7×10102+119 = (7)1019<102> is prime. は素数です。 (Makoto Kamada / PPSIQS / June 12, 2003 2003 年 6 月 12 日)
  6. 7×10386+119 = (7)3859<386> is prime. は素数です。 (Julien Peter Benney / December 6, 2004 2004 年 12 月 6 日)
  7. 7×10624+119 = (7)6239<624> is prime. は素数です。 (discovered by:発見: Makoto Kamada / PFGW / December 17, 2004 2004 年 12 月 17 日) (certified by:証明: Tyler Cadigan / PRIMO 2.2.0 beta 6 / May 29, 2006 2006 年 5 月 29 日)
  8. 7×107784+119 = (7)77839<7784> is PRP. はおそらく素数です。 (Julien Peter Benney / September 28, 2004 2004 年 9 月 28 日)
  9. 7×1018536+119 = (7)185359<18536> is PRP. はおそらく素数です。 (Serge Batalov / PFGW / January 18, 2009 2009 年 1 月 18 日)
  10. 7×10113757+119 = (7)1137569<113757> is PRP. はおそらく素数です。 (Tyler Busby / January 21, 203)
  11. 7×10135879+119 = (7)1358789<135879> is PRP. はおそらく素数です。 (Tyler Busby / January 21, 203)

2.3. Range of search 捜索範囲

  1. n≤30000 / Completed 終了 / Ray Chandler / September 30, 2010 2010 年 9 月 30 日
  2. n≤50000 / Completed 終了 / Erik Branger / March 5, 2013 2013 年 3 月 5 日
  3. n≤200000 / Completed 終了 / Tyler Busby / January 26, 2023 2023 年 1 月 26 日

2.4. Prime factors that appear periodically 周期的に現れる素因数

Cofactors are written verbosely to clarify that they are integers. 補因子はそれらが整数であることを明確にするために冗長に書かれています。

  1. 7×103k+1+119 = 3×(7×101+119×3+7×10×103-19×3×k-1Σm=0103m)
  2. 7×105k+3+119 = 41×(7×103+119×41+7×103×105-19×41×k-1Σm=0105m)
  3. 7×106k+5+119 = 13×(7×105+119×13+7×105×106-19×13×k-1Σm=0106m)
  4. 7×1013k+2+119 = 79×(7×102+119×79+7×102×1013-19×79×k-1Σm=01013m)
  5. 7×1015k+5+119 = 31×(7×105+119×31+7×105×1015-19×31×k-1Σm=01015m)
  6. 7×1016k+12+119 = 17×(7×1012+119×17+7×1012×1016-19×17×k-1Σm=01016m)
  7. 7×1018k+3+119 = 19×(7×103+119×19+7×103×1018-19×19×k-1Σm=01018m)
  8. 7×1021k+10+119 = 43×(7×1010+119×43+7×1010×1021-19×43×k-1Σm=01021m)
  9. 7×1022k+15+119 = 23×(7×1015+119×23+7×1015×1022-19×23×k-1Σm=01022m)
  10. 7×1028k+17+119 = 29×(7×1017+119×29+7×1017×1028-19×29×k-1Σm=01028m)

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2.5. Difficulty of search 捜索難易度

The difficulty of search, percentage of terms that are not divisible by prime factors that appear periodically, is 12.22%. 捜索難易度 (周期的に現れる素因数で割り切れない項の割合) は 12.22% です。

3. Factor table of 77...779 77...779 の素因数分解表

3.1. Last updated 最終更新日

May 8, 2024 2024 年 5 月 8 日

3.2. Range of factorization 分解範囲

3.3. Terms that have not been factored yet まだ分解されていない項

n=212, 215, 226, 227, 230, 231, 235, 237, 238, 240, 244, 246, 247, 249, 250, 252, 256, 257, 260, 266, 267, 268, 273, 274, 276, 277, 278, 279, 280, 282, 284, 285, 289, 290, 291, 292, 293, 295, 296, 297, 298 (41/300)

3.4. Factor table 素因数分解表

7×101+119 = 9 = 32
7×102+119 = 79 = definitely prime number 素数
7×103+119 = 779 = 19 × 41
7×104+119 = 7779 = 3 × 2593
7×105+119 = 77779 = 13 × 31 × 193
7×106+119 = 777779 = 113 × 6883
7×107+119 = 7777779 = 3 × 2592593
7×108+119 = 77777779 = 41 × 263 × 7213
7×109+119 = 777777779 = 3319 × 234341
7×1010+119 = 7777777779<10> = 32 × 43 × 20097617
7×1011+119 = 77777777779<11> = 13 × 229 × 26126227
7×1012+119 = 777777777779<12> = 17 × 47 × 3823 × 254627
7×1013+119 = 7777777777779<13> = 3 × 41 × 3167 × 19966519
7×1014+119 = 77777777777779<14> = 5347 × 14546059057<11>
7×1015+119 = 777777777777779<15> = 23 × 79 × 1063 × 4817 × 83597
7×1016+119 = 7777777777777779<16> = 3 × 103 × 617 × 40795464943<11>
7×1017+119 = 77777777777777779<17> = 13 × 29 × 5939 × 34737683593<11>
7×1018+119 = 777777777777777779<18> = 41 × 241 × 78714480090859<14>
7×1019+119 = 7777777777777777779<19> = 33 × 499 × 1170523 × 493186601
7×1020+119 = 77777777777777777779<20> = 31 × 461 × 5442430745068769<16>
7×1021+119 = 777777777777777777779<21> = 19 × 337 × 2441 × 49762735896073<14>
7×1022+119 = 7777777777777777777779<22> = 3 × 2017 × 1285370645806937329<19>
7×1023+119 = 77777777777777777777779<23> = 132 × 41 × 984110087 × 11406207973<11>
7×1024+119 = 777777777777777777777779<24> = 691 × 1125582891140054671169<22>
7×1025+119 = 7777777777777777777777779<25> = 3 × 946801 × 2738265583361860193<19>
7×1026+119 = 77777777777777777777777779<26> = 1861 × 12115193 × 16459207 × 209589689
7×1027+119 = 777777777777777777777777779<27> = 2029 × 3967 × 28772503 × 3358409417351<13>
7×1028+119 = 7777777777777777777777777779<28> = 32 × 17 × 41 × 79 × 7741 × 51620287 × 39276761311<11>
7×1029+119 = 77777777777777777777777777779<29> = 13 × 199 × 213844493 × 140592136667541869<18>
7×1030+119 = 777777777777777777777777777779<30> = 59 × 20031313 × 369189463 × 1782562657999<13>
7×1031+119 = 7777777777777777777777777777779<31> = 3 × 43 × 6329 × 9526441932457798882929419<25>
7×1032+119 = 77777777777777777777777777777779<32> = 359 × 743 × 206553061387<12> × 1411694205399241<16>
7×1033+119 = 777777777777777777777777777777779<33> = 41 × 44316373991<11> × 428062767629625652109<21>
7×1034+119 = 7777777777777777777777777777777779<34> = 3 × 317 × 8178525528683257389881995560229<31>
7×1035+119 = 77777777777777777777777777777777779<35> = 13 × 31 × 15173 × 1495019 × 7233229733<10> × 1176251135483<13>
7×1036+119 = 777777777777777777777777777777777779<36> = 107 × 1512 × 12478536487<11> × 25547841027541018831<20>
7×1037+119 = 7777777777777777777777777777777777779<37> = 32 × 23 × 2111 × 601334980484167<15> × 29599234793674181<17>
7×1038+119 = 77777777777777777777777777777777777779<38> = 41 × 44497 × 1921869684043951<16> × 22182832808880677<17>
7×1039+119 = 777777777777777777777777777777777777779<39> = 19 × 10139 × 27164399 × 148630077983747790669982781<27>
7×1040+119 = 7777777777777777777777777777777777777779<40> = 3 × 80992284229<11> × 1569029963537<13> × 20401372755780941<17>
7×1041+119 = 77777777777777777777777777777777777777779<41> = 13 × 79 × 181 × 631244566397<12> × 662840229894836844987761<24>
7×1042+119 = 777777777777777777777777777777777777777779<42> = 8231 × 94493716167874836323384494931087082709<38>
7×1043+119 = 7777777777777777777777777777777777777777779<43> = 3 × 41 × 409 × 7014113 × 22042170270010692603540949933169<32>
7×1044+119 = 77777777777777777777777777777777777777777779<44> = 17 × 679309 × 9197162269<10> × 732293858264417905262808347<27>
7×1045+119 = 777777777777777777777777777777777777777777779<45> = 29 × 827 × 85910343643<11> × 377490969662250076905617363591<30>
7×1046+119 = 7777777777777777777777777777777777777777777779<46> = 34 × 22727 × 26733319 × 158043092466149364300081853902643<33>
7×1047+119 = 77777777777777777777777777777777777777777777779<47> = 13 × 35103544449497070623<20> × 170435951033762351387132321<27>
7×1048+119 = 777777777777777777777777777777777777777777777779<48> = 41 × 241 × 112751571950542457<18> × 698123127945271170994880387<27>
7×1049+119 = 7777777777777777777777777777777777777777777777779<49> = 3 × 29893696679130299<17> × 86727065589133394801760847169507<32>
7×1050+119 = 77777777777777777777777777777777777777777777777779<50> = 31 × 103 × 9964405567<10> × 82678712461<11> × 29567289358180253679516169<26>
7×1051+119 = (7)509<51> = 2371 × 45569 × 49993482555794381<17> × 143992903020092295997382741<27>
7×1052+119 = (7)519<52> = 3 × 43 × 61 × 11399 × 507317 × 1163119 × 97576033 × 261752155261<12> × 5753501647991<13>
7×1053+119 = (7)529<53> = 13 × 41 × 145924536168438607462997706900145924536168438607463<51>
7×1054+119 = (7)539<54> = 79 × 4909 × 2005558836076794566883811386932752752701129616689<49>
7×1055+119 = (7)549<55> = 32 × 2828219351<10> × 4546731700621516379933<22> × 67204848150880872133057<23>
7×1056+119 = (7)559<56> = 90679 × 23716248665387<14> × 1214518147884479083<19> × 29778224828242663181<20>
7×1057+119 = (7)569<57> = 19 × 257 × 5503 × 351667 × 11825780787923491<17> × 6959977771853864490905371543<28>
7×1058+119 = (7)579<58> = 3 × 41 × 47 × 1345403524957235387956716446597090084376021065175190759<55>
7×1059+119 = (7)589<59> = 13 × 23 × 1557029 × 4207867 × 39703211418937180186856243144355619500839647<44>
7×1060+119 = (7)599<60> = 17 × 709 × 1683161717<10> × 38338448118410287074745998274875962539249452779<47>
7×1061+119 = (7)609<61> = 3 × 149 × 2287 × 721891 × 6085691 × 2914860829<10> × 14174832092402629<17> × 41914527126320291<17>
7×1062+119 = (7)619<62> = 547 × 1699 × 83690242327487809021736388409762251564021182239448065243<56>
7×1063+119 = (7)629<63> = 41 × 18970189701897018970189701897018970189701897018970189701897019<62>
7×1064+119 = (7)639<64> = 32 × 2339 × 9157 × 1135019 × 483764023 × 1345497360541589<16> × 54614760836502033667211029<26>
7×1065+119 = (7)649<65> = 13 × 31 × 9749 × 19751 × 465035122630486061<18> × 2155338913912796080936751765560130087<37>
7×1066+119 = (7)659<66> = definitely prime number 素数
7×1067+119 = (7)669<67> = 3 × 79 × 4993 × 84059 × 33186618067927<14> × 2356125405132088781234951457951866862498683<43>
7×1068+119 = (7)679<68> = 41 × 739 × 7229 × 3027083 × 6793338283779041<16> × 17267976151334662612818313357988584183<38>
7×1069+119 = (7)689<69> = 321719689 × 37465746377<11> × 3517862789893<13> × 18342756684400184897051481222073434151<38>
7×1070+119 = (7)699<70> = 3 × 367 × 3234857 × 2183801322111087690060311601906743983275211821472564882791847<61>
7×1071+119 = (7)709<71> = 13 × 749809 × 7979240023667337823342885965500524674927756245798573911466661487<64>
7×1072+119 = (7)719<72> = 12943499 × 60090225817437601515461760207018038768170629732947619324401985721<65>
7×1073+119 = (7)729<73> = 33 × 29 × 41 × 43 × 109 × 761 × 704477 × 2937422377<10> × 5324653129885363399<19> × 6164612216271394312499298769<28>
7×1074+119 = (7)739<74> = 17971 × 4327960479538021132812741515651759934215000711022078781246328962093249<70>
7×1075+119 = (7)749<75> = 19 × 4987870469<10> × 735234853843486626094867723831<30> × 11162479551938454492774158872305019<35>
7×1076+119 = (7)759<76> = 3 × 17 × 4924999 × 8574379637<10> × 24256676753<11> × 15222391287470643111887<23> × 9780528378951454077501253<25>
7×1077+119 = (7)769<77> = 13 × 97 × 274889 × 790058072059<12> × 101964994488179<15> × 168118568572821468713<21> × 16567511635443917446607<23>
7×1078+119 = (7)779<78> = 41 × 241 × 979701640863643<15> × 1917052398177918453832498387<28> × 41910883145412168265560398806699<32>
7×1079+119 = (7)789<79> = 3 × 7226602109<10> × 35138657517849736111362113<26> × 10209746693369446387727602237491151269208229<44>
7×1080+119 = (7)799<80> = 31 × 79 × 15971 × 99793 × 1535726545574999<16> × 12975397336455014259504246591192261833761259722113943<53>
7×1081+119 = (7)809<81> = 23 × 2897 × 12823481453<11> × 481264954223<12> × 1891424515948856274881996326584527709795521400695483111<55>
7×1082+119 = (7)819<82> = 32 × 1019 × 28546152736411827011440249<26> × 29709220149899051178451251263694965499473714075429801<53>
7×1083+119 = (7)829<83> = 13 × 41 × 52511 × 2778932722066588095122882955954865162273969998809068532438920339062980488633<76>
7×1084+119 = (7)839<84> = 103 × 21283 × 354801511109773002554925681501475391398019832492124293457140813775787037184871<78>
7×1085+119 = (7)849<85> = 3 × 269 × 17043289283<11> × 189285111128250209182492304186003<33> × 2987528974318345625824733580150171371453<40>
7×1086+119 = (7)859<86> = definitely prime number 素数
7×1087+119 = (7)869<87> = 2903 × 4186050139<10> × 2034286372007431<16> × 393921068289402364886953<24> × 79869831857230807881732232192540609<35>
7×1088+119 = (7)879<88> = 3 × 41 × 59 × 1571 × 4469401 × 37848491 × 4032963872038252292727280889655328769818837855345441298563835762627<67>
7×1089+119 = (7)889<89> = 13 × 107 × 46861 × 6641753743<10> × 179652836810392132704368015278125769145210972892606647360380321107659903<72>
7×1090+119 = (7)899<90> = definitely prime number 素数
7×1091+119 = (7)909<91> = 32 × 261845537 × 2759479755516919012573187<25> × 1196026033347142717648420586829657022478324416064826547049<58> (Tetsuya Kobayashi / GMP-ECM 5.0.1 for P25 x P58 / May 1, 2003 2003 年 5 月 1 日)
7×1092+119 = (7)919<92> = 17 × 3299 × 762919 × 2464475490454871<16> × 737600766585283741222917455127403078741407410515836008818634688737<66>
7×1093+119 = (7)929<93> = 192 × 41 × 79 × 179 × 3733 × 5851 × 385097291 × 4289785728988631058342031<25> × 102988847453502073956683243304756698283984133<45> (Tetsuya Kobayashi / GMP-ECM 5.0.1 B1=250000 for P25 x P45 / April 30, 2003 2003 年 4 月 30 日)
7×1094+119 = (7)939<94> = 3 × 43 × 213827 × 1962911 × 143649020385728695633331069186254963458877307212311526904307124664711718336376783<81>
7×1095+119 = (7)949<95> = 13 × 31 × 1453 × 32933 × 4033235452147439633947657810082845775433789451594705791766971144442006387608034413857<85>
7×1096+119 = (7)959<96> = 431 × 21725785008173<14> × 13572168196987380335048624507<29> × 6120029981391385090617474168836130760426783106264419<52>
7×1097+119 = (7)969<97> = 3 × 167 × 1531 × 10140108780189819939191216232170248370375875000851044844051645602039262790914288702513689509<92>
7×1098+119 = (7)979<98> = 41 × 24249441384503<14> × 80779851907249382243<20> × 252053753676109370439217<24> × 3842144666861629459297265197607826961583<40>
7×1099+119 = (7)989<99> = 13676200759081543<17> × 56870895029915548666914196633930210752155368137091469599841293765275145970111592053<83>
7×10100+119 = (7)999<100> = 33 × 41659 × 48479 × 90365176815227476134521656292792169718259939<44> × 1578440334622505126220537098134695122528450663<46> (Robert Backstrom / NFSX v1.8 for P44 x P46 / June 11, 2003 2003 年 6 月 11 日)
7×10101+119 = (7)1009<101> = 132 × 29 × 558781 × 972854195557<12> × 3311106460108699603722973800347<31> × 8816745335245460996486446296361070917320003185221<49>
7×10102+119 = (7)1019<102> = definitely prime number 素数
7×10103+119 = (7)1029<103> = 3 × 23 × 41 × 1487 × 1848892303528846034734822753430111223814303384790911542732378095200890997048718866865676151669073<97>
7×10104+119 = (7)1039<104> = 47 × 617 × 69978973 × 210760031 × 444208196371557749<18> × 59994166515574279026658391<26> × 6823718435034149072712288788084746408013<40>
7×10105+119 = (7