Reading the book by Diophantus, one may be led to consider the curves like:
$y^2=x^3+1$, $y^2=x^3-1$, $y^2=x^3-2$,
the first two of which are easy (after calculating some eight curves to be solved under some certain conditions, one can directly derive the ranks) to be solved, while the last , although simple enough to be solved by some elementary consideration of factorization of algebraic integers, is at present beyond my ability, as my knowledge about the topic is so far limited to some reading of the book Rational Points On Elliptic Curves, by Silverman and Tate, where he did not investigate the case where the polynomial has no visible rational points.
By the theorem of Mordell, one can determine its structure of rational points, if the rank is at hand. So, according to my imagination, if some hints about how to compute ranks of elliptic curves of this kind were offered, it would certainly be appreciated.
Thanks in advance.

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    $\begingroup$ There is are two pretty visible rational points on $y^2 = x^3 - 2$, namely $(x,y) = (3,\pm 5)$. It is a theorem of Fermat that these are the only integral points. The double of this point is (129/100,-383/1000), so by Nagell-Lutz the point (3,5) has infinite order. That the group of rational points has rank 1, and is generated by $(3,5)$, is a much deeper result. $\endgroup$ – KCd Dec 14 '11 at 14:36
  • $\begingroup$ @KCd: I mean that the polynomial on the right-hand side has no rational roots. And indeed, on this curve I as yet can do nothing to tell the rational points, except the visible ones. If the result be deep, where could it possibly be found? Thank very much. $\endgroup$ – awllower Dec 14 '11 at 15:04

Given your interest in Mordell's equation, you really ought to buy or borrow Diophantine Equations by Mordell, then the second edition of A Course in Number Theory by H. E. Rose, see AMAZON

Rose discusses the equation starting on page 286, then gives a table of $k$ with $ -50 \leq k \leq 50$ for which there are integral solutions, a second table for which there are rational solutions. The tables are copied from J. W. S. Cassels, The rational solutions of the diophantine equation $y^2 = x^3 - D.$ Acta Arithmetica, volume 82 (1950) pages 243-273.

Other than that, you are going to need to study Silverman and Tate far more carefully than you have done so far. From what I can see, all necessary machinery is present. Still, check the four pages in the Bibliography, maybe you will prefer something else.

  • $\begingroup$ Here I genuinely thank you for those books recommended. Indeed, I am trying to study the book Arithmetic On Elliptic Curves right now. Also, I will borrow the book by Rose mentioned above; thank again. Look forward to the day I can solve this question! $\endgroup$ – awllower Dec 15 '11 at 13:52
  • $\begingroup$ And of course, but I forgot in the last comment, the book by Mordell is already at my disposal, while I still try hard to follow his steps and various methods used to solve those incredibly interesting equations. The study of elliptic curves really opened up a totally new door to me, for I was completely unaware of this ancient and fascinating subject. $\endgroup$ – awllower Dec 15 '11 at 14:01

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