Birch and Swinnerton-Dyer conjecture

In mathematics, the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory. It is known as one of the most challenging mathematical problems. 
Background.
An elliptic curve is a function of the form formula_1, where formula_2 and formula_3 are rational numbers.
The formula_4 values that satisfy the equation are called solutions of the elliptic curve. Sometimes these solutions are both rational, both irrational, or one rational and the other irrational. It turns out that the set of rational solutions is an abelian group. This means that given rational points formula_5 and formula_6, there is a way to produce the sum formula_7, and this is another rational point. 
In 1922, Louis Mordell proved that the set of rational solutions is a finitely generated abelian group. This means that any rational point formula_5 can be written as a finite combination of certain generating points. For example, the points formula_9 and formula_10 are generators of the rational points of formula_11.  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:
formula_12