{"id":6716,"date":"2022-03-02T06:00:55","date_gmt":"2022-03-02T04:00:55","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=6716"},"modified":"2022-04-15T12:04:37","modified_gmt":"2022-04-15T10:04:37","slug":"primos-equidistantes","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/primos-equidistantes\/","title":{"rendered":"Primos equidistantes"},"content":{"rendered":"<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   primosEquidistantes :: Integer -> [(Integer,Integer)]\n<\/pre>\n<p>tal que (primosEquidistantes k) es la lista de los pares de primos cuya diferencia es k. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   take 3 (primosEquidistantes 2)  ==  [(3,5),(5,7),(11,13)]\n   take 3 (primosEquidistantes 4)  ==  [(7,11),(13,17),(19,23)]\n   take 3 (primosEquidistantes 6)  ==  [(23,29),(31,37),(47,53)]\n   take 3 (primosEquidistantes 8)  ==  [(89,97),(359,367),(389,397)]\n   primosEquidistantes 4 !! (10^5) ==  (18467047,18467051)\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\nimport Data.Numbers.Primes (primes)\n\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nprimosEquidistantes1 :: Integer -> [(Integer,Integer)]\nprimosEquidistantes1 k = aux primos\n  where aux (x:y:ps) | y - x == k = (x,y) : aux (y:ps)\n                     | otherwise  = aux (y:ps)\n\n-- (primo x) se verifica si x es primo. Por ejemplo,\n--    primo 7  ==  True\n--    primo 8  ==  False\nprimo :: Integer -> Bool\nprimo x = [y | y <- [1..x], x `rem` y == 0] == [1,x]\n\n-- primos es la lista de los n\u00fameros primos. Por ejemplo,\n--    take 10 primos  ==  [2,3,5,7,11,13,17,19,23,29]\nprimos :: [Integer]\nprimos = 2 : [x | x <- [3,5..], primo x]\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\nprimosEquidistantes2 :: Integer -> [(Integer,Integer)]\nprimosEquidistantes2 k = aux primos2\n  where aux (x:y:ps) | y - x == k = (x,y) : aux (y:ps)\n                     | otherwise  = aux (y:ps)\n\nprimos2 :: [Integer]\nprimos2 = criba [2..]\n  where criba (p:ps) = p : criba [n | n <- ps, mod n p \/= 0]\n\n-- 3\u00aa soluci\u00f3n\n-- ===========\n\nprimosEquidistantes3 :: Integer -> [(Integer,Integer)]\nprimosEquidistantes3 k =\n  [(x,y) | (x,y) <- zip primos2 (tail primos2)\n         , y - x == k]\n\n-- 4\u00aa soluci\u00f3n\n-- ===========\n\nprimosEquidistantes4 :: Integer -> [(Integer,Integer)]\nprimosEquidistantes4 k = aux primes\n  where aux (x:y:ps) | y - x == k = (x,y) : aux (y:ps)\n                     | otherwise  = aux (y:ps)\n\n-- 5\u00aa soluci\u00f3n\n-- ===========\n\nprimosEquidistantes5 :: Integer -> [(Integer,Integer)]\nprimosEquidistantes5 k =\n  [(x,y) | (x,y) <- zip primes (tail primes)\n         , y - x == k]\n\n-- Comprobaci\u00f3n de equivalencia\n-- ============================\n\n-- La propiedad es\nprop_primosEquidistantes :: Int -> Integer -> Bool\nprop_primosEquidistantes n k =\n  all (== take n (primosEquidistantes1 k))\n      [take n (f k) | f <- [primosEquidistantes2,\n                            primosEquidistantes3,\n                            primosEquidistantes4,\n                            primosEquidistantes5]]\n\n-- La comprobaci\u00f3n es\n--    \u03bb> prop_primosEquidistantes 100 4\n--    True\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n--    \u03bb> primosEquidistantes1 4 !! 200\n--    (9829,9833)\n--    (2.60 secs, 1,126,458,272 bytes)\n--    \u03bb> primosEquidistantes2 4 !! 200\n--    (9829,9833)\n--    (0.44 secs, 249,622,048 bytes)\n--    \u03bb> primosEquidistantes3 4 !! 200\n--    (9829,9833)\n--    (0.36 secs, 207,549,592 bytes)\n--    \u03bb> primosEquidistantes4 4 !! 200\n--    (9829,9833)\n--    (0.02 secs, 4,012,848 bytes)\n--    \u03bb> primosEquidistantes5 4 !! 200\n--    (9829,9833)\n--    (0.01 secs, 7,085,072 bytes)\n--\n--    \u03bb> primosEquidistantes2 4 !! 600\n--    (41617,41621)\n--    (5.67 secs, 3,340,313,480 bytes)\n--    \u03bb> primosEquidistantes3 4 !! 600\n--    (41617,41621)\n--    (5.43 secs, 3,090,994,096 bytes)\n--    \u03bb> primosEquidistantes4 4 !! 600\n--    (41617,41621)\n--    (0.03 secs, 15,465,824 bytes)\n--    \u03bb> primosEquidistantes5 4 !! 600\n--    (41617,41621)\n--    (0.04 secs, 28,858,232 bytes)\n--\n--    \u03bb> primosEquidistantes4 4 !! (10^5)\n--    (18467047,18467051)\n--    (3.99 secs, 9,565,715,488 bytes)\n--    \u03bb> primosEquidistantes5 4 !! (10^5)\n--    (18467047,18467051)\n--    (7.95 secs, 18,712,469,144 bytes)\n<\/pre>\n<p>El c\u00f3digo se encuentra en <a href=\"https:\/\/github.com\/jaalonso\/Exercitium\/blob\/main\/src\/Primos_equidistantes.hs\">GitHub<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Definir la funci\u00f3n primosEquidistantes :: Integer -> [(Integer,Integer)] tal que (primosEquidistantes k) es la lista de los pares de primos cuya diferencia es k. Por ejemplo, take 3 (primosEquidistantes 2) == [(3,5),(5,7),(11,13)] take 3 (primosEquidistantes 4) == [(7,11),(13,17),(19,23)] take 3 (primosEquidistantes 6) == [(23,29),(31,37),(47,53)] take 3 (primosEquidistantes 8) == [(89,97),(359,367),(389,397)] primosEquidistantes 4 !! (10^5) ==&#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":[8,501,415,6,521],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6716"}],"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=6716"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6716\/revisions"}],"predecessor-version":[{"id":6775,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6716\/revisions\/6775"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=6716"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=6716"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=6716"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}