Abc conjecture

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

It states that for any $\varepsilon > 0$ there exists a constant $C_{\varepsilon} > 0$ , such that for every triple of coprime positive integers a, b, c satisfying $a + b = c$ , we have

c < C_{\varepsilon} \operatorname{rad}(abc)^{1+\epsilon},

where rad(n) (the radical of n) is the product of the distinct prime divisors of n.

It has not been proved as of 2006. 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 or 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:

c < \exp{(C_1  \operatorname{rad}(abc)^{15}) },

1991, C.L. Stewart and Kunrui Yu:

c < \exp{ (C_2  \operatorname{rad}(abc)^{2/3+\epsilon}) },

1996, C.L. Stewart and Kunrui Yu:

c < \exp{ (C_3  \operatorname{rad}(abc)^{1/3+\epsilon}) },

where $C_1$ is an absolute constant, $C_2$ and $C_3$ are positive effectively computable constants in terms of $\epsilon$ .

References

http://www.math.unicaen.fr/~nitaj/abc.html
http://www.math.columbia.edu/~goldfeld/ABC-Conjecture.pdf

Number theory Conjectures

Gourt Home::Gourt :: Abc conjecture

Partial results

See also

References