We solve the Diophantine equation Y 2 = X 3 + k for all nonzero integers k with |k| 10 7 . Our approach uses a classical connection between these equations and cubic Thue equations. The latter can be treated algorithmically via lower bounds for linear forms in logarithms in conjunction with lattice-basis reduction.

We exactly determine the integral solutions to a previously untreated infinite family of cubic Thue equations of the form F (x, y) = 1 with at least 5 such solutions. Our approach combines elementary arguments, with lower bounds for linear forms in logarithms and lattice-basis reduction.

In this note we find all the solutions to the equation x2 + 2α 3β 19γ = yn in nonnegative unknowns with n ≥ 3 and gcd(x, y) = 1, and nonnegative solutions to x2 + 2α 3β 13γ = yn with n ≥ 3, gcd(x, y) = 1, except when α = 0 and x. β. γ is odd.

Let D be a positive nonsquare integer, p a prime number with p ∤ D, and 0 < σ < 0.847. We show that if the equation x 2 + D = p n has a huge solution (x 0 , n 0 ) (p,σ) , then there exists an effectively computable constant Cp such that for every x > C P with x 2 + D = p n .m, we have m > x σ . As an application, we show that for x = {1015, 5}, if the equation x 2 + 76 = 101 n .m holds, we have m > x 0.14 .