以10为底的φ因子结构形式

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以10为底的φ因子结构形式
当指数是素数p时（p≥2），φp(10)都是清一色数111…1，例：
Φ2(10)=11=11
Φ3(10)=111=3×37
Φ5(10)=11111=41×271
Φ7(10)=1111111=239×4649
Φ11(10)=11111111111<11>=21649×513239
Φ13(10)=1111111111111<13>=53×79×265371653
Φ17(10)=11111111111111111<17>=2071723×5363222357<10>
Φ19(10)=1111111111111111111<19>=1111111111111111111<19>
Φ23(10)=11111111111111111111111<23>=11111111111111111111111<23>
Φ29(10)=11111111111111111111111111111<29>=3191×16763×43037×62003×77843839397<11>
Φ31(10)=1111111111111111111111111111111<31>=2791×6943319×57336415063790604359<20>
当指数p=2，19，23，317，1031，49081, 86453, 109297, 270343, 5794777, 8177207……时φ因子是素数，其余是合数。
该类清一色素数应该有无穷多个，但目前仅知道这11个。

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当指数是素数2p时（p≥3），φ2p(10)都是9090…91，其中当2p=6时只有末尾1节数字91，例：
Φ6(10)=91=7×13
Φ10(10)=9091=9091素数
Φ14(10)=909091=909091素数
Φ22(10)=9090909091<10>=11×23×4093×8779
Φ26(10)=909090909091<12>=859×1058313049<10>
Φ34(10)=9090909090909091<16>=103×4013×21993833369<11>
Φ38(10)=909090909090909091<18>=909090909090909091<18位素数>
Φ46(10)=9090909090909090909091<22>=47×139×2531×549797184491917<15>
Φ58(10)=9090909090909090909090909091<28>=59×154083204930662557781201849<27>
Φ62(10)=909090909090909090909090909091<30>=909090909090909090909090909091<30位素数>
Φ74(10)=909090909090909090909090909090909091<36>
    =7253×422650073734453<15>×296557347313446299<18>
Φ82(10)=9090909090909090909090909090909090909091<40>
    =2670502781396266997<19>×3404193829806058997303<22>
Φ86(10)=909090909090909090909090909090909090909091<42>
    =57009401×2182600451<10>×7306116556571817748755241<25>
当指数2p=10,14,38,62,106,134时是素数。
(10^2p-1)/9=(10^p-1)/9*(10^p+1)=111…11*100…01=111…11*11*9090…91，其中当2p=6时只有末尾1节数字91。

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当指数是素数4p时（p≥3），φ4p(10)都是99009900…9901，其中当4p=2时只有末尾1节数字9901，例：
Φ12(10)=9901=9901
φ20(10)=3541*27961=9900 9901
Φ28(10)=9900 9900 9901<12>=29×281×121499449
Φ44(10)=9900 9900 9900 9900 9901<20>=89×1052788969<10>×1056689261<10>
Φ52(10)=9900 9900 9900 9900 9900 9901<24>=521×1900381976777332243781<22>
Φ68(10)=9900 9900 9900 9900 9900 9900 9900 9901<32>
    =28559389×1491383821<10>×2324557465671829<16>
Φ76(10)=9900 9900 9900 9900 9900 9900 9900 9900 9901<36>
    =722817036322379041<18>×1369778187490592461<19>
Φ92(10)=9900 9900 9900 9900 9900 9900 9900 9900 9900 9900 9901<44>
    =1289×18371524594609<14>×4181003300071669867932658901<28>

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当指数是素数8p时（p≥3），φ8p(10)都是9999000099990000…99990001，例：
Φ24(10)=99990001=99990001（素数）
Φ40(10)=99990000 99990001<16>=1676321×5964848081<10>
Φ56(10)=99990000 99990000 99990001<24>=7841×127522001020150503761<21>
Φ88(10)=99990000 99990000 99990000 99990000 99990001<40>
    =617×16205834846012967584927082656402106953<38>
Φ104(10)=99990000 99990000 99990000 99990000 99990000 99990001<48>
    =1580801×632527440202150745090622412245443923049201<42>
Φ136(10)=99990000 99990000 99990000 99990000 99990000 99990000 99990000 99990001<64>
    =152533657×65552746171882583264230070868884366877803237222654400793<56>
Φ152(10)=99990000 99990000 99990000 99990000 99990000 99990000 99990000 99990000 99990001<72>
    =457×1403417×5240808656722481737<19>×297478330786365628414805305290302483555043017<45>

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当指数是素数16p时（p≥3），φ16p(10)都是99999999000000009999999900000000…9999999900000001，例：
Φ48(10)=9999999900000001<16>=9999999900000001<16位素数>
Φ80(10)=9999999900000000 9999999900000001<32>=5070721×19721061166646717498359681<26>
Φ112(10)=9999999900000000 9999999900000000 9999999900000001<48>
    =113×73765755896403138401<20>×119968369144846370226083377<27>
Φ176(10)=9999999900000000 9999999900000000 9999999900000000 9999999900000000 9999999900000001<80>
    =10100113×9900879227786858424257223656804730798555422102713108259184822982673560177<73>

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当指数是素数6p时（p≥5），φ6p(10)都是，例：
Φ30(10)=109889011=211×241×2161
Φ42(10)=109890 098901 1<13>=7×127×2689×459691
Φ66(10)=109890 10988901 098901 1<21>=599144041×183411838171<12>
Φ78(10)=109890 109890 098901 098901 1<25>=13×157×6397×216451×388847808493<12>
Φ102(10)=109890 109890 10988901 098901 098901 1<33>
    =291078844423<12>×377526955309799110357<21>
Φ114(10)=109890 109890 109890 098901 098901 098901 1<37>
    =1458973×753201806271328462547977919407<30>
Φ138(10)=109890 109890 109890 10988901 098901 098901 098901 1<45>
    =31051×143574021480139<15>×24649445347649059192745899<26>
Φ174(10)=109890 109890 109890 109890 10988901 098901 098901 098901 098901 1<57>
    =638453709757<12>×135080726389891<15>×1274194732898148471766404179653<31>
Φ186(10)=109890 109890 109890 109890 109890 098901 098901 098901 098901 098901 1<61>
    =373×44641×3590254957<10>×18381907262281244633158190677786966663091011<44>

yangchuanju

有趣的φ6p(10)
将φ6p(10)的各位数字分别从前向后和从后向前按6位一节分节（最末尾的1不计），
前段各节都是109890，其中109+890=999；后段各节都是098901，其中098+901=999；
如果有中节存在则中节是10988901，其中1098+8901=9999。
这些数字均由111…11除以111111，再除以φp(10)、φ2p(10)、φ3p(10)而得，它们之中有没有素数不详。

太阳先生的1761位大因子来自a=1265，1265/5=253=11*23，1265*6=7590；
它就是没有找到任何素因子的φ7590(10)。
Φ7590(10)= 1098890109...<1761>= [1098890109...<1761>](0.00%)
由于7590=6*1265=6*5*11*23，不是6p型数字，因而1761位大φ因子的结构形式与上述6p型数字φ因子的结构形式不尽相同。

yangchuanju

当指数是素数30p时（p≥7），φ30p(10)是：
Φ210(10)=9100091910...<48>
    =29970369241<11>×1661378260814161<16>×18276168846821336356291<23>
Φ330(10)=9100091000...<80>
    =4124507971<10>×19835636682880495867311241<26>
    ×1112314101311286003379752617807870409611285281<46>
Φ390(10)=9100091000...<96>
    =15601×925081
    ×63054129911571801941639263405768461425999152026392552531582926655
    481271974023945336611<86>
Φ510(10)=9100090999...<128>
    =102001×5516286288241<13>×4270914986978327797975291<25>
    ×216219010761333454086082249502131<33>
    ×175137725562337579790651749196120587807233668420015131<54>
Φ570(10)=9100090999...<144>
    =571×11382273828031938028891<23>×43135513763991692454656851<26>
    ×324597837765556721276090482592526284629519802190230933051716260
    0205414882257480287263152055681<94>
Φ690(10)=9100090999...<176>
    =11313931×127688571001<12>×194656286401<12>
    ×7386441591567953769105190285927615814299177561<46>
    ×438103256137395222981089917412776417471375475658780571741300754
    70872858289958280148759277150273598401<101>
Φ870(10)=9100090999...<224>
    =33931×1134434761<10>×2172376081<10>×70350656881<11>
    ×1174705187104089640352756731<28>
    ×4266662839057986102285502119162080638872058454152278425256962985792707796691<76>
    ×30863783899527944644524775944574816587961449894543876630226127471096483551640405
    68369321<88>
这些数字均由111…11除以111...1<30>，再除以φp(10)、φ2p(10)、φ3p(10)、φ5p(10)、φ6p(10)、φ10p(10)、φ15p(10)而得，它们之中有没有素数不详；
其结构形式也不再具有φ6p(10)那样的相同的小循环节：
Φ210(10)=910009191000909089989898989899909091000919100091<48>
Φ330(10)=91000910000909918099089999090081809909090909008180990899990900819009100009100091<80>
Φ390(10)=910009100000090090900899999909909099099090080990900809909009009090089999990990909910000009100091<96>
Φ510(10)=91000909999999909909099100000090090900899999909909008099090090090990990900809909008999999099090991000000900909009000000009100091<128>
Φ570(10)=910009099999999090081900910000909918099089999090081900909090900818099090909008180990909090991809908999909008190091000090991809909000000009100091<144>

yangchuanju

当指数是素数210p时（p≥11），φ210p(10)是：
p=11 Φ2310(10) = 1098890000...<481>
    = 101641 × 1081148355...<476>(100.00%)
p=13 Φ2730(10) = 1098890000...<577>
    = 2731 × 5528251 × 6097697971<10> × 5953395590224596670581957589121611<34>
    × [2004998801...<523>](9.33%)
p=17 Φ3570(10) = 1098890000...<769>
    = 3571 × 289171 × 3059491 × 218159131 × [1594362456...<745>](3.10%)
p=19 Φ3990(10) = 1098890000...<865>
    = 3686412850051<13> × 2980919513...<852>(100.00%)
p=23 Φ4830(10) = 1098890000...<1057>
    = 96601 × [1137555511...<1052>](0.47%)

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当指数是素数2310p时（p≥13），φ2310p(10)是：
p=13 Φ30030(10) = 9100091910...<5760>
    = 104864761 × 7332124801<10> × 11196595411<11> × 22012504540158822691<20>
    × 63566312584211957521<20> × [7554468171...<5693>](1.16%)
p=17 Φ39270(10) = 9100091910...<7680>
    = 13665961 × 93109171 × [7151763455...<7665>](0.20%)
p=19 Φ43890(10) = 9100091910...<8640>
    = 1623931 × [5603742960...<8634>](0.07%)
p=23 Φ53130(10) = 9100091910...<10560>
    = 28955851 × [3142747180...<10553>](0.07%)