{"id":189,"date":"2010-12-05T13:06:27","date_gmt":"2010-12-05T17:06:27","guid":{"rendered":"http:\/\/bitc.bme.emory.edu\/~lzhou\/blogs\/?p=189"},"modified":"2010-12-05T23:26:48","modified_gmt":"2010-12-06T03:26:48","slug":"recursive-prime-brother-by-brillhart-lehmer-selfridge-algorithm","status":"publish","type":"post","link":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/?p=189","title":{"rendered":"Recursive prime brother by Brillhart – Lehmer – Selfridge algorithm"},"content":{"rendered":"

Define:<\/p>\n

p[k,i]=ABS[1+2*n[k,i]*p[k-1,1]*p[k-1,2]],n[k,1] is the integer with minimum ABS[n[k,1]] that makes p[k,1] a prime number, and n[k,2] is the integer with second minimum ABS[n[k,2]] that makes p[k,2] a prime number<\/p>\n

The primality of p[k,i] can be proven using Brillhart – Lehmer – Selfridge algorithm recursively by using p[k-1,1] and p[k-1,2] as helper since n is a small integer, by reducing k to 1.<\/p>\n

With this idea, taking<\/p>\n

p[0,1]=1, p[0,2]=1
\nWe got the n[i,j] ( columns : i; rows: k):<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
k<\/b><\/td>\ni=1<\/b><\/td>\ni=2<\/b><\/td>\n<\/tr>\n
1<\/td>\n1<\/td>\n-2<\/td>\n<\/tr>\n
2<\/td>\n-1<\/td>\n1<\/td>\n<\/tr>\n
3<\/td>\n1<\/td>\n-2<\/td>\n<\/tr>\n
4<\/td>\n-3<\/td>\n-5<\/td>\n<\/tr>\n
5<\/td>\n-11<\/td>\n19<\/td>\n<\/tr>\n
6<\/td>\n51<\/td>\n94<\/td>\n<\/tr>\n
7<\/td>\n7<\/td>\n33<\/td>\n<\/tr>\n
8<\/td>\n147<\/td>\n-165<\/td>\n<\/tr>\n
9<\/td>\n15<\/td>\n-29<\/td>\n<\/tr>\n
10<\/td>\n5<\/td>\n339<\/td>\n<\/tr>\n
11<\/td>\n412<\/td>\n-1260<\/td>\n<\/tr>\n
12<\/td>\n356<\/td>\n848<\/td>\n<\/tr>\n
13<\/td>\n4809<\/td>\n-5641<\/td>\n<\/tr>\n
14<\/td>\n-5215<\/td>\n-5539<\/td>\n<\/tr>\n
15<\/td>\n37695<\/td>\n41772<\/td>\n<\/tr>\n
16<\/td>\n5343<\/td>\n-6180<\/td>\n<\/tr>\n
17<\/td>\n-31463<\/td>\n-36980<\/td>\n<\/tr>\n
18<\/td>\n181802<\/td>\n-292989<\/td>\n<\/tr>\n
19<\/td>\n70660<\/td>\n–<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The last three, p[19,1] makes top 5000 list.
\nThe proof will be posted in the reply of this one.<\/p>\n","protected":false},"excerpt":{"rendered":"

Define: p[k,i]=ABS[1+2*n[k,i]*p[k-1,1]*p[k-1,2]],n[k,1] is the integer with minimum ABS[n[k,1]] that makes p[k,1] a prime number, and n[k,2] is the integer with second minimum ABS[n[k,2]] that makes p[k,2] a prime number The primality of p[k,i] can be proven using Brillhart – Lehmer – Selfridge algorithm recursively by using p[k-1,1] and p[k-1,2] as helper since n is a […]<\/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-189","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\/189","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=189"}],"version-history":[{"count":0,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/posts\/189\/revisions"}],"wp:attachment":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=189"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=189"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=189"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}