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On Mordell's Equation

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Compositio Mathematica

Abstract

In an earlier paper we developed an algorithm for computing all integral points on elliptic curves over the rationals Q. Here we illustrate our method by applying it to Mordell's Equation y2=x3+k for 0 ≠ k ∈ Z and draw some conclusions from our numerical findings. In fact we solve Mordell's Equation in Z for all integers k within the range 0 < | k | ≤ 10 000 and partially extend the computations to 0 < | k | ≤ 100 000. For these values of k, the constant in Hall's conjecture turns out to be C=5. Some other interesting observations are made concerning large integer points, large generators of the Mordell–Weil group and large Tate–Shafarevič groups. Three graphs illustrate the distribution of integer points in dependence on the parameter k. One interesting feature is the occurrence of lines in the graphs.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Concordia University, Montréal, PQ, H5G 1M8, Canada

    J. Gebel

  2. Laboratory of Informatics, University of Medicine, Nagyerdei Krt. 98, H-4032, Debrecen, Hungary

    A. Pethö

  3. Fachbereich 9 Mathematik, Universität des Saarlandes, D-66041, Saarbrücken;, Germany

    H. G. Zimmer

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  1. J. Gebel
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  2. A. Pethö
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  3. H. G. Zimmer
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Gebel, J., Pethö, A. & Zimmer, H.G. On Mordell's Equation. Compositio Mathematica 110, 335–367 (1998). https://doi.org/10.1023/A:1000281602647

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