abc conjecture

Before stating the conjecture, the notion of the radical of an integer must be introduced: for a positive integer $n$, the radical of $n$, denoted $\backslash text\{rad\}(n)$, is the product of the distinct prime factors of $n$. For example,

$\backslash text\{rad\}(16)=\backslash text\{rad\}(2^4)=\backslash text\{rad\}(2)=2$

$\backslash text\{rad\}(17)=17$

$\backslash text\{rad\}(18)=\backslash text\{rad\}(2\backslash cdot\; 3^2)=2\backslash cdot3\; =6$

$\backslash text\{rad\}(1000000)=\backslash text\{rad\}(2^6\; \backslash cdot\; 5^6)=2\backslash cdot5=10$

If a, b, and c are coprimeWhen 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 coprimality of a, b, c. So in this case, it does not matter which concept we use. positive integers such that a + b = c, it turns out that "usually" . 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|$c\; >\; \backslash operatorname\{rad\}(abc)^\{1+\backslash varepsilon\}.$}}}}

An equivalent formulation is:

{{block indent|1=For every positive real number ε, there exists a constant K_ε such that for all triples (a, b, c) of coprime positive integers, with a + b = c:{{sfn|Waldschmidt|2015}}

{{block indent|}}}}

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^1-o(1).}}

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|$q(a,\; b,\; c)\; =\; \backslash frac\{\backslash log(c)\}\{\backslash log\backslash big(\backslash textrm\{rad\}(abc)\backslash big)\}.$}}

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 numbers. 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).

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|$a\; =\; 1,\; \backslash quad\; b\; =\; 2^\{6n\}\; -\; 1,\; \backslash quad\; c\; =\; 2^\{6n\},\; \backslash qquad\; n\; >\; 1.$}}

The integer b is divisible by 9:

{{block indent|$b\; =\; 2^\{6n\}\; -\; 1\; =\; 64^n\; -\; 1\; =\; (64\; -\; 1)\; (\backslash cdots)\; =\; 9\; \backslash cdot\; 7\; \backslash cdot\; (\backslash cdots).$}}

Using this fact, the following calculation is made:

{{block indent|$\backslash begin\{align\}$

\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}}}

By replacing the exponent 6n 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|$a\; =\; 1,\; \backslash quad\; b\; =\; 2^\{p(p-1)n\}\; -\; 1,\; \backslash quad\; c\; =\; 2^\{p(p-1)n\},\; \backslash qquad\; n\; >\; 1.$}}

Now it may be plausibly claimed that b is divisible by p²:

{{block indent|$\backslash begin\{align\}$

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}}}

The last step uses the fact that p² divides 2^p(p−1) − 1. This follows from Fermat's little theorem, which shows that, for p > 2, 2^p−1 = pk + 1 for some integer k. Raising both sides to the power of p then shows that 2^p(p−1) = p²(...) + 1.

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

{{block indent|}}

A list of the 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 = 3¹⁰·109 = {{val|6,436,341}},}}

{{block indent|1=c = 23⁵ = {{val|6,436,343}},}}

{{block indent|1=rad(abc) = {{val|15042}}.}}

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 numbers.{{sfnp|Bombieri|1994|p={{page needed|date=January 2022}}}}{{sfn|Waldschmidt|2015}}
• The 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 conjecture allowing for a finite number of counterexamples.{{sfnp|Langevin|1993}}
• The existence of infinitely many non-Wieferich primes 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 a famously difficult proof by Andrew Wiles. However it follows easily, at least for $n\; \backslash ge\; 6$, from an effective form of a weak version of the abc conjecture. The abc conjecture says the 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 $n\; \backslash ge\; 6$.{{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}}
• The Fermat–Catalan conjecture, a generalization of Fermat's Last Theorem concerning powers that are sums of powers.{{sfnp|Pomerance|2008}}
• The L-function L(s, χ_d) 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.[http://www.math.uu.nl/people/beukers/ABCpresentation.pdf The ABC-conjecture], Frits Beukers, ABC-DAY, Leiden, Utrecht University, 9 September 2005.
• A generalization of Tijdeman's theorem concerning the number of solutions of y^m = xⁿ + k (Tijdeman's theorem answers the case k = 1), and Pillai's conjecture (1931) concerning the number of solutions of Ay^m = Bxⁿ + 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

  ^n−β.{{harvtxt|Mollin|2009}}; {{harvtxt|Mollin|2010|p=297}}
• all the polynominals (x^n-1)/(x-1) have an infinity of square-free values.{{harvtxt|Browkin|2000|p=10}}
• As equivalent, the modified Szpiro conjecture, which would yield a bound of rad(abc)^1.2+ε.{{sfn|Oesterlé|1988}}
• {{harvtxt|Dąbrowski|1996}} has shown that the abc conjecture implies that the Diophantine equation n! + A = k² has only finitely many solutions for any given integer A.
• There are ~c_fN positive integers n ≤ N for which f(n)/B' is square-free, with c_f > 0 a positive constant defined as:{{sfnp|Granville|1998}}{{block indent|$c\_f\; =\; \backslash prod\_\{\backslash text\{prime\; \}p\}\; x\_i\; \backslash left\; (\; 1\; -\; \backslash frac\{\backslash omega\backslash ,\backslash !\_f\; (p)\}\{p^\{2+q\_p\}\}\; \backslash right\; ).$}}
• 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^x + B^y = C^z 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.
• Lang's conjecture, a lower bound for the 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.

{{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

}}

• An effective version of Siegel's theorem about integral points on algebraic curves.{{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}}

The abc conjecture implies that c can be bounded above by a near-linear function of the radical of abc. Bounds are known that are exponential. Specifically, the following bounds have been proven:

{{block indent| {{harv|Stewart|Tijdeman|1986}},}}

{{block indent| {{harv|Stewart|Yu|1991}}, and}}

{{block indent| {{harv|Stewart|Yu|2001}}.}}

In these bounds, K₁ and K₃ are constants that do not depend on a, b, or c, and K₂ 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|$c\; >\; \backslash operatorname\{rad\}(abc)\; \backslash exp\{\; \backslash left(k\; \backslash sqrt\{\backslash log\; c\}/\backslash log\backslash log\; c\; \backslash right)\; \}$}}

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

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.

As of May 2014, ABC@Home had found 23.8 million triples.{{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 }}

Note: the quality q(a, b, c) of the triple (a, b, c) is defined above.

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^−ω 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 $\backslash big(\backslash varepsilon^\{-\backslash omega\}\backslash operatorname\{rad\}(abc)\backslash big)^\{1+\backslash varepsilon\}$ over $\backslash varepsilon\; >\; 0$ occurs when $\backslash varepsilon\; =\; \backslash frac\{\backslash omega\}\{\backslash log\backslash big(\backslash operatorname\{rad\}(abc)\backslash big)\}.$

This inspired {{Harvtxt|Baker|2004}} to propose a sharper form of the abc conjecture, namely:

{{block indent|}}

with κ an absolute constant. After some computational experiments he found that a value of $6/5$ 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|$K^\{\backslash Omega(a\; b\; c)\}\; \backslash operatorname\{rad\}(a\; b\; c),$}}

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

{{block indent|$O\backslash big(\backslash operatorname\{rad\}(a\; b\; c)\; \backslash Theta(a\; b\; c)\backslash big),$}}

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 C₁ such that

{{block indent|}}

holds whereas there is a constant C₂ such that

{{block indent|$c\; >\; k\; \backslash exp\backslash left(4\backslash sqrt\{\backslash frac\{3\backslash log\; k\}\{\backslash log\backslash log\; k\}\}\backslash left(1+\backslash frac\{\backslash log\backslash log\backslash log\; k\}\{2\backslash log\backslash log\; k\}+\backslash frac\{C\_\{2\}\}\{\backslash log\backslash log\; k\}\backslash right)\backslash right)$}}

holds infinitely often.

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

Lucien Szpiro proposed a solution in 2007, but it was found to be incorrect shortly afterwards."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}}.

Since August 2012, Shinichi Mochizuki has claimed a proof of Szpiro's conjecture and therefore the abc conjecture. 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.{{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 }}

The papers have not been widely accepted by the mathematical community as providing a proof of abc.

{{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 |date=December 17, 2017 |access-date=March 17, 2018}} This is not only because of their length and the difficulty of understanding them,{{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}} but also because at least one specific point in the argument has been identified as a gap by some other experts. Although a few mathematicians have vouched for the correctness of the proof

{{ 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}} and have attempted to communicate their understanding via workshops on IUTT, they have failed to convince the number theory community at large.{{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}}{{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 }}

In March 2018, Peter Scholze and Jakob Stix visited Kyoto for discussions with Mochizuki.

{{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 }}

{{ 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

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";{{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 }})

Mochizuki claimed that they misunderstood vital aspects of the theory and made invalid simplifications.

{{ 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.

}}

{{ 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 }}

{{ 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 }}

On April 3, 2020, two mathematicians from the Kyoto 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. The announcement was received with skepticism by Kiran Kedlaya and Edward Frenkel, as well as being described by Nature as "unlikely to move many researchers over to Mochizuki's camp". In March 2021, Mochizuki's proof was published in RIMS.

{{ 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 }}

• List of unsolved problems in mathematics

{{reflist|group=notes}}

{{reflist|25em}}

{{refbegin|25em}}

• {{cite book | last=Baker | first=Alan | author-link=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 }}
• {{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 }}
• {{Cite journal |author-link=Jerzy Browkin |first1=Jerzy |last1=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 |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 |url=https://archive.org/details/numbertheory00bamb_636 |url-access=limited |series=Trends in Mathematics |location=Basel |publisher=Birkhäuser |isbn=3-7643-6259-6 |pages=[https://archive.org/details/numbertheory00bamb_636/page/n83 75]–106 }}
• {{Cite journal |first=Andrzej |last=Dąbrowski |title=On the diophantine equation x! + A = y² | journal=Nieuw Archief voor Wiskunde, IV. |volume=14 |pages=321–324 |year=1996 | zbl=0876.11015 }}
• {{Cite journal |last=Elkies |first=N. D. |author-link=Noam Elkies |title=ABC implies Mordell |journal= International Mathematics Research Notices|volume=1991 |year=1991 |pages=99–109 |doi=10.1155/S1073792891000144 |issue=7 |doi-access= free}}
• {{cite book |last=Frey |first=Gerhard |author-link=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 |chapter-url=https://books.google.com/books?id=Va-quzVwtMsC&pg=PA527 }}
• {{cite journal |last=Goldfeld |first=Dorian | author-link=Dorian M. Goldfeld |year=1996 |title=Beyond the last theorem |journal=Math Horizons |volume=4 |issue=September |pages=26–34 |jstor= 25678079 |doi=10.1080/10724117.1996.11974985 }}
• {{cite book | zbl=1046.11035 | last=Goldfeld | first=Dorian | author-link=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 }}
• {{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=[https://archive.org/details/princetoncompani00gowe_360/page/n672 361]–362, 681 |title-link=The Princeton Companion to Mathematics }}
• {{Cite journal |last=Granville |first=A. |author-link=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 |doi-access=free }}
• {{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 }}
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• {{cite journal | last=van Frankenhuysen | first=Machiel | title=A Lower Bound in the abc Conjecture | journal=J. Number Theory | year=2000 | pages=91–95 | volume=82 | issue=1 | mr=1755155 | doi=10.1006/jnth.1999.2484 | doi-access=free }}
• {{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 | doi-access=free }}
• {{cite book |doi=10.1007/978-3-0348-0859-0_13|chapter=Lecture on the abc Conjecture and Some of Its Consequences|title=Mathematics in the 21st Century|series=Springer Proceedings in Mathematics & Statistics|year=2015|last1=Waldschmidt|first1=Michel|volume=98|pages=211–230|isbn=978-3-0348-0858-3|chapter-url=https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/abcLahoreProceedings.pdf}}

{{refend}}

• [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