{"id":116,"date":"2010-04-03T21:56:38","date_gmt":"2010-04-04T01:56:38","guid":{"rendered":"http:\/\/blogs.williams.edu\/Morgan\/?p=116"},"modified":"2010-04-03T21:56:38","modified_gmt":"2010-04-04T01:56:38","slug":"infinitely-many-primes-by-combinatorics","status":"publish","type":"post","link":"https:\/\/sites.williams.edu\/Morgan\/2010\/04\/03\/infinitely-many-primes-by-combinatorics\/","title":{"rendered":"Infinitely Many Primes by Combinatorics"},"content":{"rendered":"<p>Here&#8217;s a proof of the infinitude of primes that occurred to me when for some reason Delta upgraded me to First Class on a flight April 2. Can anyone provide a reference?<\/p>\n<p>Suppose that the set <em>P<\/em> of primes and 1 has just <em>n<\/em>+1 elements. Now every number at most\u00a02<sup><em>k<\/em><\/sup> can be obtained by choosing <em>k<\/em> elements of <em>P<\/em> with replacement, which can be done in (<em>k+n<\/em> choose <em>n<\/em>) ways. Therefore<\/p>\n<p>2<sup><em>k<\/em><\/sup> \u2264 (<em>k<\/em>+<em>n<\/em> choose <em>n<\/em>) \u2264 (<em>k<\/em>+<em>n<\/em>)<sup><em>n<\/em><\/sup> ,<\/p>\n<p>which fails for <em>k<\/em> large.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Here&#8217;s a proof of the infinitude of primes that occurred to me when for some reason Delta upgraded me to First Class on a flight April 2. Can anyone provide a reference? Suppose that the set P of primes and 1 has just n+1 elements. Now every number at most\u00a02k can be obtained by choosing [&hellip;]<\/p>\n","protected":false},"author":269,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[14043],"tags":[],"class_list":["post-116","post","type-post","status-publish","format-standard","hentry","category-math"],"acf":[],"_links":{"self":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/116","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/users\/269"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/comments?post=116"}],"version-history":[{"count":0,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/116\/revisions"}],"wp:attachment":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/media?parent=116"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/categories?post=116"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/tags?post=116"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}