Editing Abc conjecture (section)

{{DISPLAYTITLE:''abc'' conjecture}}
{{short description|The product of distinct prime factors of a,b,c, where c is a+b, is rarely much less than c}}
{{Infobox mathematical statement
| name = ''abc'' conjecture
| image = 
| caption = 
| field = [[Number theory]]
| conjectured by = {{plainlist|
*[[Joseph Oesterlé]]
*[[David Masser]]}}
| conjecture date = 1985
| first proof by = 
| first proof date = 
| open problem = 
| known cases =
| implied by =
| equivalent to = [[Szpiro's conjecture#Modified Szpiro conjecture|Modified Szpiro conjecture]]
| generalizations = 
| consequences = {{plainlist|
*[[Beal conjecture]]
*[[Erdős–Ulam problem]]
*[[Faltings's theorem]]
*[[Fermat's Last Theorem]]
*[[Fermat–Catalan conjecture]]
*[[Roth's theorem]]
*[[Tijdeman's theorem]]
}}
}}
[[File:Oesterle Joseph.jpg|thumb|Mathematician [[Joseph Oesterlé]]]]
[[File:David Masser.jpg|thumb|Mathematician [[David Masser]]]]
The '''''abc'' conjecture''' (also known as the '''Oesterlé–Masser conjecture''') is a [[conjecture]] in [[number theory]] that arose out of a discussion of [[Joseph Oesterlé]] and [[David Masser]] in 1985.{{sfn|Oesterlé|1988}}{{sfn|Masser|1985}} It is stated in terms of three [[positive integer]]s ''<math>a, b</math>'' and ''<math>c</math>'' (hence the name) that are [[coprime integers|relatively prime]] and satisfy ''<math>a+b=c</math>''.  The conjecture essentially states that the product of the distinct [[prime factor]]s of ''<math>abc</math>'' is usually not much smaller than ''<math>c</math>''.  A number of famous conjectures and theorems in number theory would follow immediately from the ''abc'' conjecture or its versions.  Mathematician [[Dorian M. Goldfeld|Dorian Goldfeld]] described the ''abc'' conjecture as "The most important unsolved problem in [[Diophantine analysis]]".{{sfn|Goldfeld|1996}}

The ''abc'' conjecture originated as the outcome of attempts by Oesterlé and Masser to understand the [[Szpiro conjecture]] about [[elliptic curve]]s,<ref>{{cite journal |last1=Fesenko |first1=Ivan |title=Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta-functions, notes on the work of Shinichi Mochizuki |journal=European Journal of Mathematics |date=September 2015 |volume=1 |issue=3 |pages=405–440 |doi=10.1007/s40879-015-0066-0 |doi-access=free }}</ref> which involves more geometric structures in its statement than the ''abc'' conjecture. The ''abc'' conjecture was shown to be equivalent to the modified Szpiro's conjecture.{{sfn|Oesterlé|1988}}

Various attempts to prove the ''abc'' conjecture have been made, but none have gained broad acceptance. [[Shinichi Mochizuki]] claimed to have a proof in 2012, but the conjecture is still regarded as unproven by the mainstream mathematical community.<ref name="Ball">
{{cite journal |last1=Ball |first1=Peter |date= 10 September 2012|title=Proof claimed for deep connection between primes |url=https://www.nature.com/news/proof-claimed-for-deep-connection-between-primes-1.11378 |journal=Nature |doi=10.1038/nature.2012.11378 |access-date=19 March 2018|doi-access=free }}</ref><ref name="nature-2020">{{cite journal |last1=Castelvecchi |first1=Davide |title=Mathematical proof that rocked number theory will be published |journal=Nature |date=9 April 2020 |volume=580 |issue=7802 |pages=177 |doi=10.1038/d41586-020-00998-2 |pmid=32246118 |bibcode=2020Natur.580..177C |s2cid=214786566 |doi-access= }}</ref><ref>[https://www.math.columbia.edu/~woit/wordpress/?p=11709&cpage=1#comment-235940 Further comment by P. Scholze at ''Not Even Wrong''] math.columbia.edu{{self-published inline|date=January 2022}}</ref><ref>{{cite web |last1=Scholze |first1=Peter |title=Review of inter-universal Teichmüller Theory I |url=https://zbmath.org/1465.14002 |website=zbmath open |access-date=2025-02-25}}</ref>