1513位唯一循环周期大素数

yangchuanju

1513位唯一循环周期大素数

从与太阳先生的互动交流中得知，太阳先生非常希望能通过限定某些代数式的条件，找到一个或几个能够获得大素数的表达式，如希望限定[(10^n^2+1)/(10^n+1)]与[(100^5t-10^8t+1)/(100^t-10^t+1)]可整除，从而断定[(100^5t-10^8t+1)/(100^t-10^t+1)]就是素数。
现经大量计算分析得知，在[(10^n^2+1)/(10^n+1)]与[(100^5t-10^8t+1)/(100^t-10^t+1)]可整除时，[(100^5t-10^8t+1)/(100^t-10^t+1)]之中确有素数，但为数不多，仅发现2个素数：
P41=10000099999999989999899999000000000100001和
P1001=10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000099999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999989999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999899999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001；
其余都是合数。

计算之中，涉及大量的φ因子及相关计算。
OEIS网站中有许多网页给出了φ因子、φ素数因子的数字，其中网页A051627给出99个唯一循环周期素数φ因子对于的清一色数的指数n，网页A040017给出46个唯一循环周期素数φ因子的具体数字。
经核对，网页A040017中的1-45号数据与A051627中的1-45号是对应的；但A040017中的46号数据与A051627中的47号才对应；A051627中的46号应对应上面的P1001，宜将该大素数P1001插在A040017的45#和46#之间，原46#素数应改为47#。

下一个唯一循环周期素数是谁？
A051627中的48#仅给出对应清一色数的指数2667，查φ因子表φ2667是一个1513位大素数，但各表都没有给出它的具体数值。
笔者通过大量计算最终得到这个大素数是：
1109999889000000000001109999889000000000001109999889000000000001109999889000000000001109999889000000000001109999889000000000000998999900100000000000998999900100000000000998999900100000000000998999900100000000000998999900100000000000998999900100000000000998999900100000000000998999900100000000000998999900100000000000998999900100000000000998999900100000000000998999900100000000001000109899989000000001000109899989000000001000109899989000000001000109899989000000001000109899989000000001000109899989000000000999998900000100000000999998900000100000000999998900000100000000999998900000100000000999998900000100000000999998900000100000000999998900000100000000999998900000100000000999998900000100000000999998900000100000000999998900000100000000999998900000100000001000000009999989000001000000009999989000001000000009999989000001000000009999989000001000000009999989000001000000009999989000001000000009999989000001000000009999989000001000000009999989000001000000009999989000001000000009999989000001000000009999989000000999999998899990110000999999998899990110000999999998899990110000999999998899990110000999999998899990110000999999998899990110001000000000009989999001000000000009989999001000000000009989999001000000000009989999001000000000009989999001000000000009989999001000000000009989999001000000000009989999001000000000009989999001000000000009989999001000000000009989999001000000000009989999000999999999998890000110999999999998890000110999999999998890000110999999999998890000110999999999998890000110999999999998890000111—P1513
它由6个“110999988900000000000”，12个“099899990010000000000”，6个“100010989998900000000”，12个“099999890000010000000”，12个“100000000999998900000”，6个“099999999889999011000”,12个“100000000000998999900”，6个“099999999999889000011”，再加“1”组成的。每个小循环节21位，共8种72个循环节，再加上末位的“1”，共1513位。

yangchuanju

再下一个唯一循环周期素数φ因子（49#）对应的清一色数指数是4354，φ4354的位数1861，请太阳先生算一算它的表达式吧！

yangchuanju

φ2667=1109999889...<1513>

R2667 = (10^2667-1)/9 = 1111111111...<2667> = 3 × 37 × 43 × 239 × 1933 × 4649 × 18797 × 90679 × 10838689 × 14097763 × 9783462102370544239<19> × 45830377642600710169<20> × 153288409997816784961649<24> × 591082109095345424416413908752729<33> × 82548590511975869997227448819484483748713764399174373<53> × 651871849785243820958119684233142982170536938147349501049955875423544371888519632103159753836519854149224951224466197<117> × 41178786268652727777818909308744042083055383859495037243190192388737567013766369770914385034350354193614379285959651069848809053<128> × [4164693337...<C726>] × 1109999889...<P1513>(72.78%)

Φ3(10) = 111 = 3 × 37(100.00%)
Φ7(10) = 1111111 = 239 × 4649(100.00%)
Φ21(10) = 900900990991<12>= 43 × 1933 × 10838689(100.00%)
Φ127(10) = 1111111111...<127> = 18797 × 90679 × 651871849785243820958119684233142982170536938147349501049955875423544371888519632103159753836519854149224951224466197<117> (100.00%)
Φ381(10) = 9009009009...<252> = 9783462102370544239<19> × 45830377642600710169<20> × 591082109095345424416413908752729<33> × 82548590511975869997227448819484483748713764399174373<53> × 41178786268652727777818909308744042083055383859495037243190192388737567013766369770914385034350354193614379285959651069848809053<128> (100.00%)
Φ889(10) = 9000000900...<756> = 14097763 × 153288409997816784961649<24> × [4164693337...<C726>](4.01%)

2667=3*7*127，共有3,7,127,21,381,889六个因子，2667的φ因子需要（10^2667-1）/9除去它的所有因子的φ因子，剩下的才是φ2667；
φ2667=(10^2667-1)/9除以φ889*φ381*φ21*φ127*φ7*φ3

yangchuanju

尚未分解到底的726位的C726复合因子的具体数值已经计算出，它是：
416469333722106156163101192109746643677898844216882996346805998134183411340677894069477531685180083064517870156337578720004813079357909197476243265019305831696873903382521343474088812374832448075323036858594813393929223740131430658119127394919038732488139054480841500648083325299027072285920493716378249845059122083912746270507391165782783117103995696926128628057684637148865605654192545627070270731409601234913167901686584862168881986306896741708322679532246673743498357756113744962274977271090228331612348843051623541034764508419035982579022019148683439930611238714534644731271066135944103382100661000821459436911357599567930143617193589593317664887234933595123775173285841481650007382226082112234345279761734102113501710893-C 726

yangchuanju

下一个唯一循环周期素数φ因子（49#）对应的清一色数指数是4354，φ4354的位数1861。
4354=2*7*311，4354的因子有2,7,14,311,622,2177；
从R4354中依次除掉φ2,7,14,311,622,2177中的各个因子，最后只剩下一个素因子P1861，它就是φ3454。

该素数各位数字已经算出，它是1099999890000010999998900000109999989000001099999890000010999998900000109999989000001099999890000010999998900000109999989000001099999890000010999998900000109999989000001099999890000010999998900000109999989000001099999890000010999998900000109999989000001099999890000010999998900000109999989000001099999890000010988998901100109889989011001098899890110010988998901100109889989011001098899890110010988998901100109889989011001098899890110010988998901100109889989011001098899890110010988998901100109889989011001098899890110010988998901100109889989011001098899890110010988998901100109889989011001098899890110010988998901100109890099010990098900990109900989009901099009890099010990098900990109900989009901099009890099010990098900990109900989009901099009890099010990098900990109900989009901099009890099010990098900990109900989009901099009890099010990098900990109900989009901099009890099010990098900990109900989009901099009890099010990098900989009901099009890099010990098900990109900989009901099009890099010990098900990109900989009901099009890099010990098900990109900989009901099009890099010990098900990109900989009901099009890099010990098900990109900989009901099009890099010990098900990109900989009901099009890099010990098900990109900989009901099009890110010988998901100109889989011001098899890110010988998901100109889989011001098899890110010988998901100109889989011001098899890110010988998901100109889989011001098899890110010988998901100109889989011001098899890110010988998901100109889989011001098899890110010988998901100109889989011001098899890110010988998901099999890000010999998900000109999989000001099999890000010999998900000109999989000001099999890000010999998900000109999989000001099999890000010999998900000109999989000001099999890000010999998900000109999989000001099999890000010999998900000109999989000001099999890000010999998900000109999989000001099999890000011——P1861

它由22个14位“10999998900000”，22个14位“10988998901100”，1个10位“1098900990”，21个14位“10990098900990”，1个19位“1099009890098900990”，21个14位“10990098900990”,22个14位“10990098901100”，1个10位“1098899890”，22个14位“10999998900000”，再加“11”组成的。
每个小循环节14位的，共6种130个循环节；小节10位、19位、10位、2位的各一个，共1861位。

yangchuanju

φ因子的结构形式
对于清一色正整数111…1而言，都存在一个φ因子，它可能是一个素数，也可能是一个合数；这个φ因子可能是清一色数本身，但一般来说它是清一色数的一个素因子或一个复合因子。
在某种意义上说，φn（10）因子就是循环周期等于n的素数或合数，在10进制下它的倒数循环节长度等于n。
仔细观察n=1—60的φ因子构成表：
Φ1(10) = 9 = 3^2
Φ2(10)= 11 = 11
Φ3(10)= 111 = 3 × 37
Φ4(10)= 101 = 101
Φ5(10)= 11111 = 41 × 271
Φ6(10)= 91 = 7×13
Φ7(10)=1111111=239×4649
Φ8(10)=10001=73×137
Φ9(10)=1001001=3×333667
Φ10(10)=9091=9091
Φ11(10)=11111111111<11>=21649×513239
Φ12(10)=9901=9901
Φ13(10)=1111111111111<13>=53×79×265371653
Φ14(10)=909091=909091
Φ15(10)=90090991=31×2906161
Φ16(10)=100000001=17×5882353
Φ17(10)=11111111111111111<17>=2071723×5363222357<10>
Φ18(10)=999001=19×52579
Φ19(10)=1111111111111111111<19>=1111111111111111111<19>
Φ20L(10)=3541=3541
Φ20M(10)=27961=27961
Φ21(10)=900900990991<12>=43×1933×10838689
Φ22(10)=9090909091<10>=11×23×4093×8779
Φ23(10)=11111111111111111111111<23>=11111111111111111111111<23>
Φ24(10)=99990001=99990001
Φ25(10)=100001000010000100001<21>=21401×25601×182521213001<12>
Φ26(10)=909090909091<12>=859×1058313049<10>
Φ27(10)=1000000001000000001<19>=3×757×440334654777631<15>
Φ28(10)=990099009901<12>=29×281×121499449
Φ29(10)=11111111111111111111111111111<29>=3191×16763×43037×62003×77843839397<11>
Φ30(10)=109889011=211×241×2161
Φ31(10)=1111111111111111111111111111111<31>=2791×6943319×57336415063790604359<20>
Φ32(10)=10000000000000001<17>=353×449×641×1409×69857
Φ33(10)=90090090090990990991<20>=67×1344628210313298373<19>
Φ34(10)=9090909090909091<16>=103×4013×21993833369<11>
Φ35(10)=900009090090909909099991<24>=71×123551×102598800232111471<18>
Φ36(10)=999999000001<12>=999999000001<12>
Φ37(10)=1111111111111111111111111111111111111<37>=2028119×247629013×2212394296770203368013<22>
Φ38(10)=909090909090909091<18>=909090909090909091<18>
Φ39(10)=900900900900990990990991<24>=900900900900990990990991<24>
Φ40(10)=9999000099990001<16>=1676321×5964848081<10>
Φ41(10)=11111111111111111111111111111111111111111<41>=83×1231×538987×201763709900322803748657942361<30>
Φ42(10)=1098900989011<13>=7×127×2689×459691
Φ43(10)=1111111111111111111111111111111111111111111<43>=173×1527791×1963506722254397<16>×2140992015395526641<19>
Φ44(10)=99009900990099009901<20>=89×1052788969<10>×1056689261<10>
Φ45(10)=999000000999000999999001<24>=238681×4185502830133110721<19>
Φ46(10)=9090909090909090909091<22>=47×139×2531×549797184491917<15>
Φ47(10)=11111111111111111111111111111111111111111111111<47>=35121409×316362908763458525001406154038726382279<39>
Φ48(10)=9999999900000001<16>=9999999900000001<16>
Φ49(10)=1000000100000010000001000000100000010000&#172;001<43>=505885997×1976730144598190963568023014679333<34>
Φ50(10)=99999000009999900001<20>=251×5051×78875943472201<14>
Φ51(10)=90090090090090090990990990990991<32>=613×210631×52986961×13168164561429877<17>
Φ52(10)=990099009900990099009901<24>=521×1900381976777332243781<22>
Φ53(10)=11111111111111111111111111111111111111111111111111111<53>=107×1659431×1325815267337711173<19>×47198858799491425660200071<26>
Φ54(10)=999999999000000001<18>=70541929×14175966169<11>
Φ55(10)=9000090000990009900099900999009999099991<40>=1321×62921×83251631×1300635692678058358830121<25>
Φ56(10)=999900009999000099990001<24>=7841×127522001020150503761<21>
Φ57(10)=900900900900900900990990990990990991<36>=21319×10749631×3931123022305129377976519<25>
Φ58(10)=9090909090909090909090909091<28>=59×154083204930662557781201849<27>
Φ59(10)=11111111111111111111111111111111111111111111111111111111111<59>=2559647034361<13>×4340876285657460212144534289928559826755746751<46>
Φ60L(10)=255522961=61×4188901
Φ60M(10)=39526741=39526741
…………
容易发现：
一、若n是素数，则φn就等于清一色本身，它可能是素数（如φ2、φ19、φ23），但一般不是素数（如φ5=11111=41*271，φ7=1111111=239*4649）。
二、若n是奇素数的2倍，则φn就等于清一色数除以φ2*φ(n/2)，如φ6=111111/（11*111）=91，φ10=1111111111/（11*11111）=9091，φ14=11111111111111/（11*1111111）=909091，……；
其结构形式是9090…9091。
三、若n是除3以外的奇素数的3倍，则φn就等于清一色数除以φ3*φ(n/3)，如
φ15=111…1<15>/（111*11111）=90090991，
φ21=111…1<21>/(111*1111111)=900900990991<P12>，
φ33=111…1<33>/(111*111…1<11>)=90090090090990990991<C20>，……
其结构形式是900900…90990…990991（1节3数，中间夹着“90”）或900900…990991（1节3数）。
四、若n是除5以外的奇素数的5倍，则φn就等于清一色数除以φ5*φ(n/5)，如
φ15=111…1<15>/（111*11111）=90090991，
φ35=111…1<35>/（11111*1111111）=900009090090909909099991<24>，
φ55=111…1<55>/（11111*111…1<11>）=9000090000990009900099900999009999099991<40>，……
其结构形式较复杂。
五、若n是奇素数的平方，则φn等于清一色数除以φ(n的平方根)：
φ9=111111111/111=1001001=3*333667，
φ25=111…1<25>/11111=100001000010000100001<21>=21401*25601*182521213001<12>，
φ49=111…1<49>/1111111=1000000100000010000001000000100000010000001<43>=505885997×1976730144598190963568023014679333<34>，
……
六、φ20、φ60、φ100都由两个φ因子构成，一个L，一个M：
Φ20L(10)=3541=3541，    Φ20M(10)=27961=27961
Φ60L(10)=255522961=61×4188901，    Φ60M(10)=39526741=39526741

更多的规律请自行总结归纳！

cz1

不唯一，

yangchuanju

cz1 发表于 2021-5-7 07:59
不唯一，

是说这两个大素数都不是“唯一循环周期素数”吗？我是从网上得出的！
它们分别是循环周期等于2667和4354的“唯一循环周期素数”。

Φ2667(10) = 1109999889...<1513> = 1109999889...<1513> (100.00%)

Φ4354(10) = 1099999890...<1861> = 1099999890...<1861> (100.00%)

太阳

(10^2667-1)/9和(10^4354-1)/9，不是固定数列

太阳

例:9091，909091，90909091，909090909091，，，固定数列