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 ≥ 1 and c < 32N^1.6 for N ≥ 1. This sharpens an estimate of Laishram and Shorey. We also show that it implies c < ₅ ⁶N¹⁺G(N⁾ for N ≥ 3, and c < ⁶ ₅N^1+G1(N⁾ for N ≥ 297856, where G(N) and G1(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.