{"id":7026,"date":"2022-05-18T14:38:17","date_gmt":"2022-05-18T12:38:17","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=7026"},"modified":"2022-05-18T14:42:21","modified_gmt":"2022-05-18T12:42:21","slug":"sumas-de-4-primos","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/sumas-de-4-primos\/","title":{"rendered":"Sumas de 4 primos"},"content":{"rendered":"

La conjetura de Waring sobre los n\u00fameros primos<\/a> establece que todo n\u00famero impar es primo o la suma de tres primos. La conjetura de Goldbach<\/a> afirma que todo par mayor que 2 es la suma de dos n\u00fameros primos. Ambos ha estado abiertos durante m\u00e1s de 200 a\u00f1os. En este problema no se propone su soluci\u00f3n, sino una tarea m\u00e1s simple: buscar una manera de expresar los enteros mayores que 7 como suma de exactamente cuatro n\u00fameros primos; es decir, definir la funci\u00f3n<\/p>\n

\n   suma4primos :: Integer -> [(Integer,Integer,Integer,Integer)]\n<\/pre>\n
tal que (suma4primos n)<\/code> es la lista de las cu\u00e1druplas crecientes (a,b,c,d)<\/code> de n\u00fameros primos cuya suma es n<\/code> (que se supone mayor que 7). Por ejemplo,<\/p>\n
\n   suma4primos 18             == [(2,2,3,11),(2,2,7,7),(3,3,5,7),(3,5,5,5)]\n   head (suma4primos (10^14)) == (2,2,23,99999999999973)\n<\/pre>\n
Comprobar con QuickCheck que todo entero mayor que 7 se puede escribir como suma de exactamente cuatro n\u00fameros primos.<\/p>\n
Soluciones<\/h4>\n
\nimport Data.Numbers.Primes (isPrime, primes)\nimport Test.QuickCheck\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nsuma4primos1 :: Integer -> [(Integer, Integer, Integer, Integer)]\nsuma4primos1 n =\n  [(a,b,c,d) | let as = takeWhile (< n) primes,\n               a <- as,\n               let bs = takeWhile (< n-a) as,\n               b <- bs, a <= b,\n               let cs = takeWhile (< n-a-b) bs,\n               c <- cs, b <= c,\n               let d = n-a-b-c, c <= d,\n               isPrime d]\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\nsuma4primos2 :: Integer -> [(Integer, Integer, Integer, Integer)]\nsuma4primos2 n =\n  [(a,b,c,d) | let as = takeWhile (< n) primes,\n               a <- as,\n               let bs = takeWhile (< n-a) (dropWhile (< a) as),\n               b <- bs,\n               let cs = takeWhile ( Bool\nprop_suma4primos (Positive n) =\n  suma4primos1 n == suma4primos2 n\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_suma4primos\n--    +++ OK, passed 100 tests; 526 discarded.\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n--    \u03bb> length (suma4primos1 2000)\n--    90219\n--    (2.98 secs, 4,517,620,744 bytes)\n--    \u03bb> length (suma4primos2 2000)\n--    90219\n--    (2.22 secs, 4,223,251,928 bytes)\n--\n--    \u03bb> head (suma4primos1 (10^14))\n--    (2,2,23,99999999999973)\n--    (1.67 secs, 5,963,327,168 bytes)\n--    \u03bb> head (suma4primos2 (10^14))\n--    (2,2,23,99999999999973)\n--    (1.70 secs, 5,963,326,848 bytes)\n\n-- Comprobaci\u00f3n de la propiedad\n-- ============================\n\n-- La propiedad es\nprop_suma4primos2 :: Integer -> Property\nprop_suma4primos2 n =\n  n > 7 ==> not (null (suma4primos1 n))\n\n-- La comprobaci\u00f3n es\n--    \u03bb> quickCheck prop_suma4primos2\n--    +++ OK, passed 100 tests; 582 discarded.\n<\/pre>\n
El c\u00f3digo se encuentra en GitHub<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"
La conjetura de Waring sobre los n\u00fameros primos establece que todo n\u00famero impar es primo o la suma de tres primos. La conjetura de Goldbach afirma que todo par mayor que 2 es la suma de dos n\u00fameros primos. Ambos ha estado abiertos durante m\u00e1s de 200 a\u00f1os. En este problema no se propone su…<\/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\/7026"}],"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=7026"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7026\/revisions"}],"predecessor-version":[{"id":7029,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/7026\/revisions\/7029"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=7026"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=7026"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=7026"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}