Explicit bounds for the solutions of elliptic equations with rational coefficients (Q1264452)

Consider solutions \(x\in \mathbb{Z}\) of the equation \(y^2=x^3+ax+b.\) The authors prove that all solutions satisfy \[ \max(| x| ,| y|)\leq \exp\{5\cdot 10^{64} c_1\log c_1 (c_1+\log c_2)\} \] with \[ c_1=32 | \Delta_f| ^{1/2}(8+\log (\Delta_F)/2)^4/3, \quad c_2=10^4\max(16a^2,256| \Delta_f| ^{2/3}), \] where \(\Delta_f\neq 0\) is the discriminant of \(f(x)=x^3+ax+b\). The authors extend the theorem also to the \(S\)-integral case. The bounds are improvements of some earlier results. The main tool is the application of the bounds of \textit{Y. Bugeaud} and \textit{K. Győry} [Acta Arith. 74, 273-292 (1996; Zbl 0861.11024)] on Thue-Mahler equations.