ABC conjecture

In mathematics, the ABC conjecture relates the prime factors of two integers to those of their sum. It was proposed by David Masser and Joseph Oesterlé in 1985. It is connected with other problems of number theory: for example, the truth of the ABC conjecture would provide a new proof of Fermat's Last Theorem.

Statement

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 $A+B=C$ holds for positive coprime integers $A,B,C$. The conjecture asserts that for every $\epsilon >0$ there exists $\kappa (\epsilon )>0$ such that

$|A|,|B|,|C|<\kappa (\epsilon )r(ABC)^{1+\epsilon }\ .$