{"id":66999,"date":"2012-10-11T09:09:04","date_gmt":"2012-10-11T13:09:04","guid":{"rendered":"https:\/\/today.uconn.edu\/?p=66999"},"modified":"2012-10-16T08:20:12","modified_gmt":"2012-10-16T12:20:12","slug":"the-mochizuki-theorem-when-youre-so-smart-nobody-can-check-your-work","status":"publish","type":"post","link":"https:\/\/today.uconn.edu\/2012\/10\/the-mochizuki-theorem-when-youre-so-smart-nobody-can-check-your-work\/","title":{"rendered":"The Mochizuki Theorem? When You’re So Smart Nobody Can Check Your Work"},"content":{"rendered":"

 <\/p>\n

\"Jeremy<\/a>
Jeremy Teitelbaum, dean of the College of Liberal Arts and Sciences.<\/figcaption><\/figure>\n

Jeremy Teitelbaum<\/a>, dean of the College of Liberal Arts and Sciences<\/a>, is a guest contributor to UConn Today. For his previous posts, click here<\/a>.<\/em><\/p>\n

With the presidential election, tension with Iran, and trouble in Libya and Syria dominating the news this fall, it\u2019s understandable that Kyoto mathematician Shinichi Mochizuki\u2019s claim to have proved the ABC conjecture (reported in the New York Times<\/a>) may not be getting the attention that it deserves.<\/p>\n

While topics of broader interest occupy dinner table conversations, among practitioners of the subfield of pure mathematics known as number theory, Mochizuki\u2019s work is the hot topic. If his proof holds together, it\u2019s possible that 2012 will be remembered into the distant future primarily as the year that ABC was proved.<\/p>\n

The only problem is that Mochizuki\u2019s work is so esoteric that it\u2019s proving difficult for the mathematical community to check his proof.<\/p>\n

The ABC<\/a>conjecture<\/a>, proposed by Joseph Oesterle and David Masser in the 1980\u2019s, is a technical assertion about the prime divisors of three numbers, called a,b,and c, that satisfy a+b=c.\u00a0 It\u2019s interesting because it implies the truth of a whole host of other difficult problems in number theory, solving all of them at once.<\/p>\n

It might seem remarkable that we don\u2019t already know everything there is to know about such a simple equation, but in fact there are many unanswered questions about the relationships between multiplicative and additive properties of whole numbers. Two famous unsolved problems are the twin <\/a>prime<\/a> conjecture<\/a>, which asserts that there are infinitely many pairs of consecutive odd prime numbers (like 11 and 13, 17 and 19, 29 and 31, and so forth), and Goldbach<\/a>\u2019<\/a>s <\/a>conjecture<\/a>, which asserts that every even number is the sum of two primes.<\/p>\n

Unlike ABC, these two problems are more curiosities than problems of central theoretical importance.<\/p>\n

Usually when someone settles an important conjecture, it\u2019s because they\u2019ve had a truly new, brilliant idea that opens up a new approach that they can outline clearly to experts in<\/p>\n

\"Triakis<\/a>
Triakis triangular tiling<\/figcaption><\/figure>\n

the field. After the experts say \u201cAha! I see it!\u201d, then they can sit down to check the details.\u00a0 This is what happened, for example, with the biggest news items from mathematics in the last 50 years, such as\u00a0 Falting<\/a>\u2019<\/a>s<\/a> proof of the Mordel l<\/a>Conjecture<\/a>, Wiles\u2019s<\/a><\/span> proof of Fermat<\/a>\u2019<\/a>s<\/a> Last <\/a>Theorem<\/a>and Perelman<\/a>\u2019<\/a>s<\/a> proof of the Poincare<\/a> Conjecture<\/a>.<\/p>\n

What\u2019s so strange about Mochizuki\u2019s work on ABC is that there don\u2019t seem to be any experts in a position to pronounce on his work.\u00a0 Mochizuki, working for years on the problem independently, has introduced so many new ideas, and so much terminology, that he has moved far outside the pool of information shared by professional number theorists.\u00a0 He has an established reputation as a mathematician that makes it impossible to dismiss his claims, but it would be difficult for another mathematician to commit to investing substantial time — perhaps years — mastering his work in order to settle the question of its validity.<\/p>\n

The gold standard for certification of mathematical results is peer review; but peer review rests on the assumption that the community of peers has the expertise, and the will, to address the correctness of really new work.<\/p>\n

In the case of Mochizuki\u2019s potential breakthrough,\u00a0 the mathematical community is<\/p>\n

gradually mobilizing to come to grips with the four long preprints posted on his web page (Inter-universal Teichmuller Theory I<\/a><\/span>, II<\/a><\/span>, III<\/a><\/span>,<\/a> and IV<\/a><\/span>).\u00a0 Slowly, a few people are stepping forward to probe the papers, trying to test their correctness.\u00a0 If those adventurous souls report optimistically on what they find, peer review will kick into gear.<\/p>\n

If, on the other hand, the work proves too opaque for the first group of serious readers, or if they find mistakes early on, it will be difficult to find enough people to put in the effort needed to clarify the situation. Then the burden will shift to Mochizuki to explain what he\u2019s done or withdraw his claims.<\/p>\n

It might be a new world in number theory, the various consequences of ABC might be settled, and graduate students in number theory might suddenly have an entirely new domain of ideas to search through for their thesis problems.\u00a0 Or the proof might not hold up. Either way, it\u2019s possible that Mochizuki has nevertheless opened up many new promising ideas for further research.<\/p>\n

It\u2019s too soon to tell.\u00a0 Stay tuned for further developments.<\/p>\n","protected":false},"excerpt":{"rendered":"

If you are a mathematical genius, how do you prove the ABC conjecture to your peers? Hint: It’s not as simple as 1,2,3.<\/p>\n","protected":false},"author":40,"featured_media":66996,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"_crdt_document":"","wds_primary_category":0,"wds_primary_series":0,"wds_primary_attribution":0,"footnotes":""},"categories":[1],"tags":[],"magazine-issues":[],"coauthors":[66],"class_list":["post-66999","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-uncategorized"],"pp_statuses_selecting_workflow":false,"pp_workflow_action":"current","pp_status_selection":"publish","acf":[],"publishpress_future_action":{"enabled":false,"date":"2026-05-03 12:28:27","action":"change-status","newStatus":"draft","terms":[],"taxonomy":"category","extraData":[]},"publishpress_future_workflow_manual_trigger":{"enabledWorkflows":[]},"_links":{"self":[{"href":"https:\/\/today.uconn.edu\/wp-rest\/wp\/v2\/posts\/66999","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/today.uconn.edu\/wp-rest\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/today.uconn.edu\/wp-rest\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/today.uconn.edu\/wp-rest\/wp\/v2\/users\/40"}],"replies":[{"embeddable":true,"href":"https:\/\/today.uconn.edu\/wp-rest\/wp\/v2\/comments?post=66999"}],"version-history":[{"count":5,"href":"https:\/\/today.uconn.edu\/wp-rest\/wp\/v2\/posts\/66999\/revisions"}],"predecessor-version":[{"id":67300,"href":"https:\/\/today.uconn.edu\/wp-rest\/wp\/v2\/posts\/66999\/revisions\/67300"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/today.uconn.edu\/wp-rest\/wp\/v2\/media\/66996"}],"wp:attachment":[{"href":"https:\/\/today.uconn.edu\/wp-rest\/wp\/v2\/media?parent=66999"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/today.uconn.edu\/wp-rest\/wp\/v2\/categories?post=66999"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/today.uconn.edu\/wp-rest\/wp\/v2\/tags?post=66999"},{"taxonomy":"magazine-issue","embeddable":true,"href":"https:\/\/today.uconn.edu\/wp-rest\/wp\/v2\/magazine-issues?post=66999"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/today.uconn.edu\/wp-rest\/wp\/v2\/coauthors?post=66999"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}