{"id":24,"date":"2008-05-14T16:34:50","date_gmt":"2008-05-14T20:34:50","guid":{"rendered":"http:\/\/bitc.bme.emory.edu\/~lzhou\/blogs\/?p=24"},"modified":"2008-05-19T10:39:10","modified_gmt":"2008-05-19T14:39:10","slug":"generalized-cullen-and-woodall-primes-can-be-twins","status":"publish","type":"post","link":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/?p=24","title":{"rendered":"Generalized Cullen and Woodall primes can be twins"},"content":{"rendered":"<p>Generalized Cullen Primes are defined as the primes of the form <em>n<\/em><sup>.<\/sup><em>b<\/em><sup><em>n<\/em><\/sup>+1 with <em>n<\/em>+2  &gt;  <em>b<\/em>, while generalized Woodall Primes as the form <em>n<\/em><sup>.<\/sup><em>b<\/em><sup><em>n<\/em><\/sup>-1 with <em>n<\/em>+2  &gt;  <em>b. <\/em>If we loose the restraint of <em>n<\/em>+2 &gt; <em>b<\/em> to a plain <em>n<\/em> &gt; 2, with every <em>b<\/em>&gt;1, except for <em>n<\/em> = <em>k<\/em>^<em>c<\/em> and mod(n, c)=0\uff0cthere is a conjuncture<\/p>\n<p>There is always an integer of b&gt;1 that makes <em>n<\/em><sup>.<\/sup><em>b<\/em><sup><em>n<\/em><\/sup>-1and <em>n<\/em><sup>.<\/sup><em>b<\/em><sup><em>n<\/em><\/sup>+1a pair of twin primes.<\/p>\n<p>Found terms as following:<\/p>\n<pre>2\t\t3\t\t1\r\n3\t\t4\t\t2\r\n--\t\t--\t\t--\r\n5\t\t570\t\t14\r\n6\t\t1820\t\t20\r\n7\t\t1464\t\t23\r\n8\t\t54\t\t14\r\n9\t\t60\t\t16\r\n10\t\t14025\t\t42\r\n11\t\t1932\t\t37\r\n12\t\t3029\t\t42\r\n13\t\t7194\t\t51\r\n14\t\t15\t\t17\r\n15\t\t3612\t\t54\r\n--\t\t--\t\t--\r\n17\t\t4746\t\t63\r\n18\t\t3154\t\t64\r\n19\t\t540\t\t53\r\n20\t\t150\t\t44\r\n21\t\t7060\t\t82\r\n22\t\t138\t\t48\r\n23\t\t80094\t\t114\r\n24\t\t6160\t\t92\r\n25\t\t33480\t\t114\r\n26\t\t93135\t\t130\r\n--\t\t--\t\t--\r\n28\t\t366618\t\t157\r\n29\t\t26058\t\t129\r\n30\t\t13516\t\t125\r\n31\t\t90510\t\t155\r\n32\t\t16836\t\t136\r\n33\t\t9824\t\t133\r\n34\t\t418875\t\t192\r\n35\t\t57246\t\t168\r\n--\t\t--\t\t--\r\n37\t\t182394\t\t196\r\n38\t\t64077\t\t184\r\n39\t\t14178\t\t163\r\n40\t\t943410\t\t241\r\n41\t\t36078\t\t189\r\n42\t\t78389\t\t208\r\n43\t\t314520\t\t239\r\n44\t\t15870\t\t187\r\n45\t\t194942\t\t240\r\n46\t\t15044700\t332\r\n47\t\t241944\t\t255\r\n48\t\t3871\t\t174\r\n49\t\t308730\t\t271\r\n50\t\t11604\t\t205\r\n51\t\t89492\t\t255\r\n52\t\t4745196\t\t349\r\n53\t\t388626\t\t298\r\n54\t\t3905\t\t196\r\n55\t\t60648\t\t265\r\n56\t\t26625\t\t250\r\n57\t\t44240\t\t267\r\n58\t\t198240\t\t310\r\n59\t\t178290\t\t312\r\n60\t\t937143\t\t361\r\n61\t\t403488\t\t344\r\n62\t\t19605\t\t268\r\n63\t\t19716\t\t273\r\n--\t\t--\t\t--\r\n65\t\t10098\t\t263\r\n66\t\t2029430\t\t419\r\n67\t\t420174\t\t379\r\n68\t\t423\t\t181\r\n69\t\t1177568\t\t421\r\n70\t\t772764\t\t415\r\n71\t\t580338\t\t412\r\n72\t\t1285530\t\t442\r\n73\t\t2978310\t\t475\r\n74\t\t885120\t\t442\r\n75\t\t68280\t\t365\r\n76\t\t158655\t\t398\r\n77\t\t1726236\t\t483\r\n78\t\t84826329\t621\r\n79\t\t1413132\t\t488\r\n80\t\t27852\t\t358\r\n81\t\t25092\t\t359\r\n82\t\t1611528\t\t511\r\n83\t\t413856\t\t469\r\n84\t\t45\t\t141\r\n85\t\t247272\t\t461\r\n86\t\t1232580\t\t526\r\n87\t\t26550\t\t387\r\n88\t\t52847043\t682\r\n89\t\t527892\t\t512\r\n90\t\t1416870\t\t556\r\n91\t\t448380\t\t517\r\n92\t\t79209\t\t453\r\n93\t\t204470\t\t496\r\n94\t\t448020\t\t534\r\n95\t\t228144\t\t512\r\n96\t\t666875\t\t562\r\n97\t\t215154\t\t520\r\n98\t\t71727\t\t478\r\n99\t\t3162208\t\t646\r\n--\t\t--\t\t--\r\n101\t\t274560\t\t552\r\n102\t\t7119669\t\t701\r\n103\t\t232464\t\t555\r\n104\t\t2007420\t\t658\r\n105\t\t186298\t\t556\r\n106\t\t484443570\t923\r\n107\t\t2822406\t\t693\r\n108\t\t16130583\t781\r\n109\t\t591780\t\t632\r\n110\t\t68748\t\t535\r\n111\t\t162498\t\t581\r\n112\t\t6032505\t\t762\r\n113\t\t171546\t\t594\r\n114\t\t56215\t\t544\r\n115\t\t323520\t\t636\r\n116\t\t6801570\t\t795\r\n117\t\t555676\t\t675\r\n118\t\t1679421\t\t737\r\n119\t\t701982\t\t698\r\n120\t\t58266208\t934\r\n121\t\t228858\t\t651\r\n122\t\t409011\t\t687\r\n123\t\t158620\t\t642\r\n124\t\t1553460\t\t770\r\n125\t\t1145286\t\t760\r\n126\t\t27835860\t941\r\n127\t\t1552446\t\t789\r\n128\t\t546417\t\t737\r\n129\t\t170172\t\t677\r\n130\t\t74235735\t1026\r\n131\t\t1259052\t\t802\r\n132\t\t1329566\t\t811\r\n133\t\t127584\t\t682\r\n134\t\t999180\t\t807\r\n135\t\t242580\t\t730\r\n136\t\t&gt;1E9\t\t--\r\n137\t\t10893030\t967\r\n138\t\t81732139\t1095\r\n139\t\t220122\t\t745\r\n140\t\t148491\t\t721\r\n141\t\t8406692\t\t979\r\n142\t\t744168\t\t836\r\n143\t\t1352616\t\t879\r\n--\t\t--\t\t--\r\n145\t\t7313424\t\t998\r\n146\t\t797505\t\t864\r\n147\t\t66890\t\t712\r\n148\t\t21504723\t1088\r\n149\t\t1754178\t\t933\r\n150\t\t17103770\t1088\r\n151\t\t11806812\t1071\r\n152\t\t--\t\t--\r\n153\t\t--\t\t--\r\n154\t\t--\t\t--\r\n155\t\t--\t\t--\r\n156\t\t--\t\t--\r\n157\t\t--\t\t--\r\n158\t\t422547\t\t933\r\n159\t\t158632\t\t892\r\n160\t\t573291\t\t924\r\n161\t\t284418\t\t881\r\n162\t\t--\t\t--\r\n163\t\t539016\t\t937\r\n164\t\t--\t\t--\r\n165\t\t169256\t\t865\r\n166\t\t--\t\t--\r\n167\t\t1054716\t\t1009\r\n168\t\t581204\t\t971\r\n169\t\t--\t\t--\r\n170\t\t1474092\t\t1051\r\n171\t\t712508\t\t1004\r\n172\t\t--\t\t--\r\n173\t\t603744\t\t1003\r\n174\t\t--\t\t--\r\n175\t\t--\t\t--\r\n176\t\t--\t\t--\r\n177\t\t--\t\t--\r\n178\t\t--\t\t--\r\n179\t\t440418\t\t1013\r\n180\t\t--\t\t--\r\n181\t\t1012620\t\t1090\r\n182\t\t497169\t\t1040\r\n183\t\t388934\t\t1026\r\n184\t\t--\t\t--\r\n185\t\t1580046\t\t1150\r\n186\t\t99465\t\t932\r\n187\t\t--\t\t--\r\n188\t\t--\t\t--\r\n189\t\t--\t\t--\r\n190\t\t--\t\t--\r\n191\t\t--\t\t--\r\n192\t\t--\t\t--\r\n193\t\t--\t\t--\r\n194\t\t1701405\t\t1212\r\n195\t\t--\t\t--\r\n196\t\t--\t\t--\r\n197\t\t--\t\t--\r\n198\t\t--\t\t--\r\n199\t\t--\t\t--\r\n200\t\t49839\t\t942\r\n201\t\t--\t\t--\r\n202\t\t--\t\t--\r\n203\t\t859146\t\t1207\r\n204\t\t--\t\t--\r\n205\t\t239604\t\t1106\r\n<\/pre>\n<p>The dashes in the first column means that that item does not exist theoretically. The other dashes means not-yet-found terms.  Currently processed to 200.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Generalized Cullen Primes are defined as the primes of the form n.bn+1 with n+2 &gt; b, while generalized Woodall Primes as the form n.bn-1 with n+2 &gt; b. If we loose the restraint of n+2 &gt; b to a plain n &gt; 2, with every b&gt;1, except for n = k^c and mod(n, c)=0\uff0cthere is [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[18],"class_list":["post-24","post","type-post","status-publish","format-standard","hentry","category-looking-for-a-megaprime","tag-actively-updated","post-blog"],"_links":{"self":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/posts\/24","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=24"}],"version-history":[{"count":0,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/posts\/24\/revisions"}],"wp:attachment":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=24"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=24"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=24"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}