On the Elliptic Logarithm Method for Elliptic Diophantine Equations: Reflections and an Improvement

Abstract

The elliptic logarithm method for the determination of all integral solutions of a given elliptic equation is discussed for equations with associated elliptic curve of moderately large rank. Major attention is given to the question of optimizing the choice of Mordell-Weil basis for the curves in question. A speculative argument suggests that for any curve of rank larger then 8 the calculations involved are unlikely to be feasible. The arguments are illustrated by examples of curves of rank 5, 6, 7, and 8, taken from the literature.

Optimal systems of fundamental S-units for LLL-reduction

“…Finally, we mention that a similar type investigation has been performed about Mordell-Weil bases of elliptic curves by Stroeker and Tzanakis [27], to reduce the final bound for the integral solutions of elliptic equations.…”

On common factors within a series of consecutive terms of an elliptic divisibility sequence

“…Put By a simple calculation e.g. using the function IntegralPoints of Magma [1] (which is based upon the deterministic and efficient method of Stroeker and Tzanakis [33] and Gebel, Pethő and Zimmer [14]) we can easily find all integral points on the corresponding elliptic curves E, and check that the sets T B are precisely those indicated in Table 3. Thus recalling from the proof of Theorem 2.1 that g B = g(T ) and G B = G(T ), the statement instantly follows from Lemma 3.5.…”

Computing all integer solutions of a genus 1 equation

“…Numerous examples have been given (see [3], [8], [9], [15], [16], [17], [19], [20], [21]) of the way Ellog works on quite regular elliptic equations. In opposition to this, here we are interested in more provocative equations that accentuate the general applicability of the method.…”

“…Both employ David's estimate of linear forms in elliptic logarithms [7]. Since then, it has been applied by a number of authors to a variety of elliptic equations of degree 3 or 4; see [15], [21], [3], [9], [19], [20], [17]. In particular, a general treatment of the cubic elliptic equation can be found in [20].…”