Define p(0)=12^9*5^5^5+7;
\np(1)[m=466;n=78]=466*((78*(12^9*5^3125+7))^3+1)+1;
\np(2)[m=6470;n=884]=6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1;
\np(3)[m=278822;n=33410]=278822*((33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1;
\np(4)[m=145950;n=46953]=145950*((46953*( 278822*((33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1))^3+1)+1;<\/p>\n
p(4) has database ID 96540 in The List of Largest Known Primes Home Page<\/a>. The direct link is HERE<\/a>.<\/p>\n The kernel 12^9*5^5^5+1 is proven by Primo. The certificate is in the first reply of this post.<\/p>\n The recursive primes are 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)=12^9*5^5^5+7; p(1)[m=466;n=78]=466*((78*(12^9*5^3125+7))^3+1)+1; p(2)[m=6470;n=884]=6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1; p(3)[m=278822;n=33410]=278822*((33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1; p(4)[m=145950;n=46953]=145950*((46953*( 278822*((33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1))^3+1)+1; p(4) has database ID 96540 in The List of Largest Known Primes Home Page. The direct link is HERE. The kernel 12^9*5^5^5+1 is proven by Primo. The certificate is in the first reply of this post. The recursive primes are proven using OpenPFGW, by the command pfgw -t […]<\/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-186","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\/186","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=186"}],"version-history":[{"count":0,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/posts\/186\/revisions"}],"wp:attachment":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=186"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=186"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=186"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\npfgw -t (or tp) -h”p(k)” p_h(k+1); pfgw -t (or tp) -h”p_h(k+1)” p(k+1)
\nThe number
\n p_h(k+1)=(n*p(k)-1)*(n*p(k))+1=(n*p(k))^2-n*p(k)+1
\n p(k+1)=m*((n*p(k))^3+1)+1=m*(n*p(k)+1)((n*p(k))^2-n*p(k)+1)+1=m*(n*p(k)+1)*p_h(k+1)+1
\nare reformatted by Mathematica to get the short expression.<\/p>\n
\n#!\/bin\/sh
\n.\/pfgw -l”pmtup_5.4.2.cert” -t -h”p_00″ ph_01
\n.\/pfgw -l”pmtup_5.4.2.cert” -t -h”ph_01″ p_01
\n.\/pfgw -l”pmtup_5.4.2.cert” -t -h”p_01″ ph_02
\n.\/pfgw -l”pmtup_5.4.2.cert” -t -h”ph_02″ p_02
\n.\/pfgw -l”pmtup_5.4.2.cert” -t -h”p_02″ ph_03
\n.\/pfgw -l”pmtup_5.4.2.cert” -t -h”ph_03″ p_03
\n.\/pfgw -l”pmtup_5.4.2.cert” -t -h”p_03″ ph_04
\n.\/pfgw -l”pmtup_5.4.2.cert” -t -h”ph_04″ p_04<\/p>\n