On \(Y^ 2=X^ 3+k\) and the Thue rank of cubic curves (Q788011)

For each integer k, the equation (*) \(Y^ 2=X^ 3+k\) has at most finitely many solutions in integers X and Y. This is a classical result of A. Thue. By means of the chord and tangent method applied to the cubic curve with equation (*), other integral points (X,Y) may be constructed from known ones. The authors define the Thue rank T(k) of (*) as the smallest integer n for which a set of integral points S of cardinality n exists, such that all integral points on the curve can be recovered by applying repeatedly the chord and tangent method to the points of S and to the new points constructed in this way from those of S. A number of examples illustrate the method. The authors appear to be unaware of the fact that complete sets of integral solutions to (*) are known for all k in the range \(| k| \leq 100\) [cf. \textit{F. B. Coghlan} and \textit{N. M. Stephens}, Computers in Number theory 1969, 199-205 (1971; Zbl 0217.036); see also \textit{H. London} and \textit{R. Finkelstein}, On Mordell's equation \(y^ 2-k=x^ 3\) (1973; Zbl 0276.10009)].