It looks like I'll be sticking with something related to the Collatz conjecture for my math research project after all. When reading about something about which I know little, which is unfortunately quite often with mathematics, I usually have to keep looking up terms within whatever I am reading, and through the term "Mersenne hailstone" on one page, I ended up on a page about Mersenne numbers, and thus, Mersenne primes. As learning about Mersenne primes in seventh grade is what initially made me think I could ever make a contribution to mathematics, I find this simultaneously amusing and inspiring. My math background is embarrassingly minimal, and I don't even know how I'll find the time this semester to pull off a publishable project, but I have a sense of adventure--it is worth trying.
And so what if number theory rarely proves useful? It is fun. It is a puzzle. And my goal, I suppose, is just to try to publish something, and to have some fun doing it. Excelsior!
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