{"id":2540,"date":"2016-11-08T06:00:33","date_gmt":"2016-11-08T04:00:33","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=2540"},"modified":"2016-11-15T08:34:18","modified_gmt":"2016-11-15T06:34:18","slug":"conjetura-de-rassias","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/conjetura-de-rassias\/","title":{"rendered":"Conjetura de Rassias"},"content":{"rendered":"<p>El art\u00edculo de esta semana del blog <a href=\"http:\/\/hojaynumeros.blogspot.com.es\">N\u00fameros y hoja de c\u00e1lculo<\/a> est\u00e1 dedicado a la <a href=\"http:\/\/bit.ly\/2fadOHy\">Conjetura de Rassias<\/a>. Dicha conjetura afirma que<\/p>\n<blockquote><p>\n  Para cada n\u00famero primo p > 2 existen dos primos a y b, con a &lt; b, tales que<br \/>\n  (p-1)a = b+1\n<\/p><\/blockquote>\n<p>Dado un primo p > 2, los <strong>pares de Rassia<\/strong> de p son los pares de primos (a,b), con a &lt; b, tales que (p-1)a = b+1. Por ejemplo, (2,7) y (3,11) son pares de Rassia de 5 ya que<\/p>\n<ul>\n<li>2 y  7 son primos, 2 &lt; 7  y (5-1)\u00b72 = 7+1<\/li>\n<li>3 y 11 son primos, 3 &lt; 11 y (5-1)\u00b73 = 11+1<\/li>\n<\/ul>\n<p>Definir las siguientes funciones<\/p>\n<pre lang=\"text\"> \n   paresRassias     :: Integer -> [(Integer,Integer)]\n   conjeturaRassias :: Integer -> Bool\n<\/pre>\n<p>tales que<\/p>\n<ul>\n<li>(paresRassias p) es la lista de los pares de Rassias del primo p (que se supone que es mayor que 2). Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\"> \n     take 3 (paresRassias 5)    == [(2,7),(3,11),(5,19)]\n     take 3 (paresRassias 1229) == [(71,87187),(113,138763),(191,234547)]\n<\/pre>\n<ul>\n<li>(conjeturaRassia x) se verifica si para todos los primos menores que x (y mayores que 2) se cumple la conjetura de Rassia. Por ejemplo, <\/li>\n<\/ul>\n<pre lang=\"text\"> \n     conjeturaRassias (10^5)  ==  True\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.Numbers.Primes (primes, isPrime)\n\nparesRassias :: Integer -> [(Integer,Integer)]\nparesRassias p =\n  [(a,b) | a <- primes\n         , let b = (p - 1) * a - 1\n         , isPrime b]\n\nconjeturaRassias :: Integer -> Bool\nconjeturaRassias x =\n  and [(not . null . paresRassias) p | p <- tail (takeWhile (<x) primes)]\n<\/pre>\n<h4>Referencias<\/h4>\n<ul>\n<li><a href=\"http:\/\/bit.ly\/2fadOHy\">Conjetura de Rassias<\/a> por Antonio Rold\u00e1n en el blog \"N\u00fameros y hoja de c\u00e1lculo\"<\/li>\n<li><a href=\"http:\/\/bit.ly\/2fabErg\">Rassias' conjecture<\/a> en Wikipedia<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>El art\u00edculo de esta semana del blog N\u00fameros y hoja de c\u00e1lculo est\u00e1 dedicado a la Conjetura de Rassias. Dicha conjetura afirma que Para cada n\u00famero primo p > 2 existen dos primos a y b, con a &lt; b, tales que (p-1)a = b+1 Dado un primo p > 2, los pares de Rassia&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"footnotes":"","_jetpack_memberships_contains_paid_content":false},"categories":[7],"tags":[100,8,174,181,141,173,45,34],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/2540"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/comments?post=2540"}],"version-history":[{"count":9,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/2540\/revisions"}],"predecessor-version":[{"id":2581,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/2540\/revisions\/2581"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=2540"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=2540"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=2540"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}