The Birch and Swinnerton-Dyer Conjecture

Adriko Bosco

Abstract: This is one of the most challenging Mathematical problems. The conjecture was chosen as one of the seven millennium prize problems listed by the Clay Mathematics Institute, which has offered a $1, 000, 000 prize for the first correct proof. It is named after Mathematicians Bryan Birch and Peter Swinnerton-Dyer, who developed the conjecture during the first half of the 1960s with the help of machine computation. The conjecture is of great economic importance used in cryptology to develop the passwords in machines used for cash transactions and in the study of planetary motions in physics. The conjecture is in number theory about elliptical curves which are equations where one side you have a quadratic equation in y and the other a cubic equation in x. For example, y? = x? - 2.The challenge is to find numbers (x, y) which solve this equation. Here one could take x = 2 and y = 3.

Keywords: Birch and Swinnerton-Dyer conjecture, Elliptic curve, Boscomplex method, Quadratic equation, Concentric circles