The Birch and Swinnerton-Dyer Conjecture is one of the Clay Mathematics Institute's Millennium Prize Problems, with a prize of $1,000,000 for whoever can prove or disprove the Birch and Swinnerton-Dyer Conjecture. It describes the set of rational solutions to equations defining an elliptic curve. The equation ‘x2 + y2 = z2’ given by Euclid does not hold true in whole numbers. The Birch and Swinnerton-Dyer conjecture asserts that the size of the group of rational points is related to the behavior of an associated zeta function ?(s) near the point s=1. In particular this amazing conjecture asserts that if ?(1) is equal to 0, then there are an infinite number of rational points (solutions), and conversely, if ?(1) is not equal to 0, then there is only a finite number of such points. As of 2017, only special cases of the conjecture have been proved.
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