Recursive prime p(k+1)=p(k)*(p(k)+/-m)+/-1

Define p(1)=2
p(2)[m=-1; +1] = p(1)*(p(1)-1)+1 = 3
p(3)[m=-1; -1] = p(2)*(p(2)-1)-1 = 5
p(4)[m=-1; -1] = p(3)*(p(3)-1)-1 = 19
p(5)[m=+1; -1] = p(4)*(p(4)+1)-1 = 379
p(6)[m=-1; -1] = p(5)*(p(5)-1)-1 = 143261
p(7)[m=-11; -1] = p(6)*(p(6)-11)-1 = 20522138249
p(8)[m=-11; +1] = p(7)*(p(7)-11)+1 = 421158158085325265263
p(9)[m=-13; +1] = p(8)*(p(8)-13)+1 = 421158158085325265256.5^2-165/4
p(10)[m=+59; -1] = p(9)*(p(9)+59)-1 = (421158158085325265256.5^2-47/4)^2-3485/4
p(11)[m=+45; +1] = p(10)*(p(10)+45)+1
= ((421158158085325265256.5^2-47/4)^2-3575/4)^2-2021/4
p(12)[m=-441; +1] = p(11)*(p(11)-441)+1
= (((421158158085325265256.5^2-47/4)^2-3575/4)^2-2903/4)^2-194477/4
p(13)[m=+127; -1] = p(12)*(p(12)+127)-1
= ((((421158158085325265256.5^2-47/4)^2-3575/4)^2-2903/4)^2-194223/4)^2-16133/4
p(14)[m=-269; +1] = p(13)*(p(13)-269)+1
= (((((421158158085325265256.5^2-47/4)^2-3575/4)^2-2903/4)^2-194223/4)^2-16671/4)^2-72357/4
p(15)[m=+973; -1] = p(14)*(p(14)+973)-1
= ((((((421158158085325265256.5^2-47/4)^2-3575/4)^2-2903/4)^2-194223/4)^2-16671/4)^2-70411/4)^2-946733/4
p(16)[m=-55; -1] = p(15)*(p(15)-55)-1
= ((((((((1/2+421158158085325265256)^2-47/4)^2-3575/4)^2-2903/4)^2-194223/4)^2-16671/4)^2-70411/4)^2-946843/4)^2-3029/4
p(17)[m=-7939; -1] = p(16)*(p(16)-7939)-1
= ((((((((421158158085325265256.5^2-47/4)^2-3575/4)^2-2903/4)^2-194223/4)^2-16671/4)^2-70411/4)^2-946843/4)^2-18907/4)^2-63027725/4
p(18)[m=+17897; +1] = p(17)*(p(17)+17897)+1
= (((((((((421158158085325265256.5^2-47/4)^2-3575/4)^2-2903/4)^2-194223/4)^2-16671/4)^2-70411/4)^2-946843/4)^2-18907/4)^2-62991931/4)^2-320302605/4
p(19)[m=+9881; -1] = p(18)*(p(18)+9881)-1
= ((((((((((421158158085325265256.5^2-47/4)^2-3575/4)^2-2903/4)^2-194223/4)^2-16671/4)^2-70411/4)^2-946843/4)^2-18907/4)^2-62991931/4)^2-320282843/4)^2-97634165/4
p(20)[m=+5017; -1] = p(19)*(p(19)+5017)-1
= (((((((((((421158158085325265256.5^2-47/4)^2-3575/4)^2-2903/4)^2-194223/4)^2-16671/4)^2-70411/4)^2-946843/4)^2-18907/4)^2-62991931/4)^2-320282843/4)^2-97624131/4)^2-25170293/4
p(21)[m=-300019; -1] = p(20)*(p(20)-300019)-1
= (((((((((((((842316316170650530513/2)^2-47/4)^2-3575/4)^2-2903/4)^2-194223/4)^2-16671/4)^2-70411/4)^2-946843/4)^2-18907/4)^2-62991931/4)^2-320282843/4)^2-97624131/4)^2-25770331/4)^2-90011400365/4

p(21) has database ID 94526 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(k+1)

The final certification code is
#!/bin/sh
#!/bin/sh
./pfgw -l”pmdance.21.cert” -t p_07
./pfgw -l”pmdance.21.cert” -t -h”p_07″ p_08
./pfgw -l”pmdance.21.cert” -t -h”p_08″ p_09
./pfgw -l”pmdance.21.cert” -tp -h”p_09″ p_10
./pfgw -l”pmdance.21.cert” -t -h”p_10″ p_11
./pfgw -l”pmdance.21.cert” -t -h”p_11″ p_12
./pfgw -l”pmdance.21.cert” -tp -h”p_12″ p_13
./pfgw -l”pmdance.21.cert” -t -h”p_13″ p_14
./pfgw -l”pmdance.21.cert” -tp -h”p_14″ p_15
./pfgw -l”pmdance.21.cert” -tp -h”p_15″ p_16
./pfgw -l”pmdance.21.cert” -tp -h”p_16″ p_17
./pfgw -l”pmdance.21.cert” -t -h”p_17″ p_18
./pfgw -l”pmdance.21.cert” -tp -h”p_18″ p_19
./pfgw -l”pmdance.21.cert” -tp -h”p_19″ p_20
./pfgw -l”pmdance.21.cert” -tp -h”p_20″ p_21

The certificate is the following:
Primality testing 20522138249 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Calling Brillhart-Lehmer-Selfridge with factored part 91.18%
20522138249 is prime! (0.0013s+0.0001s)
Primality testing 421158158085325265263 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Calling Brillhart-Lehmer-Selfridge with factored part 50.00%
421158158085325265263 is prime! (0.0020s+0.0002s)
Primality testing (842316316170650530513^2-165)/4 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Calling Brillhart-Lehmer-Selfridge with factored part 49.64%
(842316316170650530513^2-165)/4 is prime! (0.0024s+0.0002s)
Primality testing (((842316316170650530513^2-47)/2)^2-3485)/4 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 3, base 1+sqrt(3)
Calling Brillhart-Lehmer-Selfridge with factored part 50.00%
(((842316316170650530513^2-47)/2)^2-3485)/4 is prime! (0.0051s+0.0002s)
Primality testing (((((842316316170650530513^2-47)/2)^2-3575)/2)^2-2021)/4 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Calling Brillhart-Lehmer-Selfridge with factored part 50.00%
(((((842316316170650530513^2-47)/2)^2-3575)/2)^2-2021)/4 is prime! (0.0067s+0.0003s)
Primality testing (((((((842316316170650530513^2-47)/2)^2-3575)/2)^2-2903)/2)^2-194477)/4 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Calling Brillhart-Lehmer-Selfridge with factored part 50.00%
(((((((842316316170650530513^2-47)/2)^2-3575)/2)^2-2903)/2)^2-194477)/4 is prime! (0.0142s+0.0003s)
Primality testing (((((((((842316316170650530513^2-47)/2)^2-3575)/2)^2-2903)/2)^2-194223)/2)^2-16133)/4 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 11, base 1+sqrt(11)
Calling Brillhart-Lehmer-Selfridge with factored part 50.00%
(((((((((842316316170650530513^2-47)/2)^2-3575)/2)^2-2903)/2)^2-194223)/2)^2-16133)/4 is prime! (0.1029s+0.0003s)
Primality testing (((((((((((842316316170650530513^2-47)/2)^2-3575)/2)^2-2903)/2)^2-194223)/2)^2-16671)/2)^2-72357)/4 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Calling Brillhart-Lehmer-Selfridge with factored part 50.00%
(((((((((((842316316170650530513^2-47)/2)^2-3575)/2)^2-2903)/2)^2-194223)/2)^2-16671)/2)^2-72357)/4 is prime! (0.1451s+0.0004s)
Primality testing (((((((((((((842316316170650530513^2-47)/2)^2-3575)/2)^2-2903)/2)^2-194223)/2)^2-16671)/2)^2-70411)/2)^2-946733)/4 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 3, base 7+sqrt(3)
Calling Brillhart-Lehmer-Selfridge with factored part 49.99%
(((((((((((((842316316170650530513^2-47)/2)^2-3575)/2)^2-2903)/2)^2-194223)/2)^2-16671)/2)^2-70411)/2)^2-946733)/4 is prime! (1.5795s+0.0004s)
Primality testing (((((((((((((((842316316170650530513^2-47)/2)^2-3575)/2)^2-2903)/2)^2-194223)/2)^2-16671)/2)^2-70411)/2)^2-946843)/2)^2-3029)/4 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 5, base 1+sqrt(5)
Calling Brillhart-Lehmer-Selfridge with factored part 50.00%
(((((((((((((((842316316170650530513^2-47)/2)^2-3575)/2)^2-2903)/2)^2-194223)/2)^2-16671)/2)^2-70411)/2)^2-946843)/2)^2-3029)/4 is prime! (6.6442s+0.0006s)
Primality testing (((((((((((((((((842316316170650530513^2-47)/2)^2-3575)/2)^2-2903)/2)^2-194223)/2)^2-16671)/2)^2-70411)/2)^2-946843)/2)^2-18907)/2)^2-63027725)/4 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 3, base 1+sqrt(3)
Calling Brillhart-Lehmer-Selfridge with factored part 50.00%
(((((((((((((((((842316316170650530513^2-47)/2)^2-3575)/2)^2-2903)/2)^2-194223)/2)^2-16671)/2)^2-70411)/2)^2-946843)/2)^2-18907)/2)^2-63027725)/4 is prime! (29.5119s+0.0009s)
Primality testing (((((((((((((((((((842316316170650530513^2-47)/2)^2-3575)/2)^2-2903)/2)^2-194223)/2)^2-16671)/2)^2-70411)/2)^2-946843)/2)^2-18907)/2)^2-62991931)/2)^2-320302605)/4 [N-1, Brillhart-Lehmer-Selfrid
ge]
Running N-1 test using base 13
Calling Brillhart-Lehmer-Selfridge with factored part 50.00%
(((((((((((((((((((842316316170650530513^2-47)/2)^2-3575)/2)^2-2903)/2)^2-194223)/2)^2-16671)/2)^2-70411)/2)^2-946843)/2)^2-18907)/2)^2-62991931)/2)^2-320302605)/4 is prime! (44.2155s+0.0017s)
Primality testing (((((((((((((((((((((842316316170650530513^2-47)/2)^2-3575)/2)^2-2903)/2)^2-194223)/2)^2-16671)/2)^2-70411)/2)^2-946843)/2)^2-18907)/2)^2-62991931)/2)^2-320282843)/2)^2-97634165)/4 [N+1, Brillha
rt-Lehmer-Selfridge]
Running N+1 test using discriminant 3, base 3+sqrt(3)
Calling Brillhart-Lehmer-Selfridge with factored part 50.00%
(((((((((((((((((((((842316316170650530513^2-47)/2)^2-3575)/2)^2-2903)/2)^2-194223)/2)^2-16671)/2)^2-70411)/2)^2-946843)/2)^2-18907)/2)^2-62991931)/2)^2-320282843)/2)^2-97634165)/4 is prime! (518.2041s+0.0039s)
Primality testing (((((((((((((((((((((((842316316170650530513^2-47)/2)^2-3575)/2)^2-2903)/2)^2-194223)/2)^2-16671)/2)^2-70411)/2)^2-946843)/2)^2-18907)/2)^2-62991931)/2)^2-320282843)/2)^2-97624131)/2)^2-25170293
)/4 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 3, base 3+sqrt(3)
Calling Brillhart-Lehmer-Selfridge with factored part 50.00%
(((((((((((((((((((((((842316316170650530513^2-47)/2)^2-3575)/2)^2-2903)/2)^2-194223)/2)^2-16671)/2)^2-70411)/2)^2-946843)/2)^2-18907)/2)^2-62991931)/2)^2-320282843)/2)^2-97624131)/2)^2-25170293)/4 is prime! (215
3.5087s+0.0071s)
Primality testing (((((((((((((((((((((((((842316316170650530513^2-47)/2)^2-3575)/2)^2-2903)/2)^2-194223)/2)^2-16671)/2)^2-70411)/2)^2-946843)/2)^2-18907)/2)^2-62991931)/2)^2-320282843)/2)^2-97624131)/2)^2-257703
31)/2)^2-90011400365)/4 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 7, base 1+sqrt(7)
Calling Brillhart-Lehmer-Selfridge with factored part 50.00%
(((((((((((((((((((((((((842316316170650530513^2-47)/2)^2-3575)/2)^2-2903)/2)^2-194223)/2)^2-16671)/2)^2-70411)/2)^2-946843)/2)^2-18907)/2)^2-62991931)/2)^2-320282843)/2)^2-97624131)/2)^2-25770331)/2)^2-900114003
65)/4 is prime! (9575.0905s+0.0157s)