### Abstract:

In David Hilbert’s lecture in 1900, he presented 15 diﬃcult problems that guided much of the research of mathematicians in the last century. Today, only 12 of these 15 problems have been solved. In honor of this great mathematician and to celebrate mathematics of the new millennium, the Clay Mathematics Institute of Cambridge, Massachusetts is oﬀering a one million dollar prize for the solution to any one of the “Millennium Prize Problems” [7]. These problems, like Hilbert’s problems, are long-standing mathematical questions that still have not been solved after many years of serious attempts by diﬀerent experts. For my honors thesis project, I chose to research and study three of these Millennium Prize Problems: the Riemann Hypothesis, the P versus NP problem, and the Birch and Swinnerton-Dyer Conjecture. My entire honors thesis investigation of these very diﬃcult problems is meant to explain the statements of each problem, provide background information, and explore related examples to establish a foundation about some of most signiﬁcant and interesting mathematical problems of the new millennium.