{"id":155,"date":"2010-08-30T20:16:02","date_gmt":"2010-08-31T00:16:02","guid":{"rendered":"http:\/\/bitc.bme.emory.edu\/~lzhou\/blogs\/?p=155"},"modified":"2010-08-31T08:27:15","modified_gmt":"2010-08-31T12:27:15","slug":"recursive-prime-pk1mnpk311","status":"publish","type":"post","link":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/?p=155","title":{"rendered":"Recursive prime p(k+1)=m*((n*p(k))^3+1)+1"},"content":{"rendered":"

Define p(0)=1;
\np(1)[m=2;n=2]=2*((2*p(0))^3+1)+1=19;
\np(2)[m=6;n=4]=6*((4*p(1))^3+1)+1=2633863;
\np(3)[m=14;n=1]=14*((1*2633863)^3+1)+1=14*2633863^3+15;
\np(4)[m=354;n=74]=354*((74*(14*2633863^3+15 ))^3+1)+1
\n =354*(1036*2633863^3+1110)^3+355;
\np(5)[m=155;n=115]=155*((115*(354*(1036*2633863^3+1110)^3+355 ))^3+1)+1
\n =155*(40710*(1036*2633863^3+1110)^3+40825)^3+156;
\np(6)[m=146;n=629]=146*((629*(155*(40710*(1036*2633863^3+1110)^3+40825)^3+156 ))^3+1)+1
\n =146*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+147;
\np(7)[m=440;n=1754]=440*((1754*(146*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+147 ))^3+1)+1
\n =440*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+441;
\np(8)[m=8385;n=185]=8385*((185*(440*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+441 ))^3+1)+1
\n =8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386;
\np(9)[m=16182;n=2988]=16182*((2988*(8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386 ))^3+1)+1;
\np(10)[m=79194;n=97326]=79194*((97326*(16182*((2988*(8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386))^3+1)+1))^3+1)+1;
\np(11)=[m=232497;n=176845]=232497*((176845*(79194*((97326*(16182*((2988*(8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386))^3+1)+1))^3+1)+1))^3+1)+1;<\/p>\n

p(11) has database ID 94439 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
\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

The final certification code is
\n#!\/bin\/sh
\n.\/pfgw -l”pmtup.11.2.cert” -tp p_03
\n.\/pfgw -l”pmtup.11.2.cert” -t -h”p_03″ ph_04
\n.\/pfgw -l”pmtup.11.2.cert” -t -h”ph_04″ p_04
\n.\/pfgw -l”pmtup.11.2.cert” -t -h”p_04″ ph_05
\n.\/pfgw -l”pmtup.11.2.cert” -t -h”ph_05″ p_05
\n.\/pfgw -l”pmtup.11.2.cert” -t -h”p_05″ ph_06
\n.\/pfgw -l”pmtup.11.2.cert” -t -h”ph_06″ p_06
\n.\/pfgw -l”pmtup.11.2.cert” -t -h”p_06″ ph_07
\n.\/pfgw -l”pmtup.11.2.cert” -t -h”ph_07″ p_07
\n.\/pfgw -l”pmtup.11.2.cert” -t -h”p_07″ ph_08
\n.\/pfgw -l”pmtup.11.2.cert” -t -h”ph_08″ p_08
\n.\/pfgw -l”pmtup.11.2.cert” -t -h”p_08″ ph_09
\n.\/pfgw -l”pmtup.11.2.cert” -t -h”ph_09″ p_09
\n.\/pfgw -l”pmtup.11.2.cert” -t -h”p_09″ ph_10
\n.\/pfgw -l”pmtup.11.2.cert” -t -h”ph_10″ p_10
\n.\/pfgw -l”pmtup.11.2.cert” -t -h”p_10″ ph_11
\n.\/pfgw -l”pmtup.11.2.cert” -t -h”ph_11″ p_11<\/p>\n

The certificate will be posted when done.<\/p>\n","protected":false},"excerpt":{"rendered":"

Define p(0)=1; p(1)[m=2;n=2]=2*((2*p(0))^3+1)+1=19; p(2)[m=6;n=4]=6*((4*p(1))^3+1)+1=2633863; p(3)[m=14;n=1]=14*((1*2633863)^3+1)+1=14*2633863^3+15; p(4)[m=354;n=74]=354*((74*(14*2633863^3+15 ))^3+1)+1 =354*(1036*2633863^3+1110)^3+355; p(5)[m=155;n=115]=155*((115*(354*(1036*2633863^3+1110)^3+355 ))^3+1)+1 =155*(40710*(1036*2633863^3+1110)^3+40825)^3+156; p(6)[m=146;n=629]=146*((629*(155*(40710*(1036*2633863^3+1110)^3+40825)^3+156 ))^3+1)+1 =146*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+147; p(7)[m=440;n=1754]=440*((1754*(146*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+147 ))^3+1)+1 =440*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+441; p(8)[m=8385;n=185]=8385*((185*(440*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+441 ))^3+1)+1 =8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386; p(9)[m=16182;n=2988]=16182*((2988*(8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386 ))^3+1)+1; p(10)[m=79194;n=97326]=79194*((97326*(16182*((2988*(8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386))^3+1)+1))^3+1)+1; p(11)=[m=232497;n=176845]=232497*((176845*(79194*((97326*(16182*((2988*(8385*(81400*(256084*(97495*(40710*(1036*2633863^3+1110)^3+40825)^3+98124)^3+257838)^3+81585)^3+8386))^3+1)+1))^3+1)+1))^3+1)+1; p(11) has database ID 94439 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 […]<\/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-155","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\/155","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=155"}],"version-history":[{"count":0,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/posts\/155\/revisions"}],"wp:attachment":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=155"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=155"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=155"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}