Define p(0)=2;
\np(1)=2*(2-1)+1 = 3 is prime with m(1)=1; (+1)
\np(2)=3*(3-1)-1 = 5 is prime with m(2)=1; (-1)
\np(3)=5*(5-1)-1 = 19 is prime with m(3)=1; (-1)
\np(4)=19*(19-7)-1 = 227 is prime with m(4)=7; (-1)
\np(5)=227*(227-3)+1 = 50849 is prime with m(5)=3; (+1)
\np(6)=50849*(50849-29)-1 = 2584146179 is prime with m(6)=29; (-1)
\np(7)=2584146179*(2584146179-19)-1 = 6677811425341522639 is prime with m(7)=19; (-1)
\np(8)=(13355622850683045201^2-5933)\/4 is prime with m(8)=77; (-1)
\np(9)=((13355622850683045201^2-5983)^2\/4-629)\/4 is prime with m(9)=25; (-1)
\np(10)=(((13355622850683045201^2-5983)^2\/4-763)^2\/4-4493)\/4, m(10)=67; (-1)
\np(11)=((((13355622850683045201^2-5983)^2\/4-763)^2\/4-4639)^2\/4-5325)\/4, m(11)=73; (+1)
\np(12)=(((((13355622850683045201^2-5983)^2\/4-763)^2\/4-4639)^2\/4-5699)^2\/4-34965)\/4, m(12)=187; (+1)
\np(13)= ((((((13355622850683045201^2-5983)^2\/4-763)^2\/4-4639)^2\/4-5699)^2\/4-35519)^2\/4-76725)\/4, m(13)=277; (+1)
\np(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)
\np(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)
\np(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)
\np(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)
\np(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)
\np(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)
\np(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>\n
p(20) has database ID 93175 in The List of Largest Known Primes Home Page<\/a>. The direct link is HERE<\/a>.<\/p>\n These primes are recursively proven using OpenPFGW, by the command The final certification code is The full certificate is<\/p>\n Primality testing 2584146179 [N-1, Brillhart-Lehmer-Selfridge] 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 […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5,6,1],"tags":[],"class_list":["post-132","post","type-post","status-publish","format-standard","hentry","category-to-entertain-myself","category-looking-for-a-megaprime","category-uncategorized","post-blog"],"_links":{"self":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/posts\/132","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=132"}],"version-history":[{"count":0,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/posts\/132\/revisions"}],"wp:attachment":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=132"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=132"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=132"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\npfgw -t (or tp) -h”p(k)” p(k+1)
\nThe number
\n p(k+1)
\n=p(k)*(p(k)-m)+\/-1
\n=(p(k)-m\/2)^2-(m\/2)^2+\/-1
\nare reformatted by Mathematica to get the short expression.<\/p>\n
\n#!\/bin\/sh
\n.\/pfgw -l”pmrtrain.21.cert” -t p_06
\n.\/pfgw -l”pmrtrain.21.cert” -tp -h”p_06″ p_07
\n.\/pfgw -l”pmrtrain.21.cert” -tp -h”p_07″ p_08
\n.\/pfgw -l”pmrtrain.21.cert” -tp -h”p_08″ p_09
\n.\/pfgw -l”pmrtrain.21.cert” -tp -h”p_09″ p_10
\n.\/pfgw -l”pmrtrain.21.cert” -t -h”p_10″ p_11
\n.\/pfgw -l”pmrtrain.21.cert” -t -h”p_11″ p_12
\n.\/pfgw -l”pmrtrain.21.cert” -t -h”p_12″ p_13
\n.\/pfgw -l”pmrtrain.21.cert” -t -h”p_13″ p_14
\n.\/pfgw -l”pmrtrain.21.cert” -t -h”p_14″ p_15
\n.\/pfgw -l”pmrtrain.21.cert” -t -h”p_15″ p_16
\n.\/pfgw -l”pmrtrain.21.cert” -t -h”p_16″ p_17
\n.\/pfgw -l”pmrtrain.21.cert” -tp -h”p_17″ p_18
\n.\/pfgw -l”pmrtrain.21.cert” -tp -h”p_18″ p_19
\n.\/pfgw -l”pmrtrain.21.cert” -t -h”p_19″ p_20<\/p>\n
\nRunning N-1 test using base 2
\nCalling Brillhart-Lehmer-Selfridge with factored part 96.77%
\n2584146179 is prime! (0.0014s+0.0002s)
\nPrimality testing 6677811425341522639 [N+1, Brillhart-Lehmer-Selfridge]
\nRunning N+1 test using discriminant 3, base 1+sqrt(3)
\nCalling Brillhart-Lehmer-Selfridge with factored part 50.00%
\n6677811425341522639 is prime! (0.0027s+0.0001s)
\nPrimality testing (13355622850683045201^2-5933)\/4 [N+1, Brillhart-Lehmer-Selfridge]
\nRunning N+1 test using discriminant 5, base 4+sqrt(5)
\nCalling Brillhart-Lehmer-Selfridge with factored part 49.60%
\n(13355622850683045201^2-5933)\/4 is prime! (0.0041s+0.0002s)
\nPrimality testing ((13355622850683045201^2-5983)^2\/4-629)\/4 [N+1, Brillhart-Lehmer-Selfridge]
\nRunning N+1 test using discriminant 5, base 1+sqrt(5)
\nCalling Brillhart-Lehmer-Selfridge with factored part 50.00%
\n((13355622850683045201^2-5983)^2\/4-629)\/4 is prime! (0.0058s+0.0002s)
\nPrimality testing (((13355622850683045201^2-5983)^2\/4-763)^2\/4-4493)\/4 [N+1, Brillhart-Lehmer-Selfridge]
\nRunning N+1 test using discriminant 3, base 1+sqrt(3)
\nCalling Brillhart-Lehmer-Selfridge with factored part 50.00%
\n(((13355622850683045201^2-5983)^2\/4-763)^2\/4-4493)\/4 is prime! (0.0170s+0.0003s)
\nPrimality testing ((((13355622850683045201^2-5983)^2\/4-763)^2\/4-4639)^2\/4-5325)\/4 [N-1, Brillhart-Lehmer-Selfridge]
\nRunning N-1 test using base 2
\nCalling Brillhart-Lehmer-Selfridge with factored part 50.00%
\n((((13355622850683045201^2-5983)^2\/4-763)^2\/4-4639)^2\/4-5325)\/4 is prime! (0.0114s+0.0003s)
\nPrimality testing (((((13355622850683045201^2-5983)^2\/4-763)^2\/4-4639)^2\/4-5699)^2\/4-34965)\/4 [N-1, Brillhart-Lehmer-Selfridge]
\nRunning N-1 test using base 11
\nCalling Brillhart-Lehmer-Selfridge with factored part 49.98%
\n(((((13355622850683045201^2-5983)^2\/4-763)^2\/4-4639)^2\/4-5699)^2\/4-34965)\/4 is prime! (0.0315s+0.0003s)
\nPrimality 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]
\nRunning N-1 test using base 2
\nCalling Brillhart-Lehmer-Selfridge with factored part 50.00%
\n((((((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)
\nPrimality 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]
\nRunning N-1 test using base 3
\nCalling Brillhart-Lehmer-Selfridge with factored part 50.00%
\n(((((((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)
\nPrimality 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]
\nRunning N-1 test using base 2
\nCalling Brillhart-Lehmer-Selfridge with factored part 50.00%
\n((((((((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)
\nPrimality 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]
\nRunning N-1 test using base 3
\nCalling Brillhart-Lehmer-Selfridge with factored part 50.00%
\n(((((((((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)
\nPrimality 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]
\nRunning N-1 test using base 2
\nCalling Brillhart-Lehmer-Selfridge with factored part 50.00%
\n((((((((((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)
\nPrimality 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]
\nRunning N+1 test using discriminant 3, base 1+sqrt(3)
\nCalling Brillhart-Lehmer-Selfridge with factored part 50.00%
\n(((((((((((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)
\nPrimality 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]
\nRunning N+1 test using discriminant 5, base 1+sqrt(5)
\nCalling Brillhart-Lehmer-Selfridge with factored part 50.00%
\n((((((((((((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)
\nPrimality 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]
\nRunning N-1 test using base 2
\nCalling Brillhart-Lehmer-Selfridge with factored part 50.00%
\n(((((((((((((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)<\/p>\n","protected":false},"excerpt":{"rendered":"