Abc conjecture

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The abc conjecture in number theory was first proposed by Joseph Oesterlé and David Masser in 1985. It 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.

[edit] 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>|a|+|b|+ |c| < K_\epsilon \operatorname{rad}(abc)^{1+\epsilon}.</math>

[edit] Some consequences

The conjecture has not been proved, 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 (proved by Klaus Roth)
• Fermat's last theorem for all sufficiently large exponents (proved in general by Andrew Wiles)
• The Mordell conjecture (proved 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
• P(x) has only finitely many perfect powers for integral x for P a polynomial with at least three simple zeros. <ref>http://www.math.uu.nl/people/beukers/ABCpresentation.pdf</ref>
• A generalization of Tijdeman's Theorem

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. (See unifying conjectures in mathematics for some comparisons.)

[edit] Refined forms

A more precise conjecture 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.

[edit] 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+\epsilon}) }, </math>

1996, C.L. Stewart and Kunrui Yu:

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

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

[edit] See also

• Greatest common divisor (gcd)

[edit] References

<references/>

[edit] External links

• Weisstein, Eric W., abc Conjecture at MathWorld.
• Abderrahmane Nitaj's ABC conjecture home page
• http://www.math.columbia.edu/~goldfeld/ABC-Conjecture.pdf
• The amazing ABC conjecture
• ABC@Home Boinc Projectde:Abc-Vermutung

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