Define p(0)=1;
\np(1)=2*P(0)^2+1=3;
\np(2)=2*p(1)^2-1=17;
\np(3)=2*p(2)^2-1=577;
\np(4)=2*p(3)^2-1=665857;
\np(5)=20*p(4)^2-1;
\np(6)=2*p(5)^2-1;
\np(7)=28*p(6)^2+1;
\np(8)=182*p(7)^2-1;
\np(9)=272*p(8)^2-1;
\np(10)=540*p(9)^2+1;
\np(11)=162*p(10)^2+1;
\np(12)=1002*p(11)^2+1;
\np(13)=112*p(12)^2+1;
\np(14)=306*p(13)^2-1;
\np(15)=1752*p(14)^2+1;
\np(16)=20564*p(15)^2-1;
\np(17)=135236*p(16)^2-1;
\np(18)=547952*p(17)^2-1;
\np(19)=282904*p(18)^2+1;<\/p>\n
p(19)=282904*(547952*(135236*(20564*(1752*(306*(112*(1002*(162*(540*(272*(182*(28*(2*8867310888979^2-1)^2+1)^2-1)^2-1)^2+1)^2+1)^2+1)^2+1)^2-1)^2+1)^2-1)^2-1)^2-1)^2+1 has database ID 94235 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 certificate will be posted when done.<\/p>\n","protected":false},"excerpt":{"rendered":" Define p(0)=1; p(1)=2*P(0)^2+1=3; p(2)=2*p(1)^2-1=17; p(3)=2*p(2)^2-1=577; p(4)=2*p(3)^2-1=665857; p(5)=20*p(4)^2-1; p(6)=2*p(5)^2-1; p(7)=28*p(6)^2+1; p(8)=182*p(7)^2-1; p(9)=272*p(8)^2-1; p(10)=540*p(9)^2+1; p(11)=162*p(10)^2+1; p(12)=1002*p(11)^2+1; p(13)=112*p(12)^2+1; p(14)=306*p(13)^2-1; p(15)=1752*p(14)^2+1; p(16)=20564*p(15)^2-1; p(17)=135236*p(16)^2-1; p(18)=547952*p(17)^2-1; p(19)=282904*p(18)^2+1; p(19)=282904*(547952*(135236*(20564*(1752*(306*(112*(1002*(162*(540*(272*(182*(28*(2*8867310888979^2-1)^2+1)^2-1)^2-1)^2+1)^2+1)^2+1)^2+1)^2-1)^2+1)^2-1)^2-1)^2-1)^2+1 has database ID 94235 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>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5,6],"tags":[],"class_list":["post-154","post","type-post","status-publish","format-standard","hentry","category-to-entertain-myself","category-looking-for-a-megaprime","post-blog"],"_links":{"self":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/posts\/154","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=154"}],"version-history":[{"count":0,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/posts\/154\/revisions"}],"wp:attachment":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=154"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=154"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=154"}],"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)=m*p(k)^2+\/-1
\nare reformatted by Mathematica to get the short expression.<\/p>\n
\n#!\/bin\/sh
\n.\/pfgw -l\u201dpmrtrain.21.cert\u201d -t p_06
\n.\/pfgw -l\u201dpmrtrain.21.cert\u201d -tp -h\u201dp_06\u2033 p_07
\n.\/pfgw -l\u201dpmrtrain.21.cert\u201d -tp -h\u201dp_07\u2033 p_08
\n.\/pfgw -l\u201dpmrtrain.21.cert\u201d -tp -h\u201dp_08\u2033 p_09
\n.\/pfgw -l\u201dpmrtrain.21.cert\u201d -tp -h\u201dp_09\u2033 p_10
\n.\/pfgw -l\u201dpmrtrain.21.cert\u201d -t -h\u201dp_10\u2033 p_11
\n.\/pfgw -l\u201dpmrtrain.21.cert\u201d -t -h\u201dp_11\u2033 p_12
\n.\/pfgw -l\u201dpmrtrain.21.cert\u201d -t -h\u201dp_12\u2033 p_13
\n.\/pfgw -l\u201dpmrtrain.21.cert\u201d -t -h\u201dp_13\u2033 p_14
\n.\/pfgw -l\u201dpmrtrain.21.cert\u201d -t -h\u201dp_14\u2033 p_15
\n.\/pfgw -l\u201dpmrtrain.21.cert\u201d -t -h\u201dp_15\u2033 p_16
\n.\/pfgw -l\u201dpmrtrain.21.cert\u201d -t -h\u201dp_16\u2033 p_17
\n.\/pfgw -l\u201dpmrtrain.21.cert\u201d -tp -h\u201dp_17\u2033 p_18
\n.\/pfgw -l\u201dpmrtrain.21.cert\u201d -tp -h\u201dp_18\u2033 p_19
\n.\/pfgw -l\u201dpmrtrain.21.cert\u201d -t -h\u201dp_19\u2033 p_20<\/p>\n