{"id":26810,"date":"2023-09-26T20:20:40","date_gmt":"2023-09-26T19:20:40","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=26810"},"modified":"2023-10-06T18:15:56","modified_gmt":"2023-10-06T17:15:56","slug":"m_82589933","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=26810","title":{"rendered":"M82589933"},"content":{"rendered":"\n<p>O maior n\u00famero primo conhecido (em setembro de 2023) \u00e9 \\({2^{82\\,589\\,933}} &#8211; 1\\), um n\u00famero que tem \\(24\\,862\\,048\\) d\u00edgitos quando escrito na base 10. Foi encontrado por meio de um computador, oferecido voluntariamente por <a href=\"https:\/\/www.theregister.com\/2019\/01\/03\/largest_prime_number\/\" target=\"_blank\" rel=\"noopener\">Patrick Laroche<\/a>, utilizando o software gratuito da <a href=\"https:\/\/en.wikipedia.org\/wiki\/Great_Internet_Mersenne_Prime_Search\" target=\"_blank\" rel=\"noopener\">Great Internet Mersenne Prime Search<\/a> (GIMPS), em dezembro de 2018.<\/p>\n<p>Muitos dos maiores n\u00fameros primos conhecidos s\u00e3o primos de Mersenne [<a href=\"https:\/\/pt.wikipedia.org\/wiki\/Primo_de_Mersenne\" target=\"_blank\" rel=\"noopener\">Primo de Mersenne<\/a> <span style=\"text-decoration: underline;\">\u00e9 um n\u00famero de Mersenne<\/span> (n\u00famero da forma \\({M_n} = {2^n} &#8211; 1\\), com &#8220;\\(n\\)&#8221; n\u00famero natural) <span style=\"text-decoration: underline;\">que tamb\u00e9m \u00e9 um n\u00famero primo<\/span>.], porque podem utilizar um <a href=\"https:\/\/pt.wikipedia.org\/wiki\/Teste_de_primalidade\" target=\"_blank\" rel=\"noopener\">teste de primalidade<\/a> especializado que \u00e9 mais r\u00e1pido que o teste geral.<\/p>\n<p>O recorde \u00e9 atualmente detido por \\({2^{82\\,589\\,933}} &#8211; 1\\) com \\(24\\,862\\,048\\) d\u00edgitos.<br \/>Os primeiros e \u00faltimos 120 d\u00edgitos deste n\u00famero s\u00e3o mostrados abaixo:<\/p>\n<table class=\" aligncenter\" style=\"width: 80%; border-collapse: collapse;\">\n<tbody>\n<tr>\n<td style=\"width: 100%;\">\n<p style=\"text-align: left;\">148894445742041325547806458472397916603026273992795324185271<br \/>289425213239361064475310309971132180337174752834401423587560 &#8230;<\/p>\n<p style=\"text-align: left;\">(24.861.808 d\u00edgitos ignorados)<\/p>\n<p style=\"text-align: left;\">&#8230;\u00a006210755794795829753159520880719269367652178218447252664007<br \/>6912114355308311969487633766457823695074037951210325217902591.<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Em caso de curiosidade, pode apreciar os \\(24\\,862\\,048\\) d\u00edgitos de \\({2^{82\\,589\\,933}} &#8211; 1\\) aqui: <a href=\"https:\/\/www.mersenne.org\/primes\/digits\/M82589933.zip\" target=\"_blank\" rel=\"noopener\">M82589933.zip<\/a><\/p>\n<ul style=\"list-style-type: disc;\">\n<li>Fonte: Wikipedia | <a href=\"https:\/\/en.wikipedia.org\/wiki\/Largest_known_prime_number\" target=\"_blank\" rel=\"noopener\">Largest known prime number<\/a><\/li>\n<\/ul>\n<p>\u00a0<\/p>\n<h2>Desafio<\/h2>\n<p>Prove que 1 \u00e9 o algarismo das unidades do n\u00famero primo \\({2^{82\\,589\\,933}} &#8211; 1\\).<\/p>\n<p><div id=\"Dica1-link-26810\" class=\"sh-link Dica1-link sh-hide\"><a href=\"#\" onclick=\"showhide_toggle('Dica1', 26810, 'Mostrar Dica', 'Ocultar Dica'); return false;\" aria-expanded=\"false\"><span id=\"Dica1-toggle-26810\">Mostrar Dica<\/span><\/a><\/div><div id=\"Dica1-content-26810\" class=\"sh-content Dica1-content sh-hide\" style=\"display: none;\">Poder\u00e1 ser \u00fatil come\u00e7ar por recordar o t\u00f3pico <strong>Pot\u00eancias de base inteira n\u00e3o negativa e expoente natural<\/strong>: <a href=\"https:\/\/www.rtp.pt\/play\/estudoemcasa\/p7799\/e503097\/matematica-5-e-6-anos\" target=\"_blank\" rel=\"noopener\">https:\/\/www.rtp.pt\/play\/estudoemcasa\/p7799\/e503097\/matematica-5-e-6-anos<\/a>\u00a0<\/p>\n<p><br \/>Se a sugest\u00e3o anterior \u00e9 insuficiente, recorde tamb\u00e9m <span style=\"font-size: revert; color: initial;\">o t\u00f3pico <\/span><strong style=\"font-size: revert; color: initial;\">Crit\u00e9rios de divisibilidade por 2, 3, 4, 5 e 9<\/strong><span style=\"font-size: revert; color: initial;\">: <a href=\"https:\/\/www.rtp.pt\/play\/estudoemcasa\/p7799\/e501590\/matematica-5-e-6-anos\" target=\"_blank\" rel=\"noopener\">https:\/\/www.rtp.pt\/play\/estudoemcasa\/p7799\/e501590\/matematica-5-e-6-anos<\/a> <\/div><\/span><\/p>\n<p>\u00a0<\/p>\n<p><strong>Hiperliga\u00e7\u00f5es relacionadas<\/strong>:<\/p>\n<ul style=\"list-style-type: square;\">\n<li><a href=\"https:\/\/www.frontiersin.org\/\" target=\"_blank\" rel=\"noopener\">Frontiers<\/a> | <a href=\"https:\/\/kids.frontiersin.org\/\" target=\"_blank\" rel=\"noopener\">Frontiers for Young Minds<\/a> | <a href=\"https:\/\/kids.frontiersin.org\/articles\/10.3389\/frym.2018.00040\" target=\"_blank\" rel=\"noopener\">Prime Numbers\u2013Why are They So Exciting?<\/a><\/li>\n<li><a href=\"https:\/\/t5k.org\/\" target=\"_blank\" rel=\"noopener\">PrimePages<\/a><\/li>\n<li><a href=\"https:\/\/www.mersenne.org\/\" target=\"_blank\" rel=\"noopener\">Great Internet Mersenne Prime Search<\/a> (GIMPS)<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>O maior n\u00famero primo conhecido (em setembro de 2023) \u00e9 \\({2^{82\\,589\\,933}} &#8211; 1\\), um n\u00famero que tem \\(24\\,862\\,048\\) d\u00edgitos quando escrito na base 10. Foi encontrado por meio de um computador, oferecido voluntariamente por&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":26827,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[4,6,5,3],"tags":[68],"series":[],"class_list":["post-26810","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-ciencia-e-tecnologia","category-desafios","category-divulgacao","category-matematica","tag-numero-primo"],"views":377,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2023\/09\/PrimeNumbers_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/26810","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=26810"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/26810\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/26827"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=26810"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=26810"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=26810"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=26810"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}