{"id":5339,"date":"2020-01-08T05:30:10","date_gmt":"2020-01-08T03:30:10","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=5339"},"modified":"2020-01-15T07:51:41","modified_gmt":"2020-01-15T05:51:41","slug":"enteros-como-sumas-de-tres-coprimos","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/enteros-como-sumas-de-tres-coprimos\/","title":{"rendered":"Enteros como sumas de tres coprimos."},"content":{"rendered":"<p>Dos n\u00fameros enteros son <a href=\"http:\/\/bit.ly\/2tNlb0L\">coprimos<\/a> (o <strong>primos entre s\u00ed<\/strong>) si no tienen ning\u00fan factor primo en com\u00fan. Por ejemplo, 4 y 15 son coprimos.<\/p>\n<p>Una terna coprima es una terna (a,b,c) tal que<\/p>\n<ul>\n<li>a y b son coprimos,<\/li>\n<li>a y c son coprimos y<\/li>\n<li>b y c son coprimos.<\/li>\n<\/ul>\n<p>Por ejemplo, (3,4,5) es una terna coprima.<\/p>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   sumas3coprimos :: Integer -> [(Integer,Integer,Integer)]\n<\/pre>\n<p>tal que (sumas3coprimos n) es la lista de las ternas coprimas cuya suma es n. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   sumas3coprimos 10  ==  [(2,3,5)]\n   sumas3coprimos 11  ==  []\n   sumas3coprimos 12  ==  [(2,3,7),(3,4,5)]\n   length (sumas3coprimos 4000)  ==  546146\n<\/pre>\n<p>Comprobar con QuickCheck que todo n\u00famero entero mayor que 17 se puede escribir como suma de alguna terna coprima; es decir, para todo entero n, (sumas3coprimos2 (18 + abs n)) tiene alg\u00fan elemento.<\/p>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Test.QuickCheck\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nsumas3coprimos :: Integer -> [(Integer,Integer,Integer)]\nsumas3coprimos n =\n  [(a,b,c) | a <- [2..n]\n           , b <- [a+1..n]\n           , c <- [b+1..n]\n           , a + b + c == n\n           , gcd a b == 1\n           , gcd a c == 1\n           , gcd b c == 1]\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\nsumas3coprimos2 :: Integer -> [(Integer,Integer,Integer)]\nsumas3coprimos2 n =\n  [(a,b,c) | a <- [2..n `div` 3]\n           , b <- [a+1..(n - a) `div` 2]\n           , gcd a b == 1\n           , let c = n - a - b  \n           , gcd a c == 1\n           , gcd b c == 1]\n\n-- Equivalencia de las definiciones\n-- ================================\n\n-- La propiedad de equivalencia es\nprop_sumas3coprimos_equiv :: Integer -> Property\nprop_sumas3coprimos_equiv n =\n  n > 0 ==> sumas3coprimos n == sumas3coprimos2 n\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_sumas3coprimos_equiv\n--    +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n--    \u03bb> length (sumas3coprimos 400)\n--    5345\n--    (4.16 secs, 2,894,799,744 bytes)\n--    \u03bb> length (sumas3coprimos2 400)\n--    5345\n--    (0.06 secs, 16,565,136 bytes)\n\n-- Propiedad\n-- =========\n\n-- La propiedad\nprop_sumas3coprimos :: Integer -> Bool\nprop_sumas3coprimos n =\n  not (null (sumas3coprimos2 (18 + abs n)))\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_sumas3coprimos\n--    +++ OK, passed 100 tests.\n<\/pre>\n<h4>Referencias<\/h4>\n<ul>\n<li>Basado en <a href=\"http:\/\/bit.ly\/2skxBgl\">Integers as sum of three pairwise coprime integers<\/a> de ProofWiki.<\/li>\n<\/ul>\n<h4>Pensamiento<\/h4>\n<blockquote><p>\nTodo amor es fantas\u00eda;<br \/>\n\u00e9l inventa el a\u00f1o, el d\u00eda,<br \/>\nla hora y su melod\u00eda;<br \/>\ninventa el amante y, m\u00e1s<br \/>\nla amada. No prueba nada,<br \/>\ncontra el amor, que la amada<br \/>\nno haya existido jam\u00e1s.<\/p>\n<p>Antonio Machado\n<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>Dos n\u00fameros enteros son coprimos (o primos entre s\u00ed) si no tienen ning\u00fan factor primo en com\u00fan. Por ejemplo, 4 y 15 son coprimos. Una terna coprima es una terna (a,b,c) tal que a y b son coprimos, a y c son coprimos y b y c son coprimos. Por ejemplo, (3,4,5) es una terna&#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":[8,30,155,146],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5339"}],"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=5339"}],"version-history":[{"count":5,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5339\/revisions"}],"predecessor-version":[{"id":5397,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5339\/revisions\/5397"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=5339"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=5339"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=5339"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}