This article explores the Birch and Swinnerton-Dyer Conjecture, one of the famous Millennium Prize Problems. In addition to providing the basic theoretic understanding necessary to understand the simplest form of the conjecture, some of the original numerical evidence used to formulate the conjecture is recreated. Recent results and current problems related to the conjecture are given at the end.

Author Bio

Brent Johnson graduated Villanova University in the spring of 2014 with a B.S. in Mathematics and he is currently pursuing his Master's degree at Villanova. This paper was completed at the end of a semester long study in elliptic curves. Brent had the opportunity to work with Dr. Robert Styer and Dr. Alice Deanin for two years before finally jumping into the research involved in this paper. The experience gained during those years was greatly influential to his mathematics career, and he is grateful for the mentorship he received. He currently enjoys studying elliptic curves and their application in cryptography, as well as new and exciting cryptographic schemes. Brent enjoys running and spending time with his dogs.