{"id":191,"date":"2010-12-13T13:09:59","date_gmt":"2010-12-13T17:09:59","guid":{"rendered":"http:\/\/bitc.bme.emory.edu\/~lzhou\/blogs\/?p=191"},"modified":"2010-12-13T13:09:59","modified_gmt":"2010-12-13T17:09:59","slug":"19th-recursive-prime-in-the-form-of-pk1lk1mk2-nk2-1-while-pkmknk","status":"publish","type":"post","link":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/?p=191","title":{"rendered":"19th Recursive prime in the form of p[k+1]=l(k+1)*(m[k]^2-n[k]^2)+\/-1, while p[k]=m[k]+n[k]"},"content":{"rendered":"<p>Define:<br \/>\np[1] = m[1] + n[1] = 2 + 1<br \/>\np[2](l=2) = 2*(m[1]^2-n[1]^2)-1 = 2*(4-1)-1 = 5 = 2^3 &#8211; 3<br \/>\np[3](l=2) = 2*(2^6-3^2)-1 = 2^7 &#8211; 19<br \/>\np[4](l=4) = 4*(2^14 &#8211; 19^2) -1 = 2^16 &#8211; 1445<br \/>\np[5](l=8) = 8*(2^32 &#8211; 1445^2) +1 = 2^35 &#8211; 16704199<br \/>\np[6](l=18) = 2^71*3^2 &#8211; (16704199^2*18+1)<br \/>\np[7](l=84) = 2^144*3^5*7 &#8211; ((16704199^2*18+1)^2*84+1)<br \/>\np[8](l=228) = 2^290*3^11*7^2*19 &#8211; ((((1445^2*8-1)^2*18+1)^2*84+1)^2*228-1)<br \/>\np[9](l=100) = 2^582*3^22*5^2*7^4*19^2 &#8211; (((((1445^2*8-1)^2*18+1)^2*84+1)^2*228-1)^2*100+1)<br \/>\np[10](l=858) = 2^1165*3^45*5^4*7^8*11*13*19^4 &#8211; ((((((1445^2*8-1)^2*18+1)^2*84+1)^2*228-1)^2*100+1)^2*858+1)<br \/>\np[11](l=790) = 2^2331*3^90*5^9*7^16*11^2*13^2*19^8*79 &#8211; (((((((1445^2*8-1)^2*18+1)^2*84+1)^2*228-1)^2*100+1)^2*858+1)^2*790+1)<br \/>\np[12](l=1110) = 2^4663*3^181*5^19*7^32*11^4*13^4*19^16*37*79^2 &#8211; ((((((((1445^2*8-1)^2*18+1)^2*84+1)^2*228-1)^2*100+1)^2*858+1)^2*790+1)^2*1110-1)<br \/>\np[13](l=1602) = 2^9327*3^364*5^38*7^64*11^8*13^8*19^32*37^2*79^4*89 &#8211; (((((((((1445^2*8-1)^2*18+1)^2*84+1)^2*228-1)^2*100+1)^2*858+1)^2*790+1)^2*1110-1)^2*1602-1)<br \/>\np[14](l=6600) = 2^18657*3^729*5^78*7^128*11^17*13^16*19^64*37^4*79^8*89^2 &#8211; ((((((((((1445^2*8-1)^2*18+1)^2*84+1)^2*228-1)^2*100+1)^2*858+1)^2*790+1)^2*1110-1)^2*1602-1)^2*6600+1)<br \/>\np[15](l=11940) = 2^37316*3^1459*5^157*7^256*11^34*13^32*19^128*37^8*79^16*89^4*199 &#8211; (((((((((((1445^2*8-1)^2*18+1)^2*84+1)^2*228-1)^2*100+1)^2*858+1)^2*790+1)^2*1110-1)^2*1602-1)^2*6600+1)^2*11940+1)<br \/>\np[16](l=52806) = 2^74633*3^2919*5^314*7^512*11^68*13^65*19^256*37^16*79^32*89^8*199^2*677 &#8211; ((((((((((((1445^2*8-1)^2*18+1)^2*84+1)^2*228-1)^2*100+1)^2*858+1)^2*790+1)^2*1110-1)^2*1602-1)^2*6600+1)^2*11940+1)^2*52806+1)<br \/>\np[17](l=64860) = 2^149268*3^5839*5^629*7^1024*11^136*13^130*19^512*23*37^32*47*79^64*89^16*199^4*677^2 &#8211; (((((((((((((1445^2*8-1)^2*18+1)^2*84+1)^2*228-1)^2*100+1)^2*858+1)^2*790+1)^2*1110-1)^2*1602-1)^2*6600+1)^2*11940+1)^2*52806+1)^2*64860-1)<br \/>\np[18](l=229682) = 2^298537*3^11678*5^1258*7^2048*11^272*13^260*19^1024*23^2*37^64*41*47^2*79^128*89^32*199^8*677^4*2801 &#8211; (((((((((((((((1445^2*8-1)^2*18+1)^2*84+1)^2*228-1)^2*100+1)^2*858+1)^2*790+1)^2*1110-1)^2*1602-1)^2*6600+1)^2*11940+1)^2*52806+1)^2*64860-1)^2*229682-1)<br \/>\np[19](l=192788) = 2^597076*3^23356*5^2516*7^4096*11^544*13^520*19^2048*23^4*37^128*41^2*47^4*79^256*89^64*199^16*677^8*2801^2*48197-(((((((((((((((1445^2*8-1)^2*18+1)^2*84+1)^2*228-1)^2*100+1)^2*858+1)^2*790+1)^2*1110-1)^2*1602-1)^2*6600+1)^2*11940+1)^2*52806+1)^2*64860-1)<br \/>\n2*229682-1)^2*192788-1)<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Define: p[1] = m[1] + n[1] = 2 + 1 p[2](l=2) = 2*(m[1]^2-n[1]^2)-1 = 2*(4-1)-1 = 5 = 2^3 &#8211; 3 p[3](l=2) = 2*(2^6-3^2)-1 = 2^7 &#8211; 19 p[4](l=4) = 4*(2^14 &#8211; 19^2) -1 = 2^16 &#8211; 1445 p[5](l=8) = 8*(2^32 &#8211; 1445^2) +1 = 2^35 &#8211; 16704199 p[6](l=18) = 2^71*3^2 &#8211; (16704199^2*18+1) p[7](l=84) = [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-191","post","type-post","status-publish","format-standard","hentry","category-uncategorized","post-blog"],"_links":{"self":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/posts\/191","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=191"}],"version-history":[{"count":0,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/posts\/191\/revisions"}],"wp:attachment":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=191"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=191"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=191"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}