几种特殊的形式的素数

yangchuanju

几种特殊的形式的素数

（一）清一色数：已知的清一色素数有11、1111111111111111111<19>、11111111111111111111111<23>、111…1<317>、111…1<1031>等9个。

（二）101型素数：已知的101型素数只有两个，它们的11和101，有没有第3个101型素数尚不知道。

（三）9901型素数：已知的9901型素数有四个，9901、99990001、999999000001、9999999900000001，有没有更多个9901型素数，尚不知道。

（四）9091型素数：已知的9091型素数有12个，它们是由(10^n+1)/11得到的，前6个是：
9091, 909091, 909090909090909091, 909090909090909090909090909091, 9090909090909090909090909090909090909090909090909091<52>, 909090909090909090909090909090909090909090909090909090909090909091<66>
12个素数对应的指数n等于：5, 7, 19, 31, 53, 67, 293, 641, 2137, 3011, 268207, 1600787；
分别含90和91的个数是：2, 3, 9, 15, 26, 33, 146, 320, 1068, 1505, 134103, 800393。
肯定还有更多个9091型素数。

（五）9001型素数：已知的900.型素数有23个，它们是：
Known (probable) prime numbers
9×10^3+1 = 9001 is prime.
9×10^4+1 = 90001 is prime.
9×10^5+1 = 900001 is prime.
9×10^9+1 = 9000000001<10> is prime.
9×10^22+1 = 9(0)21  1<23> is prime.
9×10^27+1 = 9(0)26  1<28> is prime.
9×10^36+1 = 9(0)35  1<37> is prime.
9×10^57+1 = 9(0)56  1<58> is prime.
9×10^62+1 = 9(0)61  1<63> is prime.
9×10^78+1 = 9(0)77  1<79> is prime.
9×10^201+1 = 9(0)200  1<202> is prime.
9×10^537+1 = 9(0)536  1<538> is prime.
9×10^696+1 = 9(0)695  1<697> is prime.
9×10^790+1 = 9(0)789  1<791> is prime.
9×10^905+1 = 9(0)904  1<906> is prime.
9×10^1038+1 = 9(0)1037  1<1039> is prime. (Harvey Dubner / Cruncher / December 31, 1984 )
9×10^66886+1 = 9(0)66885  1<66887> is prime. (Peter Benson / NewPGen, OpenPFGW, Proth.exe / December 31, 2004 )
9×10^70500+1 = 9(0)70499  1<70501> is prime. (Peter Benson / NewPGen, OpenPFGW, Proth.exe / March 10, 2005 )
9×10^91836+1 = 9(0)91835  1<91837> is prime. (Dmitry Domanov / Prime95 v25.11, pfgw / March 8, 2010 )
9×10^100613+1 = 9(0)100612  1<100614> is prime. (Predrag Kurtovic / September 23, 2013 )
9×10^127240+1 = 9(0)127239  1<127241> is prime. (Bob Price / PFGW / January 22, 2015 )
9×10^380734+1 = 9(0)380733  1<380735> is prime. (Predrag Kurtovic / llr64 / September 18, 2019 )
9×10^583696+1 = 9(0)583695  1<583697> is prime. (Predrag Kurtovic / Srsieve, Prime95, LLR / June 25, 2020 )

yangchuanju

以10为底的φ因子结构形式

只看总的φ因子（素数或合数），其分解式不必关心。
以10为底的复合φ因子，都是以1结尾的整数，Φ1(10) =9除外。
类型一：特例，只有φ1一个，等于3的平方，以下各种类型的φ因子皆无平方因子。
Φ1(10) = 9=3^2

类型二：素数型，皆为清一色数。
Φ2(10)=11=11
Φ3(10)=111=3×37
Φ5(10)=11111=41×271
Φ7(10)=1111111=239×4649
Φ11(10)=11111111111<11>=21649×513239
Φ13(10)=1111111111111<13>=53×79×265371653
Φ17(10)=11111111111111111<17>=2071723×5363222357<10>
Φ19(10)=1111111111111111111<19>=1111111111111111111<p19>
Φ23(10)=11111111111111111111111<23>=11111111111111111111111<p23>
Φ29(10)=11111111111111111111111111111<29>=3191×16763×43037×62003×77843839397<11>
Φ31(10)=1111111111111111111111111111111<31>=2791×6943319×57336415063790604359<20>

类型三：2p型，p为奇素数，9090…91型（k组90加一组91型）。
Φ6(10)=91=7×13
Φ10(10)=9091=9091p
Φ14(10)=909091=909091p
Φ22(10)=9090909091<10>=11×23×4093×8779
Φ26(10)=909090909091<12>=859×1058313049<10>
Φ34(10)=9090909090909091<16>=103×4013×21993833369<11>
Φ38(10)=909090909090909091<18>=909090909090909091<p18>
Φ46(10)=9090909090909090909091<22>=47×139×2531×549797184491917<15>
Φ58(10)=9090909090909090909090909091<28>=59×154083204930662557781201849<27>
Φ62(10)=909090909090909090909090909091<30>=909090909090909090909090909091<p30>

类型四：3p型，p≥5，前半部k组900，后半部k-1组990加一组991；中间有一组90或没有。
Φ15(10)=900 90 991=31×2906161
Φ21(10)=900900 990991<12>=43×1933×10838689
Φ33(10)=900900900 90 990990991<20>=67×1344628210313298373<19>
Φ39(10)=900900900900 990990990991<24>=900900900900990990990991<p24>
Φ51(10)=900900900900900 90 990990990990991<32>=613×210631×52986961×13168164561429877<17>
Φ57(10)=900900900900900900 990990990990990991<36>=21319×10749631×3931123022305129377976519<25>
Φ69(10)=900900900900900900900 90 99099099099099099&#172;0991<44>
=277×203864078068831<15>×1595352086329224644348978893<28>
Φ87(10)=900900900900900900900900900 90 990990990990990990990990991<56>
=4003×72559×310170251658029759045157793237339498342763245483<48>
Φ93(10)=900900900900900900900900900900 990990990990990990990990990991<60>
=900900900900900900900900900900990990990990990990990990990991<p60>

类型五：4p型，p为奇素数，99009900…9901型（k组9900加一组9901型）。
Φ12(10)=9901=9901
Φ20L(10)=3541=3541
Φ20M(10)=27961=27961
Φ20(10)=99009901=3541×27961
Φ28(10)=990099009901<12>=29×281×121499449
Φ44(10)=99009900990099009901<20>=89×1052788969<10>×1056689261<10>
Φ52(10)=990099009900990099009901<24>=521×1900381976777332243781<22>
Φ68(10)=99009900990099009900990099009901<32>
=28559389×1491383821<10>×2324557465671829<16>
Φ76(10)=990099009900990099009900990099009901<36>
=722817036322379041<18>×1369778187490592461<19>
Φ92(10)=99009900990099009900990099009900990099009901<44>
=1289×18371524594609<14>×4181003300071669867932658901<28>

类型六：5p型，p≥7，结构复杂，无统一规律。
Φ35(10)=900009090090909909099991<24>=71×123551×102598800232111471<18>
Φ55(10)=9000090000990009900099900999009999099991<40>
=1321×62921×83251631×1300635692678058358830121<25>
Φ65(10)=900009000090090900909009099090990909909999099991<48>
=162503518711<12>×5538396997364024056286510640780600481<37>
Φ85(10)=9000090000900009090090900909009090990909909099090999909999099991<64>
=262533041×8119594779271<13>×4222100119405530170179331190291488789678&#172;081<43>
Φ95(10)=900009000090000900099000990009900099009990099900999009990999909999099991<72>
=191×59281×63841×1289981231950849543985493631<28>×965194617121640791456070347951751<33>

类型七：6p型，内有多个6位小循环节109891，其中109+891=999；尾部两位是11，中部有不完全循环节。
Φ30(10)=109889011=211×241×2161
Φ42(10)=1098900989011<13>=7×127×2689×459691
Φ66(10)=109890109889010989011<21>=599144041×183411838171<12>
Φ78(10)=1098901098900989010989011<25>=13×157×6397×216451×388847808493<12>
Φ102(10)=109890109890109889010989010989011<33>
=291078844423<12>×377526955309799110357<21>
Φ114(10)=1098901098901098900989010989010989011<37>
=1458973×753201806271328462547977919407<30>
Φ138(10)=109890109890109890109889010989010989010989011<45>
=31051×143574021480139<15>×24649445347649059192745899<26>

类型八：7p型，p≥11，结构复杂，无统一规律。
Φ77(10)=900000090009009000900990090099009909900990999099099909999991<60>
=5237×42043×29920507×1366146685760023293714964475559157409101&#172;81043<45>
Φ91(10)=900000090000099000009900009990000999000999900099990099999009999909999991<72>
=547×14197×17837×4262077×43442141653<11>×316877365766624209<18>×110742186470530054291318013<27>
Φ119(10)=9000000900...<96>
=923441×3924966376871<13>×768736559421401249042753476963<30>
×3230129421485627516508145444373504546404&#172;48842187<48>
Φ161(10)=9000000900...<132>
=6763×472341157×11273170771131750391<20>×1626777403161656797092007877<28>
×15362898429170396757717888856328974146292496901433891193564055671816191643<74>

类型九：8p型，p为奇素数，9999000099990000…99990001型（k组99990000加一组99990001型）。
Φ24(10)=99990001=99990001
Φ40(10)=9999000099990001<16>=1676321×5964848081<10>
Φ56(10)=999900009999000099990001<24>=7841×127522001020150503761<21>
Φ88(10)=9999000099990000999900009999000099990001<40>
=617×16205834846012967584927082656402106953<38>
Φ104(10)= 999900009999000099990000999900009999000099990001<48>
=1580801×632527440202150745090622412245443923049201<42>
Φ152(10)= 999900009999000099990000999900009999000099990000999900009999000099990001<72>
=457×1403417×5240808656722481737<19>×297478330786365628414805305290302483555043017<45>
Φ184(10)= 9999000099990000999900009999000099990000999900009999000099990000999900009999000099990001<88>
=2393×4178437150016715837818641871709193476807772628503969494400330129962348520684914375257<85>

类型十：10p型，p≥7
Φ70(10)=1099988890111109888900011<25>=4147571×265212793249617641<18>
Φ110(10)= 10999890000989990100100098999009998900011<41>
=331×5171×20163494891<11>×318727841165674579776721<24>
Φ130(10)= 1099989000109888901110988901110988890109998900011<49>
=131×8396862596258693901610602298557167100076327481<46>
Φ170(10)= 10999890001099988890111098889011110988890111098889000109998900011<65>
=87211×787223761×160220794821014452066741918303580917664386555934641<51>
Φ190(10)= 1099989000109998899901000989990100099010009899901000989989000109998900011.<73>
=1812604116731<13>×121450506296081<15>×4996731930447843676185843959746621491531100801<46>

类型十一：30倍数
Φ30(10)=109889011=211×241×2161
Φ60L(10)=255522961=61×4188901
Φ60M(10)=39526741=39526741p
Φ60(10)=10099989899000101<17>
Φ90(10)=1000999998998999000001001<25>=29611×3762091×8985695684401<13>
Φ150(10)= 10000099999999989999899999000000000100001<41>
=10000099999999989999899999000000000100001<p41>

类型十二：p平方型，(p-1)小节，每小节都是1加p-1个0（10…0），末尾再加1。
Φ4(10)=101=101
Φ9(10)=1001001=3×333667
Φ25(10)=100001000010000100001<21>=21401×25601×182521213001<12>
Φ49(10)=1000000100000010000001000000100000010000001<43>
=505885997×1976730144598190963568023014679333<34>
Φ121(10)= 10000000000100000000001000000000010000000000100000000001000000000010000000000
1000000000010000000000100000000001<111>
=15973×38237×274187
×597149176209530412360795391497657340159943421992502538230831481682232969649167277637825641074323<96>
Φ169(10)= 100000000000010000000000001000000000000100000000000010000000000001000000000000
1000000000000100000000000010000000000001000000000000100000000000010000000000001<157>
=11831×2709246550432452192307438140153898730002165574677<49>
×31198234389876128002952782212916694331889770402831700346290124318006436464972357735278842644511160852523<104>

类型十三：p立方型，(p-1)小节，每小节都是1加p^2-1个0（10…0），末尾再加1。
Φ8(10)=10001=73×137
Φ27(10)=1000000001000000001<19>=3×757×440334654777631<15>
Φ125(10)= 100000000000000000000000010000000000000000000000001000000000000000000000000
10000000000000000000000001<101>
=751×1797655751<10>×176144543406001<15>
×42051775804956304559810859008305819975199677041099230574273451704628125001<74>

类型十四：20,60,100,140,180型，都分成L和M两个φ因子
Φ20L(10)=3541=3541
Φ20M(10)=27961=27961
Φ60L(10)=255522961=61×4188901
Φ60M(10)=39526741=39526741
Φ100L(10)=99004980069800499001<20>=7019801×14103673319201<14>
Φ100M(10)=101005020070200501001<21>=60101×1680588011350901<16>
Φ140L(10)=3572835377...<24>=421×848654483879497562821<21>
Φ140M(10)=2826886473...<25>=3471301×13489841×60368344121<11>
Φ180L(10)=1105097795...<25>=1105097795002994798105101<25>
Φ180M(10)=9048981950...<24>=181×4999437541453012143121<22>

类型十五：100以内其它类型
Φ16(10)=100000001=17×5882353
Φ18(10)=999001=19×52579
Φ32(10)=10000000000000001<17>=353×449×641×1409×69857
Φ36(10)=999999000001<12>=999999000001<12>
Φ45(10)=999000000999000999999001<24>=238681×4185502830133110721<19>
Φ48(10)=9999999900000001<16>=9999999900000001<16>
Φ50(10)=99999000009999900001<20>=251×5051×78875943472201<14>
Φ54(10)=999999999000000001<18>=70541929×14175966169<11>
Φ63(10)=999000000999000000999999000999999001<36>=10837×23311×45613×45121231×1921436048294281<16>
Φ64(10)=100000000000000000000000000000001<33>=19841×976193×6187457×834427406578561<15>
Φ72(10)=999999999999000000000001<24>=3169×98641×3199044596370769<16>
Φ75(10)=9999900000000009999900000999999999900001<40>=151×4201×15763985553739191709164170940063151<35>
Φ80(10)=99999999000000009999999900000001<32>=5070721×19721061166646717498359681<26>
Φ81(10)=1000000000000000000000000001000000000000000000000000001<55>
=3×163×9397×2462401×676421558270641<15>×130654897808007778425046117<27>
Φ84(10)=1009998990000999899000101<25>=226549×4458192223320340849<19>
Φ88(10)=9999000099990000999900009999000099990001<40>=617×16205834846012967584927082656402106953<38>
Φ96(10)=99999999999999990000000000000001<32>=97×206209×66554101249<11>×75118313082913<14>
Φ98(10)=9999999000000099999990000000999999900000&#172;01<42>=197×5076141624365532994918781726395939035533<40>
Φ99(10)=9990000009990000009990000009990009999990&#172;00999999000999999001<60>
=199×397×34849×362853724342990469324766235474268869786311886053883<51>

yangchuanju

π素数与e素数
如果圆周率的十进制表达中，前n位恰好组成一个素数，这样的素数就叫做π素数。3、31和314159都是π素数。下一个π素数则有38位。类似的，前三个e素数是2、271、2718281。第四个e素数则有85位。

yangchuanju

e素数
A007512
Primes of the form floor(e*10^k), i.e., formed by concatenation of an initial segment of the decimal expansion of e.
2, 271, 2718281, 2718281828459045235360287471352662497757247093699959574966967627724076630353547594571

A064118
Numbers k such that the first k digits of e form a prime.
1, 3, 7, 85, 1781, 2780, 112280, 155025
e的前1、3、7、85、……位数字是素数。

A001113
Decimal expansion of e.
网页给出了e的前50000位数字
e=2.71828182845904523536028747135266249775724709369995957496696762772407663...

yangchuanju

π素数
A005042
Primes formed by the initial digits of the decimal expansion of Pi.       
3, 31, 314159, 31415926535897932384626433832795028841

A060421
Numbers n such that the first n digits of the decimal expansion of Pi form a prime.
1, 2, 6, 38, 16208, 47577, 78073, 613373
π的前1、2、6、38、……位数字是素数。

A000796
Decimal expansion of Pi (or digits of Pi).
网页给出了π的前20000位数字
π=3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819...

yangchuanju

时钟素数
如果按照顺时针方向读出时钟上的数字，正好得到一个素数，这样的数就叫做“时钟素数”。按照定义，23、67、89、4567 都是时钟素数，还有23456789﹑23456789101112123﹑567891011121234567891011121234567891011等。

yangchuanju

yangchuanju 发表于 2023-4-13 20:06
e素数
A007512
Primes of the form floor(e*10^k), i.e., formed by concatenation of an initial segmen ...

谢谢wlcl先生多次给予好评和加分！

yangchuanju

yangchuanju 发表于 2023-4-13 20:06
π素数
A005042
Primes formed by the initial digits of the decimal expansion of Pi.       

π素数（续）
A282973
Primes in A011546.（π的四舍五入数值表）
3, 31, 314159, 314159265359
圆周率π四舍五入后数值串的前1位、2位、6位、12位数字都是素数。

A047658
Numbers k such that the initial首字母 k digits位 in decimal portion小数点 of Pi form a prime number.
5、12、281、547、6205、16350
圆周率π的小数点后的前5位、12位、281位、……数字都是素数。
在这里不算小数点前的那个3。例13149、141592653589是素数。

衷心谢谢wlc1、cw1两位老师的点赞和加分！