Y2 = x3 − x + 1. Prize: 1 million dollars. rank E(Q) = ord s=1 L(E, s)

The equation shown is $y^2 = x^3 - x + 1$. This is the equation of an elliptic curve. The graph visually represents this curve, which is symmetric about the x-axis. The yellow dots indicate specific points on the curve.

The "Prize: 1 million dollars" refers to one of the seven Millennium Prize Problems established by the Clay Mathematics Institute. Solving any of these problems, including the one presented here, earns a prize of one million U.S. dollars.

The mathematical statement

is the Birch and Swinnerton-Dyer (BSD) conjecture. It proposes that the rank of the group of rational points $E(\mathbb{Q})$ on an elliptic curve $E$ (which measures the number of independent rational solutions) is equal to the order of vanishing of its associated $L$-function $L(E, s)$ at $s=1$. This conjecture links the algebraic properties of an elliptic curve to the analytic properties of its $L$-function.