Definir la funci\u00f3n<\/p>\n
\n primosEquidistantes :: Integer -> [(Integer,Integer)]\n<\/pre>\ntal que primosEquidistantes k<\/code> es la lista de los pares de primos cuya diferencia es k<\/code>. Por ejemplo,<\/p>\n\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<\/p>\n
1. Soluciones en Haskell<\/h2>\n\nmodule Primos_equidistantes where\n\nimport Data.Numbers.Primes (primes)\nimport Test.Hspec (Spec, describe, hspec, it, shouldBe)\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-- Verificaci\u00f3n\n-- ============\n\nverifica :: IO ()\nverifica = hspec spec\n\nspecG :: (Integer -> [(Integer,Integer)]) -> Spec\nspecG primosEquidistantes = do\n it \"e1\" $\n take 3 (primosEquidistantes 2) `shouldBe` [(3,5),(5,7),(11,13)]\n it \"e2\" $\n take 3 (primosEquidistantes 4) `shouldBe` [(7,11),(13,17),(19,23)]\n it \"e3\" $\n take 3 (primosEquidistantes 6) `shouldBe` [(23,29),(31,37),(47,53)]\n it \"e4\" $\n take 3 (primosEquidistantes 8) `shouldBe` [(89,97),(359,367),(389,397)]\n\nspec :: Spec\nspec = do\n describe \"def. 1\" $ specG primosEquidistantes1\n describe \"def. 2\" $ specG primosEquidistantes2\n describe \"def. 3\" $ specG primosEquidistantes3\n describe \"def. 4\" $ specG primosEquidistantes4\n describe \"def. 5\" $ specG primosEquidistantes5\n\n-- La verificaci\u00f3n es\n-- \u03bb> verifica\n-- 20 examples, 0 failures\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>\n2. Soluciones en Python<\/h2>\n\nfrom itertools import chain, count, islice, tee\nfrom timeit import Timer, default_timer\nfrom typing import Iterator\n\nfrom sympy import isprime\n\n# 1\u00aa soluci\u00f3n\n# ===========\n\n# primo(x) se verifica si x es primo. Por ejemplo,\n# primo(7) == True\n# primo(8) == False\ndef primo(x: int) -> bool:\n return [y for y in range(1,x+1) if x % y == 0] == [1,x]\n\n# primos() es la lista de los n\u00fameros primos. Por ejemplo,\n# >>> list(islice(primos(), 10))\n# [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]\ndef primos() -> Iterator[int]:\n return chain([2], (x for x in count(3, 2) if primo(x)))\n\ndef primosEquidistantes1(k: int) -> Iterator[tuple[int,int]]:\n a, b = tee(primos())\n next(b, None)\n return ((x,y) for (x,y) in zip(a, b) if y - x == k)\n\n# 2\u00aa soluci\u00f3n\n# ===========\n\ndef primos2() -> Iterator[int]:\n return (n for n in count() if isprime(n))\n\ndef primosEquidistantes2(k: int) -> Iterator[tuple[int,int]]:\n a, b = tee(primos2())\n next(b, None)\n return ((x,y) for (x,y) in zip(a, b) if y - x == k)\n\n# Verificaci\u00f3n\n# ============\n\ndef test_primosEquidestantes() -> None:\n for primosEquidistantes in [primosEquidistantes1,\n primosEquidistantes2]:\n assert list(islice(primosEquidistantes(2), 3)) == \\\n [(3, 5), (5, 7), (11, 13)]\n assert list(islice(primosEquidistantes(4), 3)) == \\\n [(7, 11), (13, 17), (19, 23)]\n assert list(islice(primosEquidistantes(6), 3)) == \\\n [(23, 29), (31, 37), (47, 53)]\n assert list(islice(primosEquidistantes(8), 3)) == \\\n [(89, 97), (359, 367), (389, 397)]\n print(\"Verificado\")\n\n# La verificaci\u00f3n es\n# >>> test_primosEquidestantes()\n# Verificado\n\n# Comprobaci\u00f3n de equivalencia\n# ============================\n\n# La propiedad es\ndef primosEquidistantes_equiv(n: int, k: int) -> bool:\n return list(islice(primosEquidistantes1(k), n)) == \\\n list(islice(primosEquidistantes2(k), n))\n\n# La comprobaci\u00f3n es\n# >>> primosEquidistantes_equiv(100, 4)\n# True\n\n# Comparaci\u00f3n de eficiencia\n# =========================\n\ndef tiempo(e: str) -> None:\n \"\"\"Tiempo (en segundos) de evaluar la expresi\u00f3n e.\"\"\"\n t = Timer(e, \"\", default_timer, globals()).timeit(1)\n print(f\"{t:0.2f} segundos\")\n\n# La comparaci\u00f3n es\n# >>> tiempo('list(islice(primosEquidistantes1(4), 300))')\n# 3.19 segundos\n# >>> tiempo('list(islice(primosEquidistantes2(4), 300))')\n# 0.01 segundos\n<\/pre>\n","protected":false},"excerpt":{"rendered":"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) ==…<\/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":"default","_kad_post_title":"default","_kad_post_layout":"default","_kad_post_sidebar_id":"","_kad_post_content_style":"default","_kad_post_vertical_padding":"default","_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":[581],"tags":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/8541"}],"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=8541"}],"version-history":[{"count":1,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/8541\/revisions"}],"predecessor-version":[{"id":8542,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/8541\/revisions\/8542"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=8541"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=8541"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=8541"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}