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On Mordell‘s Equation

Published online by Cambridge University Press:  04 December 2007

J. GEBEL
Affiliation:
Department of Computer Science and Engineering, Concordia University, Montréal PQ H5G 1M8, Canada; e-mail: sebp@josef.gebel@mayahtt.com
A. PETHÖ
Affiliation:
University of Medicine, Laboratory of Informatics, Nagyerdei Krt. 98, H-4032 Debrecen, Hungary; e-mail: pethoe@peugeot.dote.hu
H. G. ZIMMER
Affiliation:
Universität des Saarlandes, Fachbereich 9 Mathematik, D-66041 Saarbrücken;, Germany; e-mail: zimmer@math.uni.sb.de
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Abstract

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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 y$^2$=x$^3+$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 | [les ] 10 000 and partially extend the computations to 0 < | k | [les ] 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.

Type
Research Article
Copyright
© 1998 Kluwer Academic Publishers