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Computing integral points on Mordell’s elliptic curves. (English) Zbl 0865.11084
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 R. J. Stroeker and 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.

MSC:
11Y16 Number-theoretic algorithms; complexity
11G05 Elliptic curves over global fields
14H52 Elliptic curves
14Q05 Computational aspects of algebraic curves
11D25 Cubic and quartic Diophantine equations
Software:
SIMATH
Full Text: EuDML