On Mordell's equation

J. Gebel, A. Pethő, H. G. Zimmer

Research output: Contribution to journalArticle

34 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)335-367
Number of pages33
JournalCompositio Mathematica
Volume110
Issue number3
Publication statusPublished - 1998

Fingerprint

Integer Points
Integral Points
Graph in graph theory
Elliptic Curves
Generator
Integer
Computing
Line
Range of data
Observation

Keywords

  • Birch & Swinnerton-Dyer conjecture
  • Elliptic curve
  • Elliptic logarithm
  • Height
  • Integral points
  • L-series
  • Mordell-Weil group
  • Rank
  • Regulator
  • Torsion

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

Gebel, J., Pethő, A., & Zimmer, H. G. (1998). On Mordell's equation. Compositio Mathematica, 110(3), 335-367.

On Mordell's equation. / Gebel, J.; Pethő, A.; Zimmer, H. G.

In: Compositio Mathematica, Vol. 110, No. 3, 1998, p. 335-367.

Research output: Contribution to journalArticle

Gebel, J, Pethő, A & Zimmer, HG 1998, 'On Mordell's equation', Compositio Mathematica, vol. 110, no. 3, pp. 335-367.
Gebel J, Pethő A, Zimmer HG. On Mordell's equation. Compositio Mathematica. 1998;110(3):335-367.
Gebel, J. ; Pethő, A. ; Zimmer, H. G. / On Mordell's equation. In: Compositio Mathematica. 1998 ; Vol. 110, No. 3. pp. 335-367.
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