Soluciones<\/h4>\n\nimport Graphics.Gnuplot.Simple\n\n-- Definici\u00f3n de ternasPitagoricas\n-- ===============================\n\nternasPitagoricas :: Integer -> [(Integer,Integer,Integer)]\nternasPitagoricas x =\n [(x,y,z) | y <- [1..(x^ 2 - 1) `div` 2 ]\n , z <- raizCuadrada (x^2 + y^2)]\n\n-- La justificaci\u00f3n de la cota es\n-- x > 2\n-- x^2 + y^2 >= (y+1)^2\n-- x^2 + y^2 >= y^2 + 2*y + 1\n-- y =< (x^ 2 - 1) `div` 2 \n\n-- (raizCuadrada x) es la lista formada por la ra\u00edz cuadrada entera de\n-- x, si existe y la lista vac\u00eda, en caso contrario. Por ejemplo, \n-- raizCuadrada 25 == [5]\n-- raizCuadrada 26 == []\nraizCuadrada :: Integer -> [Integer]\nraizCuadrada x =\n [y | y <- [(round . sqrt . fromIntegral) x]\n , y^2 == x]\n\n\n-- 1\u00aa definici\u00f3n de mayorTernaPitagorica\n-- =====================================\n\nmayorTernaPitagorica :: Integer -> (Integer,Integer,Integer)\nmayorTernaPitagorica =\n last . ternasPitagoricas\n\n-- 2\u00aa definici\u00f3n de mayorTernaPitagorica\n-- =====================================\n\nmayorTernaPitagorica2 :: Integer -> (Integer,Integer,Integer)\nmayorTernaPitagorica2 x =\n head [(x,y,z) | y <- [k, k-1 .. 1]\n , z <- raizCuadrada (x^2 + y^2)]\n where k = (x^2 - 1) `div` 2\n\n \n-- 3\u00aa definici\u00f3n de mayorTernaPitagorica\n-- =====================================\n\n-- Se supone que x > 2. Se consideran dos casos:\n-- \n-- Primer caso: Supongamos que x es par. Entonces x^2 > 4 y es divisible\n-- por 4. Por tanto, existe un y tal que x^2 = 4*y + 4; luego,\n-- x^2 + y^2 = 4*y + 4 + y^2\n-- = (y + 2)^2\n-- La terna es (x,y,y+2) donde y = (x^2 - 4) \/ 4.\n--\n-- Segundo caso: Supongamos que x es impar. Entonces x^2 es impar. Por\n-- tanto, existe un y tal que x^2 = 2*y + 1; luego,\n-- x^2 + y^2 = 2*y + 1 + y^2\n-- = (y+1)^2\n-- La terna es (x,y,y+1) donde y = (x^2 - 1) \/ 2.\n\nmayorTernaPitagorica3 :: Integer -> (Integer,Integer,Integer)\nmayorTernaPitagorica3 x\n | even x = (x, y1, y1 + 2)\n | otherwise = (x, y2, y2 + 1)\n where y1 = (x^2 - 4) `div` 4\n y2 = (x^2 - 1) `div` 2 \n\n-- Comparaci\u00f3n de eficiencia\n-- \u03bb> mayorTernaPitagorica 1006\n-- (1006,253008,253010)\n-- (7.36 secs, 1,407,793,992 bytes)\n-- \u03bb> mayorTernaPitagorica2 1006\n-- (1006,253008,253010)\n-- (3.76 secs, 704,007,456 bytes)\n-- \u03bb> mayorTernaPitagorica3 1006\n-- (1006,253008,253010)\n-- (0.01 secs, 157,328 bytes)\n\ngraficaMayorHipotenusa :: Integer -> IO ()\ngraficaMayorHipotenusa n =\n plotList [ Key Nothing\n , PNG \"Terna_pitagorica_a_partir_de_un_lado.png\"\n ]\n [(x,z) | x <- [3..n]\n , let (_,_,z) = mayorTernaPitagorica3 x]\n<\/pre>\n","protected":false},"excerpt":{"rendered":"Una terna pitag\u00f3rica con primer lado x es una terna (x,y,z) tal que x^2 + y^2 = z^2. Por ejemplo, las ternas pitag\u00f3ricas con primer lado 16 son (16,12,20), (16,30,34) y (16,63,65). Definir las funciones ternasPitagoricas :: Integer -> [(Integer,Integer,Integer)] mayorTernaPitagorica :: Integer -> (Integer,Integer,Integer) graficaMayorHipotenusa :: Integer -> IO () tales que (ternasPitgoricas x)…<\/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":[4],"tags":[8,30,91,183,376,71,134,309,184,236],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3641"}],"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=3641"}],"version-history":[{"count":5,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3641\/revisions"}],"predecessor-version":[{"id":3774,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/3641\/revisions\/3774"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=3641"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=3641"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=3641"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
![\"Terna_pitagorica_a_partir_de_un_lado\"](\"https:\/\/i0.wp.com\/www.glc.us.es\/~jalonso\/exercitium\/wp-content\/uploads\/2018\/01\/Terna_pitagorica_a_partir_de_un_lado.png\")
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