质数公式议论

太阳

kanyikan

那个天下第一的昌建又回来了？

wlc1

yangchuanju

依题意，k是偶数，且是3的倍数，这样k只能是6,12,18,24,30……
(10^2k-10^k+1)是(10^3k+1)的一个因子，(10^3k+1)去掉(10^k+1)就得(10^2k-10^k+1)；
(10^62k-10^31k+1)是(10^93k+1)的一个因子，(10^93k+1)去掉(10^31k+1)就得(10^62k-10^31k+1)。
姑且只管分母，不用顾及分子；因为若令分子(10^186a-1)/9中的a=k，分子指数186是6的倍数，将分子进行两次分解后，分母的第1个因子(10^62k-10^31k+1)一定是分子的一个大因子；再将分母的第2个因子(10^2k-10^k+1)转移到分子上，整除总是能够保证的。

k=6，3k=18，31k=186，93k=558；
k=12，3k=36，31k=372，93k=1116；
k=18，3k=54，58k=558，93k=1674；
k=24，3k=72，31k=744，93k=2232；
k=30，3k=90，31k=930，93k=2790；
……

令k=6，有：
10^6+1=1000001=101×9901(100.00%)
10^18+1=100…001<19>=101×9901×999999000001<12>(100.00%)
10^12+1-10^6=999999000001<12>

10^186+1=1000000000...<187>=
101×5209×9901×2049349×139716865184144864008269344660199946429<39>×483128549554512237305554588359039822397307149685578249<54>×1387770812537486172424567765413032718466560432631753219117445147282909788805041<79> (100.00%)
10^558+1=1000000000...<559>=
101×5209×9901×2049349×4967159761<10>×28975286089<11>×999999000001<12>×206251228717780861<18>×139716865184144864008269344660199946429<39>×483128549554512237305554588359039822397307149685578249<54>×1387770812537486172424567765413032718466560432631753219117445147282909788805041<79>×3368744151...<323>(100.00%)
10^372+1-10^186=
4967159761<10>×28975286089<11>×999999000001<12>×206251228717780861<18>×3368744151...<323>

[10^62k+1-10^31k]/[ 10^2k+1-10^k]= [10^372+1-10^186]/[ 10^12+1-10^6]
= 4967159761<10>×28975286089<11>×206251228717780861<18>×3368744151...<323>
不是素数呀！

yangchuanju

令k=12，有：
10^12+1=100…001<13>=73×137×99990001(100.00%)
10^36+1=100…001<37>=73×137×3169×98641×99990001×3199044596370769<16>(100.00%)
10^24+1-10^12=3169×98641×3199044596370769<16>

10^372+1=1000000000...<373>=
73×137×1489×700849×11110153×99990001×5419392721<10>×640543322297<12>×1220725699657<13>×27908132670449<14>×42367299139993<14>×384705444182230291105649<24>×16584440161215846282167330487128069170776821649169<50>×23140616853203983900922551785166946660605063239337678349130406077337<68>×97645954668018846467287180866355758374263120864803042536883990817097<68>×1194053240550935343606131291791034479414833114386116350011658511441777011841<76> (100.00%)
10^1116+1=1000000000...<1117>=
73×137×1489×3169×98641×700849×11110153×99990001×5419392721<10>×640543322297<12>×1220725699657<13>×27908132670449<14>×42367299139993<14>×3199044596370769<16>×384705444182230291105649<24>×16584440161215846282167330487128069170776821649169<50>×23140616853203983900922551785166946660605063239337678349130406077337<68>×97645954668018846467287180866355758374263120864803042536883990817097<68>×1194053240550935343606131291791034479414833114386116350011658511441777011841<76>×1000000000...<721>(100.00%)
10^744+1-10^372=3169×98641×3199044596370769<16>×1000000000...<721>

[10^62k+1-10^31k]/[ 10^2k+1-10^k]= [10^744+1-10^372]/[ 10^24+1-10^12]= 1000000000...<721> ，倒是一个素数。

yangchuanju

令k=18，有：
10^18+1=100…001<19>=101×9901×999999000001<12>(100.00%)
10^54+1=100…001<55>=101×109×9901×153469×999999000001<12>×59779577156334533866654838281<29>(100.00%)
10^36+1-10^18=109×153469×59779577156334533866654838281<29>

10^558+1=1000000000...<559>=
101×5209×9901×2049349×4967159761<10>×28975286089<11>×999999000001<12>×206251228717780861<18>×139716865184144864008269344660199946429<39>×483128549554512237305554588359039822397307149685578249<54>×1387770812537486172424567765413032718466560432631753219117445147282909788805041<79>×3368744151...<323>(100.00%)
10^1674+1=1000000000...<1675>=
101×109×5209×9901×153469×2049349×4967159761<10>×28975286089<11>×999999000001<12>×206251228717780861<18>×59779577156334533866654838281<29>×139716865184144864008269344660199946429<39>×483128549554512237305554588359039822397307149685578249<54>×1387770812537486172424567765413032718466560432631753219117445147282909788805041<79>×3368744151...<323>×[1000000000...<1081>](35.48%)
10^1116+1-10^558=109×153469×59779577156334533866654838281<29>×[1000000000...<1081>](35.48%)

[10^62k+1-10^31k]/[ 10^2k+1-10^k]= [10^1116+1-10^558]/[10^36+1-10^18]= [1000000000...<1081>]，

这个1081位的大数虽未能分解开，但已经确定它不是素数。
奉劝太阳先生，别在苦思苦想、狂费心机了；
死了那条心吧，世上不存在简单的“素数公式”！

yangchuanju

令k=24，有：
10^24+1=100…001<25>=17×5882353×9999999900000001<16>(100.00%)
10^72+1=100…001<73>=17×8929×5882353×9999999900000001<16>×111994624258035614290513943330720125433979169<45>(100.00%)
10^48+1-10^24=8929×111994624258035614290513943330720125433979169<45>

10^744+1=1000000000...<745>=
17×5882353×7573921×789592121167489<15>×9999999900000001<16>×866353530992835497979626513<27>×2212893713887918222645451169509853038253459532292347577233<58>×660604434404260846910281367365053154708765216915398862085932105728073399727051255954484746381837884289669103543541094033006584881480446026721<141>×[1320320095...<474>](36.41%)
10^2232+1=1000000000...<2233>=
17×8929×5882353×7573921×789592121167489<15>×9999999900000001<16>×866353530992835497979626513<27>×111994624258035614290513943330720125433979169<45>×2212893713887918222645451169509853038253459532292347577233<58>×660604434404260846910281367365053154708765216915398862085932105728073399727051255954484746381837884289669103543541094033006584881480446026721<141>×[1320320095...<474>]×[1000000000...<1441>](14.29%)
10^1488+1-10^744=8929×111994624258035614290513943330720125433979169<45>×[1000000000...<1441>]

[10^62k+1-10^31k]/[ 10^2k+1-10^k]= [10^1488+1-10^744]/[10^48+1-10^24]= [1000000000...<1441>]
请太阳先生看清网页上的这个1441位的大数是黄色的，并用方括号括了起来，表明它是一个没有分解开的合数！

yangchuanju

再令k=30，有：
10^30+1=100…001<31>=61×101×3541×9901×27961×4188901×39526741(100.00%)
10^90+1=100…001<91>=61×101×181×3541×9901×27961×4188901×39526741×999999000001<12>×4999437541453012143121<22>×1105097795002994798105101<25>(100.00%)
10^60+1-10^30=181×999999000001<12>×4999437541453012143121<22>×1105097795002994798105101<25>

10^930+1=1000000000...<931>=
61×101×1861×3541×5209×9901×27961×459421×2049349×4188901×39526741×26482637738395021<17>×105966383429834881<18>×306993468333255153181<21>×22390769664507427965690714361<29>×139716865184144864008269344660199946429<39>×37677420311908637019589121589205677966961<41>×295188553361058681059167605734159744284801<42>×483128549554512237305554588359039822397307149685578249<54>×1891014345342089253412148254805092014919223493735583702021<58>×1387770812537486172424567765413032718466560432631753219117445147282909788805041<79>×1329715195291029266573799498844265388019920296298263850406876176165288274222191365801336476705156746969347581<109>×357640999964236257644576374235721183076427881763885411823611483852508851614756267658524373235935767676406423590000357641<120>×2824060999717591075967240892404687941259531205309234679469076729737362327026207286077379271406382377859361759390002824061<121>×2146644890017820932371428594631788567438802210652314972433604192953392329460076667088695100922984711653519922081763954795399076366216530380621<142>(100.00%)

10^2790+1=1000000000...<2791>=
61×101×181×1861×3541×5209×5581×9901×27961×50221×459421×2049349×4188901×39526741×4967159761<10>×28975286089<11>×999999000001<12>×26482637738395021<17>×105966383429834881<18>×206251228717780861<18>×306993468333255153181<21>×4999437541453012143121<22>×1105097795002994798105101<25>×22390769664507427965690714361<29>×139716865184144864008269344660199946429<39>×37677420311908637019589121589205677966961<41>×295188553361058681059167605734159744284801<42>×483128549554512237305554588359039822397307149685578249<54>×1891014345342089253412148254805092014919223493735583702021<58>×1387770812537486172424567765413032718466560432631753219117445147282909788805041<79>×1329715195291029266573799498844265388019920296298263850406876176165288274222191365801336476705156746969347581<109>×357640999964236257644576374235721183076427881763885411823611483852508851614756267658524373235935767676406423590000357641<120>×2824060999717591075967240892404687941259531205309234679469076729737362327026207286077379271406382377859361759390002824061<121>×2146644890017820932371428594631788567438802210652314972433604192953392329460076667088695100922984711653519922081763954795399076366216530380621<142>×3368744151...<323>×[3228508312...<712>]×[1105096689...<721>](48.69%)

10^1860+1-10^930=
181×5581×50221×459421×4967159761<10>×28975286089<11>×999999000001<12>×206251228717780861<18>×4999437541453012143121<22>×1105097795002994798105101<25>×3368744151...<323>×[3228508312...<712>]×[1105096689...<721>]

[10^62k+1-10^31k]/[ 10^2k+1-10^k]= [10^1860+1-10^930]/[10^60+1-10^30]= 5581×50221×459421×4967159761<10>×28975286089<11>×206251228717780861<18>×3368744151...<323>×[3228508312...<712>]×[1105096689...<721>]
这个1861位的大数，虽未完全分解开，但已分解出6个较小的素因子，1个323位的大素因子，还有2个未分解开的合数因子。它不是素数吧！

yangchuanju

太阳先生在（10^62k+1-10^31k）÷（10^2k+1-10^k）中找到了多少个素数，请通融一下好吗？式中k是正整数。

yangchuanju

顶上去，放到一块，让大家一同欣赏！