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 < N1.72 for N ≥ 1 and c < 32N1.6 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 5N1+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.
ASJC Scopus subject areas
- Mathematik (insg.)