On the resolution of equations $Ax^n-By^n=C$ in integers $x,y$ and $n\geq 3$. II.

Abstract

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 Frey 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.

On binomial Thue equations and ternary equations with $S$-unit coefficients

“…Moreover, they explicitly solved (1) for max {A, B, C} ≤ 10, for C = 1 and max {A, B} ≤ 20 and for A = C = 1 and B ≤ 70, respectively. The latter results were recently generalized by Bazsó, Bérczes, Győry and Pintér [3] for the cases C = 1 and max {A, B} ≤ 50 and for A = C = 1 and B < 235. Further related results can also be found in [3] concerning (1) with bounded coefficients.…”

“…The latter results were recently generalized by Bazsó, Bérczes, Győry and Pintér [3] for the cases C = 1 and max {A, B} ≤ 50 and for A = C = 1 and B < 235. Further related results can also be found in [3] concerning (1) with bounded coefficients.…”

“…Let n ≥ 3 be a prime, S = {p, q} with primes 2 ≤ p, q ≤ 71 and let A, B be coprime integer S-units with A < B. If (A, p, q) = (1, 2, 31) and (p, q, n) / ∈ { (3,37,19), (5,37,19), (3,61,31), (17,61,31), (43, 61, 31)}, then for every integer solution (x, y, A, B, n) of equation (2) with |xy| > 1 we have n ≤ 17.…”

“…Suppose that AB = 2 α q β , where q is a prime with 3 ≤ q ≤ 29 and α, β are nonnegative integers. If n > 11 is a prime, then equation (4) has no solutions in integers (x, y, z) with |xy| > 1 and Ax, By and z pairwise coprime, unless, possibly, (q, α) ∈ {(3, 1), (3, 2), (3, 3), (5, 2), (5, 3), (7, 2), (7,3), (17,4)} and xy is odd.…”

“…If n > 11 is a prime, then equation (5) has no solutions in integers (x, y, z) with |xy| > 1, xy even, and Ax, By and z pairwise coprime, unless, possibly (p, q, n) ∈ { (3,23,13), (5,19,13), (5,23,23), (5,29,13), (5,29,23), (7,17,17), (7,17,19), (7,19,13), (11,13,13), (11,17,23), (11,19,13), (11,19,31), (13,17,17), (13, 19, 13}. Proof. This is Theorem 5 in [17].…”