On the resolution of equations Ax~n - By~n = C in integers x, y and n ≥ 3, II

摘要

In Part I (cf. [13]) of this paper, the title equation was solved in x, y, n ? Z with |xy| > 1, n ≥ 3 for a collection of positive integers A, B, C under certain bounds. In the present paper we extend these results to much larger ranges of A, B, C. We give among other things all the solutions for A = C = 1, B < 235 (cf. Theorem 1), and for C= 1, A, B ≤ 50, with six explicitly given exceptions (A, B, n) (cf. Theorem 3). The equations under consideration are solved by combining powerful techniques, including Prey curves and associated modular forms, lower bounds for linear forms in logarithms, the hypergeometric method of Thue and Siegel, local methods, classical cyclotomy and computational approaches to Thue equations of low degree. Along the way, we derive a new result on the solvability of binomial Thue equations (cf. Theorem 6) which is crucial in the proof of our Theorems 1 and 2. Some important applications of our theorems will be given in a forthcoming paper.