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{{Infobox mathematical statement
| name = abc conjecture
| image = 
| caption = 
| field = [[Number theory]]
| conjectured by = [[Joseph  Oesterlé]]<br>[[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]]
*[[Faltings's theorem]]
*[[Fermat's Last Theorem]]
*[[Fermat–Catalan conjecture]]
*[[Roth's theorem]]
*[[Tijdeman's theorem]]}}
}}
{{DISPLAYTITLE:''abc'' conjecture}}
[[File:Oesterle Joseph.jpg|thumb|French mathematician Joseph  Oesterlé]]
[[File:David Masser.jpg|thumb|British mathematician David Masser]]
The '''''abc'' conjecture''' (also known as the '''Oesterlé–Masser conjecture''') is a [[conjecture]] in [[number theory]], first proposed by {{harvs|txt|authorlink=Joseph  Oesterlé|first=Joseph|last=Oesterlé|year=1988}} and {{harvs|txt|authorlink=David Masser|first=David|last=Masser|year=1985}}. It is stated in terms of three positive integers, ''a'', ''b'' and ''c'' (hence the name) that are [[coprime integers|relatively prime]] and satisfy ''a'' + ''b'' = ''c''. If ''d'' denotes the product of the distinct [[prime factor]]s of ''abc'', the conjecture essentially states that ''d'' is usually not much smaller than ''c''. In other words: if ''a'' and ''b'' are composed from large powers of primes, then ''c'' is usually not divisible by large powers of primes. The precise statement is given below.
The ''abc'' conjecture originated as the outcome of attempts by Oesterlé and Masser to understand the [[Szpiro conjecture]] about elliptic curves.<ref>{{citation|title=Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki|journal= European Journal of Mathematics|first=Ivan|last=Fesenko|volume=1 |issue=3| pages=405–440 | year=2015 |url=https://www.maths.nottingham.ac.uk/personal/ibf/notesoniut.pdf |doi=10.1007/s40879-015-0066-0}}.</ref> The latter conjecture has more geometric structures involved in its statement in comparison with the ''abc'' conjecture.

The ''abc'' conjecture and its versions express, in concentrated form, some fundamental feature of various problems in Diophantine geometry. A number of famous conjectures and theorems in number theory would follow immediately from the ''abc'' conjecture or its versions. {{harvtxt|Goldfeld|1996}} described the ''abc'' conjecture as "the most important unsolved problem in [[Diophantine analysis]]".

Various attempts to prove the abc conjecture were made, but none have been accepted by the mainstream mathematical community so far and the conjecture is still largely recognized to be unproven as of 2019.

==Formulations==
Before we state the conjecture we introduce the notion of the  [[radical of an integer]]: for a [[positive integer]] ''n'', the radical of ''n'', denoted rad(''n''), is the product of the distinct [[prime factor]]s of ''n''. For example

:rad(16) = rad(2<sup>4</sup>) = rad(2) = 2,
:rad(17) = 17,
:rad(18) = rad(2 ⋅ 3<sup>2</sup>) = 2 · 3 = 6,
:rad(1000000) = rad(2<sup>6</sup> ⋅ 5<sup>6</sup>) = 2 ⋅ 5 = 10.

If ''a'', ''b'', and ''c'' are [[coprime]]<ref group = notes>When ''a'' + ''b'' = ''c'', coprimeness of ''a'', ''b'', ''c'' implies [[pairwise coprime]]ness 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" ''c'' < rad(''abc''). The ''abc conjecture'' deals with the exceptions. Specifically, it states that:

:'''ABC conjecture.''' 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
::<math>c > \operatorname{rad}(abc)^{1+\varepsilon}.</math>

An equivalent formulation:

:'''ABC conjecture II.''' 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'':
::<math>c < K_{\varepsilon} \cdot \operatorname{rad}(abc)^{1+\varepsilon}.</math>

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

: <math> q(a, b, c) = \frac{ \log(c) }{ \log\big(\operatorname{rad}(abc)\big) }.</math>

For example:

:''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.

:'''ABC conjecture III.''' 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 rad(''abc'') < ''c''. For example, let

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

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

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

Using this fact we calculate:

:<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} \\
&= 2 \tfrac{b}{3} \\
&< \tfrac{2}{3} c.
\end{align}</math>

By replacing the exponent 6''n'' by 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

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

Now we claim that ''b'' is divisible by ''p''<sup>2</sup>:

:<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 we have

:<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
:''a'' = 2,
:''b'' = 3<sup>10</sup>·109 = {{val|6,436,341}},
:''c'' = 23<sup>5</sup> = {{val|6,436,343}},
: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 since the conjecture has been stated) and conjectures for which it gives a [[conditional proof]]. While an earlier proof of the conjecture would have been more significant in terms of consequences, the ''abc'' conjecture itself remains of interest for the other conjectures it would prove, together with its numerous links with deep questions in number theory. The consequences include:

* [[Roth's theorem]] on diophantine approximation of algebraic numbers.{{sfnp|Bombieri|1994}}
* 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}}
* 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 fields, 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 zeros.<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>
* As equivalent, the modified [[Szpiro conjecture]], which would yield a bound of rad(''abc'')<sup>1.2+''ε''</sup>.{{sfnp|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}}
::<math>c_f = \prod_{\text{prime }p} x_i \left ( 1 - \frac{\omega\,\!_f (p)}{p^{2+q_p}} \right ).</math>
*[[Fermat's Last Theorem]] has 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 = http://www.ams.org/notices/200210/fea-granville.pdf | journal = Notices of the AMS | volume = 49 | issue = 10| pages = 1224–1231 }}</ref>
* 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.

==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:

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

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

==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 |accessdate=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 |accessdate=April 30, 2014 |url-status=dead |archiveurl=https://web.archive.org/web/20140515021303/http://abcathome.com/data/ |archivedate=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

:''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 incited {{Harvtxt|Baker|2004}} to propose a sharper form of the ''abc'' conjecture, namely:
:<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 "explicit ''abc'' conjecture".

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

:<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

:<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

:<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

:<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 of abc==

[[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|authorlink=Peter Woit|work=Not Even Wrong|url=http://www.math.columbia.edu/~woit/wordpress/?p=561|date=May 26, 2007}}.</ref>

In August 2012 [[Shinichi Mochizuki]] claimed a proof of the abc conjecture.<ref>
{{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 |volume= |issue= |pages= |doi=10.1038/nature.2012.11378 |access-date=19 March 2018}}</ref>
He released a series of four preprints developing a new theory called [[Inter-universal Teichmüller theory]] (IUTT) which is then applied to prove several famous conjectures in number theory, including the abc conjecture but also [[Szpiro's conjecture]], the hyperbolic [[Vojta's conjecture]].<ref name=Mochizukiweb>
Mochizuki, Shinichi (May 2015). ''Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations'', available at http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html</ref>
The papers have not been 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 |author=<!--Not stated--> |date=December 17, 2017 |access-date=March 17, 2018}}</ref> This is not only because of their impenetrability,<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>
{{cite web |url=https://galoisrepresentations.wordpress.com/2017/12/17/the-abc-conjecture-has-still-not-been-proved/#comment-4624 |title=The ABC conjecture has still not been proved, comment by Bcnrd |author=<!-- Not stated --> |date=December 22, 2017 |access-date=March 18, 2017}}</ref>
Though a few mathematicians have vouched for the correctness of the proof,<ref>
{{ cite web | url=http://www.inference-review.com/article/fukugen |first= Ivan |last= Fesenko |authorlink= Ivan Fesenko | title=Fukugen | website = Inference | access-date=19 March 2018}}</ref>
and have attempted to communicate their understanding via workshops on inter-universal Teichmüller theory, 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 |authorlink=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 }}</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 |authorlink= 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 concluded that the gap was "so severe that … small modifications will not rescue the proof strategy";<ref>
{{ 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 |authorlink1= Peter Scholze
 |first2= Jakob |last2= Stix    |authorlink2= Jakob Stix
 | access-date=September 23, 2018 }} (updated version of their [http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-05.pdf May report])</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 |authorlink=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= http://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 |authorlink=Shinichi Mochizuki | access-date=October 2, 2018 }}</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 |authorlink=Shinichi Mochizuki | access-date=October 2, 2018 }}</ref>

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

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

==Citations==
{{reflist}}

==References==
{{Refbegin|2}}
*{{cite book | last=Baker | first=Alan | authorlink=Alan Baker (mathematician) | chapter=Logarithmic forms and the ''abc''-conjecture | editor-last=Győry | editor-first=Kálmán | title=Number theory. Diophantine, computational and algebraic aspects. Proceedings of the international conference, Eger, Hungary, July 29-August 2, 1996 | location=Berlin | publisher=de Gruyter | pages=37–44 | year=1998 | isbn=3-11-015364-5 | zbl=0973.11047 |ref=harv }}
*{{cite journal | last=Baker | first=Alan | authorlink=Alan Baker (mathematician) | title=Experiments on the ''abc''-conjecture | journal=Publ. Math. Debrecen | volume=65 | pages=253–260 | year=2004 | ref=harv }}
*{{cite journal | first1=Enrico | last=Bombieri | title=Roth's theorem and the abc-conjecture | journal=Preprint | year=1994 | publisher=ETH Zürich | ref=harv}}
*{{cite book | first1=Enrico | last1=Bombieri | authorlink1=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 | doi=10.2277/0521846153|ref=harv }}
*{{Cite journal |authorlink=Jerzy Browkin |first=Jerzy |last=Browkin |first2=Juliusz |last2=Brzeziński | title=Some remarks on the ''abc''-conjecture | journal=Math. Comp. | volume=62 | pages=931–939 | year=1994 | doi=10.2307/2153551 | jstor=2153551 | issue=206 |ref=harv |bibcode=1994MaCom..62..931B }}
* {{cite book |last=Browkin |first=Jerzy |chapter=The ''abc''-conjecture |editor1-last=Bambah |editor1-first=R. P. |editor2-last=Dumir |editor2-first=V. C. |editor3-last=Hans-Gill |editor3-first=R. J. |year=2000 |title=Number Theory |series=Trends in Mathematics |location=Basel |publisher=Birkhäuser |isbn=3-7643-6259-6 |pages=75–106 |ref=harv  }}
*{{Cite journal |first=Andrzej |last=Dąbrowski |title=On the diophantine equation ''x''! + ''A'' = ''y''<sup>2</sup> | journal=Nieuw Archief voor Wiskunde, IV. |volume=14 |pages=321–324 |year=1996 | zbl=0876.11015 | ref=harv }}
*{{Cite journal |last=Elkies |first=N. D. |authorlink=Noam Elkies |title=ABC implies Mordell |journal=  International Mathematics Research Notices|volume=1991 |year=1991 |pages=99–109 |doi=10.1155/S1073792891000144 |issue=7 |ref=harv }}
* {{cite book |last=Frey |first=Gerhard |authorlink=Gerhard Frey |chapter=On Ternary Equations of Fermat Type and Relations with Elliptic Curves |title=Modular Forms and Fermat's Last Theorem |location=New York |publisher=Springer |year=1997 |isbn=0-387-94609-8 |pages=527–548 |chapterurl=https://books.google.com/books?id=Va-quzVwtMsC&pg=PA527 }}
*{{cite journal |last=Goldfeld |first=Dorian | authorlink=Dorian M. Goldfeld |year=1996 |title=Beyond the last theorem |journal=[[Math Horizons]] |volume=4 |issue=September |pages=26–34 |jstor= 25678079 |ref=harv |doi=10.1080/10724117.1996.11974985 }}
* {{cite book | zbl=1046.11035 | last=Goldfeld | first=Dorian | authorlink=Dorian M. Goldfeld | chapter=Modular forms, elliptic curves and the abc-conjecture | editor-last=Wüstholz | editor-first=Gisbert | editor-link=Gisbert Wüstholz | title=A panorama in number theory or The view from Baker's garden. Based on a conference in honor of Alan Baker's 60th birthday, Zürich, Switzerland, 1999 | location=Cambridge | publisher=[[Cambridge University Press]] | pages=128–147 | year=2002 | isbn=0-521-80799-9 | ref=harv }}
* {{cite book |editor1-last=Gowers |editor1-first=Timothy |editor1-link=Timothy Gowers |editor2-last=Barrow-Green |editor2-first=June |editor3-last=Leader |editor3-first=Imre |title=The Princeton Companion to Mathematics |year=2008 |publisher=Princeton University Press |location=Princeton |isbn=978-0-691-11880-2 |pages=361–362, 681 |ref=harv  |title-link=The Princeton Companion to Mathematics }}
*{{Cite journal |last=Granville |first=A. |authorlink=Andrew Granville |year=1998 |title=ABC Allows Us to Count Squarefrees |url=http://www.dms.umontreal.ca/~andrew/PDF/polysq3.pdf |journal=International Mathematics Research Notices |volume=1998 |issue=19 |pages=991–1009 |doi=10.1155/S1073792898000592 |ref=harv}}
*{{cite journal |last1=Granville |first1=Andrew | authorlink=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 |pages=509–523 |doi=10.1007/s002229900036 |ref=harv  |bibcode=2000InMat.139..509G }}
*{{cite journal |last1=Granville |first1=Andrew | authorlink=Andrew Granville |last2=Tucker |first2=Thomas |year=2002 |  title=It's As Easy As abc | url= http://www.ams.org/notices/200210/fea-granville.pdf |journal=[[Notices of the AMS]] |volume= 49 | issue=10 |pages=1224–1231 |ref=harv  }}
* {{cite book |first=Richard K. |last=Guy |authorlink=Richard K. Guy |title=Unsolved Problems in Number Theory |publisher=[[Springer-Verlag]] |location=Berlin |year=2004 |isbn=0-387-20860-7 |ref=harv }}
* {{cite book |last=Lando |first=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 |ref=harv }}
*{{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 |doi= |ref=harv }}
*{{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|ref=harv}}
*{{cite journal | last=Mollin | first=R.A. | title=A note on the ABC-conjecture | journal=Far East J. Math. Sci. | volume=33 | number=3 | pages=267–275 | year=2009 | issn=0972-0871 | url=http://people.ucalgary.ca/~ramollin/abcconj.pdf | zbl=1241.11034 | ref=harv | 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 }}
*{{cite book | zbl=1200.11002 | last=Mollin | first=Richard A. | title=Advanced number theory with applications | location=Boca Raton, FL | publisher=CRC Press | year=2010 | isbn=978-1-4200-8328-6 |ref=harv}}
*{{Cite journal |last=Nitaj |first=Abderrahmane |title=La conjecture ''abc''|language=fr |journal=Enseign. Math. |volume=42 |issue=1–2 |pages=3–24 |year=1996 |ref=harv }}
*{{Citation | last1=Oesterlé | first1=Joseph | authorlink=Joseph Oesterlé | title=Nouvelles approches du "théorème" de Fermat | url= http://www.numdam.org/item?id=SB_1987-1988__30__165_0 | series=Séminaire Bourbaki exp 694 |mr=992208 | year=1988 | journal=Astérisque | issn=0303-1179 | issue=161 | pages=165–186}}
*{{Cite book |last=Pomerance |first=Carl |authorlink=Carl Pomerance |chapter=Computational Number Theory |title=The Princeton Companion to Mathematics |location= |publisher=Princeton University Press |year=2008 |pages=361–362 |ref=harv }}
*{{Cite journal |last=Silverman |first=Joseph H. | authorlink=Joseph H. Silverman | year=1988 |title=Wieferich's criterion and the ''abc''-conjecture |journal=[[Journal of Number Theory]] |volume=30 |issue=2 |pages=226–237 |doi=10.1016/0022-314X(88)90019-4 | zbl=0654.10019  |ref=harv }}
*{{Cite journal |last=Robert|first1=Olivier |last2 =Stewart|first2=Cameron L.|author2-link=Cameron Leigh Stewart |last3=Tenenbaum|first3=Gérald|author3-link=Gérald Tenenbaum |year=2014 |title=A refinement of the abc conjecture |journal=[[Bulletin of the London Mathematical Society]] |volume=46 |issue=6 |pages=1156–1166 |doi=10.1112/blms/bdu069|ref=harv |url=https://hal.archives-ouvertes.fr/hal-01281526/file/abc.pdf }}
*{{Cite journal |last=Robert|first1=Olivier|last2=Tenenbaum|first2=Gérald|author2-link=Gérald Tenenbaum|title=Sur la répartition du noyau d'un entier|journal=Indag. Math.|volume= 24|issue=4|year=2013|pages=802–914|doi=10.1016/j.indag.2013.07.007|ref=harv|url=https://hal.archives-ouvertes.fr/hal-01281342/document}}
*{{Cite journal |last=Stewart |first=C. L. |authorlink=Cameron Leigh Stewart |last2=Tijdeman |first2=R. |authorlink2=Robert Tijdeman |year=1986 |title=On the Oesterlé-Masser conjecture |journal=Monatshefte für Mathematik |volume=102 |issue=3 |pages=251–257 |doi=10.1007/BF01294603 |ref=harv }}
*{{Cite journal |last=Stewart |first=C. L. |authorlink=Cameron Leigh Stewart|authorlink2=Kunrui Yu |first2=Kunrui |last2=Yu |year=1991 |title=On the ''abc'' conjecture |journal=[[Mathematische Annalen]] |volume=291 |issue=1 |pages=225–230 |doi=10.1007/BF01445201 |ref=harv }}
*{{Cite journal |last=Stewart |first=C. L. |authorlink=Cameron Leigh Stewart|first2=Kunrui |last2=Yu |authorlink2=Kunrui Yu |year=2001 |title=On the ''abc'' conjecture, II |journal=[[Duke Mathematical Journal]] |volume=108 |issue=1 |pages=169–181 |doi=10.1215/S0012-7094-01-10815-6 |ref=harv }}
*{{cite journal | last=Van Frankenhuijsen | first=Machiel | title=The ABC conjecture implies Vojta's height inequality for curves | journal=[[J. Number Theory]] | year=2002 | pages=289–302 | volume=95 | issue=2 | mr=1924103 | doi=10.1006/jnth.2001.2769 | ref=harv}}
{{Refend}}

==External links==
* [http://abcathome.com/ 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://web.archive.org/web/20000819203144/http://www.math.unicaen.fr/~nitaj/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:Number theory]]
[[Category:Unsolved problems in mathematics]]