The number of equations c=a+b satisfying the abc-conjecture

Abstract

We prove that for a positive integer $c$ and any given $\varepsilon$, $0<\varepsilon<1$, the number $N(c)$ of equations $c=a+b$, $a<b$, with positive coprime integers $a$ and $b$, which satisfy the inequality $$c < R(c)^{\frac{\varepsilon}{1+\varepsilon}}R(a)^{\frac{1}{1+\varepsilon}}R(b)^{\frac{1}{1+\varepsilon}},$$ where R(n) is the radical of $n$, is for $c\to\infty$ $$N(c)=(1-\varepsilon)\frac{\phi(c)}{2}+O\Bigl(\frac{\phi(c)}{2}\Bigr).$$ An analogue for the abc-conjecture inequality $c<R(abc)^{1+\varepsilon}$ (without a constant factor) will also be proved.