Abstract
A positive integer n is called congruent if it appears as the area of a right-angle triangle with rational sides. There are known to be infinite congruent numbers, the smallest being 5. However, given n, it is not straightforward to know if it is congruent. This poses what is called the congruent number problem: Given n, is there an algorithm that can determine whether n is congruent in a finite number of steps? It turns out that this problem is equivalent to checking where the elliptic curve y^2=x^3-n^2x has a rational point of infinite order, so it can be studied through the theory of elliptic curves. This thesis provides an overview of some of the tools that are used to study the rank of elliptic curves, and construct rational points on them, and concludes with an exposition of the paper Mock Heegner points and congruent numbers by Paul Monsky, which builds up on this theory to prove among other things that primes p congruent to 5 mod 8 are congruent numbers, and so are numbers of the form 2p when p is congruent to 3 mod 8.