{"id":5541,"date":"2020-02-11T05:30:31","date_gmt":"2020-02-11T03:30:31","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=5541"},"modified":"2022-03-26T11:29:02","modified_gmt":"2022-03-26T09:29:02","slug":"conjetura-de-lemoine","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/conjetura-de-lemoine\/","title":{"rendered":"Conjetura de Lemoine"},"content":{"rendered":"<p>La <a href=\"http:\/\/bit.ly\/2ue1S1i\">conjetura de Lemoine<\/a> afirma que<\/p>\n<blockquote><p>\nTodos los n\u00fameros impares mayores que 5 se pueden escribir de la forma p + 2q donde p y q son n\u00fameros primos. Por ejemplo, 47 = 13 + 2 x 17\n<\/p><\/blockquote>\n<p>Definir las funciones<\/p>\n<pre lang=\"text\">\n   descomposicionesLemoine :: Integer -> [(Integer,Integer)]\n   graficaLemoine :: Integer -> IO ()\n<\/pre>\n<p>tales que<\/p>\n<ul>\n<li>(descomposicionesLemoine n) es la lista de pares de primos (p,q) tales que n = p + 2q. Por ejemplo,<\/li>\n<\/ul>\n<pre lang=\"text\"> \n     descomposicionesLemoine 5   ==  []\n     descomposicionesLemoine 7   ==  [(3,2)]\n     descomposicionesLemoine 9   ==  [(5,2),(3,3)]\n     descomposicionesLemoine 21  ==  [(17,2),(11,5),(7,7)]\n     descomposicionesLemoine 47  ==  [(43,2),(41,3),(37,5),(13,17)]\n     descomposicionesLemoine 33  ==  [(29,2),(23,5),(19,7),(11,11),(7,13)]\n     length (descomposicionesLemoine 2625)  ==  133\n<\/pre>\n<ul>\n<li>(graficaLemoine n) dibuja la gr\u00e1fica de los n\u00fameros de descomposiciones de Lemoine para los n\u00fameros impares menores o iguales que n. Por ejemplo, (graficaLemoine n 400) dibuja<br \/>\n<a href=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2020\/02\/Conjetura_de_Lemoine.png\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2020\/02\/Conjetura_de_Lemoine.png?resize=640%2C480\" alt=\"\" width=\"640\" height=\"480\" class=\"aligncenter size-full wp-image-5542\" srcset=\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2020\/02\/Conjetura_de_Lemoine.png?w=640&amp;ssl=1 640w, https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2020\/02\/Conjetura_de_Lemoine.png?resize=300%2C225&amp;ssl=1 300w\" sizes=\"(max-width: 640px) 100vw, 640px\" data-recalc-dims=\"1\" \/><\/a><\/li>\n<\/ul>\n<p>Comprobar con QuickCheck la conjetura de Lemoine.<\/p>\n<p><strong>Nota<\/strong>: Basado en <a href=\"http:\/\/bit.ly\/2ue1S1i\">Lemoine&#8217;s conjecture<\/a><\/p>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.Numbers.Primes (isPrime, primes)\nimport Graphics.Gnuplot.Simple\nimport Test.QuickCheck\n\ndescomposicionesLemoine :: Integer -> [(Integer,Integer)]\ndescomposicionesLemoine n =\n  [(p,q) | q <- takeWhile (<=(n-2) `div` 2) primes\n         , let p = n - 2 * q\n         , isPrime p]\n\ngraficaLemoine :: Integer -> IO ()\ngraficaLemoine n = do\n  plotList [ Key Nothing\n           , Title \"Conjetura de Lemoine\"\n           , PNG \"Conjetura_de_Lemoine.png\"\n           ]\n           [(k,length (descomposicionesLemoine k)) | k <- [1,3..n]]\n\n-- La conjetura es\nprop_conjeturaLemoine :: Integer -> Bool\nprop_conjeturaLemoine n =\n  not (null (descomposicionesLemoine n'))\n  where n' = 7 + 2 * abs n\n\n-- Su comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_conjeturaLemoine\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\u00abTodo el mundo sabe lo que es una curva, hasta que ha estudiado suficientes matem\u00e1ticas para confundirse a trav\u00e9s del incontable n\u00famero de posibles excepciones.\u00bb <\/p>\n<p><a href=\"https:\/\/en.wikipedia.org\/wiki\/Felix_Klein\">Felix Klein<\/a>.\n<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>La conjetura de Lemoine afirma que Todos los n\u00fameros impares mayores que 5 se pueden escribir de la forma p + 2q donde p y q son n\u00fameros primos. Por ejemplo, 47 = 13 + 2 x 17 Definir las funciones descomposicionesLemoine :: Integer -> [(Integer,Integer)] graficaLemoine :: Integer -> IO () tales que (descomposicionesLemoine&#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":[130,8,30,376,174,28,181,141,11,309,173,34,146],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5541"}],"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=5541"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5541\/revisions"}],"predecessor-version":[{"id":5605,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/5541\/revisions\/5605"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=5541"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=5541"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=5541"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}