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

Posted in categories: Fun Stuffs, Prime Search, Uncategorized

Define p(0)=2;
p(1)=2*(2-1)+1 = 3 is prime with m(1)=1; (+1)
p(2)=3*(3-1)-1 = 5 is prime with m(2)=1; (-1)
p(3)=5*(5-1)-1 = 19 is prime with m(3)=1; (-1)
p(4)=19*(19-7)-1 = 227 is prime with m(4)=7; (-1)
p(5)=227*(227-3)+1 = 50849 is prime with m(5)=3; (+1)
p(6)=50849*(50849-29)-1 = 2584146179 is prime with m(6)=29; (-1)
p(7)=2584146179*(2584146179-19)-1 = 6677811425341522639 is prime with m(7)=19; (-1)
p(8)=(13355622850683045201^2-5933)/4 is prime with m(8)=77; (-1)
p(9)=((13355622850683045201^2-5983)^2/4-629)/4 is prime with m(9)=25; (-1)
p(10)=(((13355622850683045201^2-5983)^2/4-763)^2/4-4493)/4, m(10)=67; (-1)
p(11)=((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5325)/4, m(11)=73; (+1)
p(12)=(((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5699)^2/4-34965)/4, m(12)=187; (+1)
p(13)= ((((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5699)^2/4-35519)^2/4-76725)/4, m(13)=277; (+1)
p(14)=(((((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5699)^2/4-35519)^2/4-82547)^2/4-8473917)/4, m(14)=2911; (+1)
p(15)=((((((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5699)^2/4-35519)^2/4-82547)^2/4-8476499)^2/4-1666677)/4, m(15)=1291; (+1)
p(16)=(((((((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5699)^2/4-35519)^2/4-82547)^2/4-8476499)^2/4-1678383)^2/4-34257605)/4, m(16)=5853; (+1)
p(17)=((((((((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5699)^2/4-35519)^2/4-82547)^2/4-8476499)^2/4-1678383)^2/4-34308675)^2/4-652036221)/4, m(17)=25535; (+1)
p(18)=(((((((((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5699)^2/4-35519)^2/4-82547)^2/4-8476499)^2/4-1678383)^2/4-34308675)^2/4-652040831)^2/4-5313029)/4, m(18)=2305; (-1)
p(19)=((((((((((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5699)^2/4-35519)^2/4-82547)^2/4-8476499)^2/4-1678383)^2/4-34308675)^2/4-652040831)^2/4-5471199)^2/4-6254437229)/4, m(19)=79085; (-1)
p(20)=(((((((((((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5699)^2/4-35519)^2/4-82547)^2/4-8476499)^2/4-1678383)^2/4-34308675)^2/4-652040831)^2/4-5471199)^2/4-6254699427)^2/4-17186947797)/4, m(20)=131099; (+1)

p(20) has database ID 93175 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 number
p(k+1)
=p(k)*(p(k)-m)+/-1
=(p(k)-m/2)^2-(m/2)^2+/-1
are reformatted by Mathematica to get the short expression.

The final certification code is
#!/bin/sh
./pfgw -l”pmrtrain.21.cert” -t p_06
./pfgw -l”pmrtrain.21.cert” -tp -h”p_06″ p_07
./pfgw -l”pmrtrain.21.cert” -tp -h”p_07″ p_08
./pfgw -l”pmrtrain.21.cert” -tp -h”p_08″ p_09
./pfgw -l”pmrtrain.21.cert” -tp -h”p_09″ p_10
./pfgw -l”pmrtrain.21.cert” -t -h”p_10″ p_11
./pfgw -l”pmrtrain.21.cert” -t -h”p_11″ p_12
./pfgw -l”pmrtrain.21.cert” -t -h”p_12″ p_13
./pfgw -l”pmrtrain.21.cert” -t -h”p_13″ p_14
./pfgw -l”pmrtrain.21.cert” -t -h”p_14″ p_15
./pfgw -l”pmrtrain.21.cert” -t -h”p_15″ p_16
./pfgw -l”pmrtrain.21.cert” -t -h”p_16″ p_17
./pfgw -l”pmrtrain.21.cert” -tp -h”p_17″ p_18
./pfgw -l”pmrtrain.21.cert” -tp -h”p_18″ p_19
./pfgw -l”pmrtrain.21.cert” -t -h”p_19″ p_20

The full certificate is

Primality testing 2584146179 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 96.77%
2584146179 is prime! (0.0014s+0.0002s)
Primality testing 6677811425341522639 [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%
6677811425341522639 is prime! (0.0027s+0.0001s)
Primality testing (13355622850683045201^2-5933)/4 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 5, base 4+sqrt(5)
Calling Brillhart-Lehmer-Selfridge with factored part 49.60%
(13355622850683045201^2-5933)/4 is prime! (0.0041s+0.0002s)
Primality testing ((13355622850683045201^2-5983)^2/4-629)/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%
((13355622850683045201^2-5983)^2/4-629)/4 is prime! (0.0058s+0.0002s)
Primality testing (((13355622850683045201^2-5983)^2/4-763)^2/4-4493)/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%
(((13355622850683045201^2-5983)^2/4-763)^2/4-4493)/4 is prime! (0.0170s+0.0003s)
Primality testing ((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5325)/4 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 50.00%
((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5325)/4 is prime! (0.0114s+0.0003s)
Primality testing (((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5699)^2/4-34965)/4 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 11
Calling Brillhart-Lehmer-Selfridge with factored part 49.98%
(((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5699)^2/4-34965)/4 is prime! (0.0315s+0.0003s)
Primality testing ((((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5699)^2/4-35519)^2/4-76725)/4 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 50.00%
((((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5699)^2/4-35519)^2/4-76725)/4 is prime! (0.1134s+0.0004s)
Primality testing (((((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5699)^2/4-35519)^2/4-82547)^2/4-8473917)/4 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Calling Brillhart-Lehmer-Selfridge with factored part 50.00%
(((((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5699)^2/4-35519)^2/4-82547)^2/4-8473917)/4 is prime! (0.4749s+0.0004s)
Primality testing ((((((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5699)^2/4-35519)^2/4-82547)^2/4-8476499)^2/4-1666677)/4 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 50.00%
((((((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5699)^2/4-35519)^2/4-82547)^2/4-8476499)^2/4-1666677)/4 is prime! (1.7934s+0.0006s)
Primality testing (((((((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5699)^2/4-35519)^2/4-82547)^2/4-8476499)^2/4-1678383)^2/4-34257605)/4 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Calling Brillhart-Lehmer-Selfridge with factored part 50.00%
(((((((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5699)^2/4-35519)^2/4-82547)^2/4-8476499)^2/4-1678383)^2/4-34257605)/4 is prime! (7.9507s+0.0009s)
Primality testing ((((((((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5699)^2/4-35519)^2/4-82547)^2/4-8476499)^2/4-1678383)^2/4-34308675)^2/4-652036221)/4 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 50.00%
((((((((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5699)^2/4-35519)^2/4-82547)^2/4-8476499)^2/4-1678383)^2/4-34308675)^2/4-652036221)/4 is prime! (33.1867s+0.0013s)
Primality testing (((((((((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5699)^2/4-35519)^2/4-82547)^2/4-8476499)^2/4-1678383)^2/4-34308675)^2/4-652040831)^2/4-5313029)/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%
(((((((((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5699)^2/4-35519)^2/4-82547)^2/4-8476499)^2/4-1678383)^2/4-34308675)^2/4-652040831)^2/4-5313029)/4 is prime! (478.3195s+0.0035s)
Primality testing ((((((((((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5699)^2/4-35519)^2/4-82547)^2/4-8476499)^2/4-1678383)^2/4-34308675)^2/4-652040831)^2/4-5471199)^2/4-6254437229)/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%
((((((((((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5699)^2/4-35519)^2/4-82547)^2/4-8476499)^2/4-1678383)^2/4-34308675)^2/4-652040831)^2/4-5471199)^2/4-6254437229)/4 is prime! (2091.5206s+0.0084s)
Primality testing (((((((((((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5699)^2/4-35519)^2/4-82547)^2/4-8476499)^2/4-1678383)^2/4-34308675)^2/4-652040831)^2/4-5471199)^2/4-6254699427)^2/4-17186947797)/4 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 50.00%
(((((((((((((13355622850683045201^2-5983)^2/4-763)^2/4-4639)^2/4-5699)^2/4-35519)^2/4-82547)^2/4-8476499)^2/4-1678383)^2/4-34308675)^2/4-652040831)^2/4-5471199)^2/4-6254699427)^2/4-17186947797)/4 is prime! (3036.1728s+0.0172s)

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