Editing Abc conjecture (section)

==Claimed proofs==

[[Lucien Szpiro]] proposed a solution in 2007, but it was found to be incorrect shortly afterwards.<ref>"Finiteness Theorems for Dynamical Systems", Lucien Szpiro, talk at  Conference on L-functions and Automorphic Forms (on the occasion of Dorian Goldfeld's 60th Birthday), Columbia University, May 2007. See {{citation|title=Proof of the abc Conjecture?|first=Peter|last=Woit|author-link=Peter Woit|work=Not Even Wrong|url=http://www.math.columbia.edu/~woit/wordpress/?p=561|date=May 26, 2007}}.</ref>

Since August 2012, [[Shinichi Mochizuki]] has claimed a proof of Szpiro's conjecture and therefore the ''abc'' conjecture.<ref name = "Ball"/> He released a series of four preprints developing a new theory he called [[inter-universal Teichmüller theory]] (IUTT), which is then applied to prove the ''abc'' conjecture.<ref name=Mochizukiweb>{{cite journal |last1=Mochizuki |first1=Shinichi |title=Inter-universal Teichmüller Theory IV: Log-Volume Computations and Set-Theoretic Foundations |journal=Publications of the Research Institute for Mathematical Sciences |date=4 March 2021 |volume=57 |issue=1 |pages=627–723 |doi=10.4171/PRIMS/57-1-4 |s2cid=3135393 }}</ref>
The papers have not been widely accepted by the mathematical community as providing a proof of ''abc''.<ref>
{{cite web |url=https://galoisrepresentations.wordpress.com/2017/12/17/the-abc-conjecture-has-still-not-been-proved/ |title=The ABC conjecture has (still) not been proved |last=Calegari |first=Frank |author-link=Frank Calegari<!-- stated author is "Persiflage", who can be identified by other refs on the blog to his students --> |date=December 17, 2017 |access-date=March 17, 2018}}</ref> This is not only because of their length and the difficulty of understanding them,<ref>{{cite magazine|magazine=[[New Scientist]]|title=Baffling ABC maths proof now has impenetrable 300-page 'summary'|url=https://www.newscientist.com/article/2146647-baffling-abc-maths-proof-now-has-impenetrable-300-page-summary/|first=Timothy|last=Revell|date=September 7, 2017}}</ref> but also because at least one specific point in the argument has been identified as a gap by some other experts.<ref name=stillConj/> Although a few mathematicians have vouched for the correctness of the proof<ref>
{{ cite journal | url=http://www.inference-review.com/article/fukugen |first= Ivan |last= Fesenko |author-link= Ivan Fesenko | title=Fukugen | journal = Inference |date= 28 September 2016 | volume = 2 | number = 3 | access-date=30 October 2021}}</ref> and have attempted to communicate their understanding via workshops on IUTT, they have failed to convince the number theory community at large.<ref>{{cite web | url=https://mathbabe.org/2015/12/15/notes-on-the-oxford-iut-workshop-by-brian-conrad/ |first = Brian |last=Conrad |author-link=Brian Conrad| date=December 15, 2015 | title=Notes on the Oxford IUT workshop by Brian Conrad | access-date=March 18, 2018}}</ref><ref>{{cite journal |last1=Castelvecchi |first1=Davide |date=8 October 2015 |title=The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof |journal=Nature |volume=526 |issue= 7572|pages=178–181 |doi=10.1038/526178a |bibcode=2015Natur.526..178C |pmid=26450038|doi-access=free }}</ref>

In March 2018, [[Peter Scholze]] and [[Jakob Stix]] visited [[Kyoto University|Kyoto]] for discussions with Mochizuki.<ref>
{{cite magazine|url=https://www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/ |title=Titans of Mathematics Clash Over Epic Proof of ABC Conjecture
 |magazine= [[Quanta Magazine]] |date=September 20, 2018 |first= Erica |last= Klarreich |author-link= Erica Klarreich }}</ref><ref>
{{ cite web | url=http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTch-discussions-2018-03.html | title=March 2018 Discussions on IUTeich | access-date=October 2, 2018 }} Web-page by Mochizuki describing discussions and linking consequent publications and supplementary material</ref>
While they did not resolve the differences, they brought them into clearer focus.
Scholze and Stix wrote a report asserting and explaining an error in the logic of the proof and claiming that the resulting gap was "so severe that ... small modifications will not rescue the proof strategy";<ref name=stillConj>{{cite web |url= http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pdf |title= Why abc is still a conjecture |first1= Peter |last1= Scholze |author-link1= Peter Scholze |first2= Jakob |last2= Stix |author-link2= Jakob Stix |access-date= September 23, 2018 |archive-date= February 8, 2020 |archive-url= https://web.archive.org/web/20200208075321/http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pdf |url-status= dead }} (updated version of their [http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-05.pdf May report] {{Webarchive|url=https://web.archive.org/web/20200208075318/http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-05.pdf |date=2020-02-08 }})</ref>
Mochizuki claimed that they misunderstood vital aspects of the theory and made invalid simplifications.<ref>
{{ cite web | url= http://www.kurims.kyoto-u.ac.jp/~motizuki/Rpt2018.pdf | title= Report on Discussions, Held during the Period March 15 – 20, 2018, Concerning Inter-Universal Teichmüller Theory
 |first= Shinichi |last= Mochizuki |author-link=Shinichi Mochizuki | access-date=February 1, 2019
 |quote = the ... discussions ... constitute the first detailed, ... substantive discussions concerning negative positions ... IUTch.
 }}</ref><ref>
{{ cite web | url= https://www.kurims.kyoto-u.ac.jp/~motizuki/Cmt2018-05.pdf | title= Comments on the manuscript by Scholze-Stix concerning Inter-Universal Teichmüller Theory
 |first= Shinichi |last= Mochizuki |author-link=Shinichi Mochizuki | access-date=October 2, 2018 |date=July 2018 |s2cid=174791744 }}</ref><ref>
{{ cite web | url= http://www.kurims.kyoto-u.ac.jp/~motizuki/Cmt2018-08.pdf | title= Comments on the manuscript (2018-08 version) by Scholze-Stix concerning Inter-Universal Teichmüller Theory
 |first= Shinichi |last= Mochizuki |author-link=Shinichi Mochizuki | access-date=October 2, 2018 }}</ref>

On April 3, 2020, two mathematicians from the Kyoto [[Research Institute for Mathematical Sciences|research institute]] where Mochizuki works announced that his claimed proof would be published in ''Publications of the Research Institute for Mathematical Sciences'', the institute's journal.  Mochizuki is chief editor of the journal but recused himself from the review of the paper.<ref name="nature-2020"/> The announcement was received with skepticism by [[Kiran Kedlaya]] and [[Edward Frenkel]], as well as being described by [[Nature (journal)|''Nature'']] as "unlikely to move many researchers over to Mochizuki's camp".<ref name="nature-2020"/> In March 2021, Mochizuki's proof was published in RIMS.<ref>
{{ cite web | url= https://www.ems-ph.org/journals/show_issue.php?issn=0034-5318&vol=57&iss=1 | title= Mochizuki's proof of ABC conjecture
 |first= Shinichi |last= Mochizuki |author-link=Shinichi Mochizuki | access-date=July 13, 2021 }}</ref>