{"id":8415,"date":"2024-01-24T06:00:35","date_gmt":"2024-01-24T04:00:35","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=8415"},"modified":"2024-02-17T21:23:57","modified_gmt":"2024-02-17T19:23:57","slug":"24-ene-24","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/24-ene-24\/","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 <code>phi n<\/code> 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.<br \/>\n<!--more--><\/p>\n<p><b>Soluciones<\/b><\/p>\n<p>A continuaci\u00f3n se muestran las <a href=\"#haskell\">soluciones en Haskell<\/a> y las <a href=\"#python\">soluciones en Python<\/a>.<\/p>\n<p><a name=\"haskell\"><\/a><br \/>\n<b>Soluciones en Haskell<\/b><\/p>\n<pre lang=\"haskell\">\nmodule La_funcion_indicatriz_de_Euler where\n\nimport Data.List (genericLength, group)\nimport Data.Numbers.Primes (primeFactors)\nimport Math.NumberTheory.ArithmeticFunctions (totient)\nimport Test.QuickCheck (Positive (Positive), quickCheck)\nimport Test.Hspec (Spec, describe, hspec, it, shouldBe)\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-- Verificaci\u00f3n\n-- ============\n\nverifica :: IO ()\nverifica = hspec spec\n\nspecG :: (Integer -> Integer) -> Spec\nspecG phi = do\n  it \"e1\" $\n    phi 36 `shouldBe` 12\n  it \"e2\" $\n    map phi [10..20] `shouldBe` [4,10,4,12,6,8,8,16,6,18,8]\n\nspec :: Spec\nspec = do\n  describe \"def. 1\" $ specG phi1\n  describe \"def. 2\" $ specG phi2\n  describe \"def. 3\" $ specG phi3\n  describe \"def. 4\" $ specG phi4\n\n-- La verificaci\u00f3n es\n--    \u03bb> verifica\n--\n--    8 examples, 0 failures\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><a name=\"python\"><\/a><br \/>\n<b>Soluciones en Python<\/b><\/p>\n<pre lang=\"python\">\nfrom functools import reduce\nfrom math import gcd\nfrom operator import mul\nfrom sys import set_int_max_str_digits\nfrom timeit import Timer, default_timer\n\nfrom hypothesis import given\nfrom hypothesis import strategies as st\nfrom sympy import factorint, totient\n\nset_int_max_str_digits(10**6)\n\n# 1\u00aa soluci\u00f3n\n# ===========\n\ndef phi1(n: int) -> int:\n    return len([x for x in range(1, n+1) if gcd(x, n) == 1])\n\n# 2\u00aa soluci\u00f3n\n# ===========\n\ndef producto(xs: list[int]) -> int:\n    return reduce(mul, xs, 1)\n\ndef phi2(n: int) -> int:\n    factores = factorint(n)\n    return producto([(p-1)*p**(e-1) for p, e in factores.items()])\n\n# 3\u00aa soluci\u00f3n\n# =============\n\ndef phi3(n: int) -> int:\n    return totient(n)\n\n# Verificaci\u00f3n\n# ============\n\ndef test_phi() -> None:\n    for phi in [phi1, phi2, phi3]:\n        assert phi(36) == 12\n        assert list(map(phi, range(10, 21))) == [4,10,4,12,6,8,8,16,6,18,8]\n    print(\"Verificado\")\n\n# La verificaci\u00f3n es\n#    >>> test_phi()\n#    Verificado\n\n# Comprobaci\u00f3n de equivalencia\n# ============================\n\n# La propiedad es\n@given(st.integers(min_value=1, max_value=1000))\ndef test_phi_equiv(n: int) -> None:\n    r = phi1(n)\n    assert phi2(n) == r\n    assert phi3(n) == r\n\n# La comprobaci\u00f3n es\n#    >>> test_phi_equiv()\n#    >>>\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('phi1(9*10**6)')\n#    2.09 segundos\n#    >>> tiempo('phi2(9*10**6)')\n#    0.00 segundos\n#    >>> tiempo('phi3(9*10**6)')\n#    0.00 segundos\n#\n#    >>> tiempo('phi2(10**1000000)')\n#    3.55 segundos\n#    >>> tiempo('phi3(10**1000000)')\n#    3.37 segundos\n\n# Verificaci\u00f3n de la propiedad\n# ============================\n\n# La propiedad es\n@given(st.integers(min_value=1, max_value=1000))\ndef test_phi_prop(n: int) -> None:\n    assert len(str(phi2(10**n))) == n\n\n# La comprobaci\u00f3n es\n#    >>> test_phi_prop()\n#    >>>\n\n# Comprobaci\u00f3n de todas las propiedades\n# =====================================\n\n# La comprobaci\u00f3n es\n#    src> poetry run pytest -v La_funcion_indicatriz_de_Euler.py\n#    ===== test session starts =====\n#       test_phi PASSED\n#       test_phi_equiv PASSED\n#       test_phi_prop PASSED\n#    ===== passed in 0.55s =====\n<\/pre>\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":[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\/8415"}],"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=8415"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/8415\/revisions"}],"predecessor-version":[{"id":8431,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/8415\/revisions\/8431"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=8415"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=8415"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=8415"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}