{"id":7160,"date":"2022-07-29T06:00:06","date_gmt":"2022-07-29T04:00:06","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=7160"},"modified":"2022-07-22T15:14:00","modified_gmt":"2022-07-22T13:14:00","slug":"sumas-de-dos-primos","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/sumas-de-dos-primos\/","title":{"rendered":"Sumas de dos primos"},"content":{"rendered":"<p>Definir la sucesi\u00f3n<\/p>\n<pre lang=\"text\">\n   sumasDeDosPrimos :: [Integer]\n<\/pre>\n<p>cuyos elementos son los n\u00fameros que se pueden escribir como suma de dos n\u00fameros primos. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   \u03bb> take 23 sumasDeDosPrimos\n   [4,5,6,7,8,9,10,12,13,14,15,16,18,19,20,21,22,24,25,26,28,30,31]\n   \u03bb> sumasDeDosPrimos !! (5*10^5)\n   862878\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.Numbers.Primes (isPrime, primes)\nimport Test.QuickCheck\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nsumasDeDosPrimos1 :: [Integer]\nsumasDeDosPrimos1 =\n  [n | n <- [1..], not (null (sumaDeDosPrimos1 n))]\n\n-- (sumaDeDosPrimos1 n) es la lista de pares de primos cuya suma es\n-- n. Por ejemplo,\n--    sumaDeDosPrimos  9  ==  [(2,7),(7,2)]\n--    sumaDeDosPrimos 16  ==  [(3,13),(5,11),(11,5),(13,3)]\n--    sumaDeDosPrimos 17  ==  []\nsumaDeDosPrimos1 :: Integer -> [(Integer,Integer)]\nsumaDeDosPrimos1 n = \n  [(x,n-x) | x <- primosN, isPrime (n-x)]\n  where primosN = takeWhile (< n) primes\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\nsumasDeDosPrimos2 :: [Integer]\nsumasDeDosPrimos2 =\n  [n | n <- [1..], not (null (sumaDeDosPrimos2 n))]\n\n-- (sumasDeDosPrimos2 n) es la lista de pares (x,y) de primos cuya suma\n-- es n y tales que x <= y. Por ejemplo,\n--    sumaDeDosPrimos2  9  ==  [(2,7)]\n--    sumaDeDosPrimos2 16  ==  [(3,13),(5,11)]\n--    sumaDeDosPrimos2 17  ==  []\nsumaDeDosPrimos2 :: Integer -> [(Integer,Integer)]\nsumaDeDosPrimos2 n = \n  [(x,n-x) | x <- primosN, isPrime (n-x)]\n  where primosN = takeWhile (<= (n `div` 2)) primes\n\n-- 3\u00aa soluci\u00f3n\n-- ===========\n\nsumasDeDosPrimos3 :: [Integer]\nsumasDeDosPrimos3 = filter esSumaDeDosPrimos3 [4..]\n\n-- (esSumaDeDosPrimos3 n) se verifica si n es suma de dos primos. Por\n-- ejemplo, \n--    esSumaDeDosPrimos3  9  ==  True\n--    esSumaDeDosPrimos3 16  ==  True\n--    esSumaDeDosPrimos3 17  ==  False\nesSumaDeDosPrimos3 :: Integer -> Bool\nesSumaDeDosPrimos3 n\n  | odd n     = isPrime (n-2)\n  | otherwise = any isPrime [n-x | x <- takeWhile (<= (n `div` 2)) primes]\n\n-- 4\u00aa soluci\u00f3n\n-- ===========\n\n-- Usando la conjetura de Goldbach que dice que \"Todo n\u00famero par mayor\n-- que 2 puede escribirse como suma de dos n\u00fameros primos\" .\n\nsumasDeDosPrimos4 :: [Integer]\nsumasDeDosPrimos4 = filter esSumaDeDosPrimos4 [4..]\n\n-- (esSumaDeDosPrimos4 n) se verifica si n es suma de dos primos. Por\n-- ejemplo, \n--    esSumaDeDosPrimos4  9  ==  True\n--    esSumaDeDosPrimos4 16  ==  True\n--    esSumaDeDosPrimos4 17  ==  False\nesSumaDeDosPrimos4 :: Integer -> Bool\nesSumaDeDosPrimos4 n = even n || isPrime (n-2)\n\n-- Comprobaci\u00f3n de equivalencia\n-- ============================\n\n-- La propiedad es\nprop_sumasDeDosPrimos :: NonNegative Int -> Bool\nprop_sumasDeDosPrimos (NonNegative n) =\n  all (== sumasDeDosPrimos1 !! n)\n      [sumasDeDosPrimos2 !! n,\n       sumasDeDosPrimos3 !! n,\n       sumasDeDosPrimos4 !! n]\n  \n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_sumasDeDosPrimos\n--    +++ OK, passed 100 tests.\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n--    \u03bb> sumasDeDosPrimos1 !! 5000\n--    7994\n--    (2.61 secs, 9,299,106,792 bytes)\n--    \u03bb> sumasDeDosPrimos2 !! 5000\n--    7994\n--    (1.48 secs, 5,190,651,760 bytes)\n--    \u03bb> sumasDeDosPrimos3 !! 5000\n--    7994\n--    (0.12 secs, 351,667,104 bytes)\n--    \u03bb> sumasDeDosPrimos4 !! 5000\n--    7994\n--    (0.04 secs, 63,464,320 bytes)\n--\n--    \u03bb> sumasDeDosPrimos3 !! (5*10^4)\n--    83674\n--    (2.23 secs, 7,776,049,264 bytes)\n--    \u03bb> sumasDeDosPrimos4 !! (5*10^4)\n--    83674\n--    (0.34 secs, 1,183,604,984 bytes)\n<\/pre>\n<p>El c\u00f3digo se encuentra en <a href=\"https:\/\/github.com\/jaalonso\/Exercitium\/blob\/main\/src\/Sumas_de_dos_primos.hs\">GitHub<\/a>.<\/p>\n<h4>Referencia<\/h4>\n<ul>\n<li>N.J.A. Sloane, <a href=\"http:\/\/oeis.org\/A014091\">Sucesi\u00f3n A014091 en OEIS<\/a>.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Definir la sucesi\u00f3n sumasDeDosPrimos :: [Integer] cuyos elementos son los n\u00fameros que se pueden escribir como suma de dos n\u00fameros primos. Por ejemplo, \u03bb> take 23 sumasDeDosPrimos [4,5,6,7,8,9,10,12,13,14,15,16,18,19,20,21,22,24,25,26,28,30,31] \u03bb> sumasDeDosPrimos !! (5*10^5) 862878 Soluciones import Data.Numbers.Primes (isPrime, primes) import Test.QuickCheck &#8212; 1\u00aa soluci\u00f3n &#8212; =========== sumasDeDosPrimos1 :: [Integer] sumasDeDosPrimos1 = [n | n [(Integer,Integer)] sumaDeDosPrimos1&#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\/7160"}],"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=7160"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7160\/revisions"}],"predecessor-version":[{"id":7161,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7160\/revisions\/7161"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=7160"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=7160"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=7160"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}