abc conjecture

The abc conjecture is a conjecture in number theory. It was first proposed by Joseph Oesterlé and David Masser in 1985. The conjecture is stated in terms of simple properties of three integers, one of which is the sum of the other two. 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 problem 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?

More unsolved problems in mathematics

Formulation

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, define the quality q(a, b, c) of the triple (a, b, c) by the following formula:

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

For example

• q(3, 125, 128) = log(128) / log(rad(3·125·128)) = log(128) / log(30) = 1.426565...
• q(4, 127, 131) = log(131) / log(rad(4·127·131)) = log(131) / log(2·127·131) = 0.46820...

A typical triple (a, b, c) of coprime positive integers with a + b = c will have q(a, b, c) < 1. Triples with q > 1 such as in the first 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...

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

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

holds.

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 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 $n!+A=k^{2}\,$ 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.

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—a version of the abc conjecture involving $n>2$ integers.

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:

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.

References

• Gowers, Timothy; Barrow-Green, June; Leader, Imre, eds. (2008). The Princeton Companion to Mathematics. Princeton University Press. pp. 361–362, 681. ISBN 978-0691118802.
• Richard K. Guy (2004). Unsolved Problems in Number Theory. Springer-Verlag. ISBN 0-387-20860-7.
• Lando, Sergei K. (2004). Graphs on Surfaces and Their Applications. Encyclopaedia of Mathematical Sciences: Lower-Dimensional Topology II. Vol. 141. Springer-Verlag. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (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