Graduation Year


Date of Submission


Document Type

Open Access Senior Thesis

Degree Name

Bachelor of Arts



Reader 1

David Krumm

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© 2015 Jonathan S. Star


In this paper we explain the congruent number problem and its connection to elliptic curves. We begin with a brief history of the problem and some early attempts to understand congruent numbers. We then introduce elliptic curves and many of their basic properties, as well as explain a few key theorems in the study of elliptic curves. Following this, we prove that determining whether or not a number n is congruent is equivalent to determining whether or not the algebraic rank of a corresponding elliptic curve En is 0. We then introduce L-functions and explain the Birch and Swinnerton- Dyer (BSD) Conjecture. We then explain the machinery needed to understand an algorithm by Tim Dokchitser for evaluating L-functions at 1. We end by computing whether or not a given number n is congruent by implementing Dokchitser’s algorithm with Sage and by using Tunnel’s Theorem.

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Number Theory Commons