寻找大素数

太阳

太阳

大素数：(10^250000+1)/(10^50000+1)
(10^625000+1)/(10^125000+1)
(10^5000000+1)/(10^1000000+1)

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

太阳 发表于 2021-4-18 22:04
大素数：(10^250000+1)/(10^50000+1)
(10^625000+1)/(10^125000+1)
(10^5000000+1)/(10^1000000+1)

太阳先生给定的3个代数式无现成数表可查对，现将分子、分母指数各缩小100倍，算一算看看：
（一）(10^2500+1)/(10^500+1)：
10^2500+1=1000000000...<2501>=73×137×401×1201×1601×24001×1378001×1676321×130290001×5964848081<10>×469620340001<12>×2336633447080001<16>×485632702361659071076259666970001<33>×129694419029057750551385771184564274499075700947656757821537291527196801<72>×3023577677...<390>×[1440268626...<1933>](22.71%)
10^500+1=1000000000...<501>=73×137×401×1201×1601×24001×1378001×1676321×5964848081<10>×129694419029057750551385771184564274499075700947656757821537291527196801<72>×3023577677...<390>(100.00%)
相除后(10^2500+1)/(10^500+1) =130290001×469620340001<12>×2336633447080001<16>×485632702361659071076259666970001<33>×[1440268626...<1933>]

（二）(10^6250+1)/(10^1250+1)：
10^6250+1=1000000000...<6251>=101×3541×4001×27961×37501×60101×76001×670001×1610501×7019801×756100001×170774437501<12>×604991362501<12>×630240900001<12>×14103673319201<14>×26216896777501<14>×1680588011350901<16>×3701291672812501<16>×339320469104212501<18>×11727340168618688631353490001<29>×7355683307782160268476171662501<31>×32886082501657187247904557195788749020689459942486319578774270656490822624175142646095920001<92>×6209247929687718293872530970176361270188593487368651121607499902204345107547278766048583639501<94>×[4854491469...<453>]×[9291173132...<477>]×[8758676893...<2431>]×[7204499923...<2480>](6.55%)
10^1250+1=1000000000...<1251>=101×3541×4001×27961×60101×76001×670001×1610501×7019801×170774437501<12>×630240900001<12>×14103673319201<14>×26216896777501<14>×1680588011350901<16>×11727340168618688631353490001<29>×32886082501657187247904557195788749020689459942486319578774270656490822624175142646095920001<92>×6209247929687718293872530970176361270188593487368651121607499902204345107547278766048583639501<94>×[4854491469...<453>]×[9291173132...<477>](25.63%)
相除后(10^6250+1)/(10^1250+1) =37501×756100001×604991362501<12>×3701291672812501<16>×339320469104212501<18>×7355683307782160268476171662501<31>×[8758676893...<C2431>]×[7204499923...<C2480>]

（三）(10^50000+1)/(10^10000+1)：
10^50000+1=1000000000...<50001>=353×449×641×1409×4801×43201×69857×544001×980801×1634881×12600001×18453761×188080001×245832001×24657552001<11>×66489400001<11>×947147262401<12>×37750052632001<14>×10667937242260969298800001<26>×349954396040122577928041596214187605761<39>×[4915793084...<306>]×[8033307043...<1562>]×[4219750778...<7985>]×[1409831253...<39965>](0.37%)
10^10000+1=1000000000...<10001>=353×449×641×1409×4801×43201×69857×544001×980801×1634881×12600001×18453761×188080001×245832001×24657552001<11>×947147262401<12>×37750052632001<14>×349954396040122577928041596214187605761<39>×[4915793084...<306>]×[8033307043...<1562>]×[4219750778...<7985>](1.48%)
相除后(10^50000+1)/(10^10000+1) =66489400001<11>×10667937242260969298800001<26>×[1409831253...<C39965>]

太阳

yangchuanju 发表于 2021-5-6 09:43
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 ...

位数1861，素数大了，不好找到