Recursive prime p(k+1)=p(k)*(p(k)-m)+/-1 with minimum m
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)
15 Comments
Leave a Reply
You must be logged in to post a comment.
The very heart of your writing whilst sounding agreeable originally, did not work perfectly with me personally after some time. Someplace within the sentences you were able to make me a believer unfortunately just for a very short while. I still have a problem with your jumps in logic and you would do nicely to fill in all those gaps. In the event you actually can accomplish that, I could undoubtedly be impressed.
I find myself coming to your blog more and more often to the point where my visits are almost daily now!
I consider something genuinely special in this internet site .
You made some important points there. I’ve done a lot of searching on the topic and found almost all people will agree with your post. Thanks again
Your blog provides a fresh look at the subject. Great job mate!
Couldnt agree more with that, very attractive article
Get two plants, and set both of them in two different locations. One plant should be put in a box with a light so it can grow. The other plant should be put in a dark box so it will not grow.
Keep posting stuff like this i really like it
Hey, I enjoyed the articles and excellent design you have here! I have your blog bookmarked to check out fresh material you post. I would like to thank you for taking the time to share your thoughts and time into the stuff you post!! 2 Thumbs up!
This was really interesting. I loved reading it
I found your blog on google and read a few of your other posts. I just added you to my Google News Reader. Keep up the good work Look forward to reading more from you in the future.
Tremendous post I Agree with your major tips and think the title says it all, I loved reading and I will certainly come from time to time to read more of your writing. carry on the good work!
Interesting article and one which should be more widely known about in my view. Your level of detail is good and the clarity of writing is excellent. I have bookmarked it for you so that others will be able to see what you have to say.
Interesting article and one which should be more widely known about in my view. Your level of detail is good and the clarity of writing is excellent. I have bookmarked it for you so that others will be able to see what you have to say.
You made some good points there. I did a search on the topic and found most people will agree with your blog.