Congruent number problem and elliptic curves

This paper explores the congruent number problem, which asks which rational numbers are areas of triangles with rational side lengths. We define a rational non-zero number that is the area of a rational triangle as a congruent number and seek a method for determining if a number is congruent or not.For some numbers such as 6 it is relatively simple to check that this is the area of a rational triangle, but as possible congruent numbers increase in size, it gets increasingly difficult to obtain via trial and error. Thus, we seek a more efficient method in determining if a number is congruent or not. The method of choice establishes a connection between congruent numbers and elliptic curves, an algebraic curve useful far beyond its apparent properties.After describing some essential properties of the elliptic curve, such as the algebraic operations it respects and how to add points on the curve, we move into a discussion of the rank of an elliptic curve. As it turns out, the rank of the elliptic curve determines whether a number is congruent or not. After a brief discussion of linearly independence and torsion points, we describe a way to determine the rank of an elliptic curve.After a brief discussion of the L-Series, which is connected to the rank of the elliptic curve, we connect the rank to the sides of a rational triangle via a bijective function. Finally, we state Tunnell's theorem. This theorem relies on the Birch-Swinnerton Dyer conjecture and states two conditions, the first being that a number is congruent. If the second condition is satisfied, then so is the first and vice-versa. We arrive at a somewhat straightforward algebraic computation using elliptic curves to determine if a number is congruent and thus the area of a rational triangle