Hajdu, L.; Herendi, T. Explicit bounds for the solutions of elliptic equations with rational coefficients. (English) Zbl 0923.11048 J. Symb. Comput. 25, No. 3, 361-366 (1998). 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 Y. Bugeaud and K. Győry [Acta Arith. 74, 273-292 (1996; Zbl 0861.11024)] on Thue-Mahler equations. Reviewer: I.Gaál (Debrecen) Cited in 2 ReviewsCited in 6 Documents MSC: 11D25 Cubic and quartic Diophantine equations 11D75 Diophantine inequalities 11Y50 Computer solution of Diophantine equations 68W30 Symbolic computation and algebraic computation Keywords:elliptic equation; bounds for solutions Citations:Zbl 0861.11024 PDFBibTeX XMLCite \textit{L. Hajdu} and \textit{T. Herendi}, J. Symb. Comput. 25, No. 3, 361--366 (1998; Zbl 0923.11048) Full Text: DOI Link