Explicit estimates of solutions of some Diophantine equations (Q1032758)

Let \(k\) be a fixed non-zero integer, and let \(x\) and \(y\) be integers such that \[ y^2=x^3+k. \] Then the author proves that \[ \log \max \{|x|, |y|\} < c_d |k| \log^d (|k|+1) \] for \(d\in\{4,5,6\}\), where \(c_d\) is an explicit constant. The proof combines ideas of E. Bombieri and P. Cohen and a very precise use of lower bounds for linear forms in several logarithms and in two logarithms.