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

Abstract

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