Are there an infinite number of positive rational solutions to $x^3+y^3=19$? By transforming the equation to Weierstrass form, I have established that the rational solutions to $x^3+y^3=19$ can be constructed by two generators of the elliptic curve group: $(3,-2)$ and $(\frac{5}{2},\frac{3}{2})$ , and that the group is isomorphic to $\mathbb{Z}^2$
However, I can't see how to establish whether there are an infinite number of solutions where $x$ and $y$ are both positive rational numbers.
 A: Based on the comments from @rogerl and @JyrkiLahtonen, I think I can answer the question.
As the group of rational points on the curve is infinite, it has an accumulation point, either at a finite point on the curve, or at infinity.
If there is an accumulation point at infinity, i.e. there are infinitely many such solutions with increasing large $x$, then the point of intersection (point B in the diagram) of lines joining these points with $(\frac{5}{2}, \frac{3}{2})$ lies on the curve between $(\frac{3}{2}, \frac{5}{2})$ and $(\frac{5}{2}, \frac{3}{2})$, and therefore has positive $x$ and $y$

If there is a finite accumulation point, point C on the diagram, construct the point D as the intersection of the curve with the line joining C to $(\frac{5}{2}, \frac{3}{2})$.
Then there are infinitely many points sufficiently close to C (e.g. F and G) where the  curve and the lines FD or GD intersect the curve at a point close to $(\frac{5}{2}, \frac{3}{2})$, i.e. a point with positive $x$ and $y$

Edit 26 Jul 2021
A more satisfactory proof that for an elliptic curve of rank at least 1, there are inﬁnitely many rational solutions in every neighbourhood of any one of them is provided by Zachary Scherr, The real topology of rational points on elliptic curves, 2012, http://eg.bucknell.edu/~zls002/papers/rationalpoints.pdf
