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 attack on the problem, it has already become well known for the number of interesting consequences it entails.

Unsolved problems in mathematics: For every ε > 0, does there exist a K > 0 such that for every triple of coprime positive integers a + b = c, with product d of their distinct prime factors, |a|+|b|+|c| < Kd^1 + ε?

Formulation

Let

a + b = c

be three coprime positive integers, and

rad(abc),

called the radical of abc, be the square-free product of their distinct prime factors. In other words, the product of all the unique prime factors of the three numbers, never raising a factor to a power greater than 1. The abc conjecture states that, for any ε > 0, there exists a finite K_ε such that, for all coprime positive integers a + b = c,

<math>c < K_\varepsilon\, \operatorname{rad}(abc)^{1+\varepsilon}.</math>

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.

• Roth's 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 set of consecutive triples of powerful numbers is finite
• 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. ^[1]
• A generalization of Tijdeman's Theorem
• It is equivalent to the Granville-Langevin conjecture.

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

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.

Partial results

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

<math>c < \exp{(K_1 \operatorname{rad}(abc)^{15}) }, </math>

1991, C.L. Stewart and Kunrui Yu:

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

1996, C.L. Stewart and Kunrui Yu:

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

where 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 c 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.^[2]

Grid-computing program

In 2006, the Mathematics Department of Leiden University in the Netherlands, together with the Dutch Kennislink science institute, launched a public grid computing project that aims to discover so-called "a-b-c triples" which would fulfill the conjecture. The ABC@home software runs under the University of California, Berkeley's BOINC open grid computing platform.

See also

• Greatest common divisor (gcd)

References

1. ^ http://www.math.uu.nl/people/beukers/ABCpresentation.pdf
2. ^ Lando, Sergei K. & Zvonkin, Alexander K. (2004), Graphs on Surfaces and Their Applications, vol. 141, Encyclopaedia of Mathematical Sciences: Lower-Dimensional Topology II, Springer-Verlag, p. 137.

External links

• Eric W. Weisstein, abc Conjecture at 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