From: pethoe@MATH.KLTE.HU (Attila Pethoe)
Newsgroups: sci.math.numberthy
Subject: Re: P.S. re |x^3-y^2|
Date: 17 Sep 98 04:29:29 GMT

I have some comments to Noam Elkies message from 18 Aug 1998.
More exactly to the part:
 It is not quite true that [GPZ] tried all k of absolute
 value <10^5:  for about 1000 such k, the analytic
 rank is 1 but the generator [assuming trivial
 Tate-Shafarevic group] has height at least 70.  Most
 likely none of these curves E_k has an integral point,
 because such a point would have |x|>10^28
 [and r>10^9 !], but this cannot be proved until a
 Heegner point computation finds the generator.
Recently Klaus Wildanger solved y^2=x^3 + k for those k, which
remain left in [GPZ]. He proved that there are indeed no
solutions. He used a completely different method, as the method
of [GPZ]. His technique  is a combination of the method for
the solution of y^2=x^3 + k described in [DL] and the method
for the solution of cubic index form equations [GSch]. It
does not depend on conjectures. You find details in his
thesis [W]. One can analyze the running time of Wildanger's
method probably much easier, as that of [GPZ].

Hence for the solution of y^2=x^3 + k, |k| \le 10^5 we do not
need Heegner point computation. On the other hand, there were
computed in [GPZ] a basis of the Mordell-Weil group for the
curves y^2=x^3 + k, |k| \le 10^5 with the exceptions, mentioned
in Noam Elkies's message. It would be interesting to know the
generator of the Mordell-Weil group in the remaining cases too.
The smallest value remaining left in our paper is k=7823. Josef
Gebel (the G in [GPZ]) tried Heegner point computation for the
curve y^2 = x^3 + 7823, but without success.

Finally, I computed all integer points on the record holder
Mordell curves of Elkies, and for the curve #11 in his table.
The integer points are given in one of the basis of the
Mordell-Weil group. The computation was done by SIMATH, where
the method of [GPZ] is implemented.

  all nontrivial integral points on EC(0, -1641843) modulo negation :

  PT(123, 468, 1) = PT(123, 468, 1) + 0*PT(519, 11754, 1) +
      0*PT(1257/2, 43371/4, 2)
  PT(519, 11754, 1) = 0*PT(123, 468, 1) + PT(519, 11754, 1) +
      0*PT(1257/2, 43371/4, 2)
  PT(5853886516781223, 447884928428402042307918, 1) =
 3*PT(123, 468, 1) + 2*PT(519, 11754, 1) + PT(1257/2, 43371/4, 2)

  all nontrivial integral points on EC(0, -30032270) modulo negation :

  PT(311, 219, 1) = PT(311, 219, 1) + 0*PT(479, 8937, 1) +
  0*PT(20551191/251, 34205098051/63001, 251) +
  0*PT(177377639/391, 2339549066493/152881, 391)
  PT(479, 8937, 1) = 0*PT(311, 219, 1) + PT(479, 8937, 1) +
  0*PT(20551191/251, 34205098051/63001, 251) +
  0*PT(177377639/391, 2339549066493/152881, 391)
  PT(38115991067861271, 7441505802879036345061579, 1) = 0*PT(311, 219, 1) -
  PT(479, 8937, 1) - PT(20551191/251, 34205098051/63001, 251) -
  PT(177377639/391, 2339549066493/152881, 391)

  all nontrivial integral points on EC(0, 117073) modulo negation :

  PT(146, 1797, 1) = PT(146, 1797, 1) + 0*PT(587, 14226, 1)
  PT(587, 14226, 1) = 0*PT(146, 1797, 1) + PT(587, 14226, 1)
  PT(65589428378, 16797736678114635, 1) =  - PT(146, 1797, 1) +
                                          2*PT(587, 14226, 1)

References:

[DF] Delone, B.N., Faddeev, D.K.: The theory of irrationalities of the
third degree, Amer. Math. Soc. Transl. of Math. Monographs, 10,
Providence, RI, 1964.
[GSch] Ga\'al, I., Schulte, N.: Computing all power integer bases of
cubic number fields, Math. Comp. #53 (1989), 689--696.
[GPZ] Gebel, J., Peth\H{o}, A., and Zimmer, H.G.: On Mordell's equation,
 _Compositio Math._ #110 (1998), 335--367.
[W] Wildanger, K.: \"Uber das L\"osen von Einheiten- und
Indexformgleichungen in algebraischen Zahlk\"orpern mit einer Anwendung
auf die Bestimmung aller ganzen Punkte einer Mordellschen Kurve, PhD
Thesis, TU Berlin, 1997.