{"id":157,"date":"2010-09-06T09:39:39","date_gmt":"2010-09-06T13:39:39","guid":{"rendered":"http:\/\/bitc.bme.emory.edu\/~lzhou\/blogs\/?p=157"},"modified":"2010-09-06T09:39:39","modified_gmt":"2010-09-06T13:39:39","slug":"recursive-prime-pk1pkpk-m-1","status":"publish","type":"post","link":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/?p=157","title":{"rendered":"Recursive prime p(k+1)=p(k)*(p(k)+\/-m)+\/-1"},"content":{"rendered":"<p>Define p(1)=2<br \/>\np(2)[m=-1; +1] = p(1)*(p(1)-1)+1 = 3<br \/>\np(3)[m=-1; -1] = p(2)*(p(2)-1)-1 = 5<br \/>\np(4)[m=-1; -1] = p(3)*(p(3)-1)-1 = 19<br \/>\np(5)[m=+1; -1] = p(4)*(p(4)+1)-1 = 379<br \/>\np(6)[m=-1; -1] = p(5)*(p(5)-1)-1 = 143261<br \/>\np(7)[m=-11; -1] = p(6)*(p(6)-11)-1 = 20522138249<br \/>\np(8)[m=-11; +1] = p(7)*(p(7)-11)+1 = 421158158085325265263<br \/>\np(9)[m=-13; +1] = p(8)*(p(8)-13)+1 = 421158158085325265256.5^2-165\/4<br \/>\np(10)[m=+59; -1] = p(9)*(p(9)+59)-1 = (421158158085325265256.5^2-47\/4)^2-3485\/4<br \/>\np(11)[m=+45; +1] = p(10)*(p(10)+45)+1<br \/>\n         = ((421158158085325265256.5^2-47\/4)^2-3575\/4)^2-2021\/4<br \/>\np(12)[m=-441; +1] = p(11)*(p(11)-441)+1<br \/>\n         = (((421158158085325265256.5^2-47\/4)^2-3575\/4)^2-2903\/4)^2-194477\/4<br \/>\np(13)[m=+127; -1] = p(12)*(p(12)+127)-1<br \/>\n         = ((((421158158085325265256.5^2-47\/4)^2-3575\/4)^2-2903\/4)^2-194223\/4)^2-16133\/4<br \/>\np(14)[m=-269; +1] = p(13)*(p(13)-269)+1<br \/>\n         = (((((421158158085325265256.5^2-47\/4)^2-3575\/4)^2-2903\/4)^2-194223\/4)^2-16671\/4)^2-72357\/4<br \/>\np(15)[m=+973; -1] = p(14)*(p(14)+973)-1<br \/>\n         = ((((((421158158085325265256.5^2-47\/4)^2-3575\/4)^2-2903\/4)^2-194223\/4)^2-16671\/4)^2-70411\/4)^2-946733\/4<br \/>\np(16)[m=-55; -1] = p(15)*(p(15)-55)-1<br \/>\n         = ((((((((1\/2+421158158085325265256)^2-47\/4)^2-3575\/4)^2-2903\/4)^2-194223\/4)^2-16671\/4)^2-70411\/4)^2-946843\/4)^2-3029\/4<br \/>\np(17)[m=-7939; -1] = p(16)*(p(16)-7939)-1<br \/>\n         = ((((((((421158158085325265256.5^2-47\/4)^2-3575\/4)^2-2903\/4)^2-194223\/4)^2-16671\/4)^2-70411\/4)^2-946843\/4)^2-18907\/4)^2-63027725\/4<br \/>\np(18)[m=+17897; +1] = p(17)*(p(17)+17897)+1<br \/>\n         = (((((((((421158158085325265256.5^2-47\/4)^2-3575\/4)^2-2903\/4)^2-194223\/4)^2-16671\/4)^2-70411\/4)^2-946843\/4)^2-18907\/4)^2-62991931\/4)^2-320302605\/4<br \/>\np(19)[m=+9881; -1] = p(18)*(p(18)+9881)-1<br \/>\n         = ((((((((((421158158085325265256.5^2-47\/4)^2-3575\/4)^2-2903\/4)^2-194223\/4)^2-16671\/4)^2-70411\/4)^2-946843\/4)^2-18907\/4)^2-62991931\/4)^2-320282843\/4)^2-97634165\/4<br \/>\np(20)[m=+5017; -1] = p(19)*(p(19)+5017)-1<br \/>\n         = (((((((((((421158158085325265256.5^2-47\/4)^2-3575\/4)^2-2903\/4)^2-194223\/4)^2-16671\/4)^2-70411\/4)^2-946843\/4)^2-18907\/4)^2-62991931\/4)^2-320282843\/4)^2-97624131\/4)^2-25170293\/4<br \/>\np(21)[m=-300019; -1] = p(20)*(p(20)-300019)-1<br \/>\n         = (((((((((((((842316316170650530513\/2)^2-47\/4)^2-3575\/4)^2-2903\/4)^2-194223\/4)^2-16671\/4)^2-70411\/4)^2-946843\/4)^2-18907\/4)^2-62991931\/4)^2-320282843\/4)^2-97624131\/4)^2-25770331\/4)^2-90011400365\/4<\/p>\n<p>p(21) has database ID 94526 in <a href=\"http:\/\/primes.utm.edu\/primes\/home.php\">The List of Largest Known Primes Home Page<\/a>.  The direct link is <a href=\"http:\/\/primes.utm.edu\/primes\/page.php?id=94526\">HERE<\/a>.<\/p>\n<p>These primes are recursively proven using OpenPFGW, by the command<br \/>\npfgw -t (or tp) -h&#8221;p(k)&#8221; p(k+1)<\/p>\n<p>The final certification code is<br \/>\n#!\/bin\/sh<br \/>\n#!\/bin\/sh<br \/>\n.\/pfgw -l&#8221;pmdance.21.cert&#8221; -t p_07<br \/>\n.\/pfgw -l&#8221;pmdance.21.cert&#8221; -t -h&#8221;p_07&#8243; p_08<br \/>\n.\/pfgw -l&#8221;pmdance.21.cert&#8221; -t -h&#8221;p_08&#8243; p_09<br \/>\n.\/pfgw -l&#8221;pmdance.21.cert&#8221; -tp -h&#8221;p_09&#8243; p_10<br \/>\n.\/pfgw -l&#8221;pmdance.21.cert&#8221; -t -h&#8221;p_10&#8243; p_11<br \/>\n.\/pfgw -l&#8221;pmdance.21.cert&#8221; -t -h&#8221;p_11&#8243; p_12<br \/>\n.\/pfgw -l&#8221;pmdance.21.cert&#8221; -tp -h&#8221;p_12&#8243; p_13<br \/>\n.\/pfgw -l&#8221;pmdance.21.cert&#8221; -t -h&#8221;p_13&#8243; p_14<br \/>\n.\/pfgw -l&#8221;pmdance.21.cert&#8221; -tp -h&#8221;p_14&#8243; p_15<br \/>\n.\/pfgw -l&#8221;pmdance.21.cert&#8221; -tp -h&#8221;p_15&#8243; p_16<br \/>\n.\/pfgw -l&#8221;pmdance.21.cert&#8221; -tp -h&#8221;p_16&#8243; p_17<br \/>\n.\/pfgw -l&#8221;pmdance.21.cert&#8221; -t -h&#8221;p_17&#8243; p_18<br \/>\n.\/pfgw -l&#8221;pmdance.21.cert&#8221; -tp -h&#8221;p_18&#8243; p_19<br \/>\n.\/pfgw -l&#8221;pmdance.21.cert&#8221; -tp -h&#8221;p_19&#8243; p_20<br \/>\n.\/pfgw -l&#8221;pmdance.21.cert&#8221; -tp -h&#8221;p_20&#8243; p_21<\/p>\n<p>The certificate is the following:<br \/>\nPrimality testing 20522138249 [N-1, Brillhart-Lehmer-Selfridge]<br \/>\nRunning N-1 test using base 3<br \/>\nCalling Brillhart-Lehmer-Selfridge with factored part 91.18%<br \/>\n20522138249 is prime! (0.0013s+0.0001s)<br \/>\nPrimality testing 421158158085325265263 [N-1, Brillhart-Lehmer-Selfridge]<br \/>\nRunning N-1 test using base 3<br \/>\nCalling Brillhart-Lehmer-Selfridge with factored part 50.00%<br \/>\n421158158085325265263 is prime! (0.0020s+0.0002s)<br \/>\nPrimality testing (842316316170650530513^2-165)\/4 [N-1, Brillhart-Lehmer-Selfridge]<br \/>\nRunning N-1 test using base 3<br \/>\nCalling Brillhart-Lehmer-Selfridge with factored part 49.64%<br \/>\n(842316316170650530513^2-165)\/4 is prime! (0.0024s+0.0002s)<br \/>\nPrimality testing (((842316316170650530513^2-47)\/2)^2-3485)\/4 [N+1, Brillhart-Lehmer-Selfridge]<br \/>\nRunning N+1 test using discriminant 3, base 1+sqrt(3)<br \/>\nCalling Brillhart-Lehmer-Selfridge with factored part 50.00%<br \/>\n(((842316316170650530513^2-47)\/2)^2-3485)\/4 is prime! (0.0051s+0.0002s)<br \/>\nPrimality testing (((((842316316170650530513^2-47)\/2)^2-3575)\/2)^2-2021)\/4 [N-1, Brillhart-Lehmer-Selfridge]<br \/>\nRunning N-1 test using base 3<br \/>\nCalling Brillhart-Lehmer-Selfridge with factored part 50.00%<br \/>\n(((((842316316170650530513^2-47)\/2)^2-3575)\/2)^2-2021)\/4 is prime! (0.0067s+0.0003s)<br \/>\nPrimality testing (((((((842316316170650530513^2-47)\/2)^2-3575)\/2)^2-2903)\/2)^2-194477)\/4 [N-1, Brillhart-Lehmer-Selfridge]<br \/>\nRunning N-1 test using base 3<br \/>\nCalling Brillhart-Lehmer-Selfridge with factored part 50.00%<br \/>\n(((((((842316316170650530513^2-47)\/2)^2-3575)\/2)^2-2903)\/2)^2-194477)\/4 is prime! (0.0142s+0.0003s)<br \/>\nPrimality testing (((((((((842316316170650530513^2-47)\/2)^2-3575)\/2)^2-2903)\/2)^2-194223)\/2)^2-16133)\/4 [N+1, Brillhart-Lehmer-Selfridge]<br \/>\nRunning N+1 test using discriminant 11, base 1+sqrt(11)<br \/>\nCalling Brillhart-Lehmer-Selfridge with factored part 50.00%<br \/>\n(((((((((842316316170650530513^2-47)\/2)^2-3575)\/2)^2-2903)\/2)^2-194223)\/2)^2-16133)\/4 is prime! (0.1029s+0.0003s)<br \/>\nPrimality testing (((((((((((842316316170650530513^2-47)\/2)^2-3575)\/2)^2-2903)\/2)^2-194223)\/2)^2-16671)\/2)^2-72357)\/4 [N-1, Brillhart-Lehmer-Selfridge]<br \/>\nRunning N-1 test using base 3<br \/>\nCalling Brillhart-Lehmer-Selfridge with factored part 50.00%<br \/>\n(((((((((((842316316170650530513^2-47)\/2)^2-3575)\/2)^2-2903)\/2)^2-194223)\/2)^2-16671)\/2)^2-72357)\/4 is prime! (0.1451s+0.0004s)<br \/>\nPrimality testing (((((((((((((842316316170650530513^2-47)\/2)^2-3575)\/2)^2-2903)\/2)^2-194223)\/2)^2-16671)\/2)^2-70411)\/2)^2-946733)\/4 [N+1, Brillhart-Lehmer-Selfridge]<br \/>\nRunning N+1 test using discriminant 3, base 7+sqrt(3)<br \/>\nCalling Brillhart-Lehmer-Selfridge with factored part 49.99%<br \/>\n(((((((((((((842316316170650530513^2-47)\/2)^2-3575)\/2)^2-2903)\/2)^2-194223)\/2)^2-16671)\/2)^2-70411)\/2)^2-946733)\/4 is prime! (1.5795s+0.0004s)<br \/>\nPrimality testing (((((((((((((((842316316170650530513^2-47)\/2)^2-3575)\/2)^2-2903)\/2)^2-194223)\/2)^2-16671)\/2)^2-70411)\/2)^2-946843)\/2)^2-3029)\/4 [N+1, Brillhart-Lehmer-Selfridge]<br \/>\nRunning N+1 test using discriminant 5, base 1+sqrt(5)<br \/>\nCalling Brillhart-Lehmer-Selfridge with factored part 50.00%<br \/>\n(((((((((((((((842316316170650530513^2-47)\/2)^2-3575)\/2)^2-2903)\/2)^2-194223)\/2)^2-16671)\/2)^2-70411)\/2)^2-946843)\/2)^2-3029)\/4 is prime! (6.6442s+0.0006s)<br \/>\nPrimality testing (((((((((((((((((842316316170650530513^2-47)\/2)^2-3575)\/2)^2-2903)\/2)^2-194223)\/2)^2-16671)\/2)^2-70411)\/2)^2-946843)\/2)^2-18907)\/2)^2-63027725)\/4 [N+1, Brillhart-Lehmer-Selfridge]<br \/>\nRunning N+1 test using discriminant 3, base 1+sqrt(3)<br \/>\nCalling Brillhart-Lehmer-Selfridge with factored part 50.00%<br \/>\n(((((((((((((((((842316316170650530513^2-47)\/2)^2-3575)\/2)^2-2903)\/2)^2-194223)\/2)^2-16671)\/2)^2-70411)\/2)^2-946843)\/2)^2-18907)\/2)^2-63027725)\/4 is prime! (29.5119s+0.0009s)<br \/>\nPrimality testing (((((((((((((((((((842316316170650530513^2-47)\/2)^2-3575)\/2)^2-2903)\/2)^2-194223)\/2)^2-16671)\/2)^2-70411)\/2)^2-946843)\/2)^2-18907)\/2)^2-62991931)\/2)^2-320302605)\/4 [N-1, Brillhart-Lehmer-Selfrid<br \/>\nge]<br \/>\nRunning N-1 test using base 13<br \/>\nCalling Brillhart-Lehmer-Selfridge with factored part 50.00%<br \/>\n(((((((((((((((((((842316316170650530513^2-47)\/2)^2-3575)\/2)^2-2903)\/2)^2-194223)\/2)^2-16671)\/2)^2-70411)\/2)^2-946843)\/2)^2-18907)\/2)^2-62991931)\/2)^2-320302605)\/4 is prime! (44.2155s+0.0017s)<br \/>\nPrimality testing (((((((((((((((((((((842316316170650530513^2-47)\/2)^2-3575)\/2)^2-2903)\/2)^2-194223)\/2)^2-16671)\/2)^2-70411)\/2)^2-946843)\/2)^2-18907)\/2)^2-62991931)\/2)^2-320282843)\/2)^2-97634165)\/4 [N+1, Brillha<br \/>\nrt-Lehmer-Selfridge]<br \/>\nRunning N+1 test using discriminant 3, base 3+sqrt(3)<br \/>\nCalling Brillhart-Lehmer-Selfridge with factored part 50.00%<br \/>\n(((((((((((((((((((((842316316170650530513^2-47)\/2)^2-3575)\/2)^2-2903)\/2)^2-194223)\/2)^2-16671)\/2)^2-70411)\/2)^2-946843)\/2)^2-18907)\/2)^2-62991931)\/2)^2-320282843)\/2)^2-97634165)\/4 is prime! (518.2041s+0.0039s)<br \/>\nPrimality testing (((((((((((((((((((((((842316316170650530513^2-47)\/2)^2-3575)\/2)^2-2903)\/2)^2-194223)\/2)^2-16671)\/2)^2-70411)\/2)^2-946843)\/2)^2-18907)\/2)^2-62991931)\/2)^2-320282843)\/2)^2-97624131)\/2)^2-25170293<br \/>\n)\/4 [N+1, Brillhart-Lehmer-Selfridge]<br \/>\nRunning N+1 test using discriminant 3, base 3+sqrt(3)<br \/>\nCalling Brillhart-Lehmer-Selfridge with factored part 50.00%<br \/>\n(((((((((((((((((((((((842316316170650530513^2-47)\/2)^2-3575)\/2)^2-2903)\/2)^2-194223)\/2)^2-16671)\/2)^2-70411)\/2)^2-946843)\/2)^2-18907)\/2)^2-62991931)\/2)^2-320282843)\/2)^2-97624131)\/2)^2-25170293)\/4 is prime! (215<br \/>\n3.5087s+0.0071s)<br \/>\nPrimality testing (((((((((((((((((((((((((842316316170650530513^2-47)\/2)^2-3575)\/2)^2-2903)\/2)^2-194223)\/2)^2-16671)\/2)^2-70411)\/2)^2-946843)\/2)^2-18907)\/2)^2-62991931)\/2)^2-320282843)\/2)^2-97624131)\/2)^2-257703<br \/>\n31)\/2)^2-90011400365)\/4 [N+1, Brillhart-Lehmer-Selfridge]<br \/>\nRunning N+1 test using discriminant 7, base 1+sqrt(7)<br \/>\nCalling Brillhart-Lehmer-Selfridge with factored part 50.00%<br \/>\n(((((((((((((((((((((((((842316316170650530513^2-47)\/2)^2-3575)\/2)^2-2903)\/2)^2-194223)\/2)^2-16671)\/2)^2-70411)\/2)^2-946843)\/2)^2-18907)\/2)^2-62991931)\/2)^2-320282843)\/2)^2-97624131)\/2)^2-25770331)\/2)^2-900114003<br \/>\n65)\/4 is prime! (9575.0905s+0.0157s)<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Define p(1)=2 p(2)[m=-1; +1] = p(1)*(p(1)-1)+1 = 3 p(3)[m=-1; -1] = p(2)*(p(2)-1)-1 = 5 p(4)[m=-1; -1] = p(3)*(p(3)-1)-1 = 19 p(5)[m=+1; -1] = p(4)*(p(4)+1)-1 = 379 p(6)[m=-1; -1] = p(5)*(p(5)-1)-1 = 143261 p(7)[m=-11; -1] = p(6)*(p(6)-11)-1 = 20522138249 p(8)[m=-11; +1] = p(7)*(p(7)-11)+1 = 421158158085325265263 p(9)[m=-13; +1] = p(8)*(p(8)-13)+1 = 421158158085325265256.5^2-165\/4 p(10)[m=+59; -1] = p(9)*(p(9)+59)-1 = [&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-157","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\/157","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=157"}],"version-history":[{"count":0,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=\/wp\/v2\/posts\/157\/revisions"}],"wp:attachment":[{"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=157"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=157"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/csic.som.emory.edu\/~lzhou\/blogs\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=157"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}