{"id":6263,"date":"2021-04-09T06:00:21","date_gmt":"2021-04-09T04:00:21","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=6263"},"modified":"2021-04-16T09:26:51","modified_gmt":"2021-04-16T07:26:51","slug":"raices-digitales-de-los-numeros-de-fermat","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/raices-digitales-de-los-numeros-de-fermat\/","title":{"rendered":"Ra\u00edces digitales de los n\u00fameros de Fermat."},"content":{"rendered":"<p>Los <a href=\"https:\/\/bit.ly\/31Qh6qB\">n\u00fameros de Fermat<\/a> son los n\u00famero de la forma F(n) = 2^(2^n) + 1, donde n es un n\u00famero natural.<\/p>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   raizDigitalFermat :: Integer -> Integer\n<\/pre>\n<p>tal que (raizDigitalFermat n) es la ra\u00edz digital del n-\u00e9simo n\u00famero de Fermat. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   raizDigitalFermat 3          ==  5\n   raizDigitalFermat (10^2021)  ==  8\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\n-- 1\u00aa soluci\u00f3n\n-- ===========\n\nraizDigitalFermat :: Integer -> Integer\nraizDigitalFermat =\n  raizDigital . fermat\n\n-- (fermat k) es el k-\u00e9simo n\u00famero de Fermat. Por ejemplo,\n--    fermat 0  ==  3\n--    fermat 1  ==  5\n--    fermat 2  ==  17\n--    fermat 3  ==  257\n--    fermat 4  ==  65537\n--    fermat 5  ==  4294967297\nfermat :: Integer -> Integer\nfermat k = 1 + 2^(2^k)\n\n-- (raizDigital n) es la ra\u00edz digital de n. Por ejemplo,\n--    raizDigital 23451  ==  6\nraizDigital :: Integer -> Integer\nraizDigital n = 1 + (n-1) `mod` 9\n\n-- 2\u00aa soluci\u00f3n\n-- ===========\n\n-- En el c\u00e1lculo\n--    \u03bb> map raizDigitalFermat [0..20]\n--    [3,5,8,5,8,5,8,5,8,5,8,5,8,5,8,5,8,5,8,5,8]\n-- se observa que se compone del 3 seguido por la repetici\u00f3n peri\u00f3dica\n-- de 5 y 8.\n\nraizDigitalFermat2 :: Integer -> Integer\nraizDigitalFermat2 0 = 3\nraizDigitalFermat2 n\n  | odd n     = 5\n  | otherwise = 8\n\n-- Comparaci\u00f3n de eficiencia\n-- =========================\n\n-- La comparaci\u00f3n es\n--    \u03bb> raizDigitalFermat 30\n--    8\n--    (6.76 secs, 536,981,128 bytes)\n--    \u03bb> raizDigitalFermat2 30\n--    8\n--    (0.01 secs, 98,400 bytes)\n<\/pre>\n<h4>Nuevas soluciones<\/h4>\n<ul>\n<li>En los comentarios se pueden escribir nuevas soluciones.\n<li>El c\u00f3digo se debe escribir entre una l\u00ednea con &#60;pre lang=&quot;haskell&quot;&#62; y otra con &#60;\/pre&#62;\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Los n\u00fameros de Fermat son los n\u00famero de la forma F(n) = 2^(2^n) + 1, donde n es un n\u00famero natural. Definir la funci\u00f3n raizDigitalFermat :: Integer -> Integer tal que (raizDigitalFermat n) es la ra\u00edz digital del n-\u00e9simo n\u00famero de Fermat. Por ejemplo, raizDigitalFermat 3 == 5 raizDigitalFermat (10^2021) == 8 Soluciones &#8212; 1\u00aa&#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":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6263"}],"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=6263"}],"version-history":[{"count":4,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6263\/revisions"}],"predecessor-version":[{"id":6305,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/6263\/revisions\/6305"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=6263"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=6263"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=6263"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}