On Baker's explicit $abc$-conjecture

Abstract

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.

The abcd conjecture, uniform boundedness, and dynamical systems

“…-In fact, Baker [1] and Laishram-Shorey [33] give empirical evidence for an explicit version of Conjecture 1.5 with ϵ = 3 4 and C 3 4 ≤ 1 (these exponents can be even further optimized, cf. [11]).…”

Explicit abc-conjecture and its applications

“…Therefore, (4.9) is fulfilled for ω = 11 as well and hence (4.9) holds. The SAGE Program of [MaKä18c] depends on new algorithms so that the running time is reduced greatly compared to that of the algorithm applied in the proof of (2.4) in [ChShSi,Section 4]. The executing time for each case of N in Table 2 is less than 2 hours.…”