Recursive prime p(k+1)=m*((n*p(k))^3+1)+1 base 12^9*5^5^5+7
Define p(0)=12^9*5^5^5+7;
p(1)[m=466;n=78]=466*((78*(12^9*5^3125+7))^3+1)+1;
p(2)[m=6470;n=884]=6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1;
p(3)[m=278822;n=33410]=278822*((33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1;
p(4)[m=145950;n=46953]=145950*((46953*( 278822*((33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1))^3+1)+1;
p(4) has database ID 96540 in The List of Largest Known Primes Home Page. The direct link is HERE.
The kernel 12^9*5^5^5+1 is proven by Primo. The certificate is in the first reply of this post.
The recursive primes are 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_5.4.2.cert” -t -h”p_00″ ph_01
./pfgw -l”pmtup_5.4.2.cert” -t -h”ph_01″ p_01
./pfgw -l”pmtup_5.4.2.cert” -t -h”p_01″ ph_02
./pfgw -l”pmtup_5.4.2.cert” -t -h”ph_02″ p_02
./pfgw -l”pmtup_5.4.2.cert” -t -h”p_02″ ph_03
./pfgw -l”pmtup_5.4.2.cert” -t -h”ph_03″ p_03
./pfgw -l”pmtup_5.4.2.cert” -t -h”p_03″ ph_04
./pfgw -l”pmtup_5.4.2.cert” -t -h”ph_04″ p_04
The certificate will be posted when done.
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ph_01
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(78*(12^9*5^3125+7)-1)*(78*(12^9*5^3125+7))+1
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ph_02
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(884*(466*((78*(12^9*5^3125+7))^3+1)+1)-1)*(884*(466*((78*(12^9*5^3125+7))^3+1)+1))+1
::::::::::::::
ph_03
::::::::::::::
(33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1)-1)*(33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))+1
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ph_04
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(46953*(278822*((33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1)-1)*(46953*(278822*((33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1))+1
::::::::::::::
p_00
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12^9*5^3125+7
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p_01
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466*((78*(12^9*5^3125+7))^3+1)+1
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p_02
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6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1
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p_03
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278822*((33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1
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p_04
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145950 *( ( 46953 *( 278822*((33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1))^3+1)+1
Certificate:
Primality testing (78*(12^9*5^3125+7)-1)*(78*(12^9*5^3125+7))+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 49.96%
(78*(12^9*5^3125+7)-1)*(78*(12^9*5^3125+7))+1 is prime! (1.6044s+0.0005s)
Primality testing 466*((78*(12^9*5^3125+7))^3+1)+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 66.64%
466*((78*(12^9*5^3125+7))^3+1)+1 is prime! (3.7508s+0.0005s)
Primality testing (884*(466*((78*(12^9*5^3125+7))^3+1)+1)-1)*(884*(466*((78*(12^9*5^3125+7))^3+1)+1))+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 49.98%
(884*(466*((78*(12^9*5^3125+7))^3+1)+1)-1)*(884*(466*((78*(12^9*5^3125+7))^3+1)+1))+1 is prime! (15.4107s+0.0012s)
Primality testing 6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 17
Calling Brillhart-Lehmer-Selfridge with factored part 66.65%
6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1 is prime! (37.4255s+0.0016s)
Primality testing (33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1)-1)*(33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Calling Brillhart-Lehmer-Selfridge with factored part 49.99%
(33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1)-1)*(33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))+1 is prime! (175.2733s+0.0046s)
Primality testing 278822*((33410*(6470*((884*(466*((78*(12^9*5^3125+7))^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%
278822*((33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1 is prime! (414.1709s+0.0079s)
Primality testing (46953*(278822*((33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1)-1)*(46953*(278822*((33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1))+1 [N-1, Brillh
art-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 50.00%
(46953*(278822*((33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1)-1)*(46953*(278822*((33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1))+1 is prime! (1687.6584s+0.0248s)
Primality testing 145950 *( ( 46953 *( 278822*((33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1))^3+1)+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 11
Calling Brillhart-Lehmer-Selfridge with factored part 66.66%
145950 *( ( 46953 *( 278822*((33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1))^3+1)+1 is prime! (4422.2311s+0.0269s)
The Primo certificate for 12^9*5^5^5+7 is at HERE