abc conjecture

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 (whence comes 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 cannot be much smaller than c.

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 paper with 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.

Formulations

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 > d^(1+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 c is not greater than d^(1+ε) for any ε and a,b,c is not such a triple. 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 > d^(1+ε).

Adding additional terminology: For a positive integer n, the radical of n, denoted rad(n), is the product of the distinct prime factors of n. For example

• rad(16) = rad(2⁴) = 2,
• rad(17) = 17,
• rad(18) = rad(2·3²) = 2·3 = 6.

If a, b, and c are coprime 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 for every ε>0 there exist only finitely many triples (a,b,c) of positive coprime integers with a + b = c such that

$c>\operatorname {rad} (abc)^{1+\varepsilon }$

An equivalent formulation states that for any ε > 0, there exists a constant K such that, for all triples of coprime positive integers (a, b, c) satisfying a + b = c, the inequality

$c<K\cdot \operatorname {rad} (abc)^{1+\varepsilon }$

holds.

A third formulation of the conjecture involves the quality q(a, b, c) of the triple (a, b, c), defined by:

$q(a,b,c)={\frac {\log(c)}{\log(\operatorname {rad} (abc))}}$

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 numbers.

The abc conjecture states that, for any 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 2^p(p-1) ≡ 1 (mod p²) and 2^p(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 (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 $\scriptstyle y^{m}\;=\;x^{n}\,+\,k$ (Tijdeman's theorem answers the case $\scriptstyle k\;=\;1$), and Pillai's conjecture (1931) concerning the number of solutions of $\scriptstyle Ay^{m}\;=\;Bx^{n}\,+\,k$
• It is equivalent to the Granville–Langevin conjecture.
• It is equivalent to the modified Szpiro conjecture, which would yield a bound of $\scriptstyle \operatorname {rad} (abc)^{{\frac {6}{5}}+\epsilon }$ (Oesterlé 1988).
• Dąbrowski has shown that the abc conjecture implies that n! + A= k² has only finitely many solutions for any given integer A. Dąbrowski (1996)

Theoretical results

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

$c<\exp {\left(K_{1}\operatorname {rad} (abc)^{15}\right)}$ (Stewart & Tijdeman 1986),

$c<\exp {\left(K_{2}\operatorname {rad} (abc)^{{\frac {2}{3}}+\varepsilon }\right)}$ (Stewart & Yu 1991), and

$c<\exp {\left(K_{3}\operatorname {rad} (abc)^{{\frac {1}{3}}+\varepsilon }\right)}$ (Stewart & Yu 2001).

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.

Distribution of triples with q > 1

	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	14,482,059	2,352,105	449,194	24,013	1,843	160

As of September 2012, ABC@Home has found 23.1 million triples, and its present goal is to obtain a complete list of all ABC triples (a,b,c) with c no more than 10²⁰.

Highest quality triples

	q	a	b	c	Discovered by
1	1.6299	2	310​·​109	235	Eric Reyssat
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

A stronger inequality proposed by Baker (1998) states that in the inequality, one can replace rad(abc) by

ε^−ωrad(abc)

where ω is the total number of distinct primes dividing a, b and c (Bombieri & Gubler 2006, p. 404). A related conjecture of Andrew Granville states that on the RHS we could also put

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 $\scriptstyle n\,>\,2$ integers.

See also

List of unsolved problems in mathematics

References

• Baker, Alan (1998). "Logarithmic forms and the abc-conjecture". In Győry, Kálmán (ed.). 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. {{cite book}}: Invalid |ref=harv (help)
• Bombieri, Enrico; Gubler, Walter (2006). Heights in Diophantine Geometry. New Mathematical Monographs. Vol. 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. {{cite journal}}: Invalid |ref=harv (help)
• Browkin, Jerzy (2000). "The abc-conjecture". In Bambah, R. P.; Dumir, V. C.; Hans-Gill, R. J. (eds.). Number Theory. Trends in Mathematics. Basel: Birkhäuser. pp. 75–106. ISBN 3-7643-6259-6. {{cite book}}: Invalid |ref=harv (help)
• Dąbrowski, Andrzej (1996). "On the diophantine equation $x!+A=y^{2}$". Nieuw Archief voor Wiskunde, IV. 14: 321–324. {{cite journal}}: Invalid |ref=harv (help)
• Elkies, N. D. (1991). "ABC implies Mordell". Intern. Math. Research Notices. 7 (7): 99–109. doi:10.1155/S1073792891000144. {{cite journal}}: Invalid |ref=harv (help)CS1 maint: unflagged free DOI (link)
• Goldfeld, Dorian (1996). "Beyond the last theorem". Math Horizons (September): 26–34. {{cite journal}}: Invalid |ref=harv (help)
• 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. {{cite book}}: Invalid |ref=harv (help)
• Granville, Andrew; Stark, H. (2000). "ABC implies no "Siegel zeros" for L-functions of characters with negative exponent" (PDF). Inventiones Mathematicae. 139: 509–523. {{cite journal}}: Invalid |ref=harv (help)
• Granville, Andrew; Tucker, Thomas (2002). "It's As Easy As abc" (PDF). Notices of the AMS. 49 (10): 1224–1231. {{cite journal}}: Invalid |ref=harv (help)
• Guy, Richard K. (2004). Unsolved Problems in Number Theory. Berlin: Springer-Verlag. ISBN 0-387-20860-7. {{cite book}}: Invalid |ref=harv (help)
• Lando, Sergei K.; Zvonkin, Alexander K. (2004). Graphs on Surfaces and Their Applications. Vol. 141. Springer-Verlag. ISBN 3-540-00203-0. {{cite book}}: |work= ignored (help); Invalid |ref=harv (help)
• Langevin, M. (1993). "[Cas d'égalité pour le théorème de Mason et applications de la conjecture abc] Error: {{Lang}}: text has italic markup (help)". Comptes rendus de l'Académie des sciences. 317 (5): 441–444. {{cite journal}}: Invalid |ref=harv (help) Template:Fr icon
• Masser, D. W. (1985), "Open problems", in Chen, W. W. L. (ed.), Proceedings of the Symposium on Analytic Number Theory, London: Imperial College
• Nitaj, Abderrahmane (1996). "[La conjecture abc] Error: {{Lang}}: text has italic markup (help)". Enseign. Math. 42 (1–2): 3–24. {{cite journal}}: Invalid |ref=harv (help) Template:Fr icon
• 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. {{cite book}}: Invalid |ref=harv (help)
• 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. {{cite journal}}: Invalid |ref=harv (help)
• Stewart, C. L.; Tijdeman, R. (1986). "On the Oesterlé-Masser conjecture". Monatshefte für Mathematik. 102 (3): 251–257. doi:10.1007/BF01294603. {{cite journal}}: Invalid |ref=harv (help)
• Stewart, C. L.; Yu, Kunrui (1991). "On the abc conjecture". Mathematische Annalen. 291 (1): 225–230. doi:10.1007/BF01445201. {{cite journal}}: Invalid |ref=harv (help)
• 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. {{cite journal}}: Invalid |ref=harv (help)

External links

• 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
• http://www.math.columbia.edu/~goldfeld/ABC-Conjecture.pdf
• 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.