Recursive Generalized Fermat Prime found
Define p(0)=1;
finding the smallest General Fermat prime in the form p(n+1)[m]=(2*m*p(n))^2+1, m is positive integer:
p(1)[1]=(2*p(0))^2+1=5;
p(2)[1]=(2*p(1))^2+1=101;
p(3)[5]=(2*5*p(2))^2+1=1020101;
p(4)[48]=(2*48*p(3))^2+1=((1020101)*96)^2+1;
p(5)[1]=(2*p(4))^2+1=((((1020101)*96)^2+1)*2)^2+1;
p(6)[30]=(2*30*p(5))^2+1=((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1;
p(7)[85]=(2*85*p(6))^2+1=((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1;
p(8)[935]=(2*935*p(7))^2+1=((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1;
p(9)[528]=(2*528*p(8))^2+1=((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1;
p(10)[2505]=(2*2505*p(9))^2+1=((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1;
p(11)[840]=(2*840*p(10))^2+1=((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1;
p(12)[1190]=(2*1190*p(11))^2+1=((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1;
p(13)[29382]=(2*29382*p(12))^2+1=((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1;
p(14)[25176]=(2*25176*p(13))^2+1=((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1;
p(15)[12685]=(2*12685*p(14))^2+1=((((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1;
p(16)[67852]=(2*67852*p(15))^2+1=((((((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1)*135704)^2+1;
p(17)[299549]=(2*299549*p(16))^2+1=((((((((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1)*135704)^2+1)*599098)^2+1;
p(18)[62406]=(2*62406*p(17))^2+1=((((((((((((((((((((((((((((97929696^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1)*135704)^2+1)*599098)^2+1)*124812)^2+1;
p(4) has database ID 96548 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(n)” p(n+1)
The number
p(n+1)[m]=(2*m*p(n))^2+1
are reformatted by Mathematica to get the short expression.
The final certification code is
#!/bin/sh
./pfgw -l”GF.19.cert” -t -h”p_03″ p_04
./pfgw -l”GF.19.cert” -t -h”p_04″ p_05
./pfgw -l”GF.19.cert” -t -h”p_05″ p_06
./pfgw -l”GF.19.cert” -t -h”p_06″ p_07
./pfgw -l”GF.19.cert” -t -h”p_07″ p_08
./pfgw -l”GF.19.cert” -t -h”p_08″ p_09
./pfgw -l”GF.19.cert” -t -h”p_09″ p_10
./pfgw -l”GF.19.cert” -t -h”p_10″ p_11
./pfgw -l”GF.19.cert” -t -h”p_11″ p_12
./pfgw -l”GF.19.cert” -t -h”p_12″ p_13
./pfgw -l”GF.19.cert” -t -h”p_13″ p_14
./pfgw -l”GF.19.cert” -t -h”p_14″ p_15
./pfgw -l”GF.19.cert” -t -h”p_15″ p_16
./pfgw -l”GF.19.cert” -t -h”p_16″ p_17
./pfgw -l”GF.19.cert” -t -h”p_17″ p_18
The certificate will be posted when done.
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Primality testing ((1020101)*96)^2+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Calling Brillhart-Lehmer-Selfridge with factored part 73.58%
((1020101)*96)^2+1 is prime! (0.0014s+0.0002s)
Primality testing ((((1020101)*96)^2+1)*2)^2+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 98.15%
((((1020101)*96)^2+1)*2)^2+1 is prime! (0.0013s+0.0002s)
Primality testing ((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 11
Calling Brillhart-Lehmer-Selfridge with factored part 94.74%
((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1 is prime! (0.0019s+0.0002s)
Primality testing ((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 96.82%
((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1 is prime! (0.0044s+0.0002s)
Primality testing ((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 97.72%
((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1 is prime! (0.0104s+0.0003s)
Primality testing ((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Calling Brillhart-Lehmer-Selfridge with factored part 98.97%
((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1 is prime! (0.0264s+0.0006s)
Primality testing ((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 99.36%
((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1 is prime! (0.0800s+0.0003s)
Primality testing ((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 11
Calling Brillhart-Lehmer-Selfridge with factored part 99.73%
((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1 is prime! (0.3280s+0.0003s)
Primality testing ((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Calling Brillhart-Lehmer-Selfridge with factored part 99.86%
((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1 is prime! (1.3586s+0.0005s)
Primality testing ((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Calling Brillhart-Lehmer-Selfridge with factored part 99.90%
((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1 is prime! (5.9510s+0.0007s)
Primality testing ((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Calling Brillhart-Lehmer-Selfridge with factored part 99.95%
((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1 is prime! (26.0812s+0.0013s)
Primality testing ((((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 99.98%
((((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1 is prime! (123.3036s+0.0027s)
Primality testing ((((((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1)*135704)^2+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Calling Brillhart-Lehmer-Selfridge with factored part 99.99%
((((((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1)*135704)^2+1 is prime! (521.0483s+0.0065s)
Primality testing ((((((((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1)*135704)^2+1)*599098)^2+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 99.99%
((((((((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1)*135704)^2+1)*599098)^2+1 is prime! (2312.8701s+0.0140s)
Primality testing ((((((((((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1)*135704)^2+1)*599098)^2+1)*124812)^2+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Calling Brillhart-Lehmer-Selfridge with factored part 100.00%
((((((((((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1)*135704)^2+1)*599098)^2+1)*124812)^2+1 is prime! (11355.7110s+0.0294s)