# How to find all rational points on the elliptic curves like $y^2=x^3-2$

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.

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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. –  KCd Dec 14 '11 at 14:36
@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. –  awllower Dec 14 '11 at 15:04

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.