The abc conjecture (also known as Oesterlé–Masser conjecture) is a conjecture in number theory, first proposed by Joseph Oesterlé (1988) and David Masser (1985) as an integer analogue of the Mason–Stothers theorem for polynomials. The conjecture is stated in terms of three positive integers, a, b and c (hence the name), which have no common factor and satisfy a + b = c. If d denotes the product of the distinct prime factors 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 has already become well known for the number of interesting consequences it entails. Many famous conjectures and theorems in number theory would follow immediately from the abc conjecture. Goldfeld (1996) described the abc conjecture as "the most important unsolved problem in Diophantine analysis".
In August 2012, Shinichi Mochizuki released a series of four preprints containing a serious claim to a proof of the abc conjecture. Mochizuki calls the theory on which this proof is based "inter-universal Teichmüller theory", and it has other applications including a proof of Szpiro's conjecture and Vojta's conjecture. Experts were expected to take months to check Mochizuki's new mathematical machinery, which was developed over decades in 500 pages of preprints and several of his prior papers. When an error in one of the articles was pointed out by Vesselin Dimitrov and Akshay Venkatesh in October 2012, Mochizuki posted a comment on his website acknowledging the mistake, stating that it would not affect the result, and promising a corrected version in the near future. He proceeded to post a series of corrected papers of which the latest dated March 2013.
The abc conjecture can be expressed as follows: For every ε > 0, there are only finitely many triples of coprime positive integers a + b = c such that c > d1+ε, where d denotes the product of the distinct prime factors of abc.
To illustrate the terms used, if
- a = 16 = 24,
- b = 17, and
- c = 16 + 17 = 33 = 3·11,
then d = 2·17·3·11 = 1122, which is greater than c. Therefore, for all ε > 0, c is not greater than d1+ε. According to the conjecture, most coprime triples where a + b = c are like the ones used in this example, and for only a few exceptions is c > d1+ε.
- rad(16) = rad(24) = 2,
- rad(17) = 17,
- rad(18) = rad(2·32) = 2·3 = 6.
ABC Conjecture. For every ε > 0, there exist only finitely many triples (a, b, c) of positive coprime integers, with a + b = c, such that
An equivalent formulation states that:
ABC Conjecture II. For every ε > 0, there exists a constant Kε such that for all triples (a, b, c) of coprime positive integers, with a + b = c, the inequality
A third formulation of the conjecture involves the quality, q(a, b, c), of the triple (a, b, c), defined by:
- 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.
ABC Conjecture III. For every ε > 0, 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.
Examples of triples with small radical 
The condition that ε > 0 is necessary for the truth of the conjecture, as there exist infinitely many triples a, b, c with rad(abc) < c. For instance, such a triple may be taken as
- a = 1
- b = 26n − 1
- c = 26n
As a and c together contribute only a factor of two to the radical, while b is divisible by 9, rad(abc) < 2c/3 for these examples. By replacing the exponent 6n by other exponents forcing b to have larger square factors, the ratio between the radical and c may be made arbitrarily small. Specifically, replacing 6n by p(p−1)n for an arbitrary prime p will make b divisible by p2, because 2p(p−1) ≡ 1 (mod p2) and 2p(p−1)−1 will be a factor of b.
A list of the highest quality triples (triples with a particularly small radical relative to c) is given below; the highest quality of these, with quality 1.6299, was found by Eric Reyssat (Lando & Zvonkin 2004, p. 137):
- a = 2
- b = 310·109 = 6,436,341
- c = 235 = 6,436,343
- rad(abc) = 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.
- Thue–Siegel–Roth theorem on diophantine approximation of algebraic numbers
- Fermat's Last Theorem for all sufficiently large exponents (already proven in general by Andrew Wiles) (Granville 2002)
- The Mordell conjecture (already proven in general by Gerd Faltings) (Elkies 1991)
- The Erdős–Woods conjecture except for a finite number of counterexamples (Langevin 1993)
- The existence of infinitely many non-Wieferich primes (Silverman 1988)
- The weak form of Marshall Hall's conjecture on the separation between squares and cubes of integers (Nitaj 1996)
- The Fermat–Catalan conjecture, a generalization of Fermat's last theorem concerning powers that are sums of powers (Pomerance 2008)
- The L function L(s,(−d/.)) formed with the Legendre symbol, has no Siegel zero (this consequence actually requires a uniform version of the abc conjecture in number fields, not only the abc conjecture as formulated above for rational integers) (Granville 2000)
- P(x) has only finitely many perfect powers for integral x for P a polynomial with at least three simple zeros.
- A generalization of Tijdeman's theorem concerning the number of solutions of ym = xn + k (Tijdeman's theorem answers the case k = 1), and Pillai's conjecture (1931) concerning the number of solutions of Aym = Bxn + k.
- It is equivalent to the Granville–Langevin conjecture.
- It is equivalent to the modified Szpiro conjecture, which would yield a bound of rad(abc)1.2+ε (Oesterlé 1988).
- Dąbrowski (1996) has shown that the abc conjecture implies that n! + A= k2 has only finitely many solutions for any given integer A.[clarification needed]
Theoretical results 
In these bounds, K1 is a constant that does not depend on a, b, or c, and K2 and K3 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.
|q > 1||q > 1.05||q > 1.1||q > 1.2||q > 1.3||q > 1.4|
|c < 102||6||4||4||2||0||0|
|c < 103||31||17||14||8||3||1|
|c < 104||120||74||50||22||8||3|
|c < 105||418||240||152||51||13||6|
|c < 106||1,268||667||379||102||29||11|
|c < 107||3,499||1,669||856||210||60||17|
|c < 108||8,987||3,869||1,801||384||98||25|
|c < 109||22,316||8,742||3,693||706||144||34|
|c < 1010||51,677||18,233||7,035||1,159||218||51|
|c < 1011||116,978||37,612||13,266||1,947||327||64|
|c < 1012||252,856||73,714||23,773||3,028||455||74|
|c < 1013||528,275||139,762||41,438||4,519||599||84|
|c < 1014||1,075,319||258,168||70,047||6,665||769||98|
|c < 1015||2,131,671||463,446||115,041||9,497||998||112|
|c < 1016||4,119,410||812,499||184,727||13,118||1,232||126|
|c < 1017||7,801,334||1,396,909||290,965||17,890||1,530||143|
|c < 1018||14,482,065||2,352,105||449,194||24,013||1,843||160|
|2||1.6260||112||32·56·73||221·23||Benne de Weger|
|3||1.6235||19·1307||7·292·318||28·322·54||Jerzy Browkin, Juliusz Brzezinski|
|4||1.5808||283||511·132||28·38·173||Jerzy Browkin, Juliusz Brzezinski, Abderrahmane Nitaj|
|5||1.5679||1||2·37||54·7||Benne de Weger|
Note: the quality q(a, b, c) of the triple (a, b, c) is defined above.
Refined forms and generalizations 
- O(rad(abc) Θ(rad(abc)))
where Θ(n) is the number of integers up to n divisible only by primes dividing n.
Browkin & Brzeziński (1994) formulated the n-conjecture—a version of the abc conjecture involving n > 2 integers.
See also 
- Mochizuki, Shinichi (August 2012). Inter-universal Teichmuller Theory I: Construction of Hodge Theaters, Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic Evaluation, Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice., 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
- Ball, Phillip (10 September 2012), "Proof claimed for deep connection between primes", Nature.
- Cipra, Barry (September 12, 2012), "ABC Proof Could Be Mathematical Jackpot", Science.
- Kevin Hartnett (3 November 2012). "An ABC proof too tough even for mathematicians". Boston Globe.
- Note that when it is given that a + b = c, coprimeness of a, b, c implies pairwise coprimeness of a, b, c. So in this case, it does not matter which concept we use.
- "Synthese resultaten", RekenMeeMetABC.nl, retrieved October 3, 2012 (Dutch).
- "Data collected sofar", ABC@Home, retrieved September 10, 2012
- "100 unbeaten triples". Reken mee met ABC. 2010-11-07.
- Baker, Alan (1998). "Logarithmic forms and the abc-conjecture". In Győry, Kálmán. Number theory. Diophantine, computational and algebraic aspects. Proceedings of the international conference, Eger, Hungary, July 29-August 2, 1996. Berlin: de Gruyter. pp. 37–44. ISBN 3-11-015364-5. Zbl 0973.11047.
- Bombieri, Enrico; Gubler, Walter (2006). Heights in Diophantine Geometry. New Mathematical Monographs 4. Cambridge University Press. doi:10.2277/0521846153. ISBN 978-0-521-71229-3. Zbl 1130.11034.
- Browkin, Jerzy; Brzeziński, Juliusz (1994). "Some remarks on the abc-conjecture". Math. Comp. 62 (206): 931–939. doi:10.2307/2153551. JSTOR 2153551.
- Browkin, Jerzy (2000). "The abc-conjecture". In Bambah, R. P.; Dumir, V. C.; Hans-Gill, R. J. Number Theory. Trends in Mathematics. Basel: Birkhäuser. pp. 75–106. ISBN 3-7643-6259-6.
- Dąbrowski, Andrzej (1996). "On the diophantine equation x! + A = y2". Nieuw Archief voor Wiskunde, IV. 14: 321–324. Zbl 0876.11015.
- Elkies, N. D. (1991). "ABC implies Mordell". Intern. Math. Research Notices 7 (7): 99–109. doi:10.1155/S1073792891000144.
- Goldfeld, Dorian (1996). "Beyond the last theorem". Math Horizons (September): 26–34.
- Gowers, Timothy; Barrow-Green, June; Leader, Imre, eds. (2008). The Princeton Companion to Mathematics. Princeton: Princeton University Press. pp. 361–362, 681. ISBN 978-0-691-11880-2.
- Granville, Andrew; Stark, H. (2000). "ABC implies no "Siegel zeros" for L-functions of characters with negative exponent". Inventiones Mathematicae 139: 509–523.
- Granville, Andrew; Tucker, Thomas (2002). "It’s As Easy As abc". Notices of the AMS 49 (10): 1224–1231.
- Guy, Richard K. (2004). Unsolved Problems in Number Theory. Berlin: Springer-Verlag. ISBN 0-387-20860-7.
- Lando, Sergei K.; Zvonkin, Alexander K. (2004). "Graphs on Surfaces and Their Applications". Encyclopaedia of Mathematical Sciences: Lower-Dimensional Topology II 141 (Springer-Verlag). ISBN 3-540-00203-0.
- Langevin, M. (1993). "Cas d'égalité pour le théorème de Mason et applications de la conjecture abc". Comptes rendus de l'Académie des sciences 317 (5): 441–444. (French)
- Masser, D. W. (1985), "Open problems", in Chen, W. W. L., Proceedings of the Symposium on Analytic Number Theory, London: Imperial College
- Nitaj, Abderrahmane (1996). "La conjecture abc". Enseign. Math. 42 (1–2): 3–24. (French)
- Oesterlé, Joseph (1988), "Nouvelles approches du "théorème" de Fermat", Astérisque, Séminaire Bourbaki exp 694 (161): 165–186, ISSN 0303-1179, MR992208
- Pomerance, Carl (2008). "Computational Number Theory". The Princeton Companion to Mathematics. Princeton University Press. pp. 361–362.
- Silverman, Joseph H. (1988). "Wieferich's criterion and the abc-conjecture". Journal of Number Theory 30 (2): 226–237. doi:10.1016/0022-314X(88)90019-4. Zbl 0654.10019.
- Stewart, C. L.; Tijdeman, R. (1986). "On the Oesterlé-Masser conjecture". Monatshefte für Mathematik 102 (3): 251–257. doi:10.1007/BF01294603.
- Stewart, C. L.; Yu, Kunrui (1991). "On the abc conjecture". Mathematische Annalen 291 (1): 225–230. doi:10.1007/BF01445201.
- Stewart, C. L.; Yu, Kunrui (2001). "On the abc conjecture, II". Duke Mathematical Journal 108 (1): 169–181. doi:10.1215/S0012-7094-01-10815-6.
- ABC@home Distributed Computing project called ABC@Home.
- Easy as ABC: Easy to follow, detailed explanation by Brian Hayes.
- Weisstein, Eric W., "abc Conjecture", MathWorld.
- Abderrahmane Nitaj's ABC conjecture home page
- Bart de Smit's ABC Triples webpage
- The amazing ABC conjecture
- The ABC's of Number Theory by Noam D. Elkies
- Questions about Number by Barry Mazur
- Philosophy behind Mochizuki’s work on the ABC conjecture on MathOverflow
- ABC Conjecture Polymath project wiki page linking to various sources of commentary on Mochizuki's papers.
- abc Conjecture Numberphile video