素数公式判断大素数

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

(10^p+1)/11分解式（第1数字是2p，随后是素因子）
φ2p(10)的φ因子型，p为奇素数，乘积为9090…91型整数

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>

yangchuanju

再送太阳先生一颗尚未发射的炸弹——
(10^7207+1)/11=259453*4799863*85229983*8565000143...43<7186>
该数的4个因子都是模18余1的，其中第4个因子为复合因子（合数）！