{"id":187,"date":"2010-11-22T19:59:20","date_gmt":"2010-11-22T23:59:20","guid":{"rendered":"http:\/\/bitc.bme.emory.edu\/~lzhou\/blogs\/?p=187"},"modified":"2010-11-23T00:05:49","modified_gmt":"2010-11-23T04:05:49","slug":"recursive-generalized-fermat-prime-found","status":"publish","type":"post","link":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/?p=187","title":{"rendered":"Recursive Generalized Fermat Prime found"},"content":{"rendered":"

Define p(0)=1;
\nfinding the smallest General Fermat prime in the form p(n+1)[m]=(2*m*p(n))^2+1, m is positive integer:
\np(1)[1]=(2*p(0))^2+1=5;
\np(2)[1]=(2*p(1))^2+1=101;
\np(3)[5]=(2*5*p(2))^2+1=1020101;
\np(4)[48]=(2*48*p(3))^2+1=((1020101)*96)^2+1;
\np(5)[1]=(2*p(4))^2+1=((((1020101)*96)^2+1)*2)^2+1;
\np(6)[30]=(2*30*p(5))^2+1=((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1;
\np(7)[85]=(2*85*p(6))^2+1=((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1;
\np(8)[935]=(2*935*p(7))^2+1=((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1;
\np(9)[528]=(2*528*p(8))^2+1=((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1;
\np(10)[2505]=(2*2505*p(9))^2+1=((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1;
\np(11)[840]=(2*840*p(10))^2+1=((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1;
\np(12)[1190]=(2*1190*p(11))^2+1=((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1;
\np(13)[29382]=(2*29382*p(12))^2+1=((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1;
\np(14)[25176]=(2*25176*p(13))^2+1=((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1;
\np(15)[12685]=(2*12685*p(14))^2+1=((((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1;
\np(16)[67852]=(2*67852*p(15))^2+1=((((((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1)*135704)^2+1;
\np(17)[299549]=(2*299549*p(16))^2+1=((((((((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1)*135704)^2+1)*599098)^2+1;
\np(18)[62406]=(2*62406*p(17))^2+1=((((((((((((((((((((((((((((97929696^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1)*135704)^2+1)*599098)^2+1)*124812)^2+1;<\/p>\n

p(4) has database ID 96548 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\u201dp(n)\u201d p(n+1)
\nThe number
\np(n+1)[m]=(2*m*p(n))^2+1
\nare reformatted by Mathematica to get the short expression.<\/p>\n

The final certification code is
\n#!\/bin\/sh
\n.\/pfgw -l”GF.19.cert” -t -h”p_03″ p_04
\n.\/pfgw -l”GF.19.cert” -t -h”p_04″ p_05
\n.\/pfgw -l”GF.19.cert” -t -h”p_05″ p_06
\n.\/pfgw -l”GF.19.cert” -t -h”p_06″ p_07
\n.\/pfgw -l”GF.19.cert” -t -h”p_07″ p_08
\n.\/pfgw -l”GF.19.cert” -t -h”p_08″ p_09
\n.\/pfgw -l”GF.19.cert” -t -h”p_09″ p_10
\n.\/pfgw -l”GF.19.cert” -t -h”p_10″ p_11
\n.\/pfgw -l”GF.19.cert” -t -h”p_11″ p_12
\n.\/pfgw -l”GF.19.cert” -t -h”p_12″ p_13
\n.\/pfgw -l”GF.19.cert” -t -h”p_13″ p_14
\n.\/pfgw -l”GF.19.cert” -t -h”p_14″ p_15
\n.\/pfgw -l”GF.19.cert” -t -h”p_15″ p_16
\n.\/pfgw -l”GF.19.cert” -t -h”p_16″ p_17
\n.\/pfgw -l”GF.19.cert” -t -h”p_17″ p_18<\/p>\n

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

Define p(0)=1; finding the smallest General Fermat prime in the form p(n+1)[m]=(2*m*p(n))^2+1, m is positive integer: p(1)[1]=(2*p(0))^2+1=5; p(2)[1]=(2*p(1))^2+1=101; p(3)[5]=(2*5*p(2))^2+1=1020101; p(4)[48]=(2*48*p(3))^2+1=((1020101)*96)^2+1; p(5)[1]=(2*p(4))^2+1=((((1020101)*96)^2+1)*2)^2+1; p(6)[30]=(2*30*p(5))^2+1=((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1; p(7)[85]=(2*85*p(6))^2+1=((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1; p(8)[935]=(2*935*p(7))^2+1=((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1; p(9)[528]=(2*528*p(8))^2+1=((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1; p(10)[2505]=(2*2505*p(9))^2+1=((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1; p(11)[840]=(2*840*p(10))^2+1=((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1; p(12)[1190]=(2*1190*p(11))^2+1=((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1; p(13)[29382]=(2*29382*p(12))^2+1=((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1; p(14)[25176]=(2*25176*p(13))^2+1=((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1; p(15)[12685]=(2*12685*p(14))^2+1=((((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1; p(16)[67852]=(2*67852*p(15))^2+1=((((((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1)*135704)^2+1; p(17)[299549]=(2*299549*p(16))^2+1=((((((((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1)*135704)^2+1)*599098)^2+1; p(18)[62406]=(2*62406*p(17))^2+1=((((((((((((((((((((((((((((97929696^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1)*135704)^2+1)*599098)^2+1)*124812)^2+1; p(4) has database ID 96548 in The List of Largest Known Primes Home Page. The direct link is HERE. These primes […]<\/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-187","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\/187","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=187"}],"version-history":[{"count":0,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/posts\/187\/revisions"}],"wp:attachment":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=187"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=187"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=187"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}