Dedicated to the memory of Professor Alan Baker.Abstract. We derived from Baker's explicit abc-conjecture that (1.1) 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 < 6 5 N 1+G(N ) for N ≥ 3 and c < 6 5 N 1+G 1 (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 given by (1.4) and (1.6), respectively. Finally we give applications of our estimates on the greatest prime factor of product of consecutive positive integers, triples of consecutive powerful integers and generalized Fermat equation.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.