On recognizing congruent primes

An integer n is called congruent if it corresponds to the area of a right triangle with three rational sides. The problem of classifying congruent numbers has an extensive history, and is as yet unresolved. The most promising approach to this problem utilizes elliptic curves. In this thesis we explicitly lay out the correspondence between the congruence of a number n and the rank of the elliptic curve y^2 = x^3 - n^2x. By performing two-descents on this curve and isogenous curves for $n=p$ a prime, we are able to obtain a simple and unified proof of the majority of the known results concerning the congruence of primes. Finally, by calculating the equations for homogeneous spaces associated to the curve when p = 1 mod 8, we position the problem for future analysis.