Birch and Swinnerton-Dyer conjecture
In 1922, Louis Mordell proved that the set of rational solutions is a finitely generated abelian group. This means that any rational point  can be written as a finite combination of certain generating points. For example, the points  and  are generators of the rational points of .  This means that all rational points on it, even very complicated points, can be written in terms of these two points. For example, we have: