{"id":6379,"date":"2021-05-11T06:00:48","date_gmt":"2021-05-11T04:00:48","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=6379"},"modified":"2021-05-18T08:39:50","modified_gmt":"2021-05-18T06:39:50","slug":"diferencias-de-potencias-congruentes-con-5-modulo-7","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/diferencias-de-potencias-congruentes-con-5-modulo-7\/","title":{"rendered":"Diferencias de potencias congruentes con 5 m\u00f3dulo 7"},"content":{"rendered":"<p>El enunciado de un problema 5 de la <a href=\"https:\/\/bit.ly\/2R4BTol\">Fase Local de la Olimpiada Matem\u00e1tica Espa\u00f1ola del 2012<\/a> es<\/p>\n<blockquote><p>\n  Consideremos el n\u00famero entero positivo <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=n+%3D+2%5Er+-+16%5Es&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"n = 2^r - 16^s\" class=\"latex\" \/>, donde r y s son tambi\u00e9n enteros positivos. Hallar las condiciones que deben cumplir r y s para que el resto de la divisi\u00f3n de n por 7 sea 5. Hallar el menor n\u00famero que cumple esta condici\u00f3n.\n<\/p><\/blockquote>\n<p>Definir la lista<\/p>\n<pre lang=\"text\">\n   exponentes :: [(Integer,Integer)]\n<\/pre>\n<p>tal que sus elementos son los pares de enteros positivos (r,s) tales que <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=2%5Er+-+16%5Es&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"2^r - 16^s\" class=\"latex\" \/> es un n\u00famero entero positivo cuyo resto al dividirlo por 7 es 5. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   head exponentes       ==  (10,2)\n   exponentes !! 23      ==  (43,8)\n   exponentes !! (10^7)  ==  (26836,1826)\n<\/pre>\n<p>Usando la funci\u00f3n exponentes, calcular la respuesta a la pregunta del problema; es decir, hallar el menor n\u00famero que cumple la condici\u00f3n.<\/p>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Test.QuickCheck (Property, (==>), quickCheck)\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nexponentes :: [(Integer,Integer)]\nexponentes =\n  [(r,s) | (r,s) <- pares,\n           let n = 2^r - 16^s,\n           n > 0,\n           n `mod` 7 == 5]\n\n-- pares el lista de pares de enteros positivos con el primero mayor que\n-- el segundo. Por ejemplo,\n--    \u03bb> take 10 pares\n--    [(2,1),(3,1),(3,2),(4,1),(4,2),(4,3),(5,1),(5,2),(5,3),(5,4)]\npares :: [(Integer,Integer)]\npares = [(a,b) | a <- [1..]\n               , b <- [1..a-1]]\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\n-- Observando los siguientes c\u00e1lculos\n--    \u03bb> take 28 exponentes\n--    [(10,2),(13,2),(16,2),(19,2),(22,2),(22,5),(25,2),(25,5),(28,2),\n--     (28,5),(31,2),(31,5),(34,2),(34,5),(34,8),(37,2),(37,5),(37,8),\n--     (40,2),(40,5),(40,8),(43,2),(43,5),(43,8),(46,2),(46,5),(46,8),\n--     (46,11)]\n--    \u03bb> [((x-1) `div` 3, (y-2) `div` 3) | (x,y) <- it]\n--    [(3,0),(4,0),(5,0),(6,0),(7,0),(7,1),(8,0),(8,1),(9,0),(9,1),\n--     (10,0),(10,1),(11,0),(11,1),(11,2),(12,0),(12,1),(12,2),(13,0),\n--     (13,1),(13,2),(14,0),(14,1),(14,2),(15,0),(15,1),(15,2),(15,3)]\n\nexponentes2 :: [(Integer,Integer)]\nexponentes2 =\n  concat [[(r,s) | s <- takeWhile (<= (r `div` 4)) ys] | r <- xs]\n  where xs = [10,13..]\n        ys = [2,5..]\n\n-- 3\u00aa soluci\u00f3n\n-- ===========\n\nexponentes3 :: [(Integer,Integer)]\nexponentes3 = [(r,s) | r <- [1,4..], s <- [2,5..r `div` 4]]\n\n-- Comprobaci\u00f3n de equivalencia\n-- ============================\n\n-- La propiedad es\nprop_exponentes :: Int -> Property\nprop_exponentes n =\n  n >= 0 ==>\n  all (== (exponentes !! n))\n      [exponentes2 !! n,\n       exponentes3 !! n]\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_exponentes\n--    +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n--    \u03bb> exponentes !! (3*10^4)\n--    (1471,332)\n--    (5.21 secs, 7,844,582,216 bytes)\n--    \u03bb> exponentes2 !! (3*10^4)\n--    (1471,332)\n--    (0.04 secs, 6,551,432 bytes)\n--    \u03bb> exponentes3 !! (3*10^4)\n--    (1471,332)\n--    (0.01 secs, 5,512,056 bytes)\n--\n--    \u03bb> exponentes2 !! (10^7)\n--    (26836,1826)\n--    (3.25 secs, 2,083,815,616 bytes)\n--    \u03bb> exponentes3 !! (10^7)\n--    (26836,1826)\n--    (2.57 secs, 1,757,467,216 bytes)\n\n-- C\u00e1lculo de la respuesta\n-- =======================\n\n-- El c\u00e1lculo es\n--    \u03bb> (r,s) = head exponentes\n--    \u03bb> 2^r - 16^s\n--    768\n-- Por tanto, el n\u00famero pedido es el 768.\n<\/pre>\n<h4>Nuevas soluciones<\/h4>\n<ul>\n<li>En los comentarios se pueden escribir nuevas soluciones.\n<li>El c\u00f3digo se debe escribir entre una l\u00ednea con &#60;pre lang=&quot;haskell&quot;&#62; y otra con &#60;\/pre&#62;\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>El enunciado de un problema 5 de la Fase Local de la Olimpiada Matem\u00e1tica Espa\u00f1ola del 2012 es Consideremos el n\u00famero entero positivo , donde r y s son tambi\u00e9n enteros positivos. Hallar las condiciones que deben cumplir r y s para que el resto de la divisi\u00f3n de n por 7 sea 5. Hallar&#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":[2],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6379"}],"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=6379"}],"version-history":[{"count":11,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6379\/revisions"}],"predecessor-version":[{"id":6472,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6379\/revisions\/6472"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=6379"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=6379"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=6379"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}