Computing integral points on Mordell's elliptic curves (Q1924844)

In an earlier paper [Acta Arith. 68, 171-192 (1994; Zbl 0816.11019)] the authors developed an algorithm for computing all integral points on elliptic curves \(E\) over the rationals \(\mathbb{Q}\). This algorithm, based on a method of Lang, Zagier, was implemented in the computer algebra system SIMATH. (A similar method was used by \textit{R. J. Stroeker} and \textit{N. Tzanakis} for computing all integral points on some elliptic curves.) In the present paper we report on an application of our procedure to Mordell's elliptic curves \(E_k: y^2= x^3+ k\): We compute all integral points for \(0\neq k\in \mathbb{Z}\) within the range \(|k|\leq 10 000\). In fact, we determine more generally all \(S\)-integral points on \(E_k\) for the set of primes \(S= \{2, 3, 5, \infty\}\). The computations can be extended to supply all integral points on \(E_k\) within the range \(|k|< 100 000\). The numerical results obtained are of interest, e.g. in view of Hall's conjecture asserting that the \(x\)-coordinate of an integer point on \(E_k\) satisfies the inequality \[ |x|^{1/ 2}< c|k| \] for some constant \(c>0\). Within the range \(|k|\leq 100 000\), the conjecture holds with \(c=5\). An extended version of this paper will appear in Compos. Math.