Generating all the rational points

It turns out that for some elliptic curves over ${\mathbb{Q}}$, such as $y^2=x^3-x$, there are only finitely many rational points, while for others, such as the example $y^2=x(x+5)(x-5)$ above, there are infinitely many.

But in any case there is a deep theorem, proved by Mordell, that says that the group of rational points on an elliptic curve $E$ is ``finitely generated.'' This means there is a finite list of rational points ${\mathcal S}$ on $E$ such that all rational points on $E$ can be generated from the points in ${\mathcal S}$ by iteratively applying $+$ to pairs of points.

On the other hand, it is not known whether there exists an algorithm that takes the equation of an elliptic curve and outputs a finite list ${\mathcal S}$ of generating points as above. Researchers in number theory have spent about 70 years trying to prove the existence of such an algorithm, but the problem is still unsolved!