{"id":5462,"date":"2020-01-31T05:30:26","date_gmt":"2020-01-31T03:30:26","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=5462"},"modified":"2020-02-07T08:50:26","modified_gmt":"2020-02-07T06:50:26","slug":"acotacion-del-primorial","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/acotacion-del-primorial\/","title":{"rendered":"Acotaci\u00f3n del primorial"},"content":{"rendered":"<p>El <a href=\"http:\/\/bit.ly\/2RC1C3Z\">primorial<\/a> de un n\u00famero natural n es el producto de todos los n\u00fameros primos menores o iguales a n. Por ejemplo, el primorial de 5 es 30 porque el producto de los primos menores o iguales que 5 es<\/p>\n<pre lang=\"text\">\n   2 * 3 * 5 = 30\n<\/pre>\n<p>La <em>propiedad de Erd\u00f6s de acotaci\u00f3n de los primoriales<\/em> afirma que<\/p>\n<blockquote><p>\nPara todo n\u00famero natural n, su primorial es menor o igual que 4\u207f.\n<\/p><\/blockquote>\n<p>Definir las funciones<\/p>\n<pre lang=\"text\">\n   primorial :: Integer -> Integer\n   primoriales :: [Integer]\n<\/pre>\n<p>tales que<\/p>\n<ul>\n<li>(primorial n) es el primorial de n. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">\n     primorial 3  ==  6\n     primorial 5  ==  30\n     primorial 8  ==  210\n<\/pre>\n<ul>\n<li>primoriales es la sucesi\u00f3n de los primoriales. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\">\n   \u03bb> take 15 primoriales\n   [1,1,2,6,6,30,30,210,210,210,210,2310,2310,30030,30030]\n<\/pre>\n<p>Comprobar con QuickCheck la propiedad de Erd\u00f6s de acotaci\u00f3n de los primoriales.<\/p>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.Numbers.Primes\nimport Test.QuickCheck\n\n-- 1\u00aa definici\u00f3n de primorial\n-- ==========================\n\nprimorial :: Integer -> Integer\nprimorial n = product (takeWhile (<= n) primes)\n\n-- 2\u00aa definici\u00f3n de primorial\n-- ==========================\n\nprimorial2 :: Integer -> Integer\nprimorial2 0 = 1\nprimorial2 n | gcd n x == 1 = n*x\n             | otherwise    = x\n  where x = primorial2 (n-1)\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n--    \u03bb> length (show (primorial (5*10^5)))\n--    216852\n--    (1.65 secs, 2,472,977,584 bytes)\n--    \u03bb> length (show (primorial2 (5*10^5)))\n--    216852\n--    (3.56 secs, 2,719,162,272 bytes)\n\n-- 1\u00aa definici\u00f3n de primoriales\n-- ============================\n\n--    \u03bb> take 15 primoriales\n--    [1,1,2,6,6,30,30,210,210,210,210,2310,2310,30030,30030]\nprimoriales :: [Integer]\nprimoriales = map primorial [0..]\n\n-- 2\u00aa definici\u00f3n de primoriales\n-- ============================\n\n--    \u03bb> take 15 primoriales2\n--    [1,1,2,6,6,30,30,210,210,210,210,2310,2310,30030,30030]\nprimoriales2 :: [Integer]\nprimoriales2 = map primorial2 [0..]\n\n-- 3\u00aa definici\u00f3n de primoriales\n-- ============================\n\n--    \u03bb> take 15 primoriales3\n--    [1,1,2,6,6,30,30,210,210,210,210,2310,2310,30030,30030]\nprimoriales3 :: [Integer]\nprimoriales3 = scanl1 f [1..]\n  where f x n | gcd n x == 1 = n*x\n              | otherwise    = x\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n--    \u03bb> minimum (take 5000 primoriales)\n--    1\n--    (1.56 secs, 4,857,760,464 bytes)\n--    \u03bb> minimum (take 5000 primoriales2)\n--    1\n--    (9.39 secs, 10,942,848,240 bytes)\n--    \u03bb> minimum (take 5000 primoriales3)\n--    1\n--    (0.01 secs, 5,575,024 bytes)\n--    \n--    \u03bb> minimum (take 6000 primoriales)\n--    1\n--    (2.22 secs, 7,013,937,248 bytes)\n--    \u03bb> minimum (take 6000 primoriales3)\n--    1\n--    (0.01 secs, 6,737,328 bytes)\n\n-- Propiedad\n-- =========\n\nprop_primorial :: Integer -> Property\nprop_primorial n =\n  n >= 0 ==> primorial n <= 4^n\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_primorial\n--    +++ OK, passed 100 tests.\n<\/pre>\n<h4>Otras soluciones<\/h4>\n<ul>\n<li>Se pueden escribir otras soluciones en los comentarios.\n<li>El c\u00f3digo se debe escribir entre una l\u00ednea con &#60;pre lang=\u00bbhaskell\u00bb&#62; y otra con &#60;\/pre&#62;\n<\/ul>\n<h4>Pensamiento<\/h4>\n<blockquote><p>\n\u00abLas matem\u00e1ticas son la reina de las ciencias y la teor\u00eda de los n\u00fameros es la reina de las matem\u00e1ticas.\u00bb <\/p>\n<p><a href=\"https:\/\/es.wikipedia.org\/wiki\/Carl_Friedrich_Gauss\">Carl Friedrich Gauss<\/a>.\n<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>El primorial de un n\u00famero natural n es el producto de todos los n\u00fameros primos menores o iguales a n. Por ejemplo, el primorial de 5 es 30 porque el producto de los primos menores o iguales que 5 es 2 * 3 * 5 = 30 La propiedad de Erd\u00f6s de acotaci\u00f3n de los&#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":[155,10,11,173,157,252,34,146],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5462"}],"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=5462"}],"version-history":[{"count":6,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5462\/revisions"}],"predecessor-version":[{"id":5534,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5462\/revisions\/5534"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=5462"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=5462"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=5462"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}