World War 1 and 2 - 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 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 accurate 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.

References

Categories: Number theory | Conjectures

Abc conjecture

See also

References