Editing Abc conjecture

{{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>

==Formulations==
Before stating the conjecture, the notion of the  [[radical of an integer]] must be introduced: for a [[positive integer]] ''<math>n</math>'', the radical of ''<math>n</math>'', denoted ''<math>\text{rad}(n)</math>'', is the product of the distinct [[prime factor]]s of ''<math>n</math>''. For example,

<math>\text{rad}(16)=\text{rad}(2^4)=\text{rad}(2)=2</math>

<math>\text{rad}(17)=17</math>

''<math>\text{rad}(18)=\text{rad}(2\cdot 3^2)=2\cdot3 =6</math>''

''<math>\text{rad}(1000000)=\text{rad}(2^6 \cdot 5^6)=2\cdot5=10</math>''

If ''a'', ''b'', and ''c'' are [[coprime]]<ref group="notes">When ''a'' + ''b'' = ''c'', any common factor of two of the values is necessarily shared by the third.  Thus, coprimality of ''a'', ''b'', ''c'' implies [[Pairwise coprime|pairwise coprimality]] of ''a'', ''b'', ''c''. So in this case, it does not matter which concept we use.</ref> positive integers such that ''a'' + ''b'' = ''c'', it turns out that "usually" ''<math>c<\text{rad}(abc)</math>''. The ''abc conjecture'' deals with the exceptions. Specifically, it states that:

{{block indent|1=For every positive [[real number]] ''ε'', there exist only finitely many triples (''a'', ''b'', ''c'') of coprime positive integers, with ''a'' + ''b'' = ''c'', such that{{sfn|Waldschmidt|2015}}
{{block indent|<math>c > \operatorname{rad}(abc)^{1+\varepsilon}.</math>}}}}

An equivalent formulation is:

{{block indent|1=For every positive real number ''ε'', there exists a constant ''K<sub>ε</sub>'' such that for all triples (''a'', ''b'', ''c'') of coprime positive integers, with ''a'' + ''b'' = ''c'':{{sfn|Waldschmidt|2015}}
{{block indent|<math>c < K_{\varepsilon} \cdot \operatorname{rad}(abc)^{1+\varepsilon}.</math>}}}}

Equivalently (using the [[little o notation]]):

{{block indent|1=For all triples (''a'', ''b'', ''c'') of coprime positive integers with ''a'' + ''b'' = ''c'', rad(''abc'') is at least ''c''<sup>1-''o''(1)</sup>.}}

A fourth equivalent formulation of the conjecture involves the ''quality'' ''q''(''a'', ''b'', ''c'') of the triple (''a'', ''b'', ''c''), which is defined as

{{block indent|<math> q(a, b, c) = \frac{\log(c)}{\log\big(\textrm{rad}(abc)\big)}.</math>}}

For example:

{{block indent|1=''q''(4, 127, 131) = log(131) / log(rad(4·127·131)) = log(131) / log(2·127·131) = 0.46820...
:''q''(3, 125, 128) = log(128) / log(rad(3·125·128)) = log(128) / log(30) = 1.426565...}}

A typical triple (''a'', ''b'', ''c'') of coprime positive integers with ''a'' + ''b'' = ''c'' will have ''c'' < rad(''abc''), i.e. ''q''(''a'', ''b'', ''c'') < 1. Triples with ''q'' > 1 such as in the second example are rather special, they consist of numbers divisible by high powers of small [[prime number]]s. The fourth formulation is:

{{block indent|1=For every positive real number ''ε'', there exist only finitely many triples (''a'', ''b'', ''c'') of coprime positive integers with ''a'' + ''b'' = ''c'' such that ''q''(''a'', ''b'', ''c'') > 1 + ''ε''.}}

Whereas it is known that there are infinitely many triples (''a'', ''b'', ''c'') of coprime positive integers with ''a'' + ''b'' = ''c'' such that ''q''(''a'', ''b'', ''c'') > 1, the conjecture predicts that only finitely many of those have ''q'' > 1.01 or ''q'' > 1.001 or even ''q'' > 1.0001, etc. In particular, if the conjecture is true, then there must exist a triple (''a'', ''b'', ''c'') that achieves the maximal possible quality ''q''(''a'', ''b'', ''c'').

==Examples of triples with small radical==
The condition that ''ε'' > 0 is necessary as there exist infinitely many triples ''a'', ''b'', ''c'' with ''c'' > rad(''abc''). For example, let

{{block indent|<math>a = 1, \quad b = 2^{6n} - 1, \quad c = 2^{6n}, \qquad n > 1.</math>}}

The integer ''b'' is divisible by 9:

{{block indent|<math> b = 2^{6n} - 1  =  64^n - 1 = (64 - 1) (\cdots) = 9 \cdot 7 \cdot (\cdots).</math>}}

Using this fact, the following calculation is made:

{{block indent|<math>\begin{align}
\operatorname{rad}(abc) &= \operatorname{rad}(a) \operatorname{rad}(b) \operatorname{rad}(c) \\
&= \operatorname{rad}(1)  \operatorname{rad} \left ( 2^{6n} -1 \right ) \operatorname{rad} \left (2^{6n} \right ) \\
&= 2 \operatorname{rad} \left ( 2^{6n} -1 \right ) \\
&= 2 \operatorname{rad} \left ( 9 \cdot \tfrac{b}{9} \right )  \\
&\leqslant 2 \cdot 3  \cdot \tfrac{b}{9} \\
&= \tfrac{2}{3} b \\
&< \tfrac{2}{3} c.
\end{align}</math>}}

By replacing the exponent 6''n'' with other exponents forcing ''b'' to have larger square factors, the ratio between the radical and ''c'' can be made arbitrarily small. Specifically, let ''p'' > 2 be a prime and consider

{{block indent|<math>a = 1, \quad b = 2^{p(p-1)n} - 1, \quad c = 2^{p(p-1)n}, \qquad n > 1.</math>}}

Now it may be plausibly claimed that ''b'' is divisible by ''p''<sup>2</sup>:

{{block indent|<math>\begin{align}
b &= 2^{p(p-1)n} - 1 \\
&= \left(2^{p(p-1)}\right)^n - 1 \\
&= \left(2^{p(p-1)} - 1\right) (\cdots) \\
&= p^2 \cdot r (\cdots).
\end{align}</math>}}

The last step uses the fact that ''p''<sup>2</sup> divides 2<sup>''p''(''p''−1)</sup> − 1. This follows from [[Fermat's little theorem]], which shows that, for ''p'' > 2, 2<sup>''p''−1</sup> = ''pk'' + 1 for some integer ''k''. Raising both sides to the power of ''p'' then shows that 2<sup>''p''(''p''−1)</sup> = ''p''<sup>2</sup>(...) + 1.

And now with a similar calculation as above, the following results:

{{block indent|<math>\operatorname{rad}(abc) < \tfrac{2}{p} c.</math>}}

A list of the [[#Highest-quality triples|highest-quality triples]] (triples with a particularly small radical relative to ''c'') is given below; the highest quality, 1.6299, was found by Eric Reyssat {{harv|Lando|Zvonkin|2004|p=137}} for
{{block indent|1=''a'' = 2,}}
{{block indent|1=''b'' = 3<sup>10</sup>·109 = {{val|6,436,341}},}}
{{block indent|1=''c'' = 23<sup>5</sup> = {{val|6,436,343}},}}
{{block indent|1=rad(''abc'') = {{val|15042}}.}}

==Some consequences==

The ''abc'' conjecture has a large number of consequences. These include both known results (some of which have been proven separately only since the conjecture has been stated) and conjectures for which it gives a [[conditional proof]]. The consequences include:

* [[Roth's theorem]] on [[Diophantine approximation]] of [[algebraic number]]s.{{sfnp|Bombieri|1994|p={{page needed|date=January 2022}}}}{{sfn|Waldschmidt|2015}}
* The [[Faltings's theorem|Mordell conjecture]] (already proven in general by [[Gerd Faltings]]).{{sfnp|Elkies|1991}}
* As equivalent, [[Vojta's conjecture]] in dimension 1.{{sfnp|Van Frankenhuijsen|2002}}
* The [[Erdős–Woods number|Erdős–Woods conjecture]] allowing for a finite number of counterexamples.{{sfnp|Langevin|1993}}
* The existence of infinitely many non-[[Wieferich prime]]s in every base ''b'' > 1.{{sfnp|Silverman|1988}}
* The weak form of [[Marshall Hall's conjecture]] on the separation between squares and cubes of integers.{{sfnp|Nitaj|1996}}
* [[Fermat's Last Theorem]] has [[Wiles's proof of Fermat's Last Theorem|a famously difficult proof by Andrew Wiles]]. However it follows easily, at least for <math>n \ge 6</math>, from an effective form of a weak version of the ''abc'' conjecture. The ''abc'' conjecture says the [[Limit superior and limit inferior|lim sup]] of the set of all qualities (defined above) is 1, which implies the much weaker assertion that there is a finite upper bound for qualities. The conjecture that 2 is such an upper bound suffices for a very short proof of Fermat's Last Theorem for <math>n \ge 6</math>.<ref>{{cite journal |last1=Granville |first1=Andrew |last2=Tucker |first2=Thomas |year=2002 |title=It's As Easy As abc |url=https://www.ams.org/notices/200210/fea-granville.pdf |journal=Notices of the AMS |volume=49 |issue=10 |pages=1224–1231}}</ref>
* The [[Fermat–Catalan conjecture]], a generalization of [[Fermat's Last Theorem]] concerning powers that are sums of powers.{{sfnp|Pomerance|2008}}
* The [[Dirichlet L-function|''L''-function]] ''L''(''s'', ''χ<sub>d</sub>'') formed with the [[Legendre symbol]], has no [[Siegel zero]], given a uniform version of the ''abc'' conjecture in [[number field]]s, not just the ''abc'' conjecture as formulated above for rational integers.{{sfnp|Granville|Stark|2000}}
* A [[polynomial]] ''P''(''x'') has only finitely many [[perfect powers]] for all [[integers]] ''x'' if ''P'' has at least three [[simple zero]]s.<ref name="Ref_a">[http://www.math.uu.nl/people/beukers/ABCpresentation.pdf The ABC-conjecture], Frits Beukers, ABC-DAY, Leiden, Utrecht University, 9 September 2005.</ref>
* A generalization of [[Tijdeman's theorem]] concerning the number of solutions of ''y<sup>m</sup>'' = ''x<sup>n</sup>'' + ''k'' (Tijdeman's theorem answers the case ''k'' = 1), and Pillai's conjecture (1931) concerning the number of solutions of ''Ay<sup>m</sup>'' = ''Bx<sup>n</sup>'' + ''k''.
* As equivalent, the Granville–Langevin conjecture, that if ''f'' is a square-free binary form of degree ''n'' > 2, then for every real ''β'' > 2 there is a constant ''C''(''f'', ''β'') such that for all coprime integers ''x'', ''y'', the radical of ''f''(''x'', ''y'') exceeds ''C'' · max{|''x''|, |''y''|}<sup>''n''−''β''</sup>.<ref>{{harvtxt|Mollin|2009}}; {{harvtxt|Mollin|2010|p=297}}</ref>
* all the polynominals (x^n-1)/(x-1) have an infinity of square-free values.<ref>{{harvtxt|Browkin|2000|p=10}}</ref>
* As equivalent, the modified [[Szpiro conjecture]], which would yield a bound of rad(''abc'')<sup>1.2+''ε''</sup>.{{sfn|Oesterlé|1988}}
* {{harvtxt|Dąbrowski|1996}} has shown that the ''abc'' conjecture implies that [[Brocard's problem|the Diophantine equation ''n''! + ''A'' = ''k''<sup>2</sup>]] has only finitely many solutions for any given integer ''A''.
* There are ~''c''<sub>''f''</sub>''N'' positive integers ''n'' ≤ ''N'' for which ''f''(''n'')/B' is square-free, with ''c''<sub>''f''</sub> > 0 a positive constant defined as:{{sfnp|Granville|1998}}{{block indent|<math>c_f = \prod_{\text{prime }p} x_i \left ( 1 - \frac{\omega\,\!_f (p)}{p^{2+q_p}} \right ).</math>}}
*The [[Beal conjecture]], a generalization of Fermat's Last Theorem proposing that if ''A'', ''B'', ''C'', ''x'', ''y'', and ''z'' are positive integers with ''A<sup>x</sup>'' + ''B<sup>y</sup>'' = ''C<sup>z</sup>'' and ''x'', ''y'', ''z'' > 2, then ''A'', ''B'', and ''C'' have a common prime factor. The ''abc'' conjecture would imply that there are only finitely many counterexamples.
*[[Néron–Tate height#Lower bounds for the Néron–Tate height|Lang's conjecture]], a lower bound for the [[height function|height]] of a non-torsion rational point of an elliptic curve.
* A negative solution to the [[Erdős–Ulam problem]] on dense sets of Euclidean points with rational distances.<ref>
{{citation
 | last1 = Pasten | first1 = Hector
 | doi = 10.1007/s00605-016-0973-2
 | issue = 1
 | journal = [[Monatshefte für Mathematik]]
 | mr = 3592123
 | pages = 99–126
 | title = Definability of Frobenius orbits and a result on rational distance sets
 | volume = 182
 | year = 2017| s2cid = 7805117
 }}
</ref>
* An effective version of [[Siegel's theorem on integral points|Siegel's theorem about integral points on algebraic curves]].<ref>{{cite journal | arxiv=math/0408168 | last1=Surroca | first1=Andrea | title=Siegel's theorem and the abc conjecture |url=https://www.rivmat.unipr.it/fulltext/2004-3s/pdf/22.pdf | date=2004 |journal= Rivista Matematica dell'Universita' di Parma, Atti del Secondo Convegno Italiano di Teoria dei Numeri |volume=3* |issue=7 |pages=323-332}}</ref>

==Theoretical results==
The ''abc'' conjecture implies that ''c'' can be [[upper bound|bounded above]] by a near-linear function of the radical of ''abc''. Bounds are known that are [[exponential function|exponential]]. Specifically, the following bounds have been proven:

{{block indent|<math>c < \exp{ \left(K_1 \operatorname{rad}(abc)^{15}\right) } </math> {{harv|Stewart|Tijdeman|1986}},}}
{{block indent|<math>c < \exp{ \left(K_2 \operatorname{rad}(abc)^{\frac{2}{3} + \varepsilon}\right) } </math> {{harv|Stewart|Yu|1991}}, and}}
{{block indent|<math>c < \exp{ \left(K_3 \operatorname{rad}(abc)^{\frac{1}{3}}\left(\log(\operatorname{rad}(abc)\right)^3\right) } </math> {{harv|Stewart|Yu|2001}}.}}

In these bounds, ''K''<sub>1</sub> and ''K''<sub>3</sub> are [[Constant (mathematics)|constants]] that do not depend on ''a'', ''b'', or ''c'', and ''K''<sub>2</sub> is a constant that depends on ''ε'' (in an [[effectively computable]] way) but not on ''a'', ''b'', or ''c''. The bounds apply to any triple for which ''c'' > 2.

There are also theoretical results that provide a lower bound on the best possible form of the ''abc'' conjecture.  In particular, {{Harvtxt|Stewart|Tijdeman|1986}} showed that there are infinitely many triples (''a'', ''b'', ''c'') of coprime integers with ''a'' + ''b'' = ''c'' and

{{block indent|<math>c > \operatorname{rad}(abc) \exp{ \left(k \sqrt{\log c}/\log\log c \right) } </math>}}

for all ''k'' < 4.  The constant ''k'' was improved to ''k'' = 6.068 by {{Harvtxt|van Frankenhuysen|2000}}.

==Computational results==
In 2006, the Mathematics Department of [[Leiden University]] in the Netherlands, together with the Dutch Kennislink science institute, launched the [[ABC@Home]] project, a [[grid computing]] system, which aims to discover additional triples ''a'', ''b'', ''c'' with rad(''abc'') < ''c''. Although no finite set of examples or counterexamples can resolve the ''abc'' conjecture, it is hoped that patterns in the triples discovered by this project will lead to insights about the conjecture and about number theory more generally.

{| class="wikitable" style="text-align:right;"
|+ Distribution of triples with ''q'' > 1<ref name="Ref_d">{{Citation|url=http://www.rekenmeemetabc.nl/?item=h_stats |archive-url=https://web.archive.org/web/20081222221716/http://rekenmeemetabc.nl/?item=h_stats |url-status=dead |archive-date=December 22, 2008 |title=Synthese resultaten |work=RekenMeeMetABC.nl |access-date=October 3, 2012 |language=nl }}.</ref>
|-
! scope="col" {{diagonal split header|''c''|''q''}}
! scope="col" | ''q'' > 1
! scope="col" | ''q'' > 1.05
! scope="col" | ''q'' > 1.1
! scope="col" | ''q'' > 1.2
! scope="col" | ''q'' > 1.3
! scope="col" | ''q'' > 1.4
|-
! scope="row" | ''c'' < 10<sup>2</sup>
| 6 || 4 || 4 || 2 || 0 || 0
|-
! scope="row" | ''c'' < 10<sup>3</sup>
| 31 || 17 || 14 || 8 || 3 || 1
|-
! scope="row" | ''c'' < 10<sup>4</sup>
| 120 || 74 || 50 || 22 || 8 || 3
|-
! scope="row" | ''c'' < 10<sup>5</sup>
| 418 || 240 || 152 || 51 || 13 || 6
|-
! scope="row" | ''c'' < 10<sup>6</sup>
| 1,268 || 667 || 379 || 102 || 29 || 11
|-
! scope="row" | ''c'' < 10<sup>7</sup>
| 3,499 || 1,669 || 856 || 210 || 60 || 17
|-
! scope="row" | ''c'' < 10<sup>8</sup>
| 8,987 || 3,869 || 1,801 || 384 || 98 || 25
|-
! scope="row" | ''c'' < 10<sup>9</sup>
| 22,316 || 8,742 || 3,693 || 706 || 144 || 34
|-
! scope="row" | ''c'' < 10<sup>10</sup>
| 51,677 || 18,233 || 7,035 || 1,159 || 218 || 51					
|-
! scope="row" | ''c'' < 10<sup>11</sup>
| 116,978 || 37,612 || 13,266 || 1,947 || 327 || 64					
|-
! scope="row" | ''c'' < 10<sup>12</sup>
| 252,856 || 73,714 || 23,773 || 3,028 || 455 || 74					
|-
! scope="row" | ''c'' < 10<sup>13</sup>
| 528,275 || 139,762 || 41,438 || 4,519 || 599 || 84					
|-
! scope="row" | ''c'' < 10<sup>14</sup>
| 1,075,319 || 258,168 || 70,047 || 6,665 || 769 || 98					
|-
! scope="row" | ''c'' < 10<sup>15</sup>
| 2,131,671 || 463,446 || 115,041 || 9,497 || 998 || 112					
|-
! scope="row" | ''c'' < 10<sup>16</sup>
| 4,119,410 || 812,499 || 184,727 || 13,118 || 1,232 || 126					
|-
! scope="row" | ''c'' < 10<sup>17</sup>
| 7,801,334 || 1,396,909 || 290,965 || 17,890 || 1,530 || 143
|-
! scope="row" | ''c'' < 10<sup>18</sup>
| 14,482,065 || 2,352,105 || 449,194 || 24,013 || 1,843 || 160
|-
|}

As of May 2014, [[ABC@Home]] had found 23.8 million triples.<ref name="Ref_c">{{Citation |url=http://abcathome.com/data/ |title=Data collected sofar |work=ABC@Home |access-date=April 30, 2014 |url-status=dead |archive-url=https://web.archive.org/web/20140515021303/http://abcathome.com/data/ |archive-date=May 15, 2014 }}</ref>

{| class="wikitable"
|+ {{visible anchor|Highest-quality triples}}<ref>{{cite web |url=http://www.math.leidenuniv.nl/~desmit/abc/index.php?set=2 |title=100 unbeaten triples |work=Reken mee met ABC |date=2010-11-07 }}</ref>
|-
! scope="col" | Rank
! scope="col" | ''q''
! scope="col" | ''a''
! scope="col" | ''b''
! scope="col" | ''c''
! scope="col" class="unsortable" | Discovered by
|-
! scope="row" | 1
| 1.6299 || 2 || 3<sup>10</sup>·109 || 23<sup>5</sup> || Eric Reyssat
|-
! scope="row" | 2
| 1.6260 || 11<sup>2</sup> || 3<sup>2</sup>·5<sup>6</sup>·7<sup>3</sup> || 2<sup>21</sup>·23 || Benne de Weger
|-
! scope="row" | 3
| 1.6235 || 19·1307 || 7·29<sup>2</sup>·31<sup>8</sup> || 2<sup>8</sup>·3<sup>22</sup>·5<sup>4</sup> || Jerzy Browkin, Juliusz Brzezinski
|-
! scope="row" | 4
| 1.5808 || 283 || 5<sup>11</sup>·13<sup>2</sup> || 2<sup>8</sup>·3<sup>8</sup>·17<sup>3</sup> || Jerzy Browkin, Juliusz Brzezinski, Abderrahmane Nitaj
|-
! scope="row" | 5
| 1.5679 || 1 || 2·3<sup>7</sup> || 5<sup>4</sup>·7 || Benne de Weger
|}

Note: the ''quality'' ''q''(''a'', ''b'', ''c'') of the triple (''a'', ''b'', ''c'') is defined [[#Formulations|above]].

==Refined forms, generalizations and related statements==
The ''abc'' conjecture is an integer analogue of the [[Mason–Stothers theorem]] for polynomials.

A strengthening, proposed by {{Harvtxt|Baker|1998}}, states that in the ''abc'' conjecture one can replace rad(''abc'') by

{{block indent|''ε''<sup>−''ω''</sup> rad(''abc''),}}

where ''ω'' is the total number of distinct primes dividing ''a'', ''b'' and ''c''.{{sfnp|Bombieri|Gubler|2006|p=404}}

[[Andrew Granville]] noticed that the minimum of the function <math>\big(\varepsilon^{-\omega}\operatorname{rad}(abc)\big)^{1+\varepsilon}</math> over <math>\varepsilon > 0</math> occurs when <math>\varepsilon = \frac{\omega}{\log\big(\operatorname{rad}(abc)\big)}.</math>

This inspired {{Harvtxt|Baker|2004}} to propose a sharper form of the ''abc'' conjecture, namely:
{{block indent|<math>c < \kappa \operatorname{rad}(abc) \frac{\Big(\log\big(\operatorname{rad}(abc)\big)\Big)^\omega}{\omega!}</math>}}
with ''κ'' an absolute constant. After some computational experiments he found that a value of <math>6/5</math> was admissible for ''κ''. This version is called the "explicit ''abc'' conjecture".

{{Harvtxt|Baker|1998}} also describes related conjectures of [[Andrew Granville]] that would give upper bounds on ''c'' of the form

{{block indent|<math>K^{\Omega(a b c)} \operatorname{rad}(a b c),</math>}}

where Ω(''n'') is the total number of prime factors of ''n'', and

{{block indent|<math>O\big(\operatorname{rad}(a b c) \Theta(a b c)\big),</math>}}

where Θ(''n'') is the number of integers up to ''n'' divisible only by primes dividing ''n''.

{{Harvtxt|Robert|Stewart|Tenenbaum|2014}} proposed a more precise inequality based on {{Harvtxt|Robert|Tenenbaum|2013}}.
Let ''k'' = rad(''abc''). They conjectured there is a constant ''C''<sub>1</sub> such that

{{block indent|<math>c < k \exp\left(4\sqrt{\frac{3\log k}{\log\log k}}\left(1+\frac{\log\log\log k}{2\log\log k}+\frac{C_{1}}{\log\log k}\right)\right)</math>}}

holds whereas there is a constant ''C''<sub>2</sub> such that

{{block indent|<math>c > k \exp\left(4\sqrt{\frac{3\log k}{\log\log k}}\left(1+\frac{\log\log\log k}{2\log\log k}+\frac{C_{2}}{\log\log k}\right)\right)</math>}}

holds infinitely often.

{{harvtxt|Browkin|Brzeziński|1994}} formulated the [[n conjecture]]—a version of the ''abc'' conjecture involving ''n'' > 2 integers.

==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>

==See also==
*[[List of unsolved problems in mathematics]]

==Notes==
{{reflist|group=notes}}

==References==
{{reflist|25em}}

==Sources==
{{refbegin|25em}}
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*{{cite journal | last=Baker | first=Alan | author-link=Alan Baker (mathematician) | title=Experiments on the ''abc''-conjecture | journal=Publicationes Mathematicae Debrecen | volume=65 |issue=3–4 | pages=253–260 | year=2004  | doi=10.5486/PMD.2004.3348 | s2cid=253834357 |url=https://publi.math.unideb.hu/load_pdf.php?p=972 | doi-access=free }}
*{{cite document | first1=Enrico | last1=Bombieri | title=Roth's theorem and the abc-conjecture |type=Preprint | year=1994 | publisher=ETH Zürich }}{{unreliable source?|still only preprint after all these years?|date=January 2022}}
*{{cite book | first1=Enrico | last1=Bombieri | author-link1=Enrico Bombieri | first2=Walter | last2=Gubler | title=Heights in Diophantine Geometry | series=New Mathematical Monographs | volume=4 | publisher=[[Cambridge University Press]] | year=2006 | isbn=978-0-521-71229-3 | zbl=1130.11034 }}
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*{{cite journal |last1=Granville |first1=Andrew | author-link=Andrew Granville |last2=Stark |first2=H. |year=2000 |  title=ABC implies no "Siegel zeros" for L-functions of characters with negative exponent | url= http://www.dms.umontreal.ca/~andrew/PDF/NoSiegelfinal.pdf |journal=[[Inventiones Mathematicae]] |volume= 139 |issue=3 |pages=509–523 |doi=10.1007/s002229900036 |bibcode=2000InMat.139..509G |s2cid=6901166 }}
*{{cite journal |last1=Granville |first1=Andrew | author-link=Andrew Granville |last2=Tucker |first2=Thomas |year=2002 |  title=It's As Easy As abc | url= https://www.ams.org/journals/notices/200210/fea-granville.pdf |journal=[[Notices of the AMS]] |volume= 49 | issue=10 |pages=1224–1231 |citeseerx=10.1.1.146.610 }}
* {{cite book |first=Richard K. |last=Guy |author-link=Richard K. Guy |title=Unsolved Problems in Number Theory |publisher=[[Springer-Verlag]] |location=Berlin |year=2004 |isbn=0-387-20860-7 }}
* {{cite encyclopedia |last1=Lando |first1=Sergei K. |first2=Alexander K. |last2=Zvonkin |title=Graphs on Surfaces and Their Applications |publisher=Springer-Verlag |encyclopedia=Encyclopaedia of Mathematical Sciences: Lower-Dimensional Topology II |volume=141 |year=2004 |isbn=3-540-00203-0 }}
*{{Cite journal |last=Langevin |first=M. |year=1993 |title=Cas d'égalité pour le théorème de Mason et applications de la conjecture ''abc''|language=fr |journal=Comptes rendus de l'Académie des sciences |volume=317 |issue=5 |pages=441–444 }}
*{{Cite book | last=Masser | first=D. W. | author-link=David Masser | editor1-last=Chen | editor1-first=W. W. L. | title=Proceedings of the Symposium on Analytic Number Theory | publisher=Imperial College | location=London | year=1985 | chapter=Open problems}}
*{{cite journal | last=Mollin | first=R.A. | title=A note on the ABC-conjecture | journal=Far East Journal of Mathematical Sciences | volume=33 | number=3 | pages=267–275 | year=2009 | issn=0972-0871 | url=http://people.ucalgary.ca/~ramollin/abcconj.pdf | zbl=1241.11034 | access-date=2013-06-14 | archive-url=https://web.archive.org/web/20160304053930/http://people.ucalgary.ca/~ramollin/abcconj.pdf | archive-date=2016-03-04 | url-status=dead }}
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{{refend}}

==External links==
* [https://web.archive.org/web/20100917032533/http://abcathome.com/apps.php ABC@home] [[Distributed computing]] project called [[ABC@Home]].
* [http://bit-player.org/2007/easy-as-abc Easy as ABC]: Easy to follow, detailed explanation by Brian Hayes.
* {{MathWorld | urlname=abcConjecture | title=abc Conjecture}}
* Abderrahmane Nitaj's [https://nitaj.users.lmno.cnrs.fr/abc.html ABC conjecture home page]
* Bart de Smit's [http://www.math.leidenuniv.nl/~desmit/abc/ ABC Triples webpage]
* http://www.math.columbia.edu/~goldfeld/ABC-Conjecture.pdf
* [http://dash.harvard.edu/bitstream/handle/1/2793857/Elkies%20-%20ABCs%20of%20Number%20Theory.pdf?sequence=2 The ABC's of Number Theory] by [[Noam D. Elkies]]
* [http://www.math.harvard.edu/~mazur/papers/scanQuest.pdf Questions about Number] by [[Barry Mazur]]
* [https://mathoverflow.net/q/106560 Philosophy behind Mochizuki’s work on the ABC conjecture] on [[MathOverflow]]
* [http://michaelnielsen.org/polymath1/index.php?title=ABC_conjecture ABC Conjecture] [[Polymath project]] wiki page linking to various sources of commentary on Mochizuki's papers.
* [https://www.youtube.com/watch?v=RkBl7WKzzRw abc Conjecture] Numberphile video
* [http://www.kurims.kyoto-u.ac.jp/~motizuki/news-english.html News about IUT by Mochizuki]

[[Category:Conjectures]]
[[Category:Abc conjecture]]
[[Category:Unsolved problems in number theory]]
[[Category:1985 introductions]]
[[Category:Number theory]]