{"id":7149,"date":"2022-07-22T06:00:59","date_gmt":"2022-07-22T04:00:59","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=7149"},"modified":"2022-07-17T16:20:38","modified_gmt":"2022-07-17T14:20:38","slug":"la-funcion-indicatriz-de-euler","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/la-funcion-indicatriz-de-euler\/","title":{"rendered":"La funci\u00f3n indicatriz de Euler"},"content":{"rendered":"<p>La <a href=\"https:\/\/bit.ly\/3yQbzA6\">indicatriz de Euler<\/a> (tambi\u00e9n  funci\u00f3n \u03c6 de Euler) es una funci\u00f3n importante en teor\u00eda de n\u00fameros. Si n es un entero positivo, entonces \u03c6(n) se define como el n\u00famero de enteros positivos menores o iguales a n y coprimos con n. Por ejemplo, \u03c6(36) = 12 ya que los n\u00fameros menores o iguales a 36 y coprimos con 36 son doce: 1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, y 35.<\/p>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   phi :: Integer -> Integer\n<\/pre>\n<p>tal que (phi n) es igual a \u03c6(n). Por ejemplo,<\/p>\n<pre lang=\"text\">\n   phi 36                          ==  12\n   map phi [10..20]                ==  [4,10,4,12,6,8,8,16,6,18,8]\n   phi (3^10^5) `mod` (10^9)       ==  681333334\n   length (show (phi (10^(10^5)))) == 100000\n<\/pre>\n<p>Comprobar con QuickCheck que, para todo n > 0, \u03c6(10^n) tiene n d\u00edgitos.<\/p>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.List (genericLength, group)\nimport Data.Numbers.Primes (primeFactors)\nimport Math.NumberTheory.ArithmeticFunctions (totient)\nimport Test.QuickCheck (Positive (Positive), quickCheck)\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nphi1 :: Integer -> Integer\nphi1 n = genericLength [x | x <- [1..n], gcd x n == 1] \n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\nphi2 :: Integer -> Integer\nphi2 n = product [(p-1)*p^(e-1) | (p,e) <- factorizacion n] \n    \nfactorizacion :: Integer -> [(Integer,Integer)]\nfactorizacion n =\n  [(head xs,genericLength xs) | xs <- group (primeFactors n)]\n\n-- 3\u00aa soluci\u00f3n\n-- =============\n\nphi3 :: Integer -> Integer\nphi3 n = \n  product [(x-1) * product xs | (x:xs) <- group (primeFactors n)]\n\n-- 4\u00aa soluci\u00f3n\n-- =============\n\nphi4 :: Integer -> Integer\nphi4 = totient \n\n-- Comprobaci\u00f3n de equivalencia\n-- ============================\n\n-- La propiedad es\nprop_phi :: Positive Integer -> Bool \nprop_phi (Positive n) =\n  all (== phi1 n)\n      [phi2 n,\n       phi3 n,\n       phi4 n]\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_phi\n--    +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n--    \u03bb> phi1 (2*10^6)\n--    800000\n--    (2.49 secs, 2,117,853,856 bytes)\n--    \u03bb> phi2 (2*10^6)\n--    800000\n--    (0.02 secs, 565,664 bytes)\n--    \n--    \u03bb> length (show (phi2 (10^100000)))\n--    100000\n--    (2.80 secs, 5,110,043,208 bytes)\n--    \u03bb> length (show (phi3 (10^100000)))\n--    100000\n--    (4.81 secs, 7,249,353,896 bytes)\n--    \u03bb> length (show (phi4 (10^100000)))\n--    100000\n--    (0.78 secs, 1,467,573,768 bytes)\n\n-- Verificaci\u00f3n de la propiedad\n-- ============================\n\n-- La propiedad es\nprop_phi2 :: Positive Integer -> Bool\nprop_phi2 (Positive n) =\n  genericLength (show (phi4 (10^n))) == n\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_phi2\n--    +++ OK, passed 100 tests.\n<\/pre>\n<p>El c\u00f3digo se encuentra en <a href=\"https:\/\/github.com\/jaalonso\/Exercitium\/blob\/main\/src\/La_funcion_indicatriz_de_Euler.hs\">GitHub<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>La indicatriz de Euler (tambi\u00e9n funci\u00f3n \u03c6 de Euler) es una funci\u00f3n importante en teor\u00eda de n\u00fameros. Si n es un entero positivo, entonces \u03c6(n) se define como el n\u00famero de enteros positivos menores o iguales a n y coprimos con n. Por ejemplo, \u03c6(36) = 12 ya que los n\u00fameros menores o iguales a&#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":[521],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7149"}],"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=7149"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7149\/revisions"}],"predecessor-version":[{"id":7150,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7149\/revisions\/7150"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=7149"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=7149"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=7149"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}