数学求证题

太阳

yangchuanju

不可思议？
10^n+1=100…01，其中已知素数只有2个，即11和101。
在(10^n+1)/9之中含有除2和5以外的所有素因子，包括它们的任意次幂；10^n+1之中的素因子少一些。
太阳先生推出贴认为：(10^5k+1)/(10^k+1)中不含有平方因子，咋看起来问题并不复杂，把10^5k+1分解式中的10^k+1的因子约去，观察剩余因子即可。(10^5k+1)/(10^k+1)=10^4k-10^3k+10^2k-10^k+1

一、复制10^n+1分解表1，仅取指数为5和5的倍数的：
10^5+1 =  11 × 9091(100.00%)
10^10+1 =  101 × 3541 × 27961(100.00%)
10^15+1 =  7 × 11 × 13 × 211 × 241 × 2161 × 9091(100.00%)
10^20+1 =  73 × 137 × 1676321 × 5964848081<10>(100.00%)
10^25+1 =  11 × 251 × 5051 × 9091 × 78875943472201<14>(100.00%)
10^30+1 =  61 × 101 × 3541 × 9901 × 27961 × 4188901 × 39526741(100.00%)
10^35+1 =  11 × 9091 × 909091 × 4147571 × 265212793249617641<18>(100.00%)
10^40+1 =  17 × 5070721 × 5882353 × 19721061166646717498359681<26>(100.00%)
10^45+1 =  7 × 11 × 13 × 19 × 211 × 241 × 2161 × 9091 × 29611 × 52579 × 3762091 × 8985695684401<13>(100.00%)
10^50+1 =  101 × 3541 × 27961 × 60101 × 7019801 × 14103673319201<14> × 1680588011350901<16>(100.00%)
10^55+1 =  11^2 × 23 × 331 × 4093 × 5171 × 8779 × 9091 × 20163494891<11> × 318727841165674579776721<24>(100.00%)
10^60+1 =  73 × 137 × 1676321 × 99990001 × 5964848081<10> × 100009999999899989999000000010001<33>(100.00%)
10^65+1 =  11 × 131 × 859 × 9091 × 1058313049<10> × 8396862596258693901610602298557167100076327481<46>(100.00%)
10^70+1 =  29 × 101 × 281 × 421 × 3541 × 27961 × 3471301 × 13489841 × 121499449 × 60368344121<11> × 848654483879497562821<21>(100.00%)
10^75+1 =  7 × 11 × 13 × 211 × 241 × 251 × 2161 × 5051 × 9091 × 78875943472201<14> × 10000099999999989999899999000000000100001<41>(100.00%)
10^80+1 =  353 × 449 × 641 × 1409 × 69857 × 1634881 × 18453761 × 947147262401<12> × 349954396040122577928041596214187605761<39>(100.00%)
10^85+1 =  11 × 103 × 4013 × 9091 × 87211 × 787223761 × 21993833369<11> × 1602207948210144520667419183035809176643&#172;86555934641<51>(100.00%)
10^90+1 =  61 × 101 × 181 × 3541 × 9901 × 27961 × 4188901 × 39526741 × 999999000001<12> × 4999437541453012143121<22> × 1105097795002994798105101<25>100.00%)
10^95+1 =  11 × 9091 × 1812604116731<13> × 121450506296081<15> × 909090909090909091<18> × 4996731930447843676185843959746621491531&#172;100801<46>(100.00%)
10^100+1 =  73 × 137 × 401 × 1201 × 1601 × 1676321 × 5964848081<10> × 1296944190290577505513857711845642744990-75700947656757821537291527196801<72>(100.00%)

对应删除表中的部分因子
10^1+1 = 11 = 11(100.00%)
10^2+1 = 101 = 101(100.00%)
10^3+1 = 1001 = 7 × 11 × 13(100.00%)
10^4+1 = 10001 = 73 × 137(100.00%)
10^5+1 = 100001 = 11 × 9091(100.00%)
10^6+1 = 1000001 = 101 × 9901(100.00%)
10^7+1 = 10000001 = 11 × 909091(100.00%)
10^8+1 = 100000001 = 17 × 5882353(100.00%)
10^9+1 = 1000000001<10> = 7 × 11 × 13 × 19 × 52579(100.00%)
10^10+1 = 10000000001<11> = 101 × 3541 × 27961(100.00%)
10^11+1 = 100000000001<12> = 11^2 × 23 × 4093 × 8779(100.00%)
10^12+1 = 1000000000001<13> = 73 × 137 × 99990001(100.00%)
10^13+1 = 10000000000001<14> = 11 × 859 × 1058313049<10>&#8195;(100.00%)
10^14+1 = 100000000000001<15> = 29 × 101 × 281 × 121499449(100.00%)
10^15+1 = 1000000000000001<16> = 7 × 11 × 13 × 211 × 241 × 2161 × 9091(100.00%)
10^16+1 = 10000000000000001<17> = 353 × 449 × 641 × 1409 × 69857(100.00%)
10^17+1 = 100000000000000001<18> = 11 × 103 × 4013 × 21993833369<11>(100.00%)
10^18+1 = 1000000000000000001<19> = 101 × 9901 × 999999000001<12>(100.00%)
10^19+1 = 10000000000000000001<20> = 11 × 909090909090909091<18>(100.00%)
10^20+1 = 100000000000000000001<21> = 73 × 137 × 1676321 × 5964848081<10>(100.00%)

即得到：
(10^5+1)/(10^1+1) =  9091
(10^10+1)/(10^2+1) =  3541 × 27961  =  99009901
(10^15+1)/(10^3+1) =  211 × 241 × 2161 × 9091  =  999000999001
(10^20+1)/(10^4+1) =  1676321 × 5964848081<10>  =  9999000099990001
(10^25+1)/(10^5+1) =  251 × 5051 × 78875943472201<14>  =  99999000009999900001
(10^30+1)/(10^6+1) =  61 ×3541 × 27961 × 4188901 × 39526741  =  999999000000999999000001
(10^35+1)/(10^7+1) =  9091 × 4147571 × 265212793249617641<18>  =  9999999000000099999990000001
(10^40+1)/(10^8+1) =  5070721 × 19721061166646717498359681<26>  =  99999999000000009999999900000001
(10^45+1)/(10^9+1) =  211 × 241 × 2161 × 9091 × 29611 × 3762091 × 8985695684401<13>
(10^50+1)/(10^10+1) =  60101 × 7019801 × 14103673319201<14> × 1680588011350901<16>
(10^55+1)/(10^11+1) =  331 × 5171 × 9091 × 20163494891<11> × 318727841165674579776721<24>
(10^60+1)/(10^12+1) =  1676321 × 5964848081<10> × 100009999999899989999000000010001<33>
(10^65+1)/(10^13+1) =  131 × 9091 × 8396862596258693901610602298557167100076327481<46>
(10^70+1)/(10^14+1) =  421 × 3541 × 27961 × 3471301 × 13489841 × 60368344121<11> × 848654483879497562821<21>
(10^75+1)/(10^15+1) =  251 × 5051 × 78875943472201<14> × 1000009999999998999989999900000000010000&#172;1<41>
(10^80+1)/(10^16+1) =  1634881 × 18453761 × 947147262401<12> × 349954396040122577928041596214187605761<39>
(10^85+1)/(10^17+1) =  9091 × 87211 × 787223761 × 1602207948210144520667419183035809176643&#172;86555934641<51>
(10^90+1)/(10^18+1) =  61 × 181 × 3541 × 27961 × 4188901 × 39526741 × 4999437541453012143121<22> × 1105097795002994798105101<25>
(10^95+1)/(10^19+1) =  9091 × 1812604116731<13> × 121450506296081<15> × 4996731930447843676185843959746621491531&#172;100801<46>
(10^100+1)/(10^20+1) =  401 × 1201 × 1601 × 1296944190290577505513857711845642744990-75700947656757821537291527196801<72>
……

yangchuanju

二、统计分析
1、本部分数表中当指数为55和11时都含有11的平方因子，相除后被约去，属于正常；
2、剩余各因子相乘得9091,99009901,999000999001，……积的数字构成整齐划一。

3、在第1页分解式表中找不到平方因子，扩大查找范围继续找。
然而发现当10^n+1的指数是5k并含11的平方因子时，指数k的分解式中也含11的平方因子；
指数是5k并含11的立方因子时，指数k的分解式中也含11的立方因子；
指数是5k并含11的四次方因子时，指数k的分解式中也含11的四次方因子；……
相除后不再含有素数11的平方因子。
初步认为，素数11是一个特殊数，它的倒数循环节是2位，是10的约数。改换成其它素数就是了。

4、将检验素数换成7、13试验，所得结论与素数11 完全相同，素数7和13的都是循环节都是6位，不是10的约数，为什么也不行？
再将检验素数换成17、19和23，试验结果与11仍然相同，即
当10^n+1的指数是5k并含p的平方因子时，指数k的分解式中也含同一个p的平方因子；
指数是5k并含p的立方因子时，指数k的分解式中也含同一个p的立方因子；
指数是5k并含p的四次方因子时，指数k的分解式中也含同一个p的四次方因子；……
相除后找不到含有素数p平方因子的数字。
需要说明地是，笔者仅检验了有限的几个素数（11,7,13,17,19,23）；其它更多的素数怎么样？

太阳先生是否进行了理论证明？

太阳

找到反例吗？a=？，有反例存在

yangchuanju

太阳 发表于 2021-4-22 12:00
找到反例吗？a=？，有反例存在

太阳先生还是错了吧？
观察《不可思议？》贴，从相除后剩余因子表看，不仅没有某素数的平方因子，而且根本没有7、11、13、17、19、23等素因子，该表中的最小素因子是61，于是我从素数61着手重新检索。
素因子61最早出现在（10^n+1）分解式数表的n=30、90、150、210……处，在n=1830、5490、9150、12810……处先后出现了61的平方因子；除9150/5=1830，在9150和1830处各有一个61的平方因子外，在1830/5=366、5490/5=1098、12810/5=2562处都没有素因子61。

下面是10^1830+1、10^366+1和(10^1830+1) / (10^366+1)分解式表：
10^1830+1 = 1000000000...<1831> = 61^2 × 101 × 3541 × 6101 × 7321 × 9901 × 21961 × 27961 × 51241 × 76129 × 190321 × 390889 × 1587221 × 4188901 × 4801921 × 9818561 × 39526741 × 2834820061<10> × 3812801341<10> × 81183810541<11> × 217345835281<12> × 555818110301<12> × 28474644365651641<17> × 8950221294967070861<19> × 17751033585336286181<20> × 17716886277230798340041<23> × 2418666002101476599410229029<28> × 101444162656037151745878558385892753596849<42> × 30177150878514090521547663054628235944221777770161<50> × 39069669288697789469488625615834711944836425801981<50> × 27186363592392725942593454290345801336551729326489701011779461<62> × 75743388768260974116327848920184337528059461788181539337429709<62> × 29396948928240214513043372181142292774711501573237257023101752921<65> × 1305704182448226909845888790598975114740359397093421223355030629345156521<73> × 1087990925195463452802538049940594096989130726971208253428665898558089768698552971751226781442226809<100> × 1642269048916396301246801744101379821902073519810571143520576971928815724659633161556699837755221303434764527770131486856721570324295821<136> × [8763343328...<474>] × 2529932837...<481>(74.10%) （37个素因子）

10^366+1 = 1000000000...<367> = 101 × 9901 × 76129 × 190321 × 390889 × 1587221 × 4801921 × 2834820061<10> × 3812801341<10> × 81183810541<11> × 2418666002101476599410229029<28> × 101444162656037151745878558385892753596849<42> × 75743388768260974116327848920184337528059461788181539337429709<62> × 1305704182448226909845888790598975114740359397093421223355030629345156521<73> × 1087990925195463452802538049940594096989130726971208253428665898558089768698552971751226781442226809<100>(100.00%)  (15个素因子)

（10^1830+1）/（10^366+1） =  61^2  × 3541 × 6101 × 7321 × 21961 × 27961 × 51241 × 4188901 × 9818561 × 39526741 × 217345835281<12> × 555818110301<12> × 28474644365651641<17> × 8950221294967070861<19> × 17751033585336286181<20> × 17716886277230798340041<23>  × 30177150878514090521547663054628235944221777770161<50> × 39069669288697789469488625615834711944836425801981<50> × 27186363592392725942593454290345801336551729326489701011779461<62> × 29396948928240214513043372181142292774711501573237257023101752921<65> × 1642269048916396301246801744101379821902073519810571143520576971928815724659633161556699837755221303434764527770131486856721570324295821<136> × [8763343328...<474>] × 2529932837...<481>(74.10%) （22个素因子）
对于素数131、181、211、241、251等也一定存在它们的平方因子！
有平方因子的最小素数是谁？有多少个素数没有平方因子，或者没有因子？待进一步探讨！

yangchuanju

注意到了吗？《不可思议》贴中的剩余各因子末尾数全是1，这里剩余的因子末尾数也全是1！为什么？仍然是不可思议的。

yangchuanju

(10^5k+1)/(10^k+1)=10^4k-10^3k+10^2k-10^k+1中的有效因子
经观察发现(10^5k+1)/(10^k+1)是99009901型正整数，其素因子尾数都是1。为什么？

现复制(10^n-1)/9分解式数表第1页第1-20行，复制10^n+1分解式数表的第1页第1-10行：
R1 = (10^1-1)/9 = 1
R2 = (10^2-1)/9 = 11 = 11(100.00%)
R3 = (10^3-1)/9 = 111 = 3 × 37(100.00%)
R4 = (10^4-1)/9 = 1111 = 11 × 101(100.00%)
R5 = (10^5-1)/9 = 11111 = 41 × 271(100.00%)
R6 = (10^6-1)/9 = 111111 = 3 × 7 × 11 × 13 × 37(100.00%)
R7 = (10^7-1)/9 = 1111111 = 239 × 4649(100.00%)
R8 = (10^8-1)/9 = 11111111 = 11 × 73 × 101 × 137(100.00%)
R9 = (10^9-1)/9 = 111111111 = 32 × 37 × 333667(100.00%)
R10 = (10^10-1)/9 = 1111111111<10> = 11 × 41 × 271 × 9091(100.00%)
R11 = (10^11-1)/9 = 11111111111<11> = 21649 × 513239(100.00%)
R12 = (10^12-1)/9 = 111111111111<12> = 3 × 7 × 11 × 13 × 37 × 101 × 9901(100.00%)
R13 = (10^13-1)/9 = 1111111111111<13> = 53 × 79 × 265371653(100.00%)
R14 = (10^14-1)/9 = 11111111111111<14> = 11 × 239 × 4649 × 909091(100.00%)
R15 = (10^15-1)/9 = 111111111111111<15> = 3 × 31 × 37 × 41 × 271 × 2906161(100.00%)
R16 = (10^16-1)/9 = 1111111111111111<16> = 11 × 17 × 73 × 101 × 137 × 5882353(100.00%)
R17 = (10^17-1)/9 = 11111111111111111<17> = 2071723 × 5363222357<10>&#8195;(100.00%)
R18 = (10^18-1)/9 = 111111111111111111<18> = 3^2 × 7 × 11 × 13 × 19 × 37 × 52579 × 333667(100.00%)
R19 = (10^19-1)/9 = 1111111111111111111<19> = 1111111111111111111<19>(100.00%)
R20 = (10^20-1)/9 = 11111111111111111111<20> = 11 × 41 × 101 × 271 × 3541 × 9091 × 27961(100.00%)

10^1+1 = 11 = 11(100.00%)
10^2+1 = 101 = 101(100.00%)
10^3+1 = 1001 = 7 × 11 × 13(100.00%)
10^4+1 = 10001 = 73 × 137(100.00%)
10^5+1 = 100001 = 11 × 9091(100.00%)
10^6+1 = 1000001 = 101 × 9901(100.00%)
10^7+1 = 10000001 = 11 × 909091(100.00%)
10^8+1 = 100000001 = 17 × 5882353(100.00%)
10^9+1 = 1000000001<10> = 7 × 11 × 13 × 19 × 52579(100.00%)
10^10+1 = 10000000001<11> = 101 × 3541 × 27961(100.00%)

经观察发现(10^n-1)/9分解式数表中R1、R3、R5、R7、R9……中的素因子3、37、41、271、239、4649、333667、21649、513239、53、79、265371653、31、2906161……均不出现在10^k+1分解式数表中，
10^k+1分解式数表中的数字仅可以是11、101、7、13、73、137、9091、9901、909091、17、5882353、19、52579、3541、27961……等素数，
10^k+1分解式数表中的数字末尾数不全是1，也不包括全部末尾数是1的全部素数。

yangchuanju

这一规则与清一色正整数的φ因子密切相关，现再复制清一色正整数φ因子的第1页，并按奇数行和偶数行分成两表：
奇数行表，10^n+1分解式数表中不会出现的素因子：（两等号之间的数字为φ因子总数，一般是合数；第2等号后的数字都是素因子）
Φ1(10)=9=3^2
Φ3(10)=111=3×37
Φ5(10)=11111=41×271
Φ7(10)=1111111=239×4649
Φ9(10)=1001001=3×333667
Φ11(10)=11111111111<11>=21649×513239
Φ13(10)=1111111111111<13>=53×79×265371653
Φ15(10)=90090991=31×2906161
Φ17(10)=11111111111111111<17>=2071723×5363222357<10>
Φ19(10)=1111111111111111111<19>=1111111111111111111<19>
Φ21(10)=900900990991<12>=43×1933×10838689
Φ23(10)=11111111111111111111111<23>=11111111111111111111111<23>
Φ25(10)=100001000010000100001<21>=21401×25601×182521213001<12>
Φ27(10)=1000000001000000001<19>=3×757×440334654777631<15>
Φ29(10)=11111111111111111111111111111<29>=3191×16763×43037×62003×77843839397<11>
Φ31(10)=1111111111111111111111111111111<31>=2791×6943319×57336415063790604359<20>
Φ33(10)=90090090090990990991<20>=67×1344628210313298373<19>
Φ35(10)=900009090090909909099991<24>=71×123551×102598800232111471<18>
Φ37(10)=1111111111111111111111111111111111111<37>=2028119×247629013×2212394296770203368013<22>
Φ39(10)=900900900900990990990991<24>=900900900900990990990991<24>
Φ41(10)=11111111111111111111111111111111111111111<41>=83×1231×538987×201763709900322803748657942361<30>
Φ43(10)=1111111111111111111111111111111111111111111<43>=173×1527791×1963506722254397<16>×2140992015395526641<19>
Φ45(10)=999000000999000999999001<24>=238681×4185502830133110721<19>
Φ47(10)=11111111111111111111111111111111111111111111111<47>=35121409×316362908763458525001406154038726382279<39>
Φ49(10)=1000000100000010000001000000100000010000001<43>=505885997×1976730144598190963568023014679333<34>
Φ51(10)=90090090090090090990990990990991<32>
=613×210631×52986961×13168164561429877<17>
Φ53(10)=11111111111111111111111111111111111111111111111111111<53>
=107×1659431×1325815267337711173<19>×47198858799491425660200071<26>
Φ55(10)=9000090000990009900099900999009999099991<40>
=1321×62921×83251631×1300635692678058358830121<25>
Φ57(10)=900900900900900900990990990990990991<36>
=21319×10749631×3931123022305129377976519<25>
Φ59(10)=11111111111111111111111111111111111111111111111111111111111<59>
=2559647034361<13>×4340876285657460212144534289928559826755746751<46>
Φ61(10)=1111111111111111111111111111111111111111111111111111111111111<61>
=733×4637×329401×974293×1360682471<10>×106007173861643<15>×7061709990156159479<19>
Φ63(10)=999000000999000000999999000999999001<36>
=10837×23311×45613×45121231×1921436048294281<16>
Φ65(10)=900009000090090900909009099090990909909999099991<48>
=162503518711<12>×5538396997364024056286510640780600481<37>
Φ67(10)=1111111111111111111111111111111111111111111111111111111111111111111<67>
=493121×79863595778924342083<20>×2821338094317666700126315366099917724567&#172;7<41>
Φ69(10)=90090090090090090090090990990990990990990991<44>
=277×203864078068831<15>×1595352086329224644348978893<28>
Φ71(10)=11111111111111111111111111111111111111111111111111111111111111111111111<71>
=241573142393627673576957439049<30>×45994811347886846310221728895223034301839<41>
Φ73(10)=1111111111111111111111111111111111111111111111111111111111111111111111111<73>
=12171337159<11>×1855193842151350117<19>×49207341634646326934001739482502131487446637<44>
Φ75(10)=9999900000000009999900000999999999900001<40>
=151×4201×15763985553739191709164170940063151<35>
Φ77(10)=900000090009009000900990090099009909900990999099099909999991<60>
=5237×42043×29920507×136614668576002329371496447555915740910181043<45>
Φ79(10)=1111111111111111111111111111111111111111111111111111111111111111111111111111111<79>
=317×6163×10271×307627×49172195536083790769<20>×3660574762725521461527140564875080461079917<43>
Φ81(10)=1000000000000000000000000001000000000000000000000000001<55>
=3×163×9397×2462401×676421558270641<15>×130654897808007778425046117<27>
Φ83(10)=11111111111111111111111111111111111111111111111111111111111111111111111111111111111<83>
=3367147378267<13>×9512538508624154373682136329<28>×346895716385857804544741137394505425384477<42>
Φ85(10)=9000090000900009090090900909009090990909909099090999909999099991<64>
=262533041×8119594779271<13>×4222100119405530170179331190291488789678081<43>
Φ87(10)=90090090090090090090090090090990990990990990990990990991<56>
=4003×72559×310170251658029759045157793237339498342763245483<48>
Φ89(10)=11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111<89>
=497867×103733951×104984505733<12>×5078554966026315671444089<25>×403513310222809053284932818475878953159<39>
Φ91(10)=900000090000099000009900009990000999000999900099990099999009999909999991<72>
=547×14197×17837×4262077×43442141653<11>×316877365766624209<18>×110742186470530054291318013<27>
Φ93(10)=900900900900900900900900900900990990990990990990990990990991<60>
=900900900900900900900900900900990990990990990990990990990991<60>
Φ95(10)=900009000090000900099000990009900099009990099900999009990999909999099991<72>
=191×59281×63841×1289981231950849543985493631<28>×965194617121640791456070347951751<33>
Φ97(10)=1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111<97>
=12004721×846035731396919233767211537899097169<36>×109399846855370537540339266842070119107662296580348039<54>
Φ99(10)=999000000999000000999000000999000999999000999999000999999001<60>
=199×397×34849×362853724342990469324766235474268869786311886053883<51>

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偶数行表，10^n+1分解式数表中可能出现的素因子：（两等号之间的数字为φ因子总数，一般是合数；第2等号后的数字都是素因子）
Φ2(10)=11=11
Φ4(10)=101=101
Φ6(10)=91=7×13
Φ8(10)=10001=73×137
Φ10(10)=9091=9091
Φ12(10)=9901=9901
Φ14(10)=909091=909091
Φ16(10)=100000001=17×5882353
Φ18(10)=999001=19×52579
Φ20L(10)=3541=3541
Φ20M(10)=27961=27961
Φ22(10)=9090909091<10>=11×23×4093×8779
Φ24(10)=99990001=99990001
Φ26(10)=909090909091<12>=859×1058313049<10>
Φ28(10)=990099009901<12>=29×281×121499449
Φ30(10)=109889011=211×241×2161
Φ32(10)=10000000000000001<17>=353×449×641×1409×69857
Φ34(10)=9090909090909091<16>=103×4013×21993833369<11>
Φ36(10)=999999000001<12>=999999000001<12>
Φ38(10)=909090909090909091<18>=909090909090909091<18>
Φ40(10)=9999000099990001<16>=1676321×5964848081<10>
Φ42(10)=1098900989011<13>=7×127×2689×459691
Φ44(10)=99009900990099009901<20>=89×1052788969<10>×1056689261<10>
Φ46(10)=9090909090909090909091<22>=47×139×2531×549797184491917<15>
Φ48(10)=9999999900000001<16>=9999999900000001<16>
Φ50(10)=99999000009999900001<20>=251×5051×78875943472201<14>
Φ52(10)=990099009900990099009901<24>=521×1900381976777332243781<22>
Φ54(10)=999999999000000001<18>=70541929×14175966169<11>
Φ56(10)=999900009999000099990001<24>=7841×127522001020150503761<21>
Φ58(10)=9090909090909090909090909091<28>=59×154083204930662557781201849<27>
Φ60L(10)=255522961=61×4188901
Φ60M(10)=39526741=39526741
Φ62(10)=909090909090909090909090909091<30>=909090909090909090909090909091<30>
Φ64(10)=100000000000000000000000000000001<33>=19841×976193×6187457×834427406578561<15>
Φ66(10)=109890109889010989011<21>=599144041×183411838171<12>
Φ68(10)=99009900990099009900990099009901<32>=28559389×1491383821<10>×2324557465671829<16>
Φ70(10)=1099988890111109888900011<25>=4147571×265212793249617641<18>
Φ72(10)=999999999999000000000001<24>=3169×98641×3199044596370769<16>
Φ74(10)=909090909090909090909090909090909091<36>=7253×422650073734453<15>×296557347313446299<18>
Φ76(10)=990099009900990099009900990099009901<36>=722817036322379041<18>×1369778187490592461<19>
Φ78(10)=1098901098900989010989011<25>=13×157×6397×216451×388847808493<12>
Φ80(10)=99999999000000009999999900000001<32>=5070721×19721061166646717498359681<26>
Φ82(10)=9090909090909090909090909090909090909091<40>=2670502781396266997<19>×3404193829806058997303<22>
Φ84(10)=1009998990000999899000101<25>=226549×4458192223320340849<19>
Φ86(10)=909090909090909090909090909090909090909091<42>=57009401×2182600451<10>×7306116556571817748755241<25>
Φ88(10)=9999000099990000999900009999000099990001<40>=617×16205834846012967584927082656402106953<38>
Φ90(10)=1000999998998999000001001<25>=29611×3762091×8985695684401<13>
Φ92(10)=99009900990099009900990099009900990099009901<44>=1289×18371524594609<14>×4181003300071669867932658901<28>
Φ94(10)=9090909090909090909090909090909090909090909091<46>=6299×4855067598095567<16>×297262705009139006771611927<27>
Φ96(10)=99999999999999990000000000000001<32>=97×206209×66554101249<11>×75118313082913<14>
Φ98(10)=999999900000009999999000000099999990000001<42>=197×5076141624365532994918781726395939035533<40>
Φ100L(10)=99004980069800499001<20>=7019801×14103673319201<14>
Φ100M(10)=101005020070200501001<21>=60101×1680588011350901<16>

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(10^5k+1)/(10^k+1)分解式只能是下列φ因子中的素数：（偶数行表中的10和10的倍数行）
Φ10(10)=9091=9091
Φ20L(10)=3541=3541
Φ20M(10)=27961=27961
Φ30(10)=109889011=211×241×2161
Φ40(10)=9999000099990001<16>=1676321×5964848081<10>
Φ50(10)=99999000009999900001<20>=251×5051×78875943472201<14>
Φ60L(10)=255522961=61×4188901
Φ60M(10)=39526741=39526741
Φ70(10)=1099988890111109888900011<25>=4147571×265212793249617641<18>
Φ80(10)=99999999000000009999999900000001<32>=5070721×19721061166646717498359681<26>
Φ90(10)=1000999998998999000001001<25>=29611×3762091×8985695684401<13>
Φ100L(10)=99004980069800499001<20>=7019801×14103673319201<14>
Φ100M(10)=101005020070200501001<21>=60101×1680588011350901<16>
更多的(10^5k+1)/(10^k+1)分解式中的素因子，请查看φ因子表的后续个页（共3000页）