{"id":5333,"date":"2020-01-06T05:30:16","date_gmt":"2020-01-06T03:30:16","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=5333"},"modified":"2020-01-13T08:45:09","modified_gmt":"2020-01-13T06:45:09","slug":"factorizaciones-de-4n1","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/factorizaciones-de-4n1\/","title":{"rendered":"Factorizaciones de 4n+1"},"content":{"rendered":"<p>Sea S el conjunto<\/p>\n<pre lang=\"text\">\n   S = {1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, ...}\n<\/pre>\n<p>de los enteros positivos congruentes con 1 m\u00f3dulo 4; es decir,<\/p>\n<pre lang=\"text\">\n   S = {4n+1 | n \u2208 N}\n<\/pre>\n<p>Un elemento n de S es irreducible si s\u00f3lo es divisible por dos elementos de S: 1 y n. Por ejemplo, 9 es irreducible; pero 45 no lo es (ya que es el proctos de 5 y 9 que son elementos de S).<\/p>\n<p>Definir las funciones<\/p>\n<pre lang=\"text\">\n   esIrreducible :: Integer -> Bool\n   factorizaciones :: Integer -> [[Integer]]\n   conFactorizacionNoUnica :: [Integer]\n<\/pre>\n<p>tales que<\/p>\n<ul>\n<li>(esIrreducible n) se verifica si n es irreducible. Por ejemplo, <\/li>\n<\/ul>\n<pre lang=\"text\">\n     esIrreducible 9   ==  True\n     esIrreducible 45  ==  False\n<\/pre>\n<ul>\n<li>(factorizaciones n) es la lista de conjuntos de elementos irreducibles de S cuyo producto es n. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">  \n     factorizaciones 9     ==  [[9]]\n     factorizaciones 693   ==  [[9,77],[21,33]]\n     factorizaciones 4389  ==  [[21,209],[33,133],[57,77]]\n     factorizaciones 2205  ==  [[5,9,49],[5,21,21]]\n<\/pre>\n<ul>\n<li>conFactorizacionNoUnica es la lista de elementos de S cuya factorizaci\u00f3n no es \u00fanica. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">  \n     \u03bb> take 10 conFactorizacionNoUnica\n     [441,693,1089,1197,1449,1617,1881,1953,2205,2277]\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nesIrreducible :: Integer -> Bool\nesIrreducible n = divisores n == [1,n]\n\n-- (divisores n) es el conjunto de elementos de S que dividen a n. Por\n-- ejemplo, \n--    divisores 9    ==  [9]\n--    divisores 693  ==  [9,21,33,77,693]\ndivisores :: Integer -> [Integer]\ndivisores n = [x | x <- [1,5..n], n `mod` x == 0] \n  \nfactorizaciones :: Integer -> [[Integer]]\nfactorizaciones 1 = [[1]]\nfactorizaciones n\n  | esIrreducible n = [[n]]\n  | otherwise       = [d:ds | d <- divisores n\n                            , esIrreducible d\n                            , ds@(e:_) <- factorizaciones (n `div` d)\n                            , d <= e]\n\nconFactorizacionNoUnica :: [Integer]\nconFactorizacionNoUnica =\n  [n | n <- [1,5..], length (factorizaciones n) > 1]\n<\/pre>\n<h4>Pensamiento<\/h4>\n<blockquote><p>\n\u00a1Qu\u00e9 bien los nombres pon\u00eda<br \/>\nquien puso Sierra Morena<br \/>\na esta serran\u00eda!<\/p>\n<p>Antonio Machado\n<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>Sea S el conjunto S = {1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, &#8230;} de los enteros positivos congruentes con 1 m\u00f3dulo 4; es decir, S = {4n+1 | n \u2208 N} Un elemento n de S es irreducible si s\u00f3lo es divisible por dos elementos de S: 1 y&#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":[5],"tags":[8,30,28,89,6],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5333"}],"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=5333"}],"version-history":[{"count":4,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5333\/revisions"}],"predecessor-version":[{"id":5390,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5333\/revisions\/5390"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=5333"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=5333"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=5333"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}