abc conjecture

The abc conjecture (also known as Oesterlé–Masser conjecture) is a conjecture in number theory, first proposed by Joseph Oesterlé and David Masser in 1985. 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 is rarely much smaller than c.

Although there is no obvious strategy for resolving the problem, it 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".

Unsolved problems in mathematics

Are there for every ε > 0, 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?

Formulations

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 large. Another triple with a particularly small radical was found by Eric Reyssat (Lando & Zvonkin 2004, p. 137):

a = 2:

b = 3¹⁰ 109 = 6436341

c = 23⁵ = 6436343

rad(abc) = 15042.

Some consequences

The conjecture has not been proven, but it has a large number of interesting consequences. These include both known results, and conjectures for which it gives a conditional proof.

• Thue–Siegel–Roth theorem on diophantine approximation of algebraic numbers
• Fermat's Last Theorem for all sufficiently large exponents (proven in general by Andrew Wiles)
• 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)
• 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
• It is equivalent to the Granville–Langevin conjecture
• It is equivalent to the modified Szpiro conjecture.
• Dąbrowski (1996) has shown that the abc conjecture implies that n! + A= k² has only finitely many solutions for any given integer A.

While the first group of these have now been proven, the abc conjecture itself remains of interest, because of its numerous links with deep questions in number theory.

Theoretical results

It remains unknown whether c can be upper bounded by a near-linear function of the radical of abc, as the abc conjecture states, or even whether it can be bounded by a polynomial of rad(abc). However, exponential bounds are known. Specifically, the following bounds have been proven:

$c < \exp{(K_1 \operatorname{rad}(abc)^{15}) }$ (Stewart & Tijdeman 1986),

$c < \exp{ (K_2 \operatorname{rad}(abc)^{2/3+\varepsilon}) }$ (Stewart & Yu 1991), and

$c < \exp{ (K_3 \operatorname{rad}(abc)^{1/3+\varepsilon}) }$ (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 March 2011, ABC@Home has found 20.9 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	1.5444	72​412​3113	1116​132​79	2·​33​523​953	Abderrahmane Nitaj
5	1.4805	522​79·​45949	32​1318​613	223​174​2512​17333	Frank Rubin

where the quality q(a, b, c) of the triple (a, b, c), defined by:

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

Refined forms and generalizations

A stronger inequality proposed in 1996 by Alan Baker 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. 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 $n>2$ integers.

See also

• Mason–Stothers theorem, an analogous statement for polynomials.

References

• 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 $x!+A=y^2$". Nieuw Archief voor Wiskunde, IV. 14: 321–324. 
• 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. 
• 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. 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)
• Nitaj, Abderrahmane (1996). "La conjecture abc". Enseign. Math. 42 (1–2): 3–24.  (French)
• 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. 
• 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. 

Further reading

• Bombieri, Enrico; Gubler, Walter (2006). Heights in Diophantine Geometry. New Mathematical Monographs. 4. Cambridge University Press. ISBN 978-0-521-71229-3. 

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" from 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