{"id":22016,"date":"2013-01-06T20:49:15","date_gmt":"2013-01-07T00:49:15","guid":{"rendered":"http:\/\/mathgradblog.williams.edu\/?p=22016"},"modified":"2013-01-06T20:49:15","modified_gmt":"2013-01-07T00:49:15","slug":"abc-conjecture","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/mathgradblog\/2013\/01\/06\/abc-conjecture\/","title":{"rendered":"ABC Conjecture"},"content":{"rendered":"<p>An exciting story has developed over the past few months. In August 2012, after months of rumors, the mathematician <a href=\"http:\/\/www.kurims.kyoto-u.ac.jp\/~motizuki\/top-english.html\">Shinichi Mochizuki<\/a> rounded out a series of papers which he claims prove the <a href=\"http:\/\/en.wikipedia.org\/wiki\/Abc_conjecture\">ABC Conjecture<\/a>:<\/p>\n<blockquote><p>\nFor every \u03b5 &gt; 0, there are only finitely many triples of coprime positive integers a + b = c such that c &gt; d^(1+\u03b5), where d denotes the product of the distinct prime factors of abc.\n<\/p><\/blockquote>\n<p><!--more--><\/p>\n<p>This easy to state conjecture has many <a href=\"http:\/\/en.wikipedia.org\/wiki\/Abc_conjecture#Some_consequences\">important consequences<\/a> in number theory. Professor Mochizuki has garnered an abundance of coverage from the popular media in the ensuing months due to the novelty of his approach; it is unclear (assuming correctness) how long it would take for his proof to be accepted by the mathematical community.<\/p>\n<p>If you would like to learn more about this story, I have selected a few articles:<\/p>\n<ul>\n<li>MathOverflow &#8211; <a href=\"http:\/\/mathoverflow.net\/questions\/106560\/philosophy-behind-mochizukis-work-on-the-abc-conjecture\">Philosophy behind Mochizuki\u2019s work on the ABC conjecture<\/a><\/li>\n<li>Nature &#8211; <a href=\"http:\/\/www.nature.com\/news\/proof-claimed-for-deep-connection-between-primes-1.11378\">Proof claimed for deep connection between primes<\/a><\/li>\n<li>Boston Globe &#8211; <a href=\"http:\/\/www.boston.com\/2012\/11\/03\/proof\/o9bja4kwPuXhDeDb2Ana2K\/story.html\">An ABC proof too tough even for mathematicians<\/a><\/li>\n<\/ul>\n<p>For complete coverage and technical details, check <a href=\"http:\/\/michaelnielsen.org\/polymath1\/index.php?title=ABC_conjecture\">this article on the PolyMath wiki<\/a>.<\/p>\n<p>You may also find interesting <a href=\"http:\/\/www.boston.com\/bostonglobe\/ideas\/brainiac\/2012\/12\/math_even_mathe.html\">this recent article from Kevin Hartnett<\/a>, author of the Boston Globe story linked above. It&#8217;s the slightly depressing account of a man who, like Professor Mochizuki, disappeared from the mathematical community to tackle a big problem&#8211;<a href=\"http:\/\/en.wikipedia.org\/wiki\/Goldbach's_conjecture\">Goldbach&#8217;s conjecture<\/a>.<\/p>\n<div style=\"margin-top: 0px; margin-bottom: 0px;\" class=\"sharethis-inline-share-buttons\" ><\/div>","protected":false},"excerpt":{"rendered":"<p>An exciting story has developed over the past few months. In August 2012, after months of rumors, the mathematician Shinichi Mochizuki rounded out a series of papers which he claims prove the ABC Conjecture: For every \u03b5 &gt; 0, there &hellip; <a href=\"https:\/\/blogs.ams.org\/mathgradblog\/2013\/01\/06\/abc-conjecture\/\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n<div style=\"margin-top: 0px; margin-bottom: 0px;\" class=\"sharethis-inline-share-buttons\" data-url=https:\/\/blogs.ams.org\/mathgradblog\/2013\/01\/06\/abc-conjecture\/><\/div>\n","protected":false},"author":31,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_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":""},"categories":[8,14,17],"tags":[],"class_list":["post-22016","post","type-post","status-publish","format-standard","hentry","category-general","category-math-in-pop-culture","category-news"],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p3gbww-5J6","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/posts\/22016","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/users\/31"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/comments?post=22016"}],"version-history":[{"count":0,"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/posts\/22016\/revisions"}],"wp:attachment":[{"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/media?parent=22016"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/categories?post=22016"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ams.org\/mathgradblog\/wp-json\/wp\/v2\/tags?post=22016"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}