Abc conjecture

From Exampleproblems

The correct title of this article is abc conjecture. The initial letter is capitalized due to technical restrictions.

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 positive integers a, b, c satisfying

$a + b = c \ \mbox{and}\ \gcd(a,b) = 1$

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 2004. 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.