On Baker's explicit $abc$-conjecture

We derived from Baker's explicit $abc$-conjecture that $a+b=c$, where $a$, $b$ and $c$ are relatively prime positive integers, implies that $c<N^{1.72}$ for $N\geq 1$ and $c<32 N^{1.6}$ for $N\geq 1$. This sharpens an estimate of Laishram and Shorey. We also show that it implies $c<\frac{6}{5}N^{1+G(N)}$ for $N\geq 3$, and $c<\frac{6}{5}N^{1+G_1(N)}$ for $N\geq 297856$, where $G(N)$ and $G_1(N)$ are explicitly given positive valued decreasing functions of $N$ tending to zero as $N$ tends to infinity. Finally, we give applications of our estimates on triples of consecutive powerful integers and generalized Fermat equation.