以142857为循环节的素数φ因子

yangchuanju

以142857为循环节的素数φ因子
众所周知，7分之一是一个纯循环小数，循环节为142857，1/7=0.142857142857142857…
然而亦有一个以142857为“循环节”的84位大素数：
142857 157 142857 142856 99999998 571428 571428 58 571428 571428
5 571428 5 571428 571428 572857143
有些遗憾的是，它的循环节142857之间夹杂着一些其它数字！

7是一个小素数，它隐含在(10^6-1)/9=111111之中；
49=7*7，完全平方数，它隐含在(10^42-1)/9=111111…<42>之中；
294=7*7*7，立方数，它隐含在(10^294-1)/9=111111…<294>之中；
2058=7*7*7*7，四次方数，它隐含在(10^2058-1)/9=111111…<2058>之中；
……

111111=3×7×11×13×37
111111…<42>=3×7^2×11×13×37×43×127×239×1933×2689×4649×459691×909091×10838689
111111…<294>=3×7^3×11×13×37×43×127×197×239×1933×2689×4649
    ×63799×459691×909091×4715467×10838689×505885997×267652966241599<15>
    ×2603941883787374089<19>×1976730144598190963568023014679333<34>
    ×4769337181464959147997704753876850429427<40>
    ×5076141624365532994918781726395939035533<40>
    ×142857157142857142856999999985714285714285857142857
      -142855714285571428571428572857143<84>
…………
注意到了吗？
111111…<294>分解式的最后一个大因子就是今天要说的以142857为“循环节”的大素数！

294=2*3*7*7，111111…<294>共含26个素因子（其中含3个7）
(10^294-1)/9=(10^147-1)*(10^147+1)=(10^49-1)/9*(10^98+10^49+1)*(10^49+1)*(10^98-10^49+1)
(10^49-1)/9=(10^7-1)*(10^42+10^35+10^28+10^21+10^14+10^7+1)/9
      =(10-1)*(10^6+10^5+10^4+10^3+10^2+10+1)*(10^42+10^35+10^28+10^21+10^14+10^7+1)/9
      =1111111*1000000100000010000001000000100000010000001
      =239*4649*505885997*1976730144598190963568023014679333
10^98+10^49+1
      =1000000000000000000000000000000000000000000000000
       10000000000000000000000000000000000000000000000001      
      =3*37*43*1933*63799*4715467*10838689*267652966241599*2603941883787374089<19>
       *4769337181464959147997704753876850429427<40>
10^49+1=(10^7+1)*(10^42-10^35+10^28-10^21+10^14-10^7+1)
      =(10+1)*(10^6-10^5+10^4-10^3+10^2-10+1)*(10^42-10^35+10^28-10^21+10^14-10^7+1)
      =11*909091*999999900000009999999000000099999990000001
      =11*909091*197*5076141624365532994918781726395939035533
10^98-10^49+1
      =9999999999999999999999999999999999999999999999999
       0000000000000000000000000000000000000000000000001      
      =7^3*13*127*2689*459691
       *142857157142857142856999999985714285714285857142857
        -142855714285571428571428572857143<84>

84位素因子142857…是111111…<294>的唯一φ因子，它是目前已知的第22大的唯一素数。
142857*7=999999，142857…<84>乘以7等于：
1000000099999999999998999999900000000000000999999999999989999999000000000000010000001
再乘以7^2*13*127*2689*459691=99999990000001<7个1+6个0+1个1>还原成10^98-10^49+1：
9999999999999999999999999999999999999999999999999
0000000000000000000000000000000000000000000000001<49个1+48个0+1个1>

yangchuanju

没有纯粹的以142857位循环节的素数，因为142857本身就不是素数
以142857为循环节后加1的整数中找到了2个素数1428571和1428571428571428571428571
=================================
1428571 is prime
cputime 0:00:00:00
=================================
1428571428571=P1 * P5 * P7
P1 = 7
P5 = 56527
P7 = 3610339
cputime 0:00:00:00
=================================
1428571428571428571=P3 * P4 * P6 * P7
P3 = 103
P4 = 9431
P6 = 155027
P7 = 9486361
cputime 0:00:00:00
=================================
1428571428571428571428571 is prime
cputime 0:00:00:15
=================================
1428571428571428571428571428571=P4 * P4 * P5 * P9 * P11
P4 = 1063
P4 = 2089
P5 = 95633
P9 = 167826727
P11 = 40083104483
cputime 0:00:00:20
=================================
1428571428571428571428571428571428571=P4 * P9 * C24
P4 = 5779
P9 = 702780737
C24 = 351746201673192590450777
cputime 0:00:02:75
=================================
1428571428571428571428571428571428571428571=P2 * P5 * P5 * P7 * P7 * P19
P2 = 17
P5 = 63473
P5 = 71861
P7 = 1004537
P7 = 6948583
P19 = 2639420403553807201
cputime 0:00:00:14
=================================
1428571428571428571428571428571428571428571428571=P15 * C34
P15 = 168557936063381
C34 = 8475254633126609105645276866978991
cputime 0:00:04:03
=================================
1428571428571428571428571428571428571428571428571428571=P1 * P2 * P4 * P49
P1 = 7
P2 = 29
P4 = 3989
P49 = 1764175903156622301759112718314562795753064064813
cputime 0:00:00:23
=================================
1428571428571428571428571428571428571428571428571428571428571=P3 * P4 * P55
P3 = 157
P4 = 1321
P55 = 6888100737095659876606563395668348970470023329997196543
cputime 0:00:00:28
=================================
1428571428571428571428571428571428571428571428571428571428571428571=P2 * P7 * C58
P2 = 83
P7 = 4462531
C58 = 3856937679243440661619922595958527677464432014006564834227
cputime 0:00:06:04
=================================
1428571428571428571428571428571428571428571428571428571428571428571428571=P8 * P65
P8 = 31820483
P65 = 44894712269811510134166455882251333879142294243975761506466492937
cputime 0:00:01:36
=================================
1428571428571428571428571428571428571428571428571428571428571428571428571428571=P2 * P3 * P8 * C67
P2 = 71
P3 = 113
P8 = 66457829
C67 = 2679285648759340150056154979023576140660020593415626029580971345113
cputime 0:00:07:53
=================================
1428571428571428571428571428571428571428571428571428571428571428571428571428571428571=P4 * P5 * C76
P4 = 4073
P5 = 64849
C76 = 5408592560364432134973348475971845374554253751256066521478444033527586198323
cputime 0:00:08:95
=================================
1428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571=P2 * P12 * P15 * P63
P2 = 17
P12 = 622802562607
P15 = 541191531367411
P63 = 249316864069717043485091626653147919697908905053883660187580519
cputime 0:00:01:14
=================================
1428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571=P1 * P4 * P8 * P15 * P70
P1 = 7
P4 = 2281
P8 = 56978387
P15 = 572041744923931
P70 = 2744989966329896961448749127879070039002143600805487375898998297932229
cputime 0:00:02:23
=================================
1428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571=P6 * C97
P6 = 692401
C97 = 2063213988095667931485615168914297598398285716761571071429087232068452488411442832363657145828171
cputime 0:00:12:81
=================================
1428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571 is not prime
cputime 0:00:15:07
=================================
1428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571=P3 * P5 * C108
P3 = 149
P5 = 16427
C108 = 583656645068063411492934748762954332194366300926012123365637366772345484344840454829615742060183054568219277
cputime 0:00:14:90
=================================
1428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571=P3 * P119
P3 = 103
P119 = 13869625520110957004160887656033287101248266296809986130374479889042995839112343966712898751733703190013869625520110957
cputime 0:00:02:89
=================================
1428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571428571 is not prime
cputime 0:00:20:37
=================================