The study term has started and I simply cannot contain the excitement. This second year is chock-full of some important building blocks that I hope will help me with my independent work on both the Riemann hypothesis and the congruent number problem. I have four courses, but it is the third year course, introduction to topology, which interests me the most. I am hopeful that some of the concepts of metric spaces will assist in the geometric proof I have been trying to put together since May.
Meanwhile, I began an investigation into modularity patterns for the elliptic curves associated with the congruent numbers, and have thus far found and proven three lemmas that have half-formulated a theorem about the relationship between cubic and quadratic residues. I have analysed curves for all n up to the field formed by 13 elements, and hope to find some useful generalization that extends between fields as they increase (thus approximating the integers), with the hypothesis that there will be some pattern emerging for values of n that correspond to congruent numbers.
It is all a little bit of undergraduate fun, which I enjoy. Not only do I get to learn lots of neat and useful things, but my innate curiosity for problem-solving is satisfied. There's nothing like taking abstract concepts and finding a practical use for them. After all, that's what mathematicians do.
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