{"id":118,"date":"2010-05-17T11:05:43","date_gmt":"2010-05-17T15:05:43","guid":{"rendered":"http:\/\/bitc.bme.emory.edu\/~lzhou\/blogs\/?p=118"},"modified":"2010-05-17T12:09:37","modified_gmt":"2010-05-17T16:09:37","slug":"recursive-prime-triplet-by-brillhart-lehmer-selfridge-algorithm","status":"publish","type":"post","link":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/?p=118","title":{"rendered":"Recursive prime triplet by Brillhart &#8211; Lehmer &#8211; Selfridge algorithm"},"content":{"rendered":"<p>Take any three primes, say p[1,0], p[2,0], and p[3,0].<\/p>\n<p>Define:<\/p>\n<p>p[i,j]=ABS[1+2*n[i,j]*p[(i+1) mod 3,j-1]*p[(i+2) mod 3,j-1]],n is the integer with minimum ABS[n] that makes p[i,j] a prime number.<\/p>\n<p>The primality of p[i,j] can be proven using Brillhart &#8211; Lehmer &#8211; Selfridge algorithm recursively by using p[(i+1) mod 3,j-1] and p[(i+2) mod 3,j-1] as helper since n is a small integer, by reducing j to 0.<\/p>\n<p>With this idea, taking<\/p>\n<p>p[1,0]=3, p[2,0]=5, p[3,0]=7<\/p>\n<p>We got the n[i,j] ( columns : j; rows: i):<\/p>\n<table border=\"1\">\n<tbody>\n<tr>\n<th>i<\/th>\n<th>j=1<\/th>\n<th>j=2<\/th>\n<th>j=3<\/th>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>1<\/td>\n<td>-1<\/td>\n<td>-1<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>-1<\/td>\n<td>2<\/td>\n<td>-1<\/td>\n<\/tr>\n<tr>\n<td>3<\/td>\n<td>-8<\/td>\n<td>-10<\/td>\n<td>7<\/td>\n<\/tr>\n<tr>\n<td>4<\/td>\n<td>-14<\/td>\n<td>-3<\/td>\n<td>-13<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>-18<\/td>\n<td>24<\/td>\n<td>46<\/td>\n<\/tr>\n<tr>\n<td>6<\/td>\n<td>24<\/td>\n<td>39<\/td>\n<td>-32<\/td>\n<\/tr>\n<tr>\n<td>7<\/td>\n<td>225<\/td>\n<td>-48<\/td>\n<td>27<\/td>\n<\/tr>\n<tr>\n<td>8<\/td>\n<td>120<\/td>\n<td>-76<\/td>\n<td>30<\/td>\n<\/tr>\n<tr>\n<td>9<\/td>\n<td>-132<\/td>\n<td>245<\/td>\n<td>-676<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>316<\/td>\n<td>-722<\/td>\n<td>65<\/td>\n<\/tr>\n<tr>\n<td>11<\/td>\n<td>55<\/td>\n<td>-1197<\/td>\n<td>-510<\/td>\n<\/tr>\n<tr>\n<td>12<\/td>\n<td>-427<\/td>\n<td>-1716<\/td>\n<td>-637<\/td>\n<\/tr>\n<tr>\n<td>13<\/td>\n<td>4651<\/td>\n<td>-1158<\/td>\n<td>3420<\/td>\n<\/tr>\n<tr>\n<td>14<\/td>\n<td>-16337<\/td>\n<td>17640<\/td>\n<td>-18426<\/td>\n<\/tr>\n<tr>\n<td>15<\/td>\n<td>-8915<\/td>\n<td>-70649<\/td>\n<td>-31489<\/td>\n<\/tr>\n<tr>\n<td>16<\/td>\n<td>-18844<\/td>\n<td>-92841<\/td>\n<td>124053<\/td>\n<\/tr>\n<tr>\n<td>17<\/td>\n<td>-144011<\/td>\n<td>-8853<\/td>\n<td>-14042<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The last three, p[i,17] makes top 5000 list.<br \/>\nThe proof will be posted in the reply of this one.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Take any three primes, say p[1,0], p[2,0], and p[3,0]. Define: p[i,j]=ABS[1+2*n[i,j]*p[(i+1) mod 3,j-1]*p[(i+2) mod 3,j-1]],n is the integer with minimum ABS[n] that makes p[i,j] a prime number. The primality of p[i,j] can be proven using Brillhart &#8211; Lehmer &#8211; Selfridge algorithm recursively by using p[(i+1) mod 3,j-1] and p[(i+2) mod 3,j-1] as helper since n [&hellip;]<\/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-118","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\/118","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=118"}],"version-history":[{"count":0,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/posts\/118\/revisions"}],"wp:attachment":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=118"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=118"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=118"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}