Take the 2-minute tour ×  Pell equations can be solved using
continued fractions. I have heard that some elliptic curves can
be "solved" using continued fractions. Is this true?  Which
Diophantine equations other than Pell equations can be solved
for rational or integer points using continued fractions? If
there are others, what are some good references?   Professor
Elkies has given an excellent response as to the role of
continued fractions in solving general Diophantine equations
including elliptic curves. What are some other methods to solve
the Diophantine equations $$X^2 - \Delta Y^2 = 4 Z^3$$ and $$18
x y + x^2 y^2 - 4 x^3 - 4 y^3 - 27 = D z^2 ?$$  share|improve
this question Samuel, could you be a little more specific about
your comment about solving $x^2 - D y^2 = 4 z^3,$ especially why
you are not satisfied with your parametrized solution? I printed
out your paper with Franz, "Arithmetic of Pell Surfaces," where
I think equation (1.1) with $n=3$ is what you are discussing
below. Also Buell's book, es