Recursive prime p(k+1)=m*((n*p(k))^3+1)+1
Define p(0)=1;
p(1)[m=2;n=2]=2*((2*p(0))^3+1)+1=19;
p(2)[m=6;n=4]=6*((4*p(1))^3+1)+1=2633863;
p(3)[m=14;n=1]=14*((1*2633863)^3+1)+1=14*2633863^3+15;
p(4)[m=354;n=74]=354*((74*(14*2633863^3+15 ))^3+1)+1
=354*(1036*2633863^3+1110)^3+355;
p(5)[m=155;n=115]=155*((115*(354*(1036*2633863^3+1110)^3+355 ))^3+1)+1
=155*(40710*(1036*2633863^3+1110)^3+40825)^3+156;
p(6)[m=146;n=629]=146*((629*(155*(40710*(1036*2633863^3+1110)^3+40825)^3+156 ))^3+1)+1
=146*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+147;
p(7)[m=440;n=1754]=440*((1754*(146*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+147 ))^3+1)+1
=440*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+441;
p(8)[m=8385;n=185]=8385*((185*(440*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+441 ))^3+1)+1
=8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386;
p(9)[m=16182;n=2988]=16182*((2988*(8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386 ))^3+1)+1;
p(10)[m=79194;n=97326]=79194*((97326*(16182*((2988*(8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386))^3+1)+1))^3+1)+1;
p(11)=[m=232497;n=176845]=232497*((176845*(79194*((97326*(16182*((2988*(8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386))^3+1)+1))^3+1)+1))^3+1)+1;
p(11) has database ID 94439 in The List of Largest Known Primes Home Page. The direct link is HERE.
These primes are recursively proven using OpenPFGW, by the command
pfgw -t (or tp) -h”p(k)” p_h(k+1); pfgw -t (or tp) -h”p_h(k+1)” p(k+1)
The number
p_h(k+1)=(n*p(k)-1)*(n*p(k))+1=(n*p(k))^2-n*p(k)+1
p(k+1)=m*((n*p(k))^3+1)+1=m*(n*p(k)+1)((n*p(k))^2-n*p(k)+1)+1=m*(n*p(k)+1)*p_h(k+1)+1
are reformatted by Mathematica to get the short expression.
The final certification code is
#!/bin/sh
./pfgw -l”pmtup.11.2.cert” -tp p_03
./pfgw -l”pmtup.11.2.cert” -t -h”p_03″ ph_04
./pfgw -l”pmtup.11.2.cert” -t -h”ph_04″ p_04
./pfgw -l”pmtup.11.2.cert” -t -h”p_04″ ph_05
./pfgw -l”pmtup.11.2.cert” -t -h”ph_05″ p_05
./pfgw -l”pmtup.11.2.cert” -t -h”p_05″ ph_06
./pfgw -l”pmtup.11.2.cert” -t -h”ph_06″ p_06
./pfgw -l”pmtup.11.2.cert” -t -h”p_06″ ph_07
./pfgw -l”pmtup.11.2.cert” -t -h”ph_07″ p_07
./pfgw -l”pmtup.11.2.cert” -t -h”p_07″ ph_08
./pfgw -l”pmtup.11.2.cert” -t -h”ph_08″ p_08
./pfgw -l”pmtup.11.2.cert” -t -h”p_08″ ph_09
./pfgw -l”pmtup.11.2.cert” -t -h”ph_09″ p_09
./pfgw -l”pmtup.11.2.cert” -t -h”p_09″ ph_10
./pfgw -l”pmtup.11.2.cert” -t -h”ph_10″ p_10
./pfgw -l”pmtup.11.2.cert” -t -h”p_10″ ph_11
./pfgw -l”pmtup.11.2.cert” -t -h”ph_11″ p_11
The certificate will be posted when done.
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Proof:
Primality testing 14*2633863^3+15 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 3, base 3+sqrt(3)
Calling Brillhart-Lehmer-Selfridge with factored part 32.84%
14*2633863^3+15 is prime! (0.0030s+0.0002s)
Primality testing (1036*2633863^3+1110)^2-1036*2633863^3-1109 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 45.27%
(1036*2633863^3+1110)^2-1036*2633863^3-1109 is prime! (0.0021s+0.0002s)
Primality testing 354*(1036*2633863^3+1110)^3+355 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 64.35%
354*(1036*2633863^3+1110)^3+355 is prime! (0.0025s+0.0002s)
Primality testing (40710*(1036*2633863^3+1110)^3+40824)*(40710*(1036*2633863^3+1110)^3+40825)+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 7
Calling Brillhart-Lehmer-Selfridge with factored part 48.52%
(40710*(1036*2633863^3+1110)^3+40824)*(40710*(1036*2633863^3+1110)^3+40825)+1 is prime! (0.0053s+0.0003s)
Primality testing 155*(40710*(1036*2633863^3+1110)^3+40825)^3+156 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 7
Calling Brillhart-Lehmer-Selfridge with factored part 65.92%
155*(40710*(1036*2633863^3+1110)^3+40825)^3+156 is prime! (0.0064s+0.0002s)
Primality testing (97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98123)*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Calling Brillhart-Lehmer-Selfridge with factored part 49.35%
(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98123)*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)+1 is prime! (0.0197s+0.0003s)
Primality testing 146*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+147 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Calling Brillhart-Lehmer-Selfridge with factored part 66.47%
146*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+147 is prime! (0.0403s+0.0002s)
Primality testing (256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257837)*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 49.74%
(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257837)*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)+1 is prime! (0.1451s+0.0004s)
Primality testing 440*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+441 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Calling Brillhart-Lehmer-Selfridge with factored part 66.58%
440*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+441 is prime! (0.3270s+0.0003s)
Primality testing (81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81584)*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 7
Calling Brillhart-Lehmer-Selfridge with factored part 49.94%
(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81584)*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)+1 is prime! (1.2551s+0.0005s)
Primality testing 8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 66.62%
8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386 is prime! (3.1125s+0.0005s)
Primality testing ((50108760*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+50114735)^2+3)/4 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 49.97%
((50108760*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+50114735)^2+3)/4 is prime! (14.1167s+0.0008s)
Primality testing 16182*((2988*(8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386))^3+1)+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Calling Brillhart-Lehmer-Selfridge with factored part 66.65%
16182*((2988*(8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386))^3+1)+1 is prime! (31.6051s+0.0015s)
Primality testing ((194652*(16182*((2988*(8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386))^3+1)+1)-1)^2+3)/4 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 49.99%
((194652*(16182*((2988*(8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386))^3+1)+1)-1)^2+3)/4 is prime! (134.7658s+0.0036s)
Primality testing 79194*((97326*(16182*((2988*(8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386))^3+1)+1))^3+1)+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 66.66%
79194*((97326*(16182*((2988*(8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386))^3+1)+1))^3+1)+1 is prime! (351.1452s+0.0066s)
Primality testing ((353690*(79194*((97326*(16182*((2988*(8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386))^3+1)+1))^3+1)+1)-1)^2+3)/4 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 50.00%
((353690*(79194*((97326*(16182*((2988*(8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386))^3+1)+1))^3+1)+1)-1)^2+3)/4 is prime! (1458.8072s+0.0134s)
Primality testing 232497*((176845*(79194*((97326*(16182*((2988*(8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386))^3+1)+1))^3+1)+1))^3+1)+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Calling Brillhart-Lehmer-Selfridge with factored part 66.66%
232497*((176845*(79194*((97326*(16182*((2988*(8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386))^3+1)+1))^3+1)+1))^3+1)+1 is prime! (3356.1599s+0.0239s)