999 111 101 901型整数的相互转换

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>

yangchuanju

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>

yangchuanju

(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）

太阳

太阳

验证好多数据，没有找到反例，不知命题是否正确？

wlc1

太阳 发表于 2021-4-22 17:57
验证好多数据，没有找到反例，不知命题是否正确？

可悲啊！真的可悲！

太阳

wlc1网友，能给出一个反例吗？