Research into the congruent number problem has taken on a fun spin after being given incentive to present my findings in a seminar this spring.
I'm starting to develop a sense of how to organize and conduct my research, and this has led to the construction of a "number lab" of sorts. As a student, I have found that organization is my saving grace. By nature, I work chaotically, so I found a way to label my pages, tabbing them for quick access; that allows me to move between problems and go with new ideas without losing myself.
I've applied this to my research. Work on the Riemann Hypothesis has its own place, and work on the congruent numbers has another, and as I am conducting an investigation over fields, I've tried to organize everything into tables. So far, I've found and proven three lemmas en route to looking for some sort of theory to explain the pattern of solutions that might be extrapolated to the whole field of integers. It's very much like being a scientist in a lab. When I got to the field of size 17, I took the time to write a computer program to generate larger fields (and verify calculations in others). This was very helpful, because I was able to quickly work on the next field, over 19, which showed there was an exception to a fourth lemma that I had struggled to prove (and now I know why, and am working on proving it prove another angle).
This is wonderful fun! I'm studying four math topics, and as I learn, not only am I enjoying the new challenges and techniques I assimilate, but the opportunity to tackle my hobby research from new angles. The best part is, I can bring my lab with me anywhere, and I don't have to worry about anything blowing up...