# ABC conjecture

Define the radical of an integer to be the product of its distinct prime factors $$r(n) = \prod_{p|n} p \ .$$ Suppose now that the equation $A + B + C = 0$ holds for coprime integers $A,B,C$. The conjecture asserts that for every $\epsilon > 0$ there exists $\kappa(\epsilon) > 0$ such that $$|A|, |B|, |C| < \kappa(\epsilon) r(ABC)^{1+\epsilon} \ .$$ A weaker form of the conjecture states that $$(|A| \cdot |B| \cdot |C|)^{1/3} < \kappa(\epsilon) r(ABC)^{1+\epsilon} \ .$$ If we define $$\kappa(\epsilon) = \inf_{A+B+C=0,\ (A,B)=1} \frac{\max\{|A|,|B|,|C|\}}{N^{1+\epsilon}} \,,$$ then it is known that $\kappa \rightarrow \infty$ as $\epsilon \rightarrow 0$.
Baker introduced a more refined version of the conjecture in 1998. Assume as before that $A + B + C = 0$ holds for coprime integers $A,B,C$. Let $N$ be the radical of $ABC$ and $\omega$ the number of distinct prime factors of $ABC$. Then there is an absolute constant$c$ such that $$|A|, |B|, |C| < c (\epsilon^{-\omega} N)^{1+\epsilon} \ .$$
It is known that there is an effectively computable $\kappa(\epsilon)$ such that $$|A|, |B|, |C| < \exp\left({ \kappa(\epsilon) N^{1/3} (\log N)^3 }\right) \ .$$