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.
[Math] How to find all rational points on the elliptic curves like $y^2=x^3-2$
diophantine equationselliptic-curvesmordell-curvesnumber theoryrational numbers
Best Answer
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.