Abc conjecture: Information from Answers.com

The abc 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.

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?

1 Formulations
2 Some consequences
3 Partial results
4 Triples with small radical
- 4.1 Grid-computing program
5 Refined forms and generalizations
6 Notes
7 Further reading
8 External links

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^[1] 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...

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 (proven by Klaus Roth)
Fermat's Last Theorem for all sufficiently large exponents (proven in general by Andrew Wiles)
The Mordell conjecture (proven by Gerd Faltings)
The Erdős–Woods conjecture except for a finite number of counterexamples
The existence of infinitely many non-Wieferich primes
The weak form of Hall's conjecture
The Fermat–Catalan conjecture^[2]
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. ^[3]
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 $n! + A = k^2 \,$ has only finitely many solutions for any given integer A. ^[4]

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.

Partial results

1986, C.L. Stewart and R. Tijdeman:

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

1991, C.L. Stewart and Kunrui Yu:

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

1996, C.L. Stewart and Kunrui Yu:

$c < \exp{ (K_3 \operatorname{rad}(abc)^{1/3+\varepsilon}) },$

where c is larger than 2, K₁ is an absolute constant, and K₂ and K₃ are positive effectively computable constants in terms of ε.

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:^[5]

a = 2:

b = 3¹⁰ 109 = 6436341

c = 23⁵ = 6436343

rad(abc) = 15042.

Grid-computing program

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.

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.

In 1994, Jerzy Browkin and Juliusz Brzeziński formulated the n-conjecture^[6]—a version of the abc conjecture involving $n > 2$ integers.

Notes

^ 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.
^ Carl Pomerance, Computational Number Theory. In The Princeton Companion to Mathematics, Princeton University Press, 2008, pp. 361–362.
^ http://www.math.uu.nl/people/beukers/ABCpresentation.pdf
^ Andrzej Dąbrowski (1996). "On the diophantine equation $x! + A = y 2$ ". Nieuw Archief voor Wiskunde, IV. 14: 321–324.
^ Lando and Zvonkin, p.137
^ J. Browkin, J. Brzeziński (1994). "Some remarks on the abc-conjecture". Math. Comp. 62: 931–939.

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

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Abc conjecture

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