{"id":87,"date":"2017-04-05T11:40:43","date_gmt":"2017-04-05T11:40:43","guid":{"rendered":"https:\/\/petermc.net\/blog\/?p=87"},"modified":"2017-04-07T08:16:40","modified_gmt":"2017-04-07T08:16:40","slug":"a-different-proof-of-the-fundamental-theorem-of-arithmetic","status":"publish","type":"post","link":"https:\/\/petermc.net\/blog\/2017\/04\/05\/a-different-proof-of-the-fundamental-theorem-of-arithmetic\/","title":{"rendered":"A different proof of the fundamental theorem of arithmetic"},"content":{"rendered":"

The fundamental theorem of arithmetic states that every integer can be uniquely written as a product of primes (i.e. \"\mathbb{Z}\" is a unique factorisation domain).<\/p>\n

The usual proof proceeds through the Euclidean algorithm. Yesterday at lunch I was surprised to learn (thanks to Ole Warnaar<\/a>) of a different proof bypassing the Euclidean algorithm which I reproduce below. Its primary attraction is its cuteness, as it provides a weaker result than the usual proof (i.e. doesn’t prove that \"\mathbb{Z}\" is a Euclidean domain, or even a principal ideal domain).<\/p>\n

Suppose that \"n=p_1^{a_1}p_2^{a_2}\cdots where \"p_1,p_2,\ldots,p_k\" are distinct primes. Consider the finite cyclic group \"C_n\". It has a composition series where the group \"C_{p_i}\" appears \"a_i\" times as a simple subquotient (and no other simple factors appear). Therefore by the Jordan-Holder theorem<\/a>, the primes \"p_i\" together with their multiplicities \"a_i\" are unique. QED.<\/p>\n","protected":false},"excerpt":{"rendered":"

The fundamental theorem of arithmetic states that every integer can be uniquely written as a product of primes (i.e. is a unique factorisation domain). The usual proof proceeds through the Euclidean algorithm. Yesterday at lunch I was surprised to learn … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"jetpack_post_was_ever_published":false},"categories":[6],"tags":[],"class_list":["post-87","post","type-post","status-publish","format-standard","hentry","category-maths"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":false,"jetpack_shortlink":"https:\/\/wp.me\/p7V6a7-1p","_links":{"self":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts\/87","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/comments?post=87"}],"version-history":[{"count":6,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts\/87\/revisions"}],"predecessor-version":[{"id":93,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/posts\/87\/revisions\/93"}],"wp:attachment":[{"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/media?parent=87"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/categories?post=87"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/petermc.net\/blog\/wp-json\/wp\/v2\/tags?post=87"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}